1 Introduction

Let \({\fancyscript{L}}\) be a second order elliptic differential operator in \({\mathbb R}^d\) given by

$$\begin{aligned} {\fancyscript{L}}f(x)=\sum _{i,j=1}^d \partial _i \left( a_{ij}(x)\, \partial _jf(x)\right) + \sum _{i=1}^d b_i(x) \partial _i f(x), \end{aligned}$$
(1.1)

where \((a_{ij}(x))_{1\leqslant i, j\leqslant d}\) is a bounded measurable (not necessarily symmetric) \(d\times d\)-matrix-valued function on \({\mathbb R}^d\) that is uniformly elliptic, and \(b_i(x)\), \(1\leqslant i\leqslant d\), are bounded measurable functions on \({\mathbb R}^d\). Here \(\partial _i f(x)\) stands for the partial derivative \(\frac{\partial f(x)}{\partial x_i}\). It is well known that there is a diffusion process \(X\) having \({\fancyscript{L}}\) as its infinitesimal generator; see [18]. The celebrated DeGiorgi–Nash–Moser–Aronson theory asserts that every bounded parabolic function of \({\fancyscript{L}}\) (or equivalently, of \(X\)) is locally Hölder continuous and the parabolic Harnack inequality holds for non-negative parabolic functions of \({\fancyscript{L}}\). Moreover, \({\fancyscript{L}}\) has a jointly continuous heat kernel (or equivalently, transition density function of \(X\)) \(p(t, x, y)\) with respect to the Lebesgue measure on \({\mathbb R}^d\) that enjoys Aronson’s Gaussian type estimates.

While second order elliptic operators have been studied extensively and found wide applications, there are many systems that should be and in fact have been modeled by non-local operators or discontinuous Markov processes. The study of fine potential theoretical properties of non-local operators and discontinuous Markov processes is relatively recent. Quite a lot progress has been made during the last decade in developing DeGiorgi-Nash-Moser-Aronson type theory for symmetric non-local operators; see, e.g., [3, 7, 8, 13, 14] and the references therein. In particular, it is shown in Chen and Kumagai [13] that, for every \(0<\alpha <2\) and for any symmetric measurable function \(c(x, y)\) on \({\mathbb R}^d\times {\mathbb R}^d\) that is bounded between two positive constants \(\kappa _0\) and \(\kappa _1\), the symmetric non-local operator

$$\begin{aligned} {\fancyscript{L}}f(x)=\lim _{\varepsilon \rightarrow 0} \int _{\{ y\in {\mathbb R}^d: |y-x|>\varepsilon \}} (f(y)-f(x)) \frac{c(x, y)}{|x-y|^{d+\alpha }} {\mathord {\mathrm{d}}}y , \end{aligned}$$
(1.2)

defined in the weak sense, admits a jointly Hölder continuous heat kernel \(p(t, x, y)\) with respect to the Lebesgue measure on \({\mathbb R}^d\), which satisfies

$$\begin{aligned} C^{-1}\, \frac{t}{(t^{1/\alpha } + |x-y|)^{d+\alpha }} \leqslant p(t, x, y) \leqslant C\, \frac{t}{(t^{1/\alpha } + |x-y|)^{d+\alpha }} \end{aligned}$$
(1.3)

for every \(t>0\) and \(x, y\in {\mathbb R}^d\), where \(C\geqslant 1\) is a constant that depends only on \(d, \alpha , \kappa _0\) and \(\kappa _1\). The operator \({\fancyscript{L}}\) in (1.2) is symmetric in the sense that

$$\begin{aligned} \int _{{\mathbb R}^d}g(x){\fancyscript{L}}f(x){\mathord {\mathrm{d}}}x=\int _{{\mathbb R}^d}f(x){\fancyscript{L}}g(x){\mathord {\mathrm{d}}}x \qquad \hbox {for } f,g\in C^\infty _c({\mathbb R}^d), \end{aligned}$$

where \(C^\infty _c({\mathbb R}^d)\) denotes the space of smooth functions on \({\mathbb R}^d\) with compact support. When \(c(x, y)\) is a positive constant, \({\fancyscript{L}}\) above is a constant multiple of the fractional Laplacian \(\Delta ^{\alpha /2}:=- (-\Delta )^{\alpha /2}\) on \({\mathbb R}^d\), which is the infinitesimal generator of a (rotationally) symmetric \(\alpha \)-stable process on \({\mathbb R}^d\). The symmetric non-local stable-like operator \({\fancyscript{L}}\) defined by (1.2) versus \(\Delta ^{\alpha /2}\) is the analog of a symmetric uniformly elliptic divergence form operator versus Laplacian \(\Delta \). Estimate (1.3) can be viewed as an Aronson type estimate for symmetric stable-like operator \({\fancyscript{L}}\) of (1.2). However, there are very limited results on heat kernels for non-symmetric non-local operators; see the two paragraphs following Corollary 1.3.

The purpose of this paper is to study heat kernels and their sharp two-sided estimates for non-symmetric and non-local stable-like operators of the following form:

$$\begin{aligned} {\fancyscript{L}}^\kappa _\alpha f(x) :=\hbox {p.v.}\int _{{\mathbb R}^d}(f(x+z)-f(x))\frac{\kappa (x,z)}{|z|^{d+\alpha }} {\mathord {\mathrm{d}}}z, \end{aligned}$$
(1.4)

where p.v. stands for the Cauchy principle value; that is

$$\begin{aligned} {\fancyscript{L}}^\kappa _\alpha f(x)=\lim _{\varepsilon \rightarrow 0} \int _{\{z\in {\mathbb R}^d: |z|>\varepsilon \}}(f(x+z)-f(x))\frac{\kappa (x,z)}{|z|^{d+\alpha }}{\mathord {\mathrm{d}}}z. \end{aligned}$$

Here \(d\geqslant 1, 0<\alpha <2\), and \(\kappa (x,z)\) is a measurable function on \({\mathbb R}^d\times {\mathbb R}^d\) satisfying

$$\begin{aligned} 0<\kappa _0\leqslant \kappa (x,z)\leqslant \kappa _1, \qquad \kappa (x,z)=\kappa (x,-z), \end{aligned}$$
(1.5)

and for some \(\beta \in (0,1)\)

$$\begin{aligned} |\kappa (x,z)-\kappa (y,z)|\leqslant \kappa _2|x-y|^\beta . \end{aligned}$$
(1.6)

That \(\kappa (x, z)\) is symmetric in \(z\) is a commonly assumed condition in the literature of non-local operators; see [7] for example. Due to this symmetric, we can rewrite \({\fancyscript{L}}^\kappa _\alpha \) as

$$\begin{aligned} {\fancyscript{L}}^\kappa _\alpha f(x)= \int _{ {\mathbb R}^d} (f(x+z)-f(x)-\mathbf{1}_{\{|z|\leqslant 1\}}z\cdot \nabla f(x))\frac{\kappa (x,z)}{|z|^{d+\alpha }}{\mathord {\mathrm{d}}}z \end{aligned}$$

for every bounded \(C^2\)-smooth function \(f\) on \({\mathbb R}^d\). Thus \({\fancyscript{L}}^\kappa _\alpha \) is the same type of operators considered in Bass and Levin [2], for which they established the Harnack inequality. We can also write \({\fancyscript{L}}^\kappa _\alpha \) by

$$\begin{aligned} {\fancyscript{L}}^\kappa _\alpha f(x)=\frac{1}{2}\int _{{\mathbb R}^d}(f(x+z)+f(x-z)-2f(x)) \frac{\kappa (x,z)}{|z|^{d+\alpha }}{\mathord {\mathrm{d}}}z. \end{aligned}$$

We point out here that, unlike the operator \({\fancyscript{L}}\) of (1.2), the operator \({\fancyscript{L}}^\kappa _\alpha \) defined by (1.4) is typically non-symmetric. Operators \({\fancyscript{L}}^\kappa _\alpha \) of (1.4) can be regarded as the non-local counterpart of elliptic operators of non-divergence form. Hölder continuity assumption (1.6) for non-symmetric operator \({\fancyscript{L}}^{\kappa , \alpha }\) is quite natural. Unlike the symmetric case, even for elliptic differential operators, certain Dini-type continuity assumption on the coefficients is needed for the existence and for the two-sided estimates of the fundamental solution of non-divergence form operators; see [28], Sections IV.11 and IV.13], [30] and [32], Theorem 19].

The following is the main result of this paper. For \(a, b\in {\mathbb R}\), \(a\wedge b:= {\mathord {\mathrm{min}}}\{a, b\}\) and \(a\vee b:=\max \{a, b\}\).

Theorem 1.1

Under (1.5) and (1.6), there exists a unique nonnegative jointly continuous function \(p^\kappa _{\alpha } (t,x,y)\) in \((t,x,y)\in (0,1]\times {\mathbb R}^d\times {\mathbb R}^d\) solving

$$\begin{aligned} \partial _tp^\kappa _{\alpha } (t,x,y)={\fancyscript{L}}^\kappa _\alpha p^\kappa _{\alpha }(t,\cdot ,y)(x), \quad x\not =y, \end{aligned}$$
(1.7)

and satisfying the following four properties:

  1. (i)

    (Upper bound) There is a constant \(c_1>0\) so that for all \(t\in (0,1]\) and \(x,y\in {\mathbb R}^d\),

    $$\begin{aligned} p^\kappa _{\alpha } (t,x,y)\leqslant c_1 t ( t^{1/\alpha }+|x-y|)^{-d-\alpha }. \end{aligned}$$
    (1.8)
  2. (ii)

    (Hölder’s estimate) For every \(\gamma \in (0,\alpha \wedge 1)\), there is a constant \(c_2>0\) so that for every \(t\in (0,1]\) and \(x,x',y\in {\mathbb R}^d\),

    $$\begin{aligned} |p^\kappa _{\alpha } (t,x,y)-p^\kappa _{\alpha } (t,x',y)|\leqslant c_2|x-x'|^\gamma t^{1-\frac{\gamma }{\alpha }}\left( t^{1/\alpha }+|x-y|\wedge |x'-y| \right) ^{-d-\alpha } . \end{aligned}$$
    (1.9)
  3. (iii)

    (Fractional derivative estimate) For all \(x\not =y\in {\mathbb R}^d\), the mapping \(t\mapsto {\fancyscript{L}}^\kappa _\alpha p^\kappa _{\alpha }(t,\cdot ,y)(x)\) is continuous on \((0,1]\), and

    $$\begin{aligned} |{\fancyscript{L}}^\kappa _\alpha p^\kappa _{\alpha } (t,\cdot ,y)(x)|\leqslant c_3(t^{1/\alpha }+|x-y|)^{-d-\alpha }. \end{aligned}$$
    (1.10)
  4. (iii)

    (Continuity) For any bounded and uniformly continuous function \(f:{\mathbb R}^d\rightarrow {\mathbb R}\),

    $$\begin{aligned} \lim _{t\downarrow 0}\sup _{x\in {\mathbb R}^d}\left| \int _{{\mathbb R}^d}p^\kappa _{\alpha } (t,x,y)f(y){\mathord {\mathrm{d}}}y-f(x)\right| =0. \end{aligned}$$
    (1.11)

Moreover, we have the following conclusions.

  1. (1)

    The constants \(c_1,\, c_2\) and \(c_3\) in (i)–(iii) above can be chosen so that they depend only on \((d, \alpha , \beta , \kappa _0, \kappa _1, \kappa _2)\), \((d, \alpha , \beta , \gamma , \kappa _0, \kappa _1, \kappa _2)\), and \((d, \alpha , \beta , \kappa _0, \kappa _1, \kappa _2)\), respectively.

  2. (2)

    (Conservativeness) For all \((t,x,y)\in (0,1]\times {\mathbb R}^d\times {\mathbb R}^d\), \(p^\kappa _\alpha (t, x, y)\geqslant 0\) and

    $$\begin{aligned} \int _{{\mathbb R}^d}p^\kappa _{\alpha } (t,x,y){\mathord {\mathrm{d}}}y=1. \end{aligned}$$
    (1.12)
  3. (3)

    (C–K equation) For all \(s,t\in (0,1]\) and \(x,y\in {\mathbb R}^d\), the following Chapman–Kolmogorov equation holds:

    $$\begin{aligned} \int _{{\mathbb R}^d}p^\kappa _{\alpha } (t,x,z)p^\kappa _{\alpha } (s,z,y){\mathord {\mathrm{d}}}z=p^\kappa _{\alpha } (t+s,x,y). \end{aligned}$$
    (1.13)
  4. (4)

    (Lower bound) There exists \(c_4=c_4(d, \alpha , \beta , \kappa _0, \kappa _1, \kappa _2)>0\) so that for all \(t\in (0,1]\) and \(x,y\in {\mathbb R}^d\),

    $$\begin{aligned} p^\kappa _{\alpha } (t,x,y)\geqslant c_4t(t^{1/\alpha }+|x-y|)^{-d-\alpha }. \end{aligned}$$
    (1.14)
  5. (5)

    (Gradient estimate) For \(\alpha \in [1,2)\), there exists \(c_5=c_5(d, \alpha , \beta , \kappa _0, \kappa _1, \kappa _2)>0 \) so that for all \(x\not = y\) in \({\mathbb R}^d\) and \(t\in (0,1]\),

    $$\begin{aligned} |\nabla _x \log p^\kappa _{\alpha } (t,x ,y)|\leqslant c_5t^{-1/\alpha } . \end{aligned}$$
    (1.15)
  6. (6)

    (Generator) For any \(f\in C^2_b({\mathbb R}^d)\), we have

    $$\begin{aligned} \lim _{t\downarrow 0}\frac{1}{t}\int _{{\mathbb R}^d}p^\kappa _{\alpha } (t,x,y)(f(y)-f(x)){\mathord {\mathrm{d}}}y={\fancyscript{L}}^\kappa _\alpha f(x), \end{aligned}$$
    (1.16)

    and the convergence is uniform.

  7. (7)

    (Analyticity) The \(C_0\)-semigroup \(( P^\kappa _t)_{t\geqslant 0}\) of \({\fancyscript{L}}^\kappa _\alpha \) defined by \(P^\kappa _t f(x):=\int _{{\mathbb R}^d} p^\kappa _\alpha \) \( (t, x, y) f(y) {\mathord {\mathrm{d}}}y\) is analytic in \(L^p({\mathbb R}^d)\) for every \(p\in [1,\infty )\).

Here \(C^2_b({\mathbb R}^d)\) is the space of bounded continuous functions on \({\mathbb R}^d\) that have bounded continuous first and second order partial derivatives. A \(C_0\)-semigroup in a Banach space means a strongly continuous semigroup. A semigroup \(\{P_t; t\geqslant 0\}\) is said to be analytic in \(L^p({\mathbb R}^d)\) if it is the restriction to the non-negative real axis of a semigroup \(\{P_z; z\in \Lambda \}\) in \(L^p({\mathbb R}^d)\) that is analytic in \(z\in \Lambda \), where \(\Lambda \) is a cone in the complex plane that contains the non-negative real axis; see [31], Definition II. 5.1].

Remark 1.2

Inequalities (1.8) and (1.14) give sharp two-sided estimates of the heat kernel \(p^\kappa _\alpha (t, x, y)\). We can restate estimate (1.10) as

$$\begin{aligned} |\partial _t p^\kappa _{\alpha } (t,x,y) |\leqslant c_3(t^{1/\alpha }+|x-y|)^{-d-\alpha } \quad \hbox {for } x\not = y \hbox { and } t\in (0, 1]. \end{aligned}$$

This together with (1.9) and (1) of Theorem 1.1 yields that for \(0<s \leqslant t\leqslant 1\) and \(x, x', y\in {\mathbb R}^d\) with either \(x\not = y\) or \(x'\not = y\),

$$\begin{aligned}&|p^\kappa _{\alpha } (s,x,y)-p^\kappa _{\alpha } (t,x',y)| \nonumber \\&\quad \leqslant \widetilde{c}_2 \left( |t-s|+|x-x'|^\gamma t^{1-\frac{\gamma }{\alpha }} \right) \left( s^{1/\alpha }+|x-y|\wedge |x'-y|\right) ^{-d-\alpha }, \end{aligned}$$
(1.17)

where \(\widetilde{c}_2=c_2+ c_3\). On the other hand, we prove in Lemma 3.7 that \(p^\kappa _\alpha (t, x, y)\) is jointly continuous on \((0, 1]\times {\mathbb R}^d \times {\mathbb R}^d\) so (1.17) holds for every \(0<s<t\leqslant 1\) and \(x, x', y\in {\mathbb R}^d\).

We now present an application of Theorem 1.1 to stochastic differential equations driven by (rotationally) symmetric stable processes. Suppose that \(A(x)=(a_{ij}(x))_{1\leqslant i, j\leqslant d}\) is a bounded continuous \(d\times d\)-matrix-valued function on \({\mathbb R}^d\) that is non-degenerate at every \(x\in {\mathbb R}^d\), and \(Y\) is a (rotationally) symmetric \(\alpha \)-stable process on \({\mathbb R}^d\) for some \(0<\alpha <2\). It is shown in Bass and Chen [1], Theorem 7.1] that for every \(x\in {\mathbb R}^d\), SDE

$$\begin{aligned} {\mathord {\mathrm{d}}}X_t = A(X_{t-}) {\mathord {\mathrm{d}}}Y_t, \qquad X_0=x, \end{aligned}$$
(1.18)

has a unique weak solution. (Although in [1] it is assumed \(d\geqslant 2\), the results there are valid for \(d=1\) as well.) The family of these weak solutions forms a strong Markov process \(\{X, {\mathbb P}_x, x\in {\mathbb R}^d\}\). Using Itô’s formula, one deduces (see the display above (7.2) in [1]) that \(X\) has generator

$$\begin{aligned} {\fancyscript{L}}f(x) = \mathrm{p.v.} \int _{{\mathbb R}^d} \left( f(x+A(x)y)-f(x)\right) \frac{c_{d, \alpha }}{|y|^{d+\alpha }} {\mathord {\mathrm{d}}}y, \end{aligned}$$
(1.19)

where \(c_{d, \alpha }\) is a positive constant that depends on \(d\) and \(\alpha \). A change of variable formula \(z=A(x) y\) yields

$$\begin{aligned} {\fancyscript{L}}f(x) = \mathrm{p.v.} \int _{{\mathbb R}^d} \left( f(x+z)-f(x)\right) \frac{\kappa (x, z)}{|z|^{d+\alpha }} {\mathord {\mathrm{d}}}z , \end{aligned}$$
(1.20)

where

$$\begin{aligned} \kappa (x, z)= \frac{c_{d,\alpha }}{|\mathrm{det} A(x)|} \left( \frac{|z|}{|A(x)^{-1}z|}\right) ^{d+\alpha }. \end{aligned}$$
(1.21)

Here \(\mathrm{det}(A(x))\) is the determinant of the matrix \(A(x)\) and \(A(x)^{-1}\) is the inverse of \(A(x)\). As an application of the main result of this paper, we have

Corollary 1.3

Suppose that \(A(x)=(a_{ij}(x))\) is uniformly bounded and elliptic (that is, there are positive constants \(\lambda _0\) and \(\lambda _1\) so that \(\lambda _0 I_{d\times d} \leqslant A(x)\leqslant \lambda _1 I_{d\times d}\) for every \(x\in {\mathbb R}^d\)) and there are constants \(\beta \in (0, 1)\) and \(\lambda _2>0\) so that

$$\begin{aligned} |a_{ij}(x)-a_{ij}(y)| \leqslant \lambda _2 |x-y|^\beta \quad \hbox {for } 1\leqslant i, j\leqslant d. \end{aligned}$$

Then the strong Markov process \(X\) formed by the unique weak solution to SDE (1.18) has a jointly continuous transition density function \(p(t, x, y)\) with respect to the Lebesgue measure on \({\mathbb R}^d\), and there is a constant \(C>0\) that depends only on \((d, \alpha , \beta , \lambda _0, \lambda _1)\) so that

$$\begin{aligned} C^{-1}\, \frac{t}{(t^{1/\alpha } + |x-y|)^{d+\alpha }} \leqslant p(t, x, y) \leqslant C\, \frac{t}{(t^{1/\alpha } + |x-y|)^{d+\alpha }} \end{aligned}$$

for every \(t\in (0, 1]\) and \(x, y\in {\mathbb R}^d\). Moreover, \(p(t, x, y)\) enjoys all the properties stated in the conclusion of Theorem 1.1 with \(\kappa _0= c_{d, \alpha } \lambda _0^{d+\alpha } \lambda _1^{-d},\, \kappa _1= c_{d, \alpha } \lambda _0^{-d} \lambda _1^{d+\alpha }\) and \(\kappa _2= \kappa _2 (d, \lambda _0, \lambda _1, \lambda _2)\).

To the best of our knowledge, Theorem 1.1 is the first result on heat kernels and their estimates for a general class of non-symmetric and non-local stable-like operators under Höder continuous condition on \(x\mapsto \kappa (x, z)\). We mention that in the framework of pseudodifferential operator theory, Kochubei [26] (see also [20]) has studied the existence of fundamental solutions for \({\fancyscript{L}}^\kappa _\alpha \) by using Levi’s method. But strong smoothness of \(\kappa (x,y)\) in \(y\) and \(\alpha \in [1,2)\) are required. In Chen and Wang [15], fractional Laplacian \(\Delta ^{\alpha /2}\) perturbed by lower order non-local operator is studied, which corresponds to the case when \(\kappa (x, z)= a + b (x, z) |z|^{\alpha -\delta }\) for some constant \(a>0\) and a bounded measurable \(b(x, z)\) with \(b(x, z)=b(x, -z)\). As a special case of the much more general results obtained in [15], it is proved there that for this type of \(\kappa (x, z)\), when there are two positive constants \(\kappa _0, \kappa _1\) so that \(\kappa _0 \leqslant \kappa (x, z)\leqslant \kappa _1\) (but no Hölder continuity is assumed in \(x\mapsto b(x, z)\)), \({\fancyscript{L}}^\kappa _\alpha \) has a unique jointly continuous heat kernel \(p^\kappa _\alpha (t, x, y)\) and it enjoys the two-sided estimates (1.8) and (1.14).

Although quite a lot is known for symmetric non-local operators, there are very limited results in literature on heat kernel estimates for non-symmetric and non-local operators. In addition to the literature [15, 20, 26] mentioned in last paragraph, Bogdan and Jakubowski [6] studied the estimates of heat kernel of \(\Delta ^{\alpha /2}\) perturbed by a gradient operator with \(\alpha \in (1,2)\) (see also [33] for some extension). Jakubowski and Szczypkowski [25] considered a time-dependent gradient perturbation of \(\Delta ^{\frac{\alpha }{2}}\), while Jakubowski [24] established a global time estimate of heat kernel of \(\Delta ^{\alpha /2}\) under small singular drifts. In [1012], Chen et al. obtained sharp two-sided estimates for the Dirichlet heat kernel of \(\Delta ^{\alpha /2}\) as well as of its gradient and Feynman–Kac perturbations. Global as well as Dirichlet heat kernel estimates for non-local operators \(\Delta +\Delta ^{\alpha /2} + b\cdot \nabla \) and for \(m-(m^{2/\alpha }-\Delta )^{\alpha /2}+b\cdot \nabla \) have been investigated in Chen and Hu [9] and Chen and Wang [16], respectively. In the critical case of \(\alpha =1\), sharp two-sided heat kernel estimates of \(\Delta ^{1/2}+b\cdot \nabla \) with Hölder continuous drift \(b\) was obtained recently in Xie and Zhang [34] using Levi’s method. In [29], Maekawa and Miura obtained upper bounds estimates for the fundamental solutions of general non-local diffusions with divergence free drift.

We next briefly describe the approach of this paper. For the construction and upper bound estimates of the heat kernel, we use a method based on Levi’s freezing coefficients argument (cf. [22, 28]). However, in contrast to the previous work [34], a new way to freeze the coefficient \(\kappa (x, z)\) is needed (see Sect. 3). This causes quite many new challenges. In particular, we need to estimate the fractional derivative of the freezing heat kernel and to prove the continuous dependence of heat kernels with respect to the kernel function \(\kappa \) (see Sects. 2.3 and 2.4). Strong stability of the heat kernels in terms of the maximal distance between jumping kernels has recently been studied in Bass and Ren [4] (see Theorem 5.3 there) for symmetric stable-like operators (1.2). But here we need a more refined stability results on the heat kernels and their derivatives; see Theorem 2.5 below. To show the uniqueness and non-negativeness of the heat kernel, we establish a maximum principle for solutions of the parabolic equation \(\partial _t u(t, x)={\fancyscript{L}}^\kappa _\alpha u(t, x)\); see Theorem 4.1. For the lower bound estimate (1.14) on the heat kernel, we use a probabilistic approach. The heat kernel \(p^\kappa _\alpha (t, x, y)\) determines a strong Feller process \(X=\{X_t, t\geqslant 0; {\mathbb P}_x, x\in {\mathbb R}^d\}\) on \({\mathbb R}^d\). We show that for each \(x\in {\mathbb R}^d,\, {\mathbb P}_x\) solves the martingale problem for \(({\fancyscript{L}}^\kappa _\alpha , C^2_b ({\mathbb R}^d))\) with initial value \(x\); see (4.24) below. We then deduce from it the Lévy system of \(X\), which tells us that \(k(x, z)|z|^{-(d+\alpha )}\) is the intensity of \(X\) making a jump from \(x\) with size \(z\). The lower bound estimate for \(p^\kappa _\alpha \) can then be obtained by a probabilistic argument involving the use of the Lévy system of \(X\).

Remark 1.4

It will be shown in a subsequent paper [17] that solution to the martingale problem for \(({\fancyscript{L}}^\kappa _\alpha , C^\infty _c({\mathbb R}^d))\) is unique. (In fact it will be established for a more general class of non-local operators.) Thus the heat kernel \(p^\kappa _\alpha (t, x, y)\) in Theorem 1.1 can also be regarded as the (unique) transition density function of the unique solution to the martingale problem for \(({\fancyscript{L}}^\kappa _\alpha , C^\infty _c({\mathbb R}^d))\).

The notion of analyticity of a \(C_0\)-semigroup plays a central role in the semigroup theory of evolution equations (cf. [21, 23, 31]). For elliptic differential operators \({\fancyscript{L}}\) of (1.1), it is well-known that its associated \(C_0\)-semigroup is analytic in \(L^p\)-spaces for every \(p\in (1,\infty )\) at least when \(a_{ij}\) are smooth (cf. [31], Chapter 7]). The proof of this fact is based upon the following deep a priori estimate:

$$\begin{aligned} \Vert \partial _i\partial _jf\Vert _p\leqslant C(\Vert {\fancyscript{L}}f\Vert _p+\Vert f\Vert _p),\quad f\in {\mathbb W}^{2,p}({\mathbb R}^d), \end{aligned}$$

which is a consequence of singular integral operator theory. For nonlocal operator \({\fancyscript{L}}^\kappa _\alpha \) of (1.2), under some additional assumptions on \(\kappa (x,z)\), it was shown in [35] and [36] that for any \(p\in (1,\infty )\) and \(\alpha \in (0,2)\),

$$\begin{aligned} c_6\Vert f\Vert _{{\mathbb H}^{\alpha ,p}}\leqslant \Vert {\fancyscript{L}}^\kappa _\alpha f\Vert _p+\Vert f\Vert _p\leqslant c_6^{-1}\Vert f\Vert _{{\mathbb H}^{\alpha ,p}}, \ \ f\in {\mathbb H}^{\alpha ,p}, \end{aligned}$$

where \({\mathbb H}^{\alpha ,p}=(I-\Delta )^{-\frac{\alpha }{2}}(L^p)\) is the usual Bessel potential space. In this case, it is possible to show the analyticity of its associated semigroup \((P^\kappa _t)_{t\geqslant 0}\) by using Agmon’s method [21]. However in this paper we are able to establish the analyticity of the semigroup \((P^\kappa _t)_{t\geqslant 0}\) without these additional assumptions. We achieve this by establishing the inequality \(\Vert {\fancyscript{L}}^\kappa _\alpha P_t f\Vert _p \leqslant c t^{-1} \Vert f\Vert _p\) for every \(t>0,\, p\in [1, \infty )\) and \(f\in L^p ({\mathbb R}^d)\).

The remainder of the paper is organized as follows. In Sect. 2, we prepare some necessary results about the estimates of the heat kernel of spatial-independent symmetric Lévy operators. In Sect. 3, we construct the heat kernel of spatial-dependent Lévy operators by using Levi’s method. Lastly, in Sect. 4 we present the proof of the main result of this paper, Theorem 1.1.

We conclude this section by introducing the following conventions. The letter \(C\) with or without subscripts will denote a positive constant, whose value is not important and may change in different places. We write \(f(x)\preceq g(x)\) to mean that there exists a constant \(C_0>0\) such that \(f(x)\leqslant C_0 g(x)\); and \(f(x)\asymp g(x)\) to mean that there exist \(C_1,C_2>0\) such that \(C_1 g(x)\leqslant f(x)\leqslant C_2 g(x)\). We will also use the abbreviation \(f(x\pm z)\) for \(f(x+z)+f(x-z)\). For \(p\geqslant 1,\, L^p\)-norm of \(L^p({\mathbb R}^d)=L^p({\mathbb R}^d; {\mathord {\mathrm{d}}}x)\) will be denoted as \(\Vert f\Vert _p\). We use “\(:=\)” to denote a definition.

2 Preliminaries

Throughout this paper, we shall fix \(\alpha \in (0,2)\) and, unless otherwise specified, assume

$$\begin{aligned} (t,x)\in (0,1]\times {\mathbb R}^d. \end{aligned}$$

For \(\gamma ,\beta \in {\mathbb R}\), we introduce the following function on \((0,1]\times {\mathbb R}^d\) for later use:

$$\begin{aligned} \varrho ^\beta _\gamma (t,x):=t^{\frac{\gamma }{\alpha }}(|x|^\beta \wedge 1)(t^{1/\alpha }+|x|)^{-d-\alpha }. \end{aligned}$$
(2.1)

2.1 Convolution inequalities

The following lemma will play an important role in the sequel, which is similar to [27], Lemma 1.4] and [34], Lemma 2.3].

Lemma 2.1

  1. (i)

    For all \(\beta \in [0,\frac{\alpha }{2}]\) and \(\gamma \in {\mathbb R}\), we have

    $$\begin{aligned} \int _{{\mathbb R}^d}\varrho ^\beta _\gamma (t,x){\mathord {\mathrm{d}}}x\preceq t^{\frac{\gamma +\beta -\alpha }{\alpha }},\ \ (t,x)\in (0,1)\times {\mathbb R}^d. \end{aligned}$$
    (2.2)
  2. (ii)

    For all \(\beta _1,\beta _2\in [0,\frac{\alpha }{4}]\), \(\gamma _1,\gamma _2\in {\mathbb R}\) and \(0<s<t\leqslant 1\), we have

    $$\begin{aligned}&\int _{{\mathbb R}^d}\varrho ^{\beta _1}_{\gamma _1}(t-s,x-z)\varrho ^{\beta _2}_{\gamma _2}(s,z) {\mathord {\mathrm{d}}}z \nonumber \\&\quad \preceq \left( (t-s)^{\frac{\gamma _1+\beta _1+\beta _2-\alpha }{\alpha }} s^{\frac{\gamma _2}{\alpha }} \right. \left. +(t-s)^{\frac{\gamma _1}{\alpha }} s^{\frac{\gamma _2+\beta _1+\beta _2-\alpha }{\alpha }}\right) \varrho ^0_0(t,x) \nonumber \\&\qquad +(t-s)^{\frac{\gamma _1+\beta _1-\alpha }{\alpha }}s^{\frac{\gamma _2}{\alpha }}\varrho ^{\beta _2}_0(t,x)+(t-s)^{\frac{\gamma _1}{\alpha }}s^{\frac{\gamma _2+\beta _2-\alpha }{\alpha }}\varrho ^{\beta _1}_0(t,x). \end{aligned}$$
    (2.3)
  3. (iii)

    If \(\beta _1,\beta _2\in [0,\frac{\alpha }{4}]\), \(\gamma _1+\beta _1>0\) and \(\gamma _2+\beta _2>0\), then

    $$\begin{aligned}&\int ^t_0\!\!\!\int _{{\mathbb R}^d}\varrho ^{\beta _1}_{\gamma _1}(t-s,x-z)\varrho ^{\beta _2}_{\gamma _2}(s,z){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s \nonumber \\&\qquad \preceq {\mathcal {B}}(\tfrac{\gamma _1+\beta _1}{\alpha },\tfrac{\gamma _2+\beta _2}{\alpha }) \left( \varrho ^0_{\gamma _1+\gamma _2+\beta _1+\beta _2} +\varrho ^{\beta _1}_{\gamma _1+\gamma _2+\beta _2} +\varrho ^{\beta _2}_{\gamma _1+\gamma _2+\beta _1}\right) (t,x), \end{aligned}$$
    (2.4)

    where \({\mathcal {B}}(\gamma ,\beta )\) is the usual Beta function defined by

    $$\begin{aligned} {\mathcal {B}}(\gamma ,\beta ):=\int ^1_0(1-s)^{\gamma -1}s^{\beta -1}{\mathord {\mathrm{d}}}s,\ \ \gamma ,\beta >0. \end{aligned}$$

    Moreover, the constants contained in “\(\preceq \)” of (2.2)–(2.4) only depend on \(d\) and \(\alpha \).

Proof

(i) Notice that

$$\begin{aligned} \int _{{\mathbb R}^d}\frac{|x|^\beta }{(t^{1/\alpha }+|x|)^{d+\alpha }}{\mathord {\mathrm{d}}}x&\preceq \int ^\infty _0\frac{r^{\beta +d-1}}{(t^{1/\alpha }+r)^{d+\alpha }}{\mathord {\mathrm{d}}}r \!=\!\left( \int ^{t^{1/\alpha }}_0\!+\!\int ^\infty _{t^{1/\alpha }}\right) \frac{r^{\beta +d-1}}{(t^{1/\alpha }+r)^{d+\alpha }}{\mathord {\mathrm{d}}}r \\&\leqslant \int ^{t^{1/\alpha }}_0\frac{r^{d+\beta -1}}{t^{(d+\alpha )/\alpha }}{\mathord {\mathrm{d}}}r\!+\!\int ^\infty _{t^{1/\alpha }}r^{\beta -1-\alpha }{\mathord {\mathrm{d}}}r \!=\!\frac{t^{(\beta -\alpha )/\alpha }}{d{+}\beta }\!{+}\!\frac{t^{(\beta -\alpha )/\alpha }}{\alpha -\beta }, \end{aligned}$$

which implies (2.2) by definition.

(ii) Let \(x, z\in {\mathbb R}^d\) and \(0<s<t\leqslant 1\). In view of

$$\begin{aligned} (t^{1/\alpha }+|x|)^{d+\alpha }\leqslant C_{d,\alpha }\left( ((t-s)^{1/\alpha }+|x-z|)^{d+\alpha }+(s^{1/\alpha }+|z|)^{d+\alpha }\right) , \end{aligned}$$

we have

$$\begin{aligned} \varrho ^0_0(t-s,x-z)\varrho ^0_0(s,z)\leqslant C_{d,\alpha }\left( \varrho ^0_0(t-s,x-z)+\varrho ^0_0(s,z)\right) \varrho ^0_0(t,x). \end{aligned}$$
(2.5)

Noticing that by \((a+b)^\beta \leqslant a^\beta +b^\beta \) for \(\beta \in (0,1)\) and \(a,b>0\),

$$\begin{aligned}&(|x-z|^{\beta _1}\wedge 1)(|z|^{\beta _2}\wedge 1) \leqslant (|x-z|^{\beta _1}\wedge 1)((|x-z|^{\beta _2}+|x|^{\beta _2})\wedge 1) \\&\quad \leqslant |x-z|^{\beta _1+\beta _2}\wedge 1+(|x-z|^{\beta _1}\wedge 1)(|x|^{\beta _2}\wedge 1), \\&\qquad (|x-z|^{\beta _1}\wedge 1)(|z|^{\beta _2}\wedge 1) \leqslant ((|z|^{\beta _1}+|x|^{\beta _1})\wedge 1)(|z|^{\beta _2}\wedge 1)\\&\quad \leqslant |z|^{\beta _1+\beta _2}\wedge 1+(|x|^{\beta _1}\wedge 1)(|z|^{\beta _2}\wedge 1), \end{aligned}$$

we have

$$\begin{aligned}&\varrho ^{\beta _1}_{\gamma _1}(t-s,x-z)\varrho ^{\beta _2}_{\gamma _2}(s,z) =(t-s)^{\frac{\gamma _1}{\alpha }}s^{\frac{\gamma _2}{\alpha }}(|x-z|^{\beta _1}\wedge 1)(|z|^{\beta _2}\wedge 1) \varrho ^0_0(t-s,x-z) \varrho ^0_0(s,z)\\&\quad \preceq (t-s)^{\frac{\gamma _1}{\alpha }}s^{\frac{\gamma _2}{\alpha }}\left\{ |x-z|^{\beta _1+\beta _2}\wedge 1+(|x-z|^{\beta _1}\wedge 1)(|x|^{\beta _2}\wedge 1)\right\} \varrho ^0_0(t-s,x-z) \varrho ^0_0(t,x)\\&\qquad +(t-s)^{\frac{\gamma _1}{\alpha }}s^{\frac{\gamma _2}{\alpha }}\left\{ |z|^{\beta _1+\beta _2}\wedge 1+(|x|^{\beta _1}\wedge 1)(|z|^{\beta _2}\wedge 1)\right\} \varrho ^0_0(s,z)\varrho ^0_0(t,x)\\&\quad \preceq s^{\frac{\gamma _2}{\alpha }}\left\{ \varrho ^{\beta _1+\beta _2}_{\gamma _1}(t-s,x-z)\varrho ^0_0(t,x) +\varrho ^{\beta _1}_{\gamma _1}(t-s,x-z)\varrho ^{\beta _2}_0(t,x)\right\} \\&\qquad +(t-s)^{\frac{\gamma _1}{\alpha }}\left\{ \varrho ^{\beta _1+\beta _2}_{\gamma _2}(s,z)\varrho ^0_0(t,x) +\varrho ^{\beta _2}_{\gamma _2}(s,z)\varrho ^{\beta _1}_0(t,x)\right\} . \end{aligned}$$

Integrating both sides with respect to \(z\) and using (i), we obtain (ii).

(iii) Observe that for \(\gamma ,\beta >0\),

$$\begin{aligned} \int ^t_0(t-s)^{\gamma -1}s^{\beta -1}{\mathord {\mathrm{d}}}s=t^{\gamma +\beta -1}{\mathcal {B}}(\gamma ,\beta ). \end{aligned}$$
(2.6)

Integrating both sides of (2.3) with respect to \(s\) from \(0\) to \(t\), we obtain

$$\begin{aligned}&\int ^t_0\!\!\!\int _{{\mathbb R}^d}\varrho ^{\beta _1}_{\gamma _1}(t-s,x-z)\varrho ^{\beta _2}_{\gamma _2}(s,z){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s\nonumber \\&\quad \preceq t^{\frac{\gamma _1+\gamma _2+\beta _1+\beta _2}{\alpha }}\left\{ {\mathcal {B}}\left( \tfrac{\gamma _1+\beta _1+\beta _2}{\alpha },\tfrac{\gamma _2+\alpha }{\alpha }\right) +{\mathcal {B}}\left( \tfrac{\gamma _2+\beta _1+\beta _2}{\alpha },\tfrac{\gamma _1+\alpha }{\alpha }\right) \right\} \varrho ^0_0(t,x)\\&\qquad +t^{\frac{\gamma _1+\gamma _2+\beta _1}{\alpha }}{\mathcal {B}}\left( \tfrac{\gamma _1+\beta _1}{\alpha },\tfrac{\gamma _2+\alpha }{\alpha }\right) \varrho ^{\beta _2}_0(t,x) +t^{\frac{\gamma _1+\gamma _2+\beta _2}{\alpha }}{\mathcal {B}}\left( \tfrac{\gamma _2+\beta _2}{\alpha },\tfrac{\gamma _1+\alpha }{\alpha }\right) \varrho ^{\beta _1}_0(t,x), \end{aligned}$$

which implies (2.4) by \(\beta _1,\beta _2<\alpha \) and that \({\mathcal {B}}(\gamma ,\beta )\) is symmetric and non-increasing with respect to variables \(\gamma \) and \(\beta \). \(\square \)

2.2 Some estimates of the heat kernel of \(\Delta ^{\frac{\alpha }{2}}\)

Let \((Z^{(\alpha )}_t)_{t\geqslant 0}\) be a rotationally invariant \(d\)-dimensional \(\alpha \)-stable process, and \(p_\alpha (t,x)\) its probability transition density function with respect to the Lebesgue measure on \({\mathbb R}^d\). By the scaling property of \(Z^{(\alpha )}_t\mathop {=}\limits ^{(d)}t^{1/\alpha }Z^{(\alpha )}_1\), it is easy to see that

$$\begin{aligned} p_\alpha (t,x)=t^{-d/\alpha }p_\alpha (1,t^{-1/\alpha } x). \end{aligned}$$
(2.7)

Let \((W_t)_{t\geqslant 0}\) be a \(d\)-dimensional standard Brownian motion, and \(S^{(\alpha )}_t\) an independent \(\alpha /2\)-stable subordinator, which is a nonnegative one-dimensional Lévy process with Laplace transform \({\mathbb E}{\mathrm {e}}^{-\lambda S^{(\alpha )}_t}={\mathrm {e}}^{-t\lambda ^{\alpha /2}}\) for \(\lambda >0\). It is well-known that \(Z^{(\alpha )}_t\) can be realized as

$$\begin{aligned} Z^{(\alpha )}_t=W_{S^{(\alpha )}_t}. \end{aligned}$$

Let \(\eta _t(s)\) be the density of \(S^{(\alpha )}_t\). By subordination, we have

$$\begin{aligned} p_\alpha (t,x)=\int ^\infty _0(2\pi s)^{-\frac{d}{2}}{\mathrm {e}}^{-\frac{|x|^2}{2s}}\eta _t(s){\mathord {\mathrm{d}}}s. \end{aligned}$$

By [5], Theorem 2.1], one knows that

$$\begin{aligned} p_\alpha (t,x)\asymp \varrho ^0_\alpha (t,x)=t(t^{1/\alpha }+|x|)^{-d-\alpha }. \end{aligned}$$
(2.8)

The following obvious inequality will be used frequently:

$$\begin{aligned} (t^{1/\alpha }+|x+z|)^{-\gamma }\leqslant 4^\gamma (t^{1/\alpha }+|x|)^{-\gamma } \quad \hbox {for } |z|\leqslant (2t^{1/\alpha })\vee (|x|/2). \end{aligned}$$
(2.9)

Below, for a function \(f\) defined on \({\mathbb R}_+\times {\mathbb R}^d\), we shall simply write

$$\begin{aligned} \delta _f(t,x;z):=f(t,x+z)+f(t,x-z)-2f(t,x). \end{aligned}$$
(2.10)

We need the following lemma.

Lemma 2.2

For each \(k\in {\mathbb N}\), there is a constant \(C=C(d, \alpha , k)>0\) so that for every \(t>0,\, x, x', z\in {\mathbb R}^d\),

$$\begin{aligned}&\quad \qquad \qquad \quad \quad \quad |\nabla ^kp_\alpha (t,x)| \leqslant C \, t(t^{1/\alpha }+|x|)^{-d-\alpha -k}, \end{aligned}$$
(2.11)
$$\begin{aligned}&\quad |p_\alpha (t,x)-p_\alpha (t,x')| \leqslant C\, ((t^{-1/\alpha }|x-x'|)\wedge 1)\left( p_\alpha (t,x)+p_\alpha (t,x')\right) , \end{aligned}$$
(2.12)
$$\begin{aligned}&\quad |\delta _{p_\alpha }(t,x;z)| \leqslant C\, ((t^{-\frac{2}{\alpha }}|z|^2)\wedge 1)\left( p_\alpha (t,x\pm z)+p_\alpha (t,x)\right) , \end{aligned}$$
(2.13)
$$\begin{aligned}&\quad |\delta _{p_\alpha }(t,x;z)-\delta _{p_\alpha }(t,x';z)| \leqslant C\, ((t^{-1/\alpha }|x-x'|)\wedge 1)((t^{-\frac{2}{\alpha }}|z|^2)\wedge 1) \nonumber \\&\qquad \times \left( p_\alpha (t,x\pm z)+p_\alpha (t,x)+p_\alpha (t,x'\pm z)+p_\alpha (t,x')\right) , \end{aligned}$$
(2.14)

where \(\nabla ^k\) stands for the \(k\)-order gradient with respect to the spatial variable \(x\).

Proof

By the scaling property (2.7), it suffices to prove these estimates for \(t=1\).

  1. (i)

    Noticing that (cf. [19], Theorem 37.1])

    $$\begin{aligned} \eta _1(s)\preceq s^{-1-\frac{\alpha }{2}}{\mathrm {e}}^{-s^{-\alpha /2}}\leqslant s^{-1-\frac{\alpha }{2}}, \end{aligned}$$

    we have for \(|x|>1\),

    $$\begin{aligned} |\nabla p_\alpha (1,x)|\preceq |x|\int ^\infty _0s^{-\frac{d}{2}-2-\frac{\alpha }{2}}{\mathrm {e}}^{-\frac{|x|^2}{2s}}{\mathord {\mathrm{d}}}s=|x|^{-d-\alpha -1}\int ^\infty _0u^{\frac{d+\alpha }{2}}{\mathrm {e}}^{-\frac{u}{2}}{\mathord {\mathrm{d}}}u. \end{aligned}$$

    Hence,

    $$\begin{aligned} |\nabla p_\alpha (1,x)|\preceq (1+|x|)^{-d-\alpha -1},\ \ x\in {\mathbb R}^d, \end{aligned}$$

    which gives (2.11) for \(k=1\). The estimates of higher order derivatives are similar.

  2. (ii)

    Observe that

    $$\begin{aligned} p_\alpha (1,x)-p_\alpha (1,x')= (x-x')\cdot \int ^1_0\nabla p_\alpha (1,x+\theta (x'-x)){\mathord {\mathrm{d}}}\theta . \end{aligned}$$
    (2.15)

    If \(|x-x'|\leqslant 1\), then by (2.11), we have

    $$\begin{aligned} | p_\alpha (1,x)-p_\alpha (1,x')|&\preceq |x-x'|\int ^1_0(1+|x+\theta (x'-x)|)^{-d-\alpha -1}{\mathord {\mathrm{d}}}\theta \\&\mathop {\preceq }\limits ^{(2.9)} |x-x'|(1+|x|)^{-d-\alpha -1}\mathop {\preceq }\limits ^{(2.8)} |x-x'|p_\alpha (1,x). \end{aligned}$$

    So,

    $$\begin{aligned} |p_\alpha (1,x)-p_\alpha (1,x')|\preceq (|x-x'|\wedge 1)\left\{ p_\alpha (1,x)+p_\alpha (1,x')\right\} . \end{aligned}$$

    Estimate (2.12) follows.

  3. (iii)

    By using (2.15) twice, we have

    $$\begin{aligned} \delta _{p_\alpha }(1,x;z)&= p_\alpha (1,x+z)+p_\alpha (1,x-z)-2p_\alpha (1,x) \nonumber \\&= z\cdot \int ^1_0(\nabla p_\alpha (1,x+\theta z)-\nabla p_\alpha (1,x-\theta z)){\mathord {\mathrm{d}}}\theta \nonumber \nonumber \\&= 2(z\otimes z)\cdot \int ^1_0\!\!\!\int ^1_0\theta \nabla ^2 p_\alpha (1,x+(1-2\theta ')\theta z){\mathord {\mathrm{d}}}\theta '{\mathord {\mathrm{d}}}\theta . \end{aligned}$$
    (2.16)

    If \(|z|>1\), then

    $$\begin{aligned} |\delta _{p_\alpha }(1,x;z)|\leqslant p_\alpha (1,x+z)+p_\alpha (1,x-z)+2p_\alpha (1,x). \end{aligned}$$

    If \(|z|\leqslant 1\), then by (2.16) and (2.11), we have

    $$\begin{aligned} |\delta _{p_\alpha }(1,x;z)|&\leqslant 2|z|^2 \int ^1_0\!\!\!\int ^1_0|\nabla ^2 p_\alpha (1,x+(1-2\theta ')\theta z)|{\mathord {\mathrm{d}}}\theta '{\mathord {\mathrm{d}}}\theta \\&\preceq |z|^2\int ^1_0\!\!\!\int ^1_0(1+|x+(1-2\theta ')\theta z )|)^{-d-\alpha -2}{\mathord {\mathrm{d}}}\theta '{\mathord {\mathrm{d}}}\theta \\&\mathop {\preceq }\limits ^{(2.9)}|z|^2(1+|x|)^{-d-\alpha -2}\mathop {\preceq }\limits ^{(2.8)}|z|^2p_\alpha (1,x). \end{aligned}$$

    Hence,

    $$\begin{aligned} |\delta _{p_\alpha }(1,x;z)|\preceq (|z|^2\wedge 1)\left\{ p_\alpha (1,x\pm z)+p_\alpha (1,x)\right\} \!, \end{aligned}$$
    (2.17)

    which yields (2.13).

  4. (iv)

    If \(|z|\leqslant 1\) and \(|x-x'|\leqslant 1\), then by (2.16) and (2.11), we have

    $$\begin{aligned}&|\delta _{p_\alpha }(1,x;z)-\delta _{p_\alpha }(1,x';z)| \nonumber \\&\quad \preceq |x-x'|\cdot |z|^2\int ^1_0\!\!\!\int ^1_0\!\!\!\int ^1_0|\nabla ^3p_\alpha |(1,x+(1-2\theta ')\theta z+\theta ''(x'-x)){\mathord {\mathrm{d}}}\theta ''{\mathord {\mathrm{d}}}\theta '{\mathord {\mathrm{d}}}\theta \nonumber \\&\quad \preceq |x-x'|\cdot |z|^2\int ^1_0\!\!\!\int ^1_0\!\!\!\int ^1_0(1+|x+(1-2\theta ')\theta z+\theta ''(x'-x)|)^{-d-\alpha -3}{\mathord {\mathrm{d}}}\theta ''{\mathord {\mathrm{d}}}\theta '{\mathord {\mathrm{d}}}\theta \nonumber \\&\quad \mathop {\preceq }\limits ^{(2.9)} |x-x'|\cdot |z|^2(1+|x|)^{-d-\alpha -3}\mathop {\preceq }\limits ^{(2.8)}|x-x'|\cdot |z|^2p_\alpha (1,x). \end{aligned}$$
    (2.18)

    If \(|z|>1\) and \(|x-x'|\leqslant 1\), then we have

    $$\begin{aligned}&|\delta _{p_\alpha }(1,x;z)-\delta _{p_\alpha }(1,x';z)| \preceq |x-x'|\int ^1_0|\nabla p_\alpha (1,x\pm z+\theta (x'-x))|{\mathord {\mathrm{d}}}\theta \nonumber \\&\qquad +|x-x'|\int ^1_0|\nabla p_\alpha (1,x+\theta (x'-x))|{\mathord {\mathrm{d}}}\theta \nonumber \\&\quad \mathop {\preceq }\limits ^{(2.9)} |x-x'|\left( (1+|x\pm z|)^{-d-\alpha -1}+(1+|x|)^{-d-\alpha -1}\right) \nonumber \\&\quad \mathop {\preceq }\limits ^{(2.8)}|x-x'|\left( p_\alpha (1,x\pm z)+p_\alpha (1,x)\right) . \end{aligned}$$
    (2.19)

    Combining (2.17), (2.18) and (2.19), we obtain

    $$\begin{aligned}&|\delta _{p_\alpha }(1,x;z)-\delta _{p_\alpha }(1,x';z)|\\&\quad \preceq (|x-x'|)\wedge 1)(|z|^2\wedge 1) \left( p_\alpha (1,x\pm z)+p_\alpha (1,x)\right. \\&\qquad \left. +p_\alpha (1,x'\pm z) +p_\alpha (1,x')\right) , \end{aligned}$$

    which implies (2.14). The proof is complete.

\(\square \)

2.3 Fractional derivative estimate of heat kernel of \({\fancyscript{L}}^\kappa _\alpha \)

Throughout this subsection and next, \(\kappa (x, z)= \kappa (z)\) is a measurable function on \({\mathbb R}^d\) with

$$\begin{aligned} \kappa (z)=\kappa (-z),\ \ 0<\kappa _0\leqslant \kappa (z)\leqslant \kappa _1. \end{aligned}$$
(2.20)

Consider the following nonlocal symmetric operator

$$\begin{aligned} {\fancyscript{L}}^\kappa _\alpha f(x):= & {} \hbox {p.v.}\int _{{\mathbb R}^d} (f(x+z)-f(x))\kappa (z)|z|^{-d-\alpha }{\mathord {\mathrm{d}}}z \nonumber \\= & {} \frac{1}{2}\int _{{\mathbb R}^d} \delta _f(x;z)\kappa (z)|z|^{-d-\alpha }{\mathord {\mathrm{d}}}z, \end{aligned}$$

where \(\delta _f(x;z)\) is defined in a similar way as in (2.10) but with function \(f\) not containing \(t\) variable. It is the infinitesimal generator of a symmetric Lévy process that is stable-like. Let \(p^\kappa _{\alpha } (t,x)\) be the heat kernel of operator \({\fancyscript{L}}^\kappa _\alpha \), i.e.,

$$\begin{aligned} \partial _t p^\kappa _{\alpha } (t,x)={\fancyscript{L}}^\kappa _\alpha p^\kappa _{\alpha } (t,x),\ \ \lim _{t\downarrow 0}p^\kappa _{\alpha } (t,x)=\delta _0(x). \end{aligned}$$

Under (2.20), it is well-known from the inverse Fourier transform that

$$\begin{aligned} p^\kappa _\alpha \in C\left( {\mathbb R}_+; C^\infty _b({\mathbb R}^d)\right) . \end{aligned}$$
(2.21)

Moreover, it follows from [13], Theorem 1.1] that

$$\begin{aligned} p^\kappa _{\alpha } (t,x)\asymp \varrho ^0_\alpha (t,x)=t(t^{1/\alpha }+|x|)^{-d-\alpha }. \end{aligned}$$
(2.22)

If we set

$$\begin{aligned} \hat{\kappa }(z):=\kappa (z)-\tfrac{\kappa _0}{2}, \end{aligned}$$

then by the construction of the Lévy process, one can write

$$\begin{aligned} p^\kappa _{\alpha } (t,x)=\int _{{\mathbb R}^d}p^{\kappa _0/2}_\alpha (t,x-y)p^{\hat{\kappa }}_\alpha (t,y){\mathord {\mathrm{d}}}y=\int _{{\mathbb R}^d}p_\alpha \left( \tfrac{\kappa _0t}{2},x-y\right) p^{\hat{\kappa }}_\alpha (t,y){\mathord {\mathrm{d}}}y.\nonumber \\ \end{aligned}$$
(2.23)

The following lemma is an easy consequence of (2.22), (2.23) and Lemma 2.2.

Lemma 2.3

Under (2.20), there exists a constant \(C=C(d,\alpha ,\kappa _0,\kappa _1, \kappa _2)>0\) such that for every \(t>0\) and \(x,x',z\in {\mathbb R}^d\),

$$\begin{aligned}&\displaystyle |p^\kappa _{\alpha } (t,x)-p^\kappa _{\alpha } (t,x')|\leqslant C((t^{-1/\alpha }|x-x'|)\wedge 1)\left( \varrho ^0_\alpha (t,x)+\varrho ^0_\alpha (t,x')\right) , \qquad \end{aligned}$$
(2.24)
$$\begin{aligned}&\displaystyle |\nabla p^\kappa _{\alpha }(t,x)|\leqslant Ct^{-1/\alpha }\varrho ^0_\alpha (t,x), \end{aligned}$$
(2.25)
$$\begin{aligned}&\displaystyle |\delta _{p^\kappa _{\alpha } }(t,x;z)|\leqslant C \left( (t^{-\frac{2}{\alpha }}|z|^2)\wedge 1 \right) \left( \varrho ^0_\alpha (t,x\pm z)+\varrho ^0_\alpha (t,x)\right) , \end{aligned}$$
(2.26)
$$\begin{aligned}&\displaystyle |\delta _{p^\kappa _{\alpha } }(t,x;z)-\delta _{p^\kappa _{\alpha } }(t,x';z)| \leqslant C \left( (t^{-1/\alpha }|x-x'|)\wedge 1\right) \left( (t^{-\frac{2}{\alpha }}|z|^2)\wedge 1\right) \nonumber \\&\displaystyle \times \left( \varrho ^0_\alpha (t,x\pm z)+\varrho ^0_\alpha (t,x)+\varrho ^0_\alpha (t,x'\pm z)+\varrho ^0_\alpha (t,x')\right) . \end{aligned}$$
(2.27)

Proof

By (2.23) and (2.12), we have

$$\begin{aligned}&|p^\kappa _{\alpha } (t,x)-p^\kappa _{\alpha } (t,x')| \preceq ((t^{-1/\alpha }|x-x'|)\wedge 1) \\&\qquad \times \int _{{\mathbb R}^d}\left\{ p_\alpha \left( \tfrac{\kappa _0t}{2},x-y\right) c+p_\alpha \left( \tfrac{\kappa _0t}{2},x'-y\right) \right\} p^{\hat{\kappa }}_\alpha (t,y){\mathord {\mathrm{d}}}y \\&\quad =((t^{-1/\alpha }|x-x'|)\wedge 1)\left\{ p^\kappa _{\alpha } (t,x)+p^\kappa _{\alpha } (t,x')\right\} \\&\quad \mathop {\preceq }\limits ^{(2.22)}((t^{-1/\alpha }|x-x'|)\wedge 1)\left\{ \varrho ^0_\alpha (t,x)+\varrho ^0_\alpha (t,x')\right\} . \end{aligned}$$

Similarly, we have (2.25), (2.26) and (2.27) by (2.23), (2.11), (2.13), (2.14) and (2.22). \(\square \)

Now, we can prove the following fractional derivative estimate of \(p^\kappa _\alpha (t,x)\).

Theorem 2.4

Under (2.20), there exists a constant \(C=C(d,\alpha ,\kappa _0,\kappa _1, \kappa _2)>0\) such that for all \(t\in (0,1]\) and \(x,x'\in {\mathbb R}^d\),

$$\begin{aligned}&\displaystyle \int _{{\mathbb R}^d}|\delta _{p^\kappa _{\alpha } }(t,x;z)|\cdot |z|^{-d-\alpha }{\mathord {\mathrm{d}}}z \leqslant C\varrho ^0_0(t,x), \end{aligned}$$
(2.28)
$$\begin{aligned}&\int _{{\mathbb R}^d}|\delta _{p^\kappa _{\alpha } }(t,x;z)-\delta _{p^\kappa _{\alpha } }(t,x';z)|\cdot |z|^{-d-\alpha }{\mathord {\mathrm{d}}}z \leqslant C((t^{-1/\alpha }|x-x'|)\wedge 1)\nonumber \\&\times \left\{ \varrho ^0_0(t,x)+\varrho ^0_0(t,x')\right\} . \end{aligned}$$
(2.29)

Proof

By (2.26), we have

$$\begin{aligned} \int _{{\mathbb R}^d}|\delta _{p^\kappa _{\alpha } }(t,x;z)|\cdot |z|^{-d-\alpha }{\mathord {\mathrm{d}}}z&\preceq \int _{{\mathbb R}^d}((t^{-\frac{2}{\alpha }}|z|^2)\wedge 1)\varrho ^0_\alpha (t,x\pm z)|z|^{-d-\alpha }{\mathord {\mathrm{d}}}z\\&\quad +\varrho ^0_\alpha (t,x)\int _{{\mathbb R}^d}((t^{-\frac{2}{\alpha }}|z|^2)\wedge 1)|z|^{-d-\alpha }{\mathord {\mathrm{d}}}z \\&=:I_1+I_2. \end{aligned}$$

For \(I_1\), we have

$$\begin{aligned} I_1\leqslant & {} t^{-\frac{2}{\alpha }}\int _{|z|\leqslant t^{1/\alpha }}\varrho ^0_\alpha (t,x\pm z)|z|^{2-d-\alpha }{\mathord {\mathrm{d}}}z\\&+\int _{|z|>t^{1/\alpha }}\varrho ^0_\alpha (t,x\pm z)|z|^{-d-\alpha }{\mathord {\mathrm{d}}}z =:I_{11}+I_{12}. \end{aligned}$$

For \(I_{11}\), by (2.9), we have

$$\begin{aligned} I_{11}&\preceq t^{1-\frac{2}{\alpha }}\int _{|z|\leqslant t^{1/\alpha }}(t^{1/\alpha }+|x\pm z|)^{-d-\alpha }|z|^{2-d-\alpha }{\mathord {\mathrm{d}}}z \\&\preceq t^{1-\frac{2}{\alpha }}(t^{1/\alpha }+|x|)^{-d-\alpha }\int _{|z|\leqslant t^{1/\alpha }}|z|^{2-d-\alpha }{\mathord {\mathrm{d}}}z\preceq \varrho ^0_0(t,x). \end{aligned}$$

For \(I_{12}\), if \(|x|\leqslant 2t^{1/\alpha }\), then

$$\begin{aligned} I_{12}&\preceq t\int _{|z|>t^{1/\alpha }}(t^{1/\alpha }+|x\pm z|)^{-d-\alpha }|z|^{-d-\alpha }{\mathord {\mathrm{d}}}z\\&\preceq t^{-d/\alpha }\int _{|z|>t^{1/\alpha }}|z|^{-d-\alpha }{\mathord {\mathrm{d}}}z\preceq t^{-\frac{d+\alpha }{\alpha }}\leqslant \varrho ^0_0(t,x); \end{aligned}$$

if \(|x|>2t^{1/\alpha }\), then

$$\begin{aligned} I_{12}&\preceq \left( \int _{\frac{|x|}{2}\geqslant |z|>t^{1/\alpha }}+\int _{|z|>\frac{|x|}{2}}\right) \varrho ^0_\alpha (t,x\pm z)\cdot |z|^{-d-\alpha }{\mathord {\mathrm{d}}}z\\&\preceq t\int _{\frac{|x|}{2}\geqslant |z|>t^{1/\alpha }} (t^{1/\alpha }{+}|x\pm z|)^{-d-\alpha }|z|^{-d-\alpha }{\mathord {\mathrm{d}}}z+|x|^{-d-\alpha }\int _{|z|>\frac{|x|}{2}}\varrho ^0_\alpha (t,x\pm z){\mathord {\mathrm{d}}}z\\&\preceq t(t^{1/\alpha }+|x|)^{-d-\alpha }\int _{|z|>t^{1/\alpha }} |z|^{-d-\alpha }{\mathord {\mathrm{d}}}z+|x|^{-d-\alpha }\int _{{\mathbb R}^d}\varrho ^0_\alpha (t,x\pm z){\mathord {\mathrm{d}}}z\\&\preceq (t^{1/\alpha }+|x|)^{-d-\alpha }+|x|^{-d-\alpha }\preceq \varrho ^0_0(t,x). \end{aligned}$$

For \(I_2\), we have

$$\begin{aligned} I_2= t^{-1}\varrho ^0_\alpha (t,x)\int _{{\mathbb R}^d}(|z|^2\wedge 1)|z|^{-d-\alpha }{\mathord {\mathrm{d}}}z\preceq \varrho ^0_0(t,x). \end{aligned}$$

Combining the above calculations, we obtain (2.28).

By (2.27), as above, we have

$$\begin{aligned}&\int _{{\mathbb R}^d}|\delta _{p^\kappa _{\alpha } }(t,x;z)-\delta _{p^\kappa _{\alpha } }(t,x';z)|\cdot |z|^{-d-\alpha }{\mathord {\mathrm{d}}}z\preceq \left( \left( t^{-1/\alpha }|x-x'|\right) \wedge 1\right) \\&\quad \times \left\{ \int _{{\mathbb R}^d}\left( \left( t^{-\frac{2}{\alpha }}|z|^2\right) \wedge 1\right) \left\{ \varrho ^0_\alpha (t,x\pm z)+\varrho ^0_\alpha (t,x'\pm z)\right\} |z|^{-d-\alpha }{\mathord {\mathrm{d}}}z\right. \\&\qquad \left. +\left\{ \varrho ^0_\alpha (t,x)+\varrho ^0_\alpha (t,x')\right\} \int _{{\mathbb R}^d}\left( \left( t^{-\frac{2}{\alpha }}|z|^2\right) \wedge 1\right) |z|^{-d-\alpha }{\mathord {\mathrm{d}}}z\right\} \\&\quad \preceq \left( \left( t^{-1/\alpha }|x-x'|\right) \wedge 1\right) \left\{ \varrho ^0_0(t,x)+\varrho ^0_0(t,x')\right\} . \end{aligned}$$

The proof is complete. \(\square \)

2.4 Continuous dependence of heat kernels with respect to \(\kappa \)

In this subsection, we prove the following continuous dependence of the heat kernel with respect to the kernel function \(\kappa \) that is a function of \(z\) only and satisfies (2.20). To the best of our knowledge, these results are new.

Theorem 2.5

Let \(\kappa \) and \(\tilde{\kappa }\) be two functions on \({\mathbb R}^d\) satisfying (2.20). For any \(\gamma \in (0,\frac{\alpha }{4} )\), there exists a constant \(C=C(d,\alpha ,\kappa _0,\kappa _1, \kappa _2,\gamma )>0\) such that for every \(t\in (0, 1]\) and \(x, y\in {\mathbb R}^d\),

$$\begin{aligned} \left| p^{\kappa }_\alpha (t,x)-p^{\tilde{\kappa }}_\alpha (t,x)\right|&\leqslant C\Vert \kappa -\tilde{\kappa }\Vert _\infty \left( \varrho ^0_{\alpha }+\varrho ^\gamma _{\alpha -\gamma }\right) (t,x), \end{aligned}$$
(2.30)
$$\begin{aligned} \left| \nabla p^{\kappa }_\alpha (t,x)-\nabla p^{\tilde{\kappa }}_\alpha (t,x)\right|&\leqslant C\Vert \kappa -\tilde{\kappa }\Vert _\infty t^{-1/\alpha }\left( \varrho ^0_{\alpha }+\varrho ^\gamma _{\alpha -\gamma }\right) (t,x), \end{aligned}$$
(2.31)

and

$$\begin{aligned} \int _{{\mathbb R}^d}|\delta _{p^{\kappa }_\alpha }(t,x;z)-\delta _{p^{\tilde{\kappa }}_\alpha }(t,x;z)|\cdot |z|^{-d-\alpha }{\mathord {\mathrm{d}}}z\leqslant C\Vert \kappa -\tilde{\kappa }\Vert _\infty \left( \varrho ^0_0+\varrho ^\gamma _{-\gamma }\right) (t,x). \end{aligned}$$
(2.32)

Proof

(i) Note that the heat kernel \(p^\kappa _\alpha (t, x)\) is an even function in \(x\). We have

$$\begin{aligned}&p^{\kappa }_\alpha (t,x)-p^{\tilde{\kappa }}_\alpha (t,x) =\int ^t_0 \frac{{\mathord {\mathrm{d}}}}{{\mathord {\mathrm{d}}}s} \left( \int _{{\mathbb R}^d} p^{\kappa }_\alpha (s,y) p^{\tilde{\kappa }}_\alpha (t-s,x-y) {\mathord {\mathrm{d}}}y\right) {\mathord {\mathrm{d}}}s \\&\quad =\int ^t_0 \left( \int _{{\mathbb R}^d} \left( {\fancyscript{L}}^{\kappa }_\alpha p^{\kappa }_\alpha (s,y) p^{\tilde{\kappa }}_\alpha (t\!-\!s,x-y)-p^{\kappa }_\alpha (s,y) {\fancyscript{L}}^{\tilde{\kappa }}_\alpha p^{\tilde{\kappa }}_\alpha (t\!-\!s,x\!-\!y) \right) {\mathord {\mathrm{d}}}y \right) {\mathord {\mathrm{d}}}s\\&\quad =\int ^t_0 \left( \int _{{\mathbb R}^d}p^{\kappa }_\alpha (s,y)\left( {\fancyscript{L}}^{\kappa }_\alpha -{\fancyscript{L}}^{\tilde{\kappa }}_\alpha \right) p^{\tilde{\kappa }}_\alpha (t-s,x-y){\mathord {\mathrm{d}}}y \right) {\mathord {\mathrm{d}}}s\\&\quad =\int ^t_0 \left( \int _{{\mathbb R}^d}\left( {\fancyscript{L}}^{\kappa }_\alpha -{\fancyscript{L}}^{\tilde{\kappa }}_\alpha \right) p^{\tilde{\kappa }}_\alpha (t-s,x-y)\left( p^{\kappa }_\alpha (s,y)-p^{\kappa }_\alpha (s,x)\right) {\mathord {\mathrm{d}}}y \right) {\mathord {\mathrm{d}}}s, \end{aligned}$$

where the third equality is due to the symmetry of the operator \({\fancyscript{L}}^{\tilde{\kappa }}_\alpha \), (2.22), (2.28) and (2.3), and the fourth equality is due to

$$\begin{aligned} \int _{{\mathbb R}^d}p^{\tilde{\kappa }}_\alpha (t-s,x-y){\mathord {\mathrm{d}}}y=1. \end{aligned}$$

Thus, by (2.24) and (2.28), we have

$$\begin{aligned} \left| p^{\kappa }_\alpha (t,x)-p^{\tilde{\kappa }}_\alpha (t,x)\right|&\leqslant \frac{1}{2}\Vert \kappa -\tilde{\kappa }\Vert _\infty \int ^t_0\!\!\!\int _{{\mathbb R}^d} \left( \int _{{\mathbb R}^d}|\delta _{p^{\tilde{\kappa }}_\alpha }(t-s,x-y;z)|\cdot |z|^{-d-\alpha }{\mathord {\mathrm{d}}}z\right) \\&\quad \times \left| p^{\kappa }_\alpha (s,y)-p^{\kappa }_\alpha (s,x)\right| {\mathord {\mathrm{d}}}y{\mathord {\mathrm{d}}}s \\&\preceq \Vert \kappa -\tilde{\kappa }\Vert _\infty \int ^t_0\!\!\!\int _{{\mathbb R}^d}\varrho ^0_0(t-s,x-y)\\&\quad \times \left( \left( s^{-1/\alpha }|x-y|\right) \wedge 1\right) \left( \varrho ^0_\alpha (s,y)+\varrho ^0_\alpha (s,x)\right) {\mathord {\mathrm{d}}}y{\mathord {\mathrm{d}}}s\\&\leqslant \Vert \kappa -\tilde{\kappa }\Vert _\infty \int ^t_0\!\!\!\int _{{\mathbb R}^d}\varrho ^0_0(t-s,x-y)\\&\quad \times \left( \left( s^{-1/\alpha }|x-y|\right) ^\gamma \wedge 1\right) \left( \varrho ^0_\alpha (s,y)+\varrho ^0_\alpha (s,x)\right) {\mathord {\mathrm{d}}}y{\mathord {\mathrm{d}}}s\\&\leqslant \Vert \kappa -\tilde{\kappa }\Vert _\infty \int ^t_0\!\!\!\int _{{\mathbb R}^d}\varrho ^\gamma _0(t-s,x-y)\left( \varrho ^0_{\alpha -\gamma }(s,y)+\varrho ^0_{\alpha -\gamma }(s,x)\right) {\mathord {\mathrm{d}}}y{\mathord {\mathrm{d}}}s \\&\mathop {\preceq }\limits ^{(2.4)}\Vert \kappa -\tilde{\kappa }\Vert _\infty \left\{ \varrho ^0_{\alpha }(t,x)+\varrho ^\gamma _{\alpha -\gamma }(t,x)\right\} , \end{aligned}$$

which gives (2.30).

(ii) By (2.23), (2.25) and (2.30), we have

$$\begin{aligned} \left| \nabla p^{\kappa }_\alpha (t,x)-\nabla p^{\tilde{\kappa }}_\alpha (t,x)\right|&=\left| \int _{{\mathbb R}^d}\nabla p_\alpha \left( \tfrac{\kappa _0t}{2},x-y\right) \left( p_{\hat{\kappa }}(t,y)-p_{\hat{ \tilde{\kappa }}}(t,y)\right) {\mathord {\mathrm{d}}}y\right| \\&\preceq \Vert \kappa -\tilde{\kappa }\Vert _\infty t^{-1/\alpha }\int _{{\mathbb R}^d}\varrho ^0_\alpha (t,x-y)\left( \varrho ^0_{\alpha }+\varrho ^\gamma _{\alpha -\gamma }\right) (t,y){\mathord {\mathrm{d}}}y\\&\mathop {\preceq }\limits ^{(2.3)}\Vert \kappa -\tilde{\kappa }\Vert _\infty t^{-1/\alpha }\left( \varrho ^0_{\alpha }+\varrho ^\gamma _{\alpha -\gamma }\right) (2t,x), \end{aligned}$$

which gives (2.31).

(iii) By (2.23), (2.26) and (2.30), we have

By using this estimate and the same argument used in deriving (2.29), we obtain (2.32). \(\square \)

3 Levi’s construction of heat kernels

In this section we consider the spatial dependent operator \({\fancyscript{L}}^{\kappa }_{\alpha }\) defined by (1.4), with the kernel function \(\kappa (x, z)\) satisfying conditions (1.5)–(1.6). In order to reflect the dependence on \(x\), we also write

$$\begin{aligned} {\fancyscript{L}}^{\kappa (x)}_{\alpha } f(x)={\fancyscript{L}}^{\kappa }_{\alpha } f(x)=\frac{1}{2}\int _{{\mathbb R}^d} \delta _f(x;z)\kappa (x,z)|z|^{-d-\alpha }{\mathord {\mathrm{d}}}z. \end{aligned}$$

For fixed \(y\in {\mathbb R}^d\), let \({\fancyscript{L}}^{\kappa (y)}_{\alpha }\) be the freezing operator

$$\begin{aligned} {\fancyscript{L}}^{\kappa (y)}_{\alpha } f(x)=\frac{1}{2}\int _{{\mathbb R}^d} \delta _f(x;z)\kappa (y,z)|z|^{-d-\alpha }{\mathord {\mathrm{d}}}z. \end{aligned}$$

Let \(p_y(t,x):=p^{\kappa (y)}_\alpha (t,x)\) be the heat kernel of operator \({\fancyscript{L}}^{\kappa (y)}_{\alpha }\), i.e.,

$$\begin{aligned} \partial _t p_y(t,x)={\fancyscript{L}}^{\kappa (y)}_{\alpha } p_y(t,x),\ \ \lim _{t\downarrow 0}p_y(t,x)=\delta _0(x), \end{aligned}$$
(3.1)

where, with a little abuse of notation, \(\delta _0(x)\) denotes the usual Dirac function.

Now, we want to seek the heat kernel \(p^\kappa _{\alpha } (t,x,y)\) of \({\fancyscript{L}}^{\kappa }_{\alpha }\) with the following form:

$$\begin{aligned} p^\kappa _{\alpha } (t,x,y)=p_y(t,x-y)+\int ^t_0\!\!\!\int _{{\mathbb R}^d}p_z(t-s,x-z)q(s,z,y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s. \end{aligned}$$
(3.2)

The classical Levi’s method suggests that \(q(t,x,y)\) solves the following integral equation:

$$\begin{aligned} q(t,x,y)=q_0(t,x,y)+\int ^t_0\!\!\!\int _{{\mathbb R}^d}q_0(t-s,x,z)q(s,z,y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s, \end{aligned}$$
(3.3)

where

$$\begin{aligned} q_0(t,x,y):= & {} \left( {\fancyscript{L}}^{\kappa (x)}_{\alpha }-{\fancyscript{L}}^{\kappa (y)}_{\alpha }\right) p_y(t,x-y) \\= & {} \int _{{\mathbb R}^d}\delta _{p_y}(t,x-y;z)(\kappa (x,z)-\kappa (y,z))|z|^{-d-\alpha }{\mathord {\mathrm{d}}}z. \end{aligned}$$

In fact, we formally have

$$\begin{aligned} \partial _tp^\kappa _{\alpha } (t,x,y)&={\fancyscript{L}}^{\kappa (y)}_{\alpha }p_y(t,x-y)+q(t,x,y) \nonumber \\&\quad +\int ^t_0\!\!\!\int _{{\mathbb R}^d}\partial _tp_z(t-s,x-z)q(s,z,y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s \nonumber \\&={\fancyscript{L}}^{\kappa (x)}_{\alpha }p_y(t,x-y)+\int ^t_0\!\!\!\int _{{\mathbb R}^d}{\fancyscript{L}}^{\kappa (x)}_{\alpha }p_z(t-s,x-z)q(s,z,y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s\nonumber \\&={\fancyscript{L}}^{\kappa (x)}_{\alpha }p^\kappa _{\alpha } (t,x,y). \end{aligned}$$
(3.4)

Thus, the main aims of this section are to solve Eq. (3.3), and to make the calculations in (3.4) rigorous.

3.1 Solving Eq. (3.3)

In this subsection, we use Picard’s iteration to solve (3.3).

Theorem 3.1

For \(n\in {\mathbb N}\), define \(q_n(t,x,y)\) recursively by

$$\begin{aligned} q_n(t,x,y):=\int ^t_0\!\!\!\int _{{\mathbb R}^d}q_0(t-s,x,z)q_{n-1}(s,z,y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s. \end{aligned}$$
(3.5)

Under (1.5) and (1.6), the series \(q(t,x,y):=\sum _{n=0}^{\infty }q_n(t,x,y)\) is absolutely and locally uniformly convergent on \((0,1]\times {\mathbb R}^d\times {\mathbb R}^d\) and solves the integral equation (3.3). Moreover, \(q(t,x,y)\) is jointly continuous in \((t,x,y)\in (0,1]\times {\mathbb R}^d\times {\mathbb R}^d\), and has the following estimates: there is a constant \(C_1=C_1 (d, \alpha , \beta , \kappa _0, \kappa _1, \kappa _2)>0\) so that

$$\begin{aligned} |q(t,x,y)| \leqslant C_1 \left( \varrho ^\beta _0+\varrho ^0_\beta \right) (t,x-y), \end{aligned}$$
(3.6)

and for any \(\gamma \in (0,\beta )\), there is a constant \(C_2=C_2 (d, \alpha , \beta , \gamma , \kappa _0, \kappa _1, \kappa _2)>0\) so that

$$\begin{aligned}&|q(t,x,y)-q(t,x',y)| \leqslant C_2 \left( |x-x'|^{\beta -\gamma }\wedge 1\right) \nonumber \\&\qquad \quad \times \left( \left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (t,x-y) +\left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (t,x'-y)\right) . \end{aligned}$$
(3.7)

Proof

Without loss of generality, we assume \(\beta \in (0,\frac{\alpha }{4}]\). We divide the proof into four steps.

Step 1. First of all, by (1.5), (1.6) and (2.28), we have

$$\begin{aligned}&|q_0(t,x,y)| \preceq \left( |x-y|^\beta \wedge 1\right) \int _{{\mathbb R}^d}|\delta _{p_y}(t,x-y;z)|\cdot |z|^{-d-\alpha }{\mathord {\mathrm{d}}}z \nonumber \\&\quad \preceq \left( |x-y|^\beta \wedge 1\right) \varrho ^0_0(t,x-y)=\varrho ^\beta _0(t,x-y). \end{aligned}$$
(3.8)

For \(n=1\), by definition (3.5) and (2.4), there exits a constant \(C_{d,\alpha }>0\) such that

$$\begin{aligned} |q_1(t,x,y)|\leqslant C_{d,\alpha } {\mathcal {B}}( {\beta }/\alpha , \beta /\alpha ) \left\{ \varrho ^0_{2\beta }+\varrho ^\beta _\beta \right\} (t,x-y). \end{aligned}$$
(3.9)

Suppose now that

$$\begin{aligned} |q_n(t,x,y)|\leqslant \gamma _n\left\{ \varrho ^0_{(n+1)\beta }+\varrho ^\beta _{n\beta }\right\} (t,x-y), \end{aligned}$$

where \(\gamma _n>0\) will be determined below. By (2.4), we have

$$\begin{aligned} |q_{n+1}(t,x,y)|&\leqslant C_{d,\alpha }\gamma _{n} {\mathcal {B}}(\beta /\alpha , (n+1)\beta /\alpha ) \left\{ \varrho ^0_{(n+2)\beta }+\varrho ^\beta _{(n+1)\beta }\right\} (t,x-y)\\&=:\gamma _{n+1}\left\{ \varrho ^0_{(n+2)\beta }+\varrho ^\beta _{(n+1)\beta }\right\} (t,x-y), \end{aligned}$$

where

$$\begin{aligned} \gamma _{n+1}=C_{d,\alpha }\gamma _{n} {\mathcal {B}}(\beta /\alpha , (n+1)\beta /\alpha ). \end{aligned}$$

Hence, by \({\mathcal {B}}(\gamma ,\beta )=\frac{\Gamma (\gamma )\Gamma (\beta )}{\Gamma (\gamma +\beta )}\), where \(\Gamma \) is the usual Gamma function, we obtain

$$\begin{aligned} \gamma _{n}= C_{d,\alpha }^{n+1}{\mathcal {B}}(\beta /\alpha ,\beta /\alpha ){\mathcal {B}}(\beta /\alpha ,2\beta /\alpha )\cdots {\mathcal {B}}(\beta /\alpha ,n\beta /\alpha )= \frac{(C_{d,\alpha }\Gamma (\beta /\alpha ))^{n+1}}{\Gamma ((n+1)\beta /\alpha )}. \end{aligned}$$

Thus,

$$\begin{aligned} |q_n(t,x,y)|\leqslant \frac{(C_{d,\alpha }\Gamma (\beta /\alpha ))^{n+1}}{\Gamma ((n+1)\beta /\alpha )} \left\{ \varrho ^0_{(n+1)\beta }+\varrho ^\beta _{n\beta }\right\} (t,x-y), \end{aligned}$$
(3.10)

which in turn implies that

$$\begin{aligned} \sum _{n=0}^{\infty }|q_n(t,x,y)|&\leqslant \sum _{n=0}^\infty \frac{(C_{d,\alpha }\Gamma (\beta /\alpha ))^{n+1}}{\Gamma ((n+1)\beta /\alpha )} \left\{ \varrho ^0_{(n+1)\beta }+\varrho ^\beta _{n\beta }\right\} (t,x-y) \\&\leqslant \sum _{n=0}^\infty \frac{(C_{d,\alpha }\Gamma (\beta /\alpha ))^{n+1}}{\Gamma ((n+1)\beta /\alpha )} \left\{ \varrho ^0_{\beta }+\varrho ^\beta _0\right\} (t,x-y)\\&\preceq \left\{ \varrho ^0_{\beta }+\varrho ^\beta _0\right\} (t,x-y). \end{aligned}$$

Thus, (3.6) is proven. Moreover, by (3.5), we have

$$\begin{aligned} \sum _{n=0}^{m+1}q_n(t,x,y)=q_0(t,x,y)+\int ^t_0\!\!\!\int _{{\mathbb R}^d}q_0(t-s,x,z)\sum _{n=0}^mq_n(s,z,y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s, \end{aligned}$$

which yields (3.3) by taking limits \(m\rightarrow \infty \) for both sides.

Step 2. By (1.6), (2.28), (2.29), (2.32) and (3.10), it is easy to see that \(q_0(t,x,y)\) is jointly continuous in \((t,x,y)\in (0,1]\times {\mathbb R}^d\times {\mathbb R}^d\). Then by definition (3.5) and induction method, for each \(n\in {\mathbb N}\), \(q_n(t,x,y)\) is also jointly continuous in \((t,x,y)\in (0,1]\times {\mathbb R}^d\times {\mathbb R}^d\). On the other hand, by (3.10), the series \(q(t,x,y)=\sum _{n=0}^{\infty }|q_n(t,x,y)|\) is locally uniformly convergent in \((t,x,y)\in (0,1]\times {\mathbb R}^d\times {\mathbb R}^d\). Thus, \(q(t,x,y)\) is a jointly continuous function on \((0,1]\times {\mathbb R}^d\times {\mathbb R}^d\).

Step 3. In this step, we prove the following estimate:

$$\begin{aligned}&|q_0(t,x,y)-q_0(t,x',y)| \preceq (|x-x'|^{\beta -\gamma }\wedge 1) \left\{ \left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (t,x-y)\right. \nonumber \\&\qquad \qquad \left. +\left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (t,x'-y)\right\} . \end{aligned}$$
(3.11)

In the case of \(|x-x'|>1\), we have

$$\begin{aligned} |q_0(t,x,y)|\preceq \varrho ^\beta _0(t,x-y)\leqslant \varrho ^\beta _{\gamma -\beta }(t,x-y) \end{aligned}$$

and

$$\begin{aligned} |q_0(t,x',y)|\preceq \varrho ^\beta _0(t,x'-y)\leqslant \varrho ^\beta _{\gamma -\beta }(t,x'-y). \end{aligned}$$

In the case of \(1\geqslant |x-x'|>t^{1/\alpha }\), by (3.8), we have

$$\begin{aligned} |q_0(t,x,y)|\preceq \varrho ^\beta _0(t,x-y)=t^{\frac{\beta -\gamma }{\alpha }}\varrho ^\beta _{\gamma -\beta }(t,x-y) \leqslant |x-x'|^{\beta -\gamma }\varrho ^\beta _{\gamma -\beta }(t,x-y), \end{aligned}$$

and also

$$\begin{aligned} |q_0(t,x',y)|\preceq |x-x'|^{\beta -\gamma }\varrho ^\beta _{\gamma -\beta }(t,x'-y). \end{aligned}$$

Suppose now that

$$\begin{aligned} |x-x'|\leqslant t^{1/\alpha }. \end{aligned}$$
(3.12)

By definition and Theorem 2.4, we have

$$\begin{aligned}&|q_0(t,x,y)-q_0(t,x',y)| =\left| \int _{{\mathbb R}^d}\delta _{p_y}(t,x-y,z)(\kappa (x,z)-\kappa (y,z))|z|^{-d-\alpha }{\mathord {\mathrm{d}}}z \right. \\&\qquad \left. -\int _{{\mathbb R}^d}\delta _{p_y}(t,x'-y,z)(\kappa (x',z)-\kappa (y,z))|z|^{-d-\alpha }{\mathord {\mathrm{d}}}z\right| \\&\quad \preceq (|x-y|^\beta \wedge 1)\int _{{\mathbb R}^d}|\delta _{p_y}(t,x-y,z)-\delta _{p_y}(t,x'-y,z)|\cdot |z|^{-d-\alpha }{\mathord {\mathrm{d}}}z\\&\qquad +(|x-x'|^\beta \wedge 1)\int _{{\mathbb R}^d}|\delta _{p_y}(t,x'-y,z)|\cdot |z|^{-d-\alpha }{\mathord {\mathrm{d}}}z\\&\quad \preceq (|x-y|^\beta \wedge 1)t^{-1/\alpha }|x-x'|\left\{ \varrho ^0_0(t,x-y)+\varrho ^0_0(t,x'-y)\right\} \\&\qquad +(|x-x'|^\beta \wedge 1)\varrho ^0_0(t,x'-y)\\&\quad \mathop {\preceq }\limits ^{(2.9)} t^{-1/\alpha }|x-x'|\varrho ^\beta _0(t,x-y)+(|x-x'|^\beta \wedge 1)\varrho ^0_0(t,x'-y)\\&\quad \mathop {\preceq }\limits ^{(3.12)} |x-x'|^{\beta -\gamma }\varrho ^\beta _{\gamma -\beta }(t,x-y)+|x-x'|^{\beta -\gamma }\varrho ^0_\gamma (t,x'-y). \end{aligned}$$

Combining the above calculations, we obtain (3.11).

Step 4. By definition (3.5) and (3.10), (3.11), we have for \(n\in {\mathbb N}\),

$$\begin{aligned}&|q_n(t,x,y)-q_n(t,x',y)| \\&\quad \preceq \int _0^t\!\!\!\int _{{\mathbb R}^d}|q_0(t-s,x,z)-q_0(t-s,x',z)|q_{n-1}(s,z,y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s\\&\quad \preceq \frac{(C_d\Gamma (\beta /\alpha ))^{n}}{\Gamma (n\beta /\alpha )} \left( |x-x'|^{\beta -\gamma }\wedge 1\right) \int _0^t\!\!\!\int _{{\mathbb R}^d}\left( \varrho _{n\beta }^{0}+\varrho _{(n-1)\beta }^{\beta }\right) (s,z-y)\\&\qquad \times \left\{ \left( \varrho _{\gamma }^0+\varrho _{\gamma -\beta }^{\beta }\right) (t,x-z)+\left( \varrho _{\gamma }^0+\varrho _{\gamma -\beta }^{\beta }\right) (t,x'-z)\right\} {\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s\\&\quad \mathop {\preceq }\limits ^{(2.4)} \frac{\left( C_d\Gamma (\beta /\alpha )\right) ^{n}}{\Gamma (n\beta /\alpha )} {\mathcal {B}}\left( n\beta /\alpha , \gamma /\alpha \right) \left( |x-x'|^{\beta -\gamma }\wedge 1\right) \left\{ \left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (t,x-y)\right. \\&\qquad \left. +\left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (t,x'-y)\right\} \\&\quad {\preceq } \frac{(C_d\Gamma (\beta /\alpha ))^{n} \Gamma (\gamma \beta /\alpha )}{\Gamma ((n\beta +\gamma ) /\alpha )} \left( |x-x'|^{\beta -\gamma }\wedge 1\right) \left( \left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (t,x-y)\right. \\&\qquad \left. +\left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (t,x'-y)\right) , \end{aligned}$$

which yields (3.7) by summing up in \(n\). \(\square \)

3.2 Some estimates about \(p_y(t,x-y)\)

In this subsection, we prepare some important estimates for later use.

Lemma 3.2

Under (1.5) and (1.6), there exists a constant \(C=C(d, \alpha , \beta , \kappa _0, \kappa _1, \kappa _2)\) \(>0\) such that for all \(\varepsilon \geqslant 0,\, x,y\in {\mathbb R}^d\) and \(t>0\),

$$\begin{aligned} \left| \int _{{\mathbb R}^d}\left( \int _{|w|>\varepsilon }\delta _{p_y}(t,x-y;w)\kappa (x,w)|w|^{-d-\alpha }{\mathord {\mathrm{d}}}w\right) {\mathord {\mathrm{d}}}y\right| \leqslant C t^{\frac{\beta }{\alpha }-1}, \end{aligned}$$
(3.13)

and

$$\begin{aligned} \left| \int _{{\mathbb R}^d}\nabla p_y(t,\cdot )(x-y){\mathord {\mathrm{d}}}y\right| \leqslant Ct^{\frac{\beta -1}{\alpha }}. \end{aligned}$$
(3.14)

Proof

Since

$$\begin{aligned} \int _{{\mathbb R}^d}p_x(t,\xi -y){\mathord {\mathrm{d}}}y=1, \ \ \forall \xi \in {\mathbb R}^d, \end{aligned}$$
(3.15)

by definition of \(\delta _{p_x}(t,x-y;w)\), we have

$$\begin{aligned} \int _{{\mathbb R}^d}\delta _{p_x}(t, x-y;w){\mathord {\mathrm{d}}}y=0,\ \ \forall w\in {\mathbb R}^d. \end{aligned}$$

Thus, by Fubini’s theorem and (2.32), we have for any \(\gamma \in (0,\alpha \wedge 1)\),

$$\begin{aligned}&\left| \int _{{\mathbb R}^d}\left( \int _{|w|>\varepsilon }\delta _{p_y}(t,x-y;w)\kappa (x,w)|w|^{-d-\alpha }{\mathord {\mathrm{d}}}w\right) {\mathord {\mathrm{d}}}y\right| \\&\quad =\left| \int _{{\mathbb R}^d}\left( \int _{|w|>\varepsilon }(\delta _{p_y}(t,x-y;w)-\delta _{p_x}(t,x-y;w))\kappa (x,w)|w|^{-d-\alpha }{\mathord {\mathrm{d}}}w\right) {\mathord {\mathrm{d}}}y\right| \\&\quad \leqslant \kappa _1\int _{{\mathbb R}^d}\left( \int _{|w|>\varepsilon }|\delta _{p_y}(t,x-y;w)-\delta _{p_x}(t,x-y;w)|\cdot |w|^{-d-\alpha }{\mathord {\mathrm{d}}}w\right) {\mathord {\mathrm{d}}}y \\&\quad \preceq \int _{{\mathbb R}^d}\Vert \kappa (y,\cdot )-\kappa (x,\cdot )\Vert _\infty \left\{ \varrho ^0_0(t,x-y)+\varrho ^\gamma _{-\gamma }(t,x-y)\right\} {\mathord {\mathrm{d}}}y \\&\quad \preceq \int _{{\mathbb R}^d}\left\{ \varrho ^\beta _0(t,x-y)+\varrho ^{\beta +\gamma }_{-\gamma }(t,x-y)\right\} {\mathord {\mathrm{d}}}y\mathop {\preceq }\limits ^{(2.2)} t^{\frac{\beta }{\alpha }-1}, \end{aligned}$$

which gives (3.13).

As for (3.14), it is similar by (3.15) and (2.31) that

$$\begin{aligned}&\left| \int _{{\mathbb R}^d}\nabla p_y(t,\cdot )(x-y){\mathord {\mathrm{d}}}y\right| =\left| \int _{{\mathbb R}^d}(\nabla p_y(t,\cdot )-\nabla p_x(t,\cdot ))(x-y){\mathord {\mathrm{d}}}y\right| \\&\quad \preceq t^{-1/\alpha }\int _{{\mathbb R}^d}\Vert \kappa (y,\cdot )-\kappa (x,\cdot )\Vert _\infty \left\{ \varrho ^0_\alpha (t,x-y)+\varrho ^\gamma _{\alpha -\gamma }(t,x-y)\right\} {\mathord {\mathrm{d}}}y\\&\quad \preceq t^{-1/\alpha }\int _{{\mathbb R}^d}\left\{ \varrho ^\beta _\alpha (t,x-y)+\varrho ^{\beta +\gamma }_{\alpha -\gamma }(t,x-y)\right\} {\mathord {\mathrm{d}}}y\mathop {\preceq }\limits ^{(2.2)} t^{\frac{\beta -1}{\alpha }}. \end{aligned}$$

The proof is complete. \(\square \)

Lemma 3.3

Under (1.5) and (1.6), there is a constant \(C=C(d, \alpha , \beta , \kappa _0, \kappa _1, \kappa _2)>0\) so that

$$\begin{aligned}&\left| \int _{{\mathbb R}^d}{\fancyscript{L}}^{\kappa (x)}_{\alpha }p_y(t,\cdot )(x-y){\mathord {\mathrm{d}}}y\right| \leqslant C t^{\frac{\beta }{\alpha }-1}, \end{aligned}$$
(3.16)
$$\begin{aligned}&\quad \left| \int _{{\mathbb R}^d}\partial _tp_y(t,x-y){\mathord {\mathrm{d}}}y\right| \leqslant C t^{\frac{\beta }{\alpha }-1}, \end{aligned}$$
(3.17)
$$\begin{aligned}&\quad \lim _{t\downarrow 0}\sup _{x\in {\mathbb R}^d}\left| \int _{{\mathbb R}^d}p_y(t,x-y){\mathord {\mathrm{d}}}y-1\right| =0. \end{aligned}$$
(3.18)

Proof

Estimate (3.16) follows by (3.13). For (3.17), by (3.1) we have

$$\begin{aligned}&\left| \int _{{\mathbb R}^d}\partial _tp_y(t,x-y){\mathord {\mathrm{d}}}y\right| =\left| \int _{{\mathbb R}^d}{\fancyscript{L}}^{\kappa (y)}_{\alpha }p_y(t,\cdot )(x-y){\mathord {\mathrm{d}}}y\right| \\&\quad \leqslant \left| \int _{{\mathbb R}^d}\left( {\fancyscript{L}}^{\kappa (x)}_{\alpha }-{\fancyscript{L}}^{\kappa (y)}_{\alpha }\right) p_y(t,\cdot )(x-y){\mathord {\mathrm{d}}}y\right| +\left| \int _{{\mathbb R}^d}{\fancyscript{L}}^{\kappa (x)}_{\alpha }p_y(t,\cdot )(x-y){\mathord {\mathrm{d}}}y\right| \\&\mathop {\preceq }\limits ^{(3.8)(3.16)}\int _{{\mathbb R}^d}\varrho ^\beta _0(t,x-y){\mathord {\mathrm{d}}}y+t^{\frac{\beta }{\alpha }-1}\preceq t^{\frac{\beta }{\alpha }-1}. \end{aligned}$$

For (3.18), by (3.15), we have for any \(\gamma \in (0,\alpha \wedge 1)\),

$$\begin{aligned}&\sup _{x\in {\mathbb R}^d}\left| \int _{{\mathbb R}^d}p_y(t,x-y){\mathord {\mathrm{d}}}y-1\right| \leqslant \sup _{x\in {\mathbb R}^d}\int _{{\mathbb R}^d}|p_y(t,x-y)-p_x(t,x-y)|{\mathord {\mathrm{d}}}y \\&\mathop {\preceq }\limits ^{(2.30)}\sup _{x\in {\mathbb R}^d}\int _{{\mathbb R}^d}\Vert \kappa (y,\cdot )-\kappa (x,\cdot )\Vert _\infty \left( \varrho _\alpha (t,x-y)+\varrho ^\gamma _{\alpha -\gamma }(t,x-y)\right) {\mathord {\mathrm{d}}}y \\&\quad \preceq \sup _{x\in {\mathbb R}^d}\int _{{\mathbb R}^d}\left( \varrho ^\beta _\alpha (t,x-y)+\varrho ^{\gamma +\beta }_{\alpha -\gamma }(t,x-y)\right) {\mathord {\mathrm{d}}}y\preceq t^{\frac{\beta }{\alpha }}\rightarrow 0, \end{aligned}$$

as \(t\rightarrow 0\). The proof is complete. \(\square \)

3.3 Smoothness of \(p^\kappa _{\alpha } (t,x,y)\)

In this subsection, we give a rigorous proof of (3.4). Below, for the simplicity of notation, we write

$$\begin{aligned} \phi _{y}(t,x,s):=\int _{{\mathbb R}^d}p_z(t-s,x-z)q(s,z,y){\mathord {\mathrm{d}}}z, \end{aligned}$$
(3.19)

and

$$\begin{aligned} \varphi _y(t,x):=\int ^t_0\phi _y(t,x,s){\mathord {\mathrm{d}}}s=\int ^t_0\!\!\!\int _{{\mathbb R}^d}p_z(t-s,x-z)q(s,z,y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s. \end{aligned}$$
(3.20)

The definition (3.2) of \(p^\kappa _a (t, x, y)\) can be rewritten as

$$\begin{aligned} p^\kappa _\alpha (t, x, y)= p_y (t, x-y) + \varphi _y(t,x). \end{aligned}$$
(3.21)

First of all, we have

Lemma 3.4

For all \(\gamma \in (0,\alpha \wedge 1)\), there is a constant \(C=C(d, \alpha , \beta , \gamma , \kappa _0, \kappa _1, \kappa _2)>0\) so that

$$\begin{aligned} |p^\kappa _{\alpha } (t,x,y)-p^\kappa _{\alpha } (t,x',y)| \leqslant C |x-x'|^\gamma \, \left( \varrho ^0_{\alpha -\gamma }(t,x-y) +\varrho ^0_{\alpha -\gamma }(t,x'-y)\right) . \end{aligned}$$

Proof

We have by (2.24)

$$\begin{aligned} |p_y(t,x-y)-p_y(t,x'-y)|\preceq & {} ((t^{-1/\alpha }|x\!-\!x'|)\wedge 1)\left\{ \varrho ^0_\alpha (t,x-y)\!+\!\varrho ^0_\alpha (t,x'-y)\right\} \\\preceq & {} |x-x'|^\gamma \left\{ \varrho ^0_{\alpha -\gamma }(t,x-y)+\varrho ^0_{\alpha -\gamma }(t,x'-y)\right\} . \end{aligned}$$

On the other hand, by (3.6) we also have

$$\begin{aligned}&|\varphi _y(t,x)-\varphi _y(t,x')| \leqslant \int ^t_0\!\!\!\int _{{\mathbb R}^d}|p_z(t\!-\!s,x\!-\!z)\!-\!p_z(t-s,x'-z)|\cdot |q(s,z,y)|{\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s\\&\quad \preceq \int ^t_0\!\!\!\int _{{\mathbb R}^d}((t-s)^{-1/\alpha }|x-x'|\wedge 1)\left\{ \varrho ^0_\alpha (t-s,x-z)+\varrho ^0_\alpha (t-s,x'-z)\right\} \\&\qquad \times \left\{ \varrho ^0_\beta (s,z-y)+\varrho ^\beta _0(s,z-y)\right\} {\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s\\&\quad \preceq |x-x'|^\gamma \int ^t_0\!\!\!\int _{{\mathbb R}^d}\left\{ \varrho ^0_{\alpha -\gamma }(t-s,x-z)+\varrho ^0_{\alpha -\gamma }(t-s,x'-z)\right\} \\&\qquad \times \left\{ \varrho ^0_\beta (s,z-y)+\varrho ^\beta _0(s,z-y)\right\} {\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s\\&\quad \preceq |x-x'|^\gamma \left\{ \left( \varrho ^0_{\alpha -\gamma +\beta }+\varrho ^\beta _{\alpha -\gamma }\right) (t,x-y)+\left( \varrho ^0_{\alpha -\gamma +\beta }+\varrho ^\beta _{\alpha -\gamma }\right) (t,x'-y)\right\} . \end{aligned}$$

Combining the above two estimations, we obtain from (3.21) the desired estimate. \(\square \)

Lemma 3.5

For all \(x\not =y\in {\mathbb R}^d\), the mapping \(t\mapsto \varphi _y(t,x)\) is absolutely continuous, and

$$\begin{aligned} \partial _t \varphi _y(t,x)=q(t,x,y)+\int ^t_0\!\!\!\int _{{\mathbb R}^d}{\fancyscript{L}}^{\kappa (z)}_\alpha p_z(t-s,\cdot )(x-z)q(s,z,y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s. \end{aligned}$$
(3.22)

Proof

We divide the proof into four steps.

Step 1. In this step we prove that for any \(s\in (0,t)\),

$$\begin{aligned} \partial _t \phi _{y}(t,x,s)=\int _{{\mathbb R}^d}\partial _tp_z(t-s,x-z)q(s,z,y){\mathord {\mathrm{d}}}z. \end{aligned}$$
(3.23)

Notice that

$$\begin{aligned} \frac{\phi _{y}(t+\varepsilon ,x,s)-\phi _{y}(t,x,s)}{\varepsilon }&=\frac{1}{\varepsilon }\int _{{\mathbb R}^d}\left( p_z(t+\varepsilon -s,x-z)-p_z(t-s,x-z)\right) \nonumber \\&\qquad \qquad \times q(s,z,y){\mathord {\mathrm{d}}}z\nonumber \\&=\int _{{\mathbb R}^d}\!\left( \!\int _0^1 \partial _t p_z(t+\theta \varepsilon -s,x,z){\mathord {\mathrm{d}}}\theta \right) q(s,z,y){\mathord {\mathrm{d}}}z. \end{aligned}$$

By (3.1) and (2.28), we have for \(|\varepsilon |<\frac{t-s}{2}\),

$$\begin{aligned} |\partial _t p_z(t+\theta \varepsilon -s,x-z)|&=\left| {\fancyscript{L}}^{\kappa (z)}_\alpha p_z(t+\theta \varepsilon -s,\cdot )(x-z)\right| \\&\quad \preceq (|x-z|+t+\theta \varepsilon -s)^{-d-\alpha }\\&\mathop {\preceq }\limits ^{(2.9)} (|x-z|+(t-s))^{-d-\alpha }\\&\quad =\varrho ^0_0(t-s,x-z), \end{aligned}$$

which together with (3.6) yields

$$\begin{aligned} |\partial _t p_z(t+\theta \varepsilon ,x;r,z)q(s,z,y)| \preceq \varrho _0^0(t-s,x-z)\left( \varrho ^0_\beta +\varrho ^\beta _0\right) (s,z-y)=:g(z). \end{aligned}$$

By (2.3), one sees that

$$\begin{aligned} \int _{{\mathbb R}^d}g(z){\mathord {\mathrm{d}}}z<+\infty . \end{aligned}$$

Hence, by the dominated convergence theorem, we have

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\frac{\phi _{y}(t+\varepsilon ,x,s)-\phi _{y}(t,x,s)}{\varepsilon } =\int _{{\mathbb R}^d}\partial _tp_z(t-s,x-z)q(s,z,y){\mathord {\mathrm{d}}}z, \end{aligned}$$

and (3.23) is proven.

Step 2. In this step we prove that for all \(x\ne y\) and \(t>0\),

$$\begin{aligned} \int ^t_0\!\!\!\int ^r_0|\partial _r \phi _{y}(r,x,s)|{\mathord {\mathrm{d}}}s{\mathord {\mathrm{d}}}r<+\infty . \end{aligned}$$
(3.24)

By (3.23), we have

$$\begin{aligned} |\partial _r \phi _{y}(r,x,s)|&\leqslant \int _{{\mathbb R}^d}|\partial _rp_z(r-s,x-z)|\cdot |q(s,z,y)-q(s,x,y)|{\mathord {\mathrm{d}}}z\nonumber \\&+|q(s,x,y)|\left| \int _{{\mathbb R}^d}\partial _rp_z(r-s,x-z){\mathord {\mathrm{d}}}z\right| \nonumber \\&=:Q^{(1)}_y(r,x,s)+Q^{(2)}_y(r,x,s). \end{aligned}$$
(3.25)

For \(Q^{(1)}_{y}(r,x,s)\), by (3.7) and (2.28), we have

$$\begin{aligned}&\int ^t_0\!\!\!\int ^r_0Q^{(1)}_{y}(r,x,s){\mathord {\mathrm{d}}}s{\mathord {\mathrm{d}}}r \preceq \int ^t_0\!\!\!\int ^r_0\!\!\!\int _{{\mathbb R}^d}\left| {\fancyscript{L}}^{\kappa (z)}_\alpha p_z(r-s,x-z)\right| \cdot (|x-z|^{\beta -\gamma }\wedge 1)\nonumber \\&\qquad \times \left\{ \left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (s,x-y)+ \left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (s,z-y)\right\} {\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s{\mathord {\mathrm{d}}}r\nonumber \\&\quad \preceq \int ^t_0\!\!\!\int ^r_0\!\!\!\int _{{\mathbb R}^d}\varrho ^{\beta -\gamma }_0(r-s,x-z) \left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (s,x-y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s{\mathord {\mathrm{d}}}r\nonumber \\&\qquad +\int ^t_0\!\!\!\int ^r_0\!\!\!\int _{{\mathbb R}^d}\varrho ^{\beta -\gamma }_0(r-s,x-z) \left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (s,z-y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s{\mathord {\mathrm{d}}}r\nonumber \\&\quad \preceq \int ^t_0\!\!\!\int ^r_0(r-s)^{\frac{\beta -\gamma }{\alpha }-1}\left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (s,x-y){\mathord {\mathrm{d}}}s{\mathord {\mathrm{d}}}r\nonumber \\&\qquad +\int ^t_0\left( \varrho ^0_\beta +\varrho ^\beta _0+\varrho ^{\beta -\gamma }_\gamma \right) (r,x-y){\mathord {\mathrm{d}}}r\nonumber \\&\quad \preceq \frac{1}{|x-y|^{d+\alpha }}\int ^t_0\!\!\!\int ^{r}_0(r-s)^{\frac{\beta -\gamma }{\alpha }-1}\left( s^{\frac{\gamma }{\alpha }}+s^{\frac{\gamma -\beta }{\alpha }}\right) {\mathord {\mathrm{d}}}s{\mathord {\mathrm{d}}}r\nonumber \\&\qquad +\frac{1}{|x-y|^{d+\alpha }}\int ^t_0\left( r^{\frac{\gamma }{\alpha }}+1+r^{\frac{\beta }{\alpha }}\right) {\mathord {\mathrm{d}}}r<+\infty . \end{aligned}$$
(3.26)

For \(Q^{(2)}_{y}(r,x,s)\), by (3.17) and (3.6) we have

$$\begin{aligned} \int ^t_0\!\!\!\int ^{r}_0Q^{(2)}_{y}(r,x,s){\mathord {\mathrm{d}}}r{\mathord {\mathrm{d}}}r \preceq \int ^t_0\!\!\!\int ^{r}_0 \left( \varrho ^0_\beta +\varrho ^\beta _0\right) (s,x-y)(r-s)^{\frac{\beta }{\alpha }-1}{\mathord {\mathrm{d}}}s{\mathord {\mathrm{d}}}r<+\infty . \end{aligned}$$
(3.27)

Combining (3.25)–(3.27), we obtain (3.24).

Step 3. For fixed \(s,x,y\), we have

$$\begin{aligned} \lim _{t\downarrow s}\phi _{y}(t,x,s)=q(s,x,y). \end{aligned}$$
(3.28)

By (3.18), it suffices to prove that

$$\begin{aligned} \lim _{t\downarrow r}\left| \int _{{\mathbb R}^d}p_z(t-s,x-z)(q(s,z,y)- q(s,x,y)){\mathord {\mathrm{d}}}z \right| =0. \end{aligned}$$

Notice that for any \(\delta >0\),

$$\begin{aligned}&\left| \int _{{\mathbb R}^d}p_z(t-s,x-z)(q(s,z,y)- q(s,x,y)){\mathord {\mathrm{d}}}z \right| \\&\quad \leqslant \int _{|x-z|\leqslant \delta }p_z(t-s,x-z)|q(s,z,y)-q(s,x,y)|{\mathord {\mathrm{d}}}z\\&\qquad +\int _{|x-z|>\delta }p_z(t-s,x-z)|q(s,z,y)-q(s,x,y)|{\mathord {\mathrm{d}}}z\\&\quad =: J_1(\delta ,t,s)+J_2(\delta ,t,s). \end{aligned}$$

For any \(\varepsilon >0\), by (3.7), there exists a \(\delta =\delta (s,x,y)>0\) such that for all \(|x-z|\leqslant \delta \),

$$\begin{aligned} |q(s,z,y)-q(s,x,y)|\leqslant \varepsilon . \end{aligned}$$

Thus,

$$\begin{aligned} J_1(\delta ,t,s)&\leqslant \varepsilon \int _{|x-z|\leqslant \delta }p_z(t-s,x-z){\mathord {\mathrm{d}}}z\\&\leqslant \varepsilon \int _{{\mathbb R}^d}p_z(t-s,x-z){\mathord {\mathrm{d}}}z\\&\preceq \varepsilon \int _{{\mathbb R}^d}\varrho ^0_\alpha (t-s,x-z){\mathord {\mathrm{d}}}z\mathop {\preceq }\limits ^{(2.2)}\varepsilon . \end{aligned}$$

On the other hand, we have

$$\begin{aligned} J_2(\delta ,t,s)&\mathop {\preceq }\limits ^{(2.22)}(t-s)\int _{|x-z|>\delta }\frac{|q(s,z,y)|+|q(s,x,y)|}{|x-z|^{d+\alpha }}{\mathord {\mathrm{d}}}z\\&\leqslant (t-s)\left( \delta ^{-d-\alpha }\int _{{\mathbb R}^d}|q(s,z,y)|{\mathord {\mathrm{d}}}z +|q(s,x,y)|\int _{|z|>\delta }|z|^{-d-\alpha }{\mathord {\mathrm{d}}}z\right) , \end{aligned}$$

which, by (3.6) and (2.2), converges to zero as \(t\downarrow r\). Thus, (3.28) is proved.

Step 4. Now, by the integration by parts formula and (3.28), we have

$$\begin{aligned} \int ^t_s\partial _{r} \phi _{y}(r,x,s){\mathord {\mathrm{d}}}r=\phi _{y}(t,x,s)-q(s,x,y). \end{aligned}$$

Integrating both sides with respect to \(s\) from \(0\) to \(t\), and then by (3.24) and Fubini’s theorem, we obtain

$$\begin{aligned}&\varphi _y(t,x)-\int ^t_0q(s,x,y){\mathord {\mathrm{d}}}s=\int ^t_0\!\!\!\int ^t_s\partial _{r} \phi _{y}(r,x,s){\mathord {\mathrm{d}}}r{\mathord {\mathrm{d}}}s \mathop {=}\limits ^{(3.24)}\int ^t_0\!\!\!\int ^r_0\partial _{r} \phi _{y}(r,x,s){\mathord {\mathrm{d}}}s{\mathord {\mathrm{d}}}r\\&\quad \mathop {=}\limits ^{(3.23) (3.1)}\int ^t_0\!\!\!\int ^r_0\!\!\!\int _{{\mathbb R}^d}{\fancyscript{L}}^{\kappa (z)}_\alpha p_z(r-s,\cdot )(x-z)q(s,z,y) {\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s{\mathord {\mathrm{d}}}r, \end{aligned}$$

which in turn implies (3.22) by the Lebesgue differential theorem. \(\square \)

For a function \(f(x, y)\) defined on \({\mathbb R}^d\times {\mathbb R}^d\), we use \(\nabla _x f(x, y)\) to denote the gradient taken in variable \(x\); that is, gradient for the function \(x\mapsto f(x, y)\).

Lemma 3.6

For all \(t>0\) and \(x\not =y\), we have

$$\begin{aligned} {\fancyscript{L}}^{\kappa (x)}_{\alpha }\varphi _y(t,x)=\int ^t_0\!\!\!\int _{{\mathbb R}^d}{\fancyscript{L}}^{\kappa (x)}_{\alpha }p_z(t-s,\cdot )(x-z)q(s,z,y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s, \end{aligned}$$
(3.29)

and if \(\beta >(1-\alpha )\vee 0\), then

$$\begin{aligned} \nabla _x \varphi _y(t,x)=\int ^t_0\!\!\!\int _{{\mathbb R}^d}\nabla p_z(t-s,\cdot )(x-z)q(s,z,y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s, \end{aligned}$$
(3.30)

where the integrals are understood in the sense of iterated integrals. Moreover, for any \(x\not =y\),

$$\begin{aligned} t\mapsto {\fancyscript{L}}^{\kappa (x)}_{\alpha }\varphi _y(t,x) \,\hbox { is continuous on } \,(0,1). \end{aligned}$$
(3.31)

Proof

We only prove (3.29), and (3.30) is analogue by using (3.14). First of all, for fixed \(s\in (0,t)\), since

$$\begin{aligned} x\mapsto p_y(t-s,x-y)\in C^\infty _b({\mathbb R}^d\times {\mathbb R}^d) \end{aligned}$$

and

$$\begin{aligned} z\mapsto q(s,z,y)\in C_b({\mathbb R}^d), \end{aligned}$$

by (2.28) and Fubini’s theorem, it is easy to see that

$$\begin{aligned} {\fancyscript{L}}^{\kappa (x)}_{\alpha }\phi _{y}(t,x,s)=\int _{{\mathbb R}^d}{\fancyscript{L}}^{\kappa (x)}_{\alpha } p_z(t-s,\cdot )(x-z)q(s,z,y){\mathord {\mathrm{d}}}z. \end{aligned}$$
(3.32)

By definition of \(\phi _y\) and Fubini’s theorem, we have for \(\varepsilon \in (0,1)\)

$$\begin{aligned} I_\varepsilon (t,x,s,y)&:=\left| \int _{|w|>\varepsilon }\delta _{\phi _{y}}(t,x,s;w)\kappa (x,w)|w|^{-d-\alpha }{\mathord {\mathrm{d}}}w\right| \\&=\left| \int _{|w|>\varepsilon }\left( \int _{{\mathbb R}^d}\delta _{p_{z}}(t-s,x-z;w) q(s,z,y){\mathord {\mathrm{d}}}z\right) \kappa (x,w)|w|^{-d-\alpha }{\mathord {\mathrm{d}}}w\right| \\&=\left| \int _{{\mathbb R}^d}\left( \int _{|w|>\varepsilon }\delta _{p_{z}}(t-s,x-z;w) \kappa (x,w)|w|^{-d-\alpha }{\mathord {\mathrm{d}}}w\right) q(s,z,y) {\mathord {\mathrm{d}}}z\right| \\&\leqslant \int _{{\mathbb R}^d}\left( \int _{|w|>\varepsilon }|\delta _{p_{z}}(t-s,x-z;w)|\cdot |w|^{-d-\alpha }{\mathord {\mathrm{d}}}w\right) |q(s,z,y)-q(s,x,y)| {\mathord {\mathrm{d}}}z\\&\quad +\left| \int _{{\mathbb R}^d}\left( \int _{|w|>\varepsilon }\delta _{p_{z}}(t-s,x-z;w) \kappa (x,w)|w|^{-d-\alpha }{\mathord {\mathrm{d}}}w\right) {\mathord {\mathrm{d}}}z\right| \cdot |q(s,x,y)|. \end{aligned}$$

Using (2.28), (3.13), (3.6) and (3.7), we further have

$$\begin{aligned}&I_\varepsilon (t,x,s,y)\preceq \int _{{\mathbb R}^d}\varrho ^{\beta -\gamma }_0(t-s,x-z)\left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (s,z-y){\mathord {\mathrm{d}}}z\\&\qquad +\left( \int _{{\mathbb R}^d}\varrho ^{\beta -\gamma }_0(t-s,x-z){\mathord {\mathrm{d}}}z\right) \left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (s,x-y)\\&\qquad +(t-s)^{\frac{\beta }{\alpha }-1}\left( \varrho ^\beta _0(s,x-y)+\varrho ^0_\beta (s,x-y)\right) \\&\quad \preceq \int _{{\mathbb R}^d}\varrho ^{\beta -\gamma }_0(t-s,x-z)\left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (s,z-y){\mathord {\mathrm{d}}}z\\&\qquad +(t-s)^{\frac{\beta -\gamma }{\alpha }-1}\left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (s,x-y)\\&\qquad +(t-s)^{\frac{\beta }{\alpha }-1}\left( \varrho ^\beta _0(s,x-y)+\varrho ^0_\beta (s,x-y)\right) , \end{aligned}$$

which implies that for some \(p>1\),

$$\begin{aligned} \sup _{\varepsilon \in (0,1)}\int ^t_0|I_\varepsilon (t,x,s,y)|^p{\mathord {\mathrm{d}}}s<+\infty . \end{aligned}$$
(3.33)

Now, by Fubini’s theorem again, we obtain

$$\begin{aligned} {\fancyscript{L}}^{\kappa (x)}_{\alpha }\varphi _y(t,x)&=\lim _{\varepsilon \downarrow 0}\int _{|w|>\varepsilon }\!\int ^t_0 \delta _{\phi _{y}}(t,x,s;w)\kappa (x,w)|w|^{-d-\alpha }{\mathord {\mathrm{d}}}s{\mathord {\mathrm{d}}}w\\&=\lim _{\varepsilon \downarrow 0}\int ^t_0\!\!\!\int _{|w|>\varepsilon } \delta _{\phi _{y}}(t,x,s;w)\kappa (x,w)|w|^{-d-\alpha }{\mathord {\mathrm{d}}}w{\mathord {\mathrm{d}}}r\\&=\int ^t_0\lim _{\varepsilon \downarrow 0}\int _{|w|>\varepsilon } \delta _{\phi _{y}}(t,x,s;w)\kappa (x,w)|w|^{-d-\alpha }{\mathord {\mathrm{d}}}w{\mathord {\mathrm{d}}}r\\&=\int ^t_0{\fancyscript{L}}^{\kappa (x)}_{\alpha }\phi _{y}(t,x,s){\mathord {\mathrm{d}}}s, \end{aligned}$$

which together with (3.32) yields (3.29).

As for (3.31), it follows by (3.29) and a direct calculation. \(\square \)

Finally, we show that

Lemma 3.7

The function \(p^\kappa _\alpha (t,x,y)\) in (3.2) is jointly continuous in \((t,x,y)\in (0,1]\times {\mathbb R}^d\times {\mathbb R}^d\).

Proof

Recall that \(p^\kappa _\alpha (t,x,y)=p_y(t,x-y)+\varphi _y(t,x)\), where \(\varphi _y(t,x)\) is defined by (3.20). Since \(p_y(t,x)=p^{\kappa (y)}_\alpha (t,x)\) is the heat kernel of the freezing operator \({\fancyscript{L}}^{\kappa (y)}_\alpha \) and \(\kappa (x, z)\) is bounded between two positive constants, we have by the Hölder estimates in [13], Theorem 4.14] that the family of functions \(\{(t, x)\mapsto p_y (t, x); y\in {\mathbb R}^d\}\) are equi-continuous in any compact subset of \((0, \infty ) \times {\mathbb R}^d\). It then follows from (2.30) and (1.6) that the function \((t, x, y)\mapsto p_y(t,x)\) is continuous on \((0,\infty ) \times {\mathbb R}^d\times {\mathbb R}^d\), and hence so does \((t, x, y)\mapsto p_y(t,x-y)\). On the other hand, by Theorem 3.1, \(q(t,x,y)\) is a jointly continuous function of \((t,x,y)\in (0,1]\times {\mathbb R}^d\times {\mathbb R}^d\). By (3.20), (2.22), (3.6), (2.4) and the dominated convergence theorem, \((t, x, y)\mapsto \varphi _y(t,x)\) is continuous on \((0, 1] \times {\mathbb R}^d\times {\mathbb R}^d\). This proves the lemma. \(\square \)

4 Proofs of Theorem 1.1 and Corollary 1.3

4.1 A nonlocal maximal principle

In this subsection, we prove a nonlocal maximal principle (cf. [35]). Notice that the current assumptions are weaker than [35].

Theorem 4.1

Let \(u(t,x)\in C_b([0,1]\times {\mathbb R}^d)\) with

$$\begin{aligned} \lim _{t\downarrow 0}\sup _{x\in {\mathbb R}^d}|u(t,x)-u(0,x)|=0. \end{aligned}$$
(4.1)

Suppose that for each \(x\in {\mathbb R}^d\),

$$\begin{aligned} t\mapsto {\fancyscript{L}}^\kappa _\alpha u(t,x) \,\hbox { is continuous on }\, (0,1], \end{aligned}$$
(4.2)

and for any \(\varepsilon \in (0,1)\) and some \(\gamma \in ((\alpha -1)\vee 0,1)\),

$$\begin{aligned} \sup _{t\in (\varepsilon ,1)}|u(t,x)-u(t,x')|\leqslant C_\varepsilon |x-x'|^\gamma ,\ \ x,x'\in {\mathbb R}^d. \end{aligned}$$
(4.3)

If \(u(t,x)\) satisfies the following equation: for all \((t,x)\in (0,1]\times {\mathbb R}^d\),

$$\begin{aligned} \partial _t u(t,x)={\fancyscript{L}}^\kappa _\alpha u(t,x), \end{aligned}$$

then for all \(t\in (0,1)\),

$$\begin{aligned} \sup _{x\in {\mathbb R}^d}u(t,x)\leqslant \sup _{x\in {\mathbb R}^d}u(0,x). \end{aligned}$$

Proof

First of all, by (4.1), it suffices to prove that for any \(\varepsilon \in (0,1)\),

$$\begin{aligned} \sup _{x\in {\mathbb R}^d}u(t,x)\leqslant \sup _{x\in {\mathbb R}^d}u(\varepsilon ,x),\ \ \forall t\in (\varepsilon ,1). \end{aligned}$$
(4.4)

Below, we shall fix \(\varepsilon \in (0,1)\). Let \(\chi (x):{\mathbb R}^d\rightarrow [0,1]\) be a smooth function with \(\chi (x)=1\) for \(|x|\leqslant 1\) and \(\chi (x)=0\) for \(|x|>2\). For \(R>0\), define the following cutoff function

$$\begin{aligned} \chi _R(x):=\chi (x/R). \end{aligned}$$

For \(R,\delta >0\), consider

$$\begin{aligned} u^\delta _R(t,x):=u(t,x)\chi _R(x)- (t-\varepsilon )\delta . \end{aligned}$$

Then

$$\begin{aligned} \partial _t u^\delta _R(t,x)={\fancyscript{L}}^\kappa _\alpha u^\delta _R(t,x)+g^\delta _R(t,x), \end{aligned}$$
(4.5)

where

$$\begin{aligned} g^\delta _R(t,x):={\fancyscript{L}}^\kappa _\alpha u(t,x)\chi _R(x)-{\fancyscript{L}}^\kappa _\alpha (u\chi _R)(t,x)-\delta . \end{aligned}$$

Our aim is to prove that for each \(\delta >0\), there exists an \(R_0\geqslant 1\) such that for all \(t\in (\varepsilon ,1)\) and \(R>R_0\),

$$\begin{aligned} \sup _{x\in {\mathbb R}^d}u^\delta _R(t,x) \leqslant \sup _{x\in {\mathbb R}^d}u^\delta _R(\varepsilon ,x)\leqslant \sup _{x\in {\mathbb R}^d}u(\varepsilon ,x). \end{aligned}$$
(4.6)

If this is proven, then taking \(R\rightarrow \infty \) and \(\delta \rightarrow 0\), we obtain (4.4).

We first prove the following claim:

Claim: For \(\beta \in (0,\alpha \wedge 1)\), there exists a constant \(C_\varepsilon >0\) such that for all \(R\geqslant 1\),

$$\begin{aligned} \sup _{(t,x)\in [\varepsilon ,1]\times {\mathbb R}^d}\left| {\fancyscript{L}}^\kappa _\alpha u(t,x)\chi _R(x)-{\fancyscript{L}}^\kappa _\alpha (u\chi _R)(t,x)\right| \leqslant \frac{C_\varepsilon }{R^\beta }. \end{aligned}$$
(4.7)

Moreover, for each \(x\in {\mathbb R}^d\),

$$\begin{aligned} t\mapsto {\fancyscript{L}}^\kappa _\alpha u^\delta _R(t,x) \,\hbox { and }\, g^\delta _R(t,x) \,\hbox { are continuous on } \, (\varepsilon ,1). \end{aligned}$$
(4.8)

Proof of Claim: Notice that by definitions,

$$\begin{aligned}&{\fancyscript{L}}^\kappa _\alpha (u\chi _R)(t,x)-{\fancyscript{L}}^\kappa _\alpha u(t,x)\chi _R(x)-u(t,x){\fancyscript{L}}^\kappa _\alpha \chi _R(x)\nonumber \\&\quad =\int _{{\mathbb R}^d}(u(t,x+z)-u(t,x))(\chi _R(x+z)-\chi _R(x))\kappa (x,z)|z|^{-d-\alpha }{\mathord {\mathrm{d}}}z. \end{aligned}$$
(4.9)

Thus,

$$\begin{aligned}&\left| {\fancyscript{L}}^\kappa _\alpha (u\chi _R)(t,x)-{\fancyscript{L}}^\kappa _\alpha u(t,x)\chi _R(x)-u(t,x){\fancyscript{L}}^\kappa _\alpha \chi _R(x)\right| \\&\quad \leqslant \Vert \kappa \Vert _\infty \int _{|z|>1}|u(t,x+z)-u(t,x)|\cdot |\chi _R(x+z)-\chi _R(x)|\cdot |z|^{-d-\alpha }{\mathord {\mathrm{d}}}z,\\&\qquad +\Vert \kappa \Vert _\infty \int _{|z|\leqslant 1}|u(t,x+z)-u(t,x)|\cdot |\chi _R(x+z)-\chi _R(x)|\cdot |z|^{-d-\alpha }{\mathord {\mathrm{d}}}z\\&\quad =I_1+I_2. \end{aligned}$$

For \(I_1\), we have

$$\begin{aligned}&I_1\leqslant 2\Vert \kappa \Vert _\infty \Vert u\Vert _\infty \int _{|z|>1}(2\Vert \chi _R\Vert _\infty )^{1-\beta }\Vert \nabla \chi _R\Vert ^{\beta }_\infty |z|^{\beta -d-\alpha }{\mathord {\mathrm{d}}}z \nonumber \\&\quad \preceq \Vert \kappa \Vert _\infty \Vert u\Vert _\infty (2\Vert \chi _R \Vert _\infty )^{1-\beta } \Vert \nabla \chi \Vert ^{\beta }_\infty /R^{\beta }, \end{aligned}$$
(4.10)

For \(I_2\), by (4.3), we have

$$\begin{aligned} I_2\leqslant \Vert \kappa \Vert _\infty C_\varepsilon \int _{|z|\leqslant 1}\Vert \nabla \chi _R\Vert _\infty |z|^{1+\gamma -d-\alpha }{\mathord {\mathrm{d}}}z \preceq \Vert \kappa \Vert _\infty C_\varepsilon \Vert \nabla \chi \Vert _\infty /R. \end{aligned}$$
(4.11)

Moreover, it is also easy to see that

$$\begin{aligned} \Vert {\fancyscript{L}}^\kappa _\alpha \chi _R\Vert _\infty \leqslant \frac{C}{R^{\beta }}. \end{aligned}$$
(4.12)

Combining (4.9)–(4.12), we obtain (4.7). As for (4.8), it follows by (4.2), (4.9) and the dominated convergence theorem.

We now use the contradiction argument to prove (4.6). Fix

$$\begin{aligned} R>(2C_\varepsilon /\delta )^{1/\beta }. \end{aligned}$$
(4.13)

Suppose that (4.6) does not hold, then there exists a \((t_0,x_0)\in (\varepsilon ,1]\times {\mathbb R}^d\) such that

$$\begin{aligned} \sup _{(t,x)\in (\varepsilon ,1)\times {\mathbb R}^d}u^\delta _R(t,x)=u^\delta _R(t_0,x_0). \end{aligned}$$
(4.14)

Thus, by (4.5), we have for any \(h\in (0,t_0-\varepsilon )\),

$$\begin{aligned} 0\leqslant \frac{u^\delta _R(t_0,x_0)-u^\delta _R(t_0-h,x_0)}{h}=\frac{1}{h}\int ^{t_0}_{t_0-h}{\fancyscript{L}}^\kappa _\alpha u^\delta _R(s,x_0){\mathord {\mathrm{d}}}s+\frac{1}{h}\int ^{t_0}_{t_0-h}g^\delta _R(s,x_0){\mathord {\mathrm{d}}}s, \end{aligned}$$

which implies by (4.8) and letting \(h\rightarrow 0\) that

$$\begin{aligned} 0\leqslant {\fancyscript{L}}^\kappa _\alpha u^\delta _R(t_0,x_0)+g^\delta _R(t_0,x_0). \end{aligned}$$
(4.15)

On the other hand, by definition of \({\fancyscript{L}}^\kappa _\alpha \) and (4.14), we have

$$\begin{aligned} {\fancyscript{L}}^\kappa _\alpha u^\delta _R(t_0,x_0)=\int _{{\mathbb R}^d}\delta _{u^\delta _R}(t_0,x_0;z)\kappa (x_0,z)|z|^{-d-\alpha }{\mathord {\mathrm{d}}}z\leqslant 0, \end{aligned}$$
(4.16)

and by the claim and (4.13),

$$\begin{aligned} g^\delta _R(t_0,x_0)\leqslant \frac{C_\varepsilon }{R^{\beta }}-\delta \leqslant -\frac{\delta }{2}. \end{aligned}$$
(4.17)

Combining (4.15)–(4.17), we obtain a contradiction, and the proof is complete. \(\square \)

4.2 Fractional derivative and gradient estimates of \(p^\kappa _\alpha \)

We prove two lemmas about the fractional derivative and gradient estimates of \(p^\kappa _\alpha \).

Lemma 4.2

For all \(t\in (0,1]\) and \(x\not =y\in {\mathbb R}^d\), we have

$$\begin{aligned} \left| {\fancyscript{L}}^\kappa _\alpha p^\kappa _{\alpha } (t,\cdot ,y)(x)\right| \preceq \varrho ^0_0(t,x-y), \end{aligned}$$
(4.18)

and if \(\alpha \in [1,2)\), then

$$\begin{aligned} \left| \nabla _x p^\kappa _{\alpha } (t,x,y)\right| \preceq t^{\frac{\alpha -1}{\alpha }}\varrho ^0_0(t,x-y). \end{aligned}$$
(4.19)

Proof

(i) First of all, by (2.28), it is easy to see that

$$\begin{aligned} \left| {\fancyscript{L}}^{\kappa }_{\alpha }p_y(t,\cdot )(x-y)\right| \preceq \varrho ^0_0(t,x-y). \end{aligned}$$

Recalling (3.20), by (3.29), we can write

$$\begin{aligned} {\fancyscript{L}}^{\kappa }_{\alpha }\varphi _y(t,x)&=\int ^t_{\frac{t}{2}}\!\!\!\int _{{\mathbb R}^d}{\fancyscript{L}}^{\kappa }_{\alpha }p_z(t-s,\cdot )(x-z)(q(s,z,y)-q(s,x,y)){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s\\&\quad +\int ^t_{\frac{t}{2}}\left( \int _{{\mathbb R}^d}{\fancyscript{L}}^{\kappa }_{\alpha }p_z(t-s,\cdot )(x-z){\mathord {\mathrm{d}}}z\right) q(s,x,y){\mathord {\mathrm{d}}}s\\&\quad +\int ^{\frac{t}{2}}_0\!\!\!\int _{{\mathbb R}^d}{\fancyscript{L}}^{\kappa }_{\alpha }p_z(t-s,\cdot )(x-z)q(s,z,y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s\\&=:Q_1(t,x,y)+Q_2(t,x,y)+Q_3(t,x,y). \end{aligned}$$

For \(Q_1(t,x,y)\), by (2.28), (3.7) and Lemma 2.1, we have for any \(\gamma \in (0,\beta )\),

$$\begin{aligned} Q_1(t,x,y)&\preceq \int ^t_{\frac{t}{2}}\!\!\!\int _{{\mathbb R}^d}\varrho ^{\beta -\gamma }_0(t-s,x-z)\left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (s,x-y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s\\&\quad +\int ^t_{\frac{t}{2}}\!\!\!\int _{{\mathbb R}^d}\varrho ^{\beta -\gamma }_0(t-s,x-z)\left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (s,z-y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s\\&\preceq \int ^t_{\frac{t}{2}}(t-s)^{\frac{\beta -\gamma }{\alpha }-1}\left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (s,x-y){\mathord {\mathrm{d}}}s\\&\quad +\left( \varrho ^0_\beta +\varrho ^{\beta -\gamma }_\gamma +\varrho ^\beta _0\right) (t,x-y)\preceq \varrho ^0_0(t,x-y). \end{aligned}$$

For \(Q_2(t,x,y)\), by (3.16) and (3.6), we have

$$\begin{aligned} Q_2(t,x,y)\preceq \int ^t_{\frac{t}{2}}(t-s)^{\frac{\beta }{\alpha }-1}\left( \varrho ^0_\beta +\varrho ^\beta _0\right) (s,x-y){\mathord {\mathrm{d}}}s\preceq \varrho ^0_0(t,x-y). \end{aligned}$$

For \(Q_3(t,x,y)\), by (2.28), (3.6) and (2.3), we have

$$\begin{aligned} Q_3(t,x,y)\preceq \int ^{\frac{t}{2}}_0\!\!\!\int _{{\mathbb R}^d}\varrho ^0_0(t-s,x-z)\left( \varrho ^0_\beta +\varrho ^\beta _0\right) (s,z-y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s\preceq \varrho ^0_0(t,x-y). \end{aligned}$$

Combining the above calculations and by (3.2), we obtain (4.18).

(ii) By (2.25), we have

$$\begin{aligned} |\nabla p_y(t,\cdot )(x-y)|\preceq t^{\frac{\alpha -1}{\alpha }} \varrho ^0_0(t,x-y). \end{aligned}$$

By (3.30), we can write

$$\begin{aligned} \nabla _x \varphi _y(t,x)&=\int ^t_{\frac{t}{2}}\!\!\!\int _{{\mathbb R}^d}\nabla p_z(t-s,\cdot )(x-z)(q(s,z,y)-q(s,x,y)){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s\\&\quad +\int ^t_{\frac{t}{2}}\left( \int _{{\mathbb R}^d}\nabla p_z(t-s,\cdot )(x-z){\mathord {\mathrm{d}}}z\right) q(s,x,y){\mathord {\mathrm{d}}}s\\&\quad +\int ^{\frac{t}{2}}_0\!\!\!\int _{{\mathbb R}^d}\nabla p_z(t-s,\cdot )(x-z)q(s,z,y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s\\&=:R_1(t,x,y)+R_2(t,x,y)+R_3(t,x,y). \end{aligned}$$

For \(R_1(t,x,y)\), by (2.25), (3.7) and Lemma 2.1, in view of \(\alpha \in [1,2)\), we have for any \(\gamma \in (\beta -\frac{\alpha }{4},\beta )\),

$$\begin{aligned}&R_1(t,x,y)\preceq \int ^t_{\frac{t}{2}}\!\!\!\int _{{\mathbb R}^d}\varrho ^{\beta -\gamma }_{\alpha -1}(t-s,x-z)\left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (s,x-y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s\\&\quad +\int ^t_{\frac{t}{2}}\!\!\!\int _{{\mathbb R}^d}\varrho ^{\beta -\gamma }_{\alpha -1}(t-s,x-z)\left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (s,z-y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s\\&\quad \preceq \int ^t_{\frac{t}{2}}(t-s)^{\frac{\beta -\gamma -1}{\alpha }}\left( \varrho ^0_\gamma +\varrho ^\beta _{\gamma -\beta }\right) (s,x-y){\mathord {\mathrm{d}}}s\\&\quad +\left( \varrho ^0_{\beta +\alpha -1}+\varrho ^{\beta -\gamma }_{\alpha +\gamma -1}+\varrho ^\beta _{\alpha -1}\right) (t,x-y)\preceq \varrho ^0_{\alpha -1}(t,x-y). \end{aligned}$$

For \(R_2(t,x,y)\), by (3.14), we have

$$\begin{aligned} R_2(t,x,y)\preceq \int ^t_{\frac{t}{2}}(t{-}s)^{\frac{\beta {-}1}{\alpha }}\left( \varrho ^0_\beta {+}\varrho ^\beta _0\right) (s,x-y){\mathord {\mathrm{d}}}s\preceq \varrho ^0_{\alpha {-}1}(t,x-y). \end{aligned}$$

For \(R_3(t,x,y)\), by (2.28), (3.6) and (2.3), we have

$$\begin{aligned} R_3(t,x,y) \preceq \int ^{\frac{t}{2}}_0\!\!\!\int _{{\mathbb R}^d}\varrho ^0_{\alpha -1}(t\!{-}\!s,x-z)\left( \varrho ^0_\beta {+}\varrho ^\beta _0\right) (s,z-y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s\preceq \varrho ^0_{\alpha -1}(t,x-y). \end{aligned}$$

Combining the above calculations and by (3.2), we obtain (4.19). \(\square \)

Below, we write

$$\begin{aligned} P^\kappa _t f(x):=\int _{{\mathbb R}^d}p^\kappa _{\alpha } (t,x,y)f(y){\mathord {\mathrm{d}}}y. \end{aligned}$$

Lemma 4.3

For any bounded and Hölder continuous function \(f\), we have

$$\begin{aligned} {\fancyscript{L}}^\kappa _\alpha \left( \int ^t_0 P^\kappa _sf(\cdot ){\mathord {\mathrm{d}}}s\right) (x)=\int ^t_0{\fancyscript{L}}^\kappa _\alpha P^\kappa _sf(x){\mathord {\mathrm{d}}}s, \ \ x\in {\mathbb R}^d. \end{aligned}$$
(4.20)

Proof

By definition of \({\fancyscript{L}}^\kappa _\alpha \) and Fubini’s theorem, we have

$$\begin{aligned} {\fancyscript{L}}^\kappa _\alpha \left( \int ^t_0 P^\kappa _sf{\mathord {\mathrm{d}}}s\right) (x)&=\lim _{\varepsilon \downarrow 0}\int _{|w|>\varepsilon }\left( \int ^t_0\delta _{ P^\kappa _sf}(x;w){\mathord {\mathrm{d}}}s\right) \kappa (x,w)|w|^{-d-\alpha }{\mathord {\mathrm{d}}}w \\&=\lim _{\varepsilon \downarrow 0}\int ^t_0I_\varepsilon (s,x){\mathord {\mathrm{d}}}s, \end{aligned}$$

where

$$\begin{aligned} I_\varepsilon (s,x):=\int _{|w|>\varepsilon }\delta _{ P^\kappa _sf}(x;w)\kappa (x,w)|w|^{-d-\alpha }{\mathord {\mathrm{d}}}w. \end{aligned}$$

Using the same argument as in proving (3.33), one can prove that for some \(p>1\),

$$\begin{aligned} \sup _{\varepsilon \in (0,1)}\int ^t_0|I_\varepsilon (s,x)|^p{\mathord {\mathrm{d}}}s<+\infty . \end{aligned}$$

Hence, we can interchange the limit and integral, and obtain (4.20). \(\square \)

Lemma 4.4

For any \(p\in [1,\infty )\) and \(f\in L^p({\mathbb R}^d)\), \((0,1)\ni t\mapsto {\fancyscript{L}}^\kappa _\alpha P^\kappa _tf\in L^p({\mathbb R}^d)\) is continuous. In the case of \(p=\infty \), i.e., if \(f\) is a bounded measurable function on \({\mathbb R}^d\), then for each \(x\in {\mathbb R}^d\), \(t\mapsto {\fancyscript{L}}^\kappa _\alpha P^\kappa _tf(x)\) is a continuous function on \((0,1)\). Moreover, for any \(p\in [1,\infty ]\), there exists a constant \(C=C(p,d,\alpha ,\beta , \kappa _0,\kappa _1, \kappa _2, p)>0\) such that for all \(f\in L^p({\mathbb R}^d)\) and \(t>0\),

$$\begin{aligned} \Vert {\fancyscript{L}}^\kappa _\alpha P^\kappa _tf\Vert _p\leqslant C t^{-1}\Vert f\Vert _p. \end{aligned}$$
(4.21)

Proof

For any \(p\in [1,\infty ]\), by Lemma 4.2 and Young’s inequality, we have

$$\begin{aligned}&\Vert {\fancyscript{L}}^\kappa _\alpha P^\kappa _tf\Vert _p\preceq \left( \int _{{\mathbb R}^d}\left| \int _{{\mathbb R}^d}\varrho ^0_0(t,x-y)|f(y)|{\mathord {\mathrm{d}}}y\right| ^p{\mathord {\mathrm{d}}}x\right) ^{1/p} \\&\quad \leqslant \Vert \varrho ^0_0(t)\Vert _1\Vert f\Vert _p\mathop {\preceq }\limits ^{(2.2)} t^{-1}\Vert f\Vert _p. \end{aligned}$$

Thus, we obtain (4.21).

On the other hand, for any \(\varepsilon \in (0,1)\), by Lemma 4.2, we have for \(x\not =y\),

$$\begin{aligned} \sup _{t\in (\varepsilon ,1)}\left| {\fancyscript{L}}^\kappa _\alpha p^\kappa _{\alpha } (t,x,y)\right| \preceq \sup _{t\in (\varepsilon ,1)}\varrho ^0_0(t,x-y)\preceq \varrho ^0_0(\varepsilon ,x-y). \end{aligned}$$

Since for fixed \(x\not =y\in {\mathbb R}^d\), the mapping \(t\mapsto {\fancyscript{L}}^\kappa _\alpha p^\kappa _{\alpha } (t,x,y)\) is continuous by (3.31), the desired continuity of \(t\mapsto {\fancyscript{L}}^\kappa _\alpha P^\kappa _{t}f(x)\) follows by the dominated convergence theorem. \(\square \)

4.3 Proof of Theorem 1.1

After the above preparation, we are now in a position to give the proof of Theorem 1.1. First of all, using Lemmas 3.5 and 3.6, one sees that the calculations in (3.4) make sense, and thus, we obtain (1.7). Moreover, by Lemma 3.7, \(p^\kappa _\alpha \) is a jointly continuous function of \((t,x,y)\in (0,1]\times {\mathbb R}^d\times {\mathbb R}^d\).

(i) Notice that by (2.22), (3.6) and (2.4),

$$\begin{aligned}&\int ^t_0\!\!\!\int _{{\mathbb R}^d}p_z(t-s,x-z)|q(s,z,y)|{\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s \nonumber \\&\quad \preceq \int ^t_0\!\!\!\int _{{\mathbb R}^d}\varrho ^0_\alpha (t-s,x-z)\left( \varrho ^0_\beta +\varrho ^\beta _0\right) (s,z-y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s\nonumber \\&\quad \preceq \left( \varrho ^0_{\alpha +\beta }+\varrho ^\beta _\alpha \right) (t,x-y), \end{aligned}$$
(4.22)

which in turn gives estimate (1.8) by Eq. (3.2) and (2.22), where the constant \(c_1\) can be chosen to depend only on \((d, \alpha , \beta , \kappa _0, \kappa _1, \kappa _2)\).

(ii) Estimate (1.9) follows by Lemma 3.4.

(iii) Estimate (1.10) follows by (4.18). The continuity of \(t\mapsto {\fancyscript{L}}^\kappa _\alpha p^\kappa _\alpha (t,\cdot ,y)(x)\) for \(x\not =y\) follows by (3.31).

(iv) Let \(f\) be a bounded and uniformly continuous function. For any \(\varepsilon >0\), there exists a \(\delta >0\) such that for all \(|x-y|\leqslant \delta \),

$$\begin{aligned} |f(x)-f(y)|\leqslant \varepsilon . \end{aligned}$$

By (3.18) and (2.22), we have

$$\begin{aligned}&\lim _{t\downarrow 0}\sup _{x\in {\mathbb R}^d}\left| \int _{{\mathbb R}^d}p_y(t,x-y)f(y){\mathord {\mathrm{d}}}y-f(x)\right| \nonumber \\&\quad \preceq \lim _{t\downarrow 0}\sup _{x\in {\mathbb R}^d}\int _{{\mathbb R}^d}\varrho ^0_\alpha (t,x-y)\cdot |f(y)-f(x)|{\mathord {\mathrm{d}}}y\nonumber \\&\quad \preceq \varepsilon +2\Vert f\Vert _\infty \lim _{t\downarrow 0}\sup _{x\in {\mathbb R}^d}\int _{|x-y|>\delta }\varrho ^0_\alpha (t,x-y){\mathord {\mathrm{d}}}y\leqslant \varepsilon , \end{aligned}$$

which implies that

$$\begin{aligned} \lim _{t\downarrow 0}\sup _{x\in {\mathbb R}^d}\left| \int _{{\mathbb R}^d}p_y(t,x-y)f(y){\mathord {\mathrm{d}}}y-f(x)\right| =0. \end{aligned}$$

Moreover, by (4.22), we also have

$$\begin{aligned}&\left| \int _{{\mathbb R}^d}\!\int _0^t\!\!\!\int _{{\mathbb R}^d}p_z(t-s,x-z)q(s,z,y)f(y){\mathord {\mathrm{d}}}z{\mathord {\mathrm{d}}}s{\mathord {\mathrm{d}}}y\right| \\&\quad \preceq \int _{{\mathbb R}^d}\left( \varrho ^0_{\alpha +\beta }+ \varrho ^\beta _\alpha \right) (t,x-y){\mathord {\mathrm{d}}}y \mathop {\preceq }\limits ^{(2.2)} t^{\frac{\beta }{\alpha }}\rightarrow 0,\ t\downarrow 0 . \end{aligned}$$

Thus, (1.11) is proven by Eq. (3.2).

We now show that kernels that satisfy (1.7)–(1.11) is unique. For this, let \(\widetilde{p}^\kappa _\alpha (t, x, y)\) be any kernel that satisfies (1.7)–(1.11) and, for \(f\in C^\infty _c({\mathbb R}^d)\), define \(\widetilde{u}_f(t,x):=\int _{{\mathbb R}^d} \widetilde{p}^\kappa _\alpha (t, x, y) f(y) {\mathord {\mathrm{d}}}y\). First of all, by (iv), one sees that

$$\begin{aligned} \widetilde{u}_f\in C_b([0,1]\times {\mathbb R}^d),\quad \lim _{t\downarrow 0}\sup _{x\in {\mathbb R}^d}| \widetilde{u}_f(t,x)-f(x)|=0. \end{aligned}$$

Secondly, by (1.9) we have for any \(\gamma \in (0,\alpha \wedge 1)\),

$$\begin{aligned} |\widetilde{u}_f(t,x)- \widetilde{u}_f(t,x')|&\leqslant \Vert f\Vert _\infty \int _{{\mathbb R}^d} \left| \widetilde{p}^\kappa _{\alpha } (t,x,y)-\widetilde{p}^\kappa _{\alpha } (t,x',y)\right| {\mathord {\mathrm{d}}}y\\&\preceq \Vert f\Vert _\infty |x-x'|^\gamma \int _{{\mathbb R}^d} \left( \varrho ^0_{\alpha {-}\gamma }(t,x-y){+}\varrho ^0_{\alpha -\gamma }(t,x'-y)\right) {\mathord {\mathrm{d}}}y \\&\mathop {\preceq }\limits ^{(2.2)}\Vert f\Vert _\infty |x-x'|^\gamma t^{-\frac{\gamma }{\alpha }}. \end{aligned}$$

The same holds for \(u_f (t, x):= \int _{{\mathbb R}^d} p^\kappa _\alpha (t, x, y) f(y) {\mathord {\mathrm{d}}}y\). Thus in view of (1.7) and (iii), \(w(t, x):= u_f(t, x)-\widetilde{u}_f (t, x)\) satisfies all the conditions of Theorem 4.1 with \(w(0, x)=0\) for every \(x\in {\mathbb R}^d\). Applying Theorem 4.1 to both \(w\) and \(-w\) yields \(w(t, x)=0\) for every \(t>0\) and \(x\in {\mathbb R}^d\). Consequently, we have \(\widetilde{p}^\kappa _\alpha (t, x,y)=p^\kappa _\alpha (t, x,y)\).

  1. (1)

    has already been proved in the above.

  2. (2)

    Applying the maximum principle Theorem 4.1 to \(u_f\) with \(f\in C^\infty _c ({\mathbb R}^d)\) and \(f\leqslant 0\) implies that \(p^\kappa _\alpha (t, x, y)\geqslant 0\). Moreover, since constant function \(u(t, x)=1\) solves the equation \(\partial _t u(t, x)={\fancyscript{L}}^\kappa _\alpha u(t, x)\) with initial value 1, we have (1.12).

  3. (3)

    This follows from the uniqueness of the solution to \(\partial _t u(t, x)={\fancyscript{L}}^\kappa _\alpha u(t, x)\), implied by Theorem 4.1, as well as the continuity of \(p^\kappa _\alpha (t,x,y)\).

  4. (4)

    will be proven in the next subsection.

  5. (5)

    If \(\alpha \in [1,2)\), then estimate (1.15) follows by (4.19).

  6. (6)

    For \(f\in C^2_b({\mathbb R}^d)\), define

    $$\begin{aligned} u(t,x):=f(x)+\int ^t_0 P^\kappa _s{\fancyscript{L}}^\kappa _\alpha f(x){\mathord {\mathrm{d}}}s. \end{aligned}$$

    By (4.20) we have

    $$\begin{aligned} {\fancyscript{L}}^\kappa _\alpha u(t,x)&={\fancyscript{L}}^\kappa _\alpha f(x)+\int ^t_0{\fancyscript{L}}^\kappa _\alpha P^\kappa _s{\fancyscript{L}}^\kappa _\alpha f(x){\mathord {\mathrm{d}}}s\\&={\fancyscript{L}}^\kappa _\alpha f(x)+\int ^t_0\partial _s( P^\kappa _s{\fancyscript{L}}^\kappa _\alpha f)(x){\mathord {\mathrm{d}}}s\\&= P^\kappa _t{\fancyscript{L}}^\kappa _\alpha f(x)=\partial _t u(t,x). \end{aligned}$$

    Moreover, it is easy to see that (4.1), (4.2) and (4.3) are satisfied for \(u\). Thus, by Theorem 4.1 we obtain

    $$\begin{aligned} P^\kappa _t f(x)=u(t,x)=f(x)+\int ^t_0 P^\kappa _s{\fancyscript{L}}^\kappa _\alpha f(x){\mathord {\mathrm{d}}}s, \end{aligned}$$
    (4.23)

    which in turn implies that

    $$\begin{aligned} \lim _{t\downarrow 0}\frac{1}{t}\left( P^\kappa _t f(x)-f(x)\right) =\lim _{t\downarrow 0}\frac{1}{t}\int ^t_0 P^\kappa _s{\fancyscript{L}}^\kappa _\alpha f(x){\mathord {\mathrm{d}}}s\mathop {=}\limits ^{(1.11)}{\fancyscript{L}}^\kappa _\alpha f(x) \end{aligned}$$

    and the convergence is uniform.

  7. (7)

    Fix \(p\in [1,\infty )\). By (iv), (2) and (4.21), it is easy to see that \(( P^\kappa _t)_{t\geqslant 0}\) is a \(C_0\)-semigroup in \(L^p({\mathbb R}^d)\). On the other hand, for any \(f\in L^p({\mathbb R}^d)\), by Eq. (1.7) and Lemma 4.4, one sees that \( P^\kappa _t f\) is differentiable in \(L^p({\mathbb R}^d)\) for any \(t>0\), i.e.,

    $$\begin{aligned}&\lim _{\varepsilon \rightarrow 0}\frac{\Vert P^\kappa _{t+\varepsilon } f- P^\kappa _t f-\varepsilon {\fancyscript{L}}^\kappa _\alpha P^\kappa _t f\Vert _p}{\varepsilon }\\&\quad \leqslant \lim _{\varepsilon \rightarrow 0}\frac{1}{\varepsilon }\int ^{t+\varepsilon }_t\left\| {\fancyscript{L}}^\kappa _\alpha P^\kappa _{t+s} f-{\fancyscript{L}}^\kappa _\alpha P^\kappa _t f\right\| _p{\mathord {\mathrm{d}}}s=0. \end{aligned}$$

    The analyticity of \(C_0\)-semigroup \(( P^\kappa _t)_{t\geqslant 0}\) follows by (4.21) and [31], Theorem II.5.2 (d)].

4.4 Proof of lower bound estimate of \(p^\kappa _\alpha (t,x,y)\)

From the previous subsection, one sees that \(( P^\kappa _t)_{t\geqslant 0}\) is a Feller semigroup. Hence, it determines a Feller process \((\Omega ,{\fancyscript{F}}, ({\mathbb P}_x)_{x\in {\mathbb R}^d}, (X_t)_{t\geqslant 0})\). For any \(f\in C^2_b({\mathbb R}^d)\), it follows from (4.23) and the Markov property of \(X\) that under \({\mathbb P}_x\), with respect to the filtration \({\fancyscript{F}}_t:=\sigma \{X_s, s\leqslant t\}\),

$$\begin{aligned} M^f_t:=f(X_t)-f(X_0)-\int ^t_0{\fancyscript{L}}^\kappa _\alpha f(X_s){\mathord {\mathrm{d}}}s\ \hbox { is a martingale}. \end{aligned}$$
(4.24)

In other words, \({\mathbb P}_x\) solves the martingale problem for \(({\fancyscript{L}}^\kappa _\alpha , C^2_b ({\mathbb R}^d))\). Thus \({\mathbb P}_x\) in particular solves the martingale problem for \(({\fancyscript{L}}^\kappa _\alpha , C^\infty _c ({\mathbb R}^d))\).

We now derive a Lévy system of \(X\) by following an approach from [11]. By (4.24), one can derive that \(X_t=(X^{1}_t, \dots , X^{d}_t)\) is a semi-martingale. By Itô’s formula, we have that, for any \(f\in C^\infty _c({\mathbb R}^d)\),

$$\begin{aligned} f(X_t)-f(X_0)=\sum ^d_{i=1}\int ^t_0{\partial }_if(X_{s-}){\mathord {\mathrm{d}}}X^{i}_s +\sum _{s\leqslant t}\eta _s(f) +\frac{1}{2} A_t(f), \end{aligned}$$
(4.25)

where

$$\begin{aligned} \eta _s(f)=f(X_s)-f(X_{s-})-\sum ^d_{i=1}{\partial }_if(X_{s-})\left( X^{i}_s-X^{i}_{s-}\right) \end{aligned}$$
(4.26)

and

$$\begin{aligned} A_t(f)=\sum ^d_{i, j=1}\int ^t_0{\partial }_i{\partial }_jf(X_{s-}){\mathord {\mathrm{d}}}\langle (X^{i})^c, (X^{j})^c\rangle _s. \end{aligned}$$
(4.27)

Now suppose that \(A\) and \(B\) are two bounded closed subsets of \({\mathbb R}^d\) having a positive distance from each other. Let \(f\in C^\infty _c({\mathbb R}^d)\) with \(f=0\) on \(A\) and \(f=1\) on \(B\). Clearly \(N^f_t:=\int ^t_0\mathbf{1}_A(X_{s-}){\mathord {\mathrm{d}}}M^f_s\) is a martingale. Define

$$\begin{aligned} J(x, y)= k(x, y-x)/|y-x|^{d+\alpha }, \end{aligned}$$
(4.28)

so \({\fancyscript{L}}^\kappa _\alpha \) can be rewritten as

$$\begin{aligned} {\fancyscript{L}}^\kappa _\alpha f(x)= \lim _{\varepsilon \rightarrow 0} \int _{\{|y-x|>\varepsilon \}} (f(y)-f(x)) J(x, y) {\mathord {\mathrm{d}}}y. \end{aligned}$$
(4.29)

We get by (4.24)–(4.27) and (4.29),

$$\begin{aligned} N^f_t&=\sum _{s\leqslant t}\mathbf{1}_A(X_{s-})(f(X_s)-f(X_{s-})) -\int ^t_0\mathbf{1}_A(X_s){\fancyscript{L}}^\kappa _\alpha f(X_s){\mathord {\mathrm{d}}}s\\&=\sum _{s\leqslant t}\mathbf{1}_A(X_{s-})f(X_s)-\int ^t_0\mathbf{1}_A(X_s)\int _{{\mathbb R}^d}f(y)J(X_s,y){\mathord {\mathrm{d}}}y{\mathord {\mathrm{d}}}s. \end{aligned}$$

By taking a sequence of functions \(f_n\in C^\infty _c({\mathbb R}^d)\) with \(f_n=0\) on \(A\), \(f_n=1\) on \(B\) and \(f_n\downarrow \mathbf{1}_B\), we get that, for any \(x\in {\mathbb R}^d\),

$$\begin{aligned} \sum _{s\leqslant t}\mathbf{1}_A(X_{s-})\mathbf{1}_B(X_s) -\int ^t_0\mathbf{1}_A(X_s)\int _BJ(X_s, y){\mathord {\mathrm{d}}}y{\mathord {\mathrm{d}}}s \end{aligned}$$

is a martingale with respect to \({\mathbb P}_x\). Thus,

$$\begin{aligned} {\mathbb E}_x\left[ \sum _{s\leqslant t}\mathbf{1}_A(X_{s-})\mathbf{1}_B(X_s)\right] = {\mathbb E}_x\left[ \int ^t_0\int _{{\mathbb R}^d} \mathbf{1}_A(X_s)\mathbf{1}_B(y)J(X_s, y){\mathord {\mathrm{d}}}y{\mathord {\mathrm{d}}}s\right] . \end{aligned}$$

Using this and a routine measure theoretic argument, we get

$$\begin{aligned} {\mathbb E}_x\left[ \sum _{s\leqslant t}f(X_{s-}, X_s) \right] ={\mathbb E}_x\left[ \int ^t_0\int _{{\mathbb R}^d}f(X_s, y)J(X_s, y){\mathord {\mathrm{d}}}y{\mathord {\mathrm{d}}}s\right] \end{aligned}$$

for any non-negative measurable function \(f\) on \({\mathbb R}^d\times {\mathbb R}^d\) vanishing on \(\{(x, y)\in {\mathbb R}^d\times {\mathbb R}^d: x=y\}\). Finally following the same arguments as in [13], Lemma 4.7] and [14], Appendix A], we get

Theorem 4.5

\(X\) has a Lévy system \((J, t)\) as \(X\), that is, for any \(x\in {\mathbb R}^d\) and any non-negative measurable function \(f\) on \({\mathbb R}_+ \times {\mathbb R}^d\times {\mathbb R}^d\) vanishing on \(\{(s, x, y)\in {\mathbb R}_+ \times {\mathbb R}^d\times {\mathbb R}^d: x=y\}\) and \(({\fancyscript{F}}_t)\)-stopping time \(T\),

$$\begin{aligned} {\mathbb E}_x \left[ \sum _{s\leqslant T} f(s,X_{s-}, X_s) \right] = {\mathbb E}_x \left[ \int _0^T \left( \int _{{\mathbb R}^d} f(s,X_s, y) J(X_s, y){\mathord {\mathrm{d}}}y \right) {\mathord {\mathrm{d}}}s \right] . \end{aligned}$$
(4.30)

For a set \(K\subset {\mathbb R}^d\), denote

$$\begin{aligned} \sigma _K:=\inf \{t\geqslant 0: X_t\in K \},\ \ \tau _K:=\inf \{t\geqslant 0: X_t\notin K\}. \end{aligned}$$

Let \(B(x,r)\) be the ball with radius \(r\) and center \(x\). We need the following lemma (see [2, 13]).

Lemma 4.6

For each \(\gamma \in (0,1)\), there exists \(A_0>0\) such that for every \(A>A_0\) and \(r\in (0,1)\),

$$\begin{aligned} {\mathbb P}_x\left( \tau _{B(x,Ar)} \leqslant r^\alpha \right) \leqslant \gamma . \end{aligned}$$
(4.31)

Proof

Without loss of generality, we assume that \(x=0\). Given \(f\in C^2_b({\mathbb R}^d)\) with \(f(0)=0\) and \(f(x)=1\) for \(|x|\geqslant 1\), we set

$$\begin{aligned} f_r(x):=f(x/r),\ \ r>0. \end{aligned}$$

By the definition of \(f_r\), we have

$$\begin{aligned}&{\mathbb P}_0\left( \tau _{B(0,Ar)}\leqslant r^\alpha \right) \nonumber \\&\quad \leqslant {\mathbb E}_0 \left[ f_{Ar} \left( X_{\tau _{B(0,Ar)}\wedge r^{\alpha }}\right) \right] \mathop {=}\limits ^{(4.24)}{\mathbb E}_0\left( \int ^{\tau _{B(0,Ar)}\wedge r^{\alpha }}_0{\fancyscript{L}}^\kappa _\alpha f_{Ar}(X_s){\mathord {\mathrm{d}}}s\right) . \end{aligned}$$
(4.32)

On the other hand, by the definition of \({\fancyscript{L}}^\kappa _\alpha \), we have for \(\lambda >0\),

$$\begin{aligned} |{\fancyscript{L}}^\kappa _\alpha f_{Ar}(x)|&=\frac{1}{2}\left| \int _{{\mathbb R}^d}(f_{Ar}(x+z)+f_{Ar}(x-z)-2f_{Ar}(x))\kappa (x,z)|z|^{-d-\alpha }{\mathord {\mathrm{d}}}z\right| \\&\leqslant \frac{\kappa _1\Vert \nabla ^2 f_{Ar}\Vert _\infty }{2}\int _{|z|\leqslant \lambda r}|z|^{2-d-\alpha }{\mathord {\mathrm{d}}}z+2\kappa _1\Vert f_{Ar}\Vert _\infty \int _{|z|\geqslant \lambda r}|z|^{-d-\alpha }{\mathord {\mathrm{d}}}z\\&=\kappa _1\frac{\Vert \nabla ^2 f\Vert _\infty }{(Ar)^2}\frac{(\lambda r)^{2-\alpha }}{2(2-\alpha )}s_1+2\kappa _1\Vert f\Vert _\infty \frac{(\lambda r)^{-\alpha }}{\alpha }s_1\\&=\kappa _1s_1\left( \frac{\Vert \nabla ^2 f\Vert _\infty }{A^2}\frac{\lambda ^{2-\alpha }}{2(2-\alpha )}+2\Vert f\Vert _\infty \frac{\lambda ^{-\alpha }}{\alpha }\right) r^{-\alpha }, \end{aligned}$$

where \(s_1\) is the sphere area of the unit ball. Substituting this into (4.32), we get

$$\begin{aligned} {\mathbb P}_0(\tau _{B(0,Ar)}\leqslant r^\alpha ) \leqslant \kappa _1s_1\left( \frac{\Vert \nabla ^2 f\Vert _\infty }{A^2}\frac{\lambda ^{2(2-\alpha )}}{2-\alpha }+2\Vert f\Vert _\infty \frac{\lambda ^{-\alpha }}{\alpha }\right) . \end{aligned}$$

Choosing first \(\lambda \) large enough and then \(A\) large enough yield the desired estimate. \(\square \)

Now we can give

Proof of lower bound of \(p^\kappa _\alpha (t,x,y)\). By Lemma 4.6, there is a constant \(\lambda \in (0,\tfrac{1}{2})\) such that for all \(t\in (0,1)\),

$$\begin{aligned} {\mathbb P}_x\left( \tau _{B(x,t^{1/\alpha }/2)}\leqslant \lambda t\right) \leqslant \tfrac{1}{2}. \end{aligned}$$
(4.33)

By (3.2), (2.22) and (4.22), there is a time \(t_0\in (0,1)\) such that

$$\begin{aligned} p^\kappa _\alpha (t,x,y)\succeq t^{-d/\alpha }\,\hbox { for all }\, t\in (0,t_0) \,\hbox { and }\, |x-y|\leqslant 3t^{1/\alpha }. \end{aligned}$$

By C–K equation (1.13) and iterating \([1/t_0]+1\) times, we conclude that

$$\begin{aligned} p^\kappa _\alpha (t,x,y)\succeq t^{-d/\alpha }\hbox { for all }\, t\in (0,1) \,\hbox { and }\, |x-y|\leqslant 3t^{1/\alpha }. \end{aligned}$$
(4.34)

Below, we assume

$$\begin{aligned} |x-y|>3t^{1/\alpha }. \end{aligned}$$
(4.35)

For the given \(\lambda \) in (4.33), by the strong Markov property, we have

$$\begin{aligned} {\mathbb P}_x\left( X_{\lambda t}\in B(y,t^{1/\alpha })\right)&\geqslant {\mathbb P}_x\left( \sigma :=\sigma _{B(y,t^{1/\alpha }/2)} \leqslant \lambda t; \sup _{s\in [\sigma ,\sigma +\lambda t]}|X_s-X_\sigma |< t^{1/\alpha }/2\right) \nonumber \\&={\mathbb E}_x\left( {\mathbb P}_z\left( \sup _{s\in [0,\lambda t]}|X_s-z| < t^{1/\alpha }/2\right) \Big |_{z=X_\sigma }; \sigma _{B(y,t^{1/\alpha }/2)} \leqslant \lambda t\right) \nonumber \\&\geqslant \inf _{z\in B(y,t^{1/\alpha }/2)} {\mathbb P}_z\left( \tau _{B(z,t^{1/\alpha }/2)}> \lambda t\right) {\mathbb P}_x\left( \sigma _{B(y,t^{1/\alpha }/2)}\leqslant \lambda t\right) \nonumber \\&\mathop {\geqslant }\limits ^{(4.33)}\tfrac{1}{2} {\mathbb P}_x\left( \sigma _{B(y,t^{1/\alpha }/2)} \leqslant \lambda t\right) \nonumber \\&\geqslant \tfrac{1}{2}{\mathbb P}_x\left( X_{\lambda t\wedge \tau _{B(x,t^{1/\alpha })}}\in B(y,t^{1/\alpha }/2)\right) . \end{aligned}$$
(4.36)

Noticing that

$$\begin{aligned} X_s\notin B\left( y,t^{1/\alpha }/2\right) \subset B\left( x,t^{1/\alpha }\right) ^c,\ \ s<\lambda t\wedge \tau _{B(x,t^{1/\alpha })}, \end{aligned}$$

we have

$$\begin{aligned} \mathbf{1}_{X_{\lambda t\wedge \tau _{B(x,t^{1/\alpha })}}\in B(y,t^{1/\alpha }/2)}=\sum _{s\leqslant \lambda t\wedge \tau _{B(x,t^{1/\alpha })}}\mathbf{1}_{X_s\in B(y,t^{1/\alpha }/2)}. \end{aligned}$$

Thus, by (4.30) we have

$$\begin{aligned}&{\mathbb P}_x\left[ X_{\lambda t\wedge \tau _{B(x,t^{1/\alpha })}}\in B(y,t^{1/\alpha }/2)\right] ={\mathbb E}_x\left[ \int ^{\lambda t\wedge \tau _{B(x,t^{1/\alpha })}}_0\!\!\int _{B(y,t^{1/\alpha }/2)} J(X_s,u){\mathord {\mathrm{d}}}u{\mathord {\mathrm{d}}}s\right] \nonumber \\&\quad \geqslant {\mathbb E}_x\left[ \int ^{\lambda t\wedge \tau _{B(x,t^{1/\alpha })}}_0\!\!\int _{B(y,t^{1/\alpha }/2)}\frac{\kappa _0}{|X_s-u|^{d+\alpha }}{\mathord {\mathrm{d}}}u{\mathord {\mathrm{d}}}s\right] \nonumber \\&\quad \geqslant {\mathbb E}_x\left[ \lambda t\wedge \tau _{B(x,t^{1/\alpha })}\right] \int _{B(y,t^{1/\alpha }/2)} \frac{\kappa _0}{(|x-y|+3t^{1/\alpha }/2)^{d+\alpha }}{\mathord {\mathrm{d}}}u \nonumber \\&\mathop {\geqslant }\limits ^{(4.35)} \lambda t\, {\mathbb P}_x\left( \tau _{B(x,t^{1/\alpha })}\geqslant \lambda t\right) \left( \int _{B(y,t^{1/\alpha }/2)}{\mathord {\mathrm{d}}}u\right) \frac{\kappa _0(2/3)^{d+\alpha }}{|x-y|^{d+\alpha }} \nonumber \\&\quad \geqslant \left( \lambda \kappa _0(2/3)^{d+\alpha }2^{-d-1}s_1\right) \frac{t^{1+d/\alpha }}{|x-y|^{d+\alpha }}, \end{aligned}$$
(4.37)

where \(s_1\) is the sphere area of the unit ball.

Now, by Chapman–Kolmogorov’s equation again, we have

$$\begin{aligned} p^\kappa _\alpha (t,x,y)&\geqslant \int _{B(y,t^{1/\alpha })}p^\kappa _\alpha (\lambda t,x,z)p^\kappa _\alpha ((1-\lambda ) t, z,y){\mathord {\mathrm{d}}}z\\&\geqslant \inf _{z\in B(y,t^{1/\alpha })}p^\kappa _\alpha ((1-\lambda ) t, z,y)\int _{B(y,t^{1/\alpha })}p^\kappa _\alpha (\lambda t,x,z){\mathord {\mathrm{d}}}z\\&\mathop {\succeq }\limits ^{(4.34)} t^{-d/\alpha }{\mathbb P}_x\left( X_{\lambda t}\in B(y,t^{1/\alpha })\right) \mathop {\succeq }\limits ^{(4.36),(4.37)} t|x-y|^{-d-\alpha }. \end{aligned}$$

which, combining with (4.34), gives the lower bound estimate of \(p^\kappa _\alpha (t,x,y)\). \(\square \)

4.5 Proof of Corollary 1.3

Since \(\lambda _0 I_{d\times d} \leqslant A(x)) \leqslant \lambda _1 I_{d\times d}\) and \(|a_{ij}(x)-a_{ij}(y)|\leqslant \lambda _2 |x-y|^\beta \) for each \(1\leqslant i,j\leqslant d\), the function \(\kappa (x, z)\) defined by (1.21) satisfies the conditions (1.5)-(1.6) with \(\kappa _i,\, i=0, 1, 2\), depend only on \(d, \alpha , \beta , \lambda _0, \lambda _1\) and \(\lambda _2\). Thus by Theorem 1.1, there is a jointly continuous heat kernel \(p(t, x, y)\) for the non-local operator \({\fancyscript{L}}={\fancyscript{L}}^\kappa _\alpha \) of (1.20) corresponding to this \(\kappa (x, z)\). Let \(\{\widetilde{X}, {\mathbb P}_x, x\in {\mathbb R}^d\}\) be the Feller process having \(p(t, x , y)\) as its transition density function. As we observed in the beginning of Sect. 4.4, \({\mathbb P}_x\) solves the martingale problem for \(({\fancyscript{L}}, C^2_b({\mathbb R}^d))\). On the other hand, it is shown in §7 of [1] (see Theorem 7.1 and its proof as well as Theorems 4.1 and 6.3 there) that the law of the unique weak solution \(X\) to SDE (1.18) is the unique solution to the martingale problem for \(({\fancyscript{L}}, C^2_b({\mathbb R}^d))\). Hence \(\widetilde{X}\) and \(X\) have the same distribution. Therefore \(p(t, x, y)\) is the transition density function of \(X\). The conclusion of the corollary now follows from Theorem 1.1. \(\square \)