Fractional Leibniz-type Rules on Spaces of Homogeneous Type

Abstract

Let (M,ρ,μ) be a space of homogeneous type satisfying the reverse doubling condition and the non-collapsing condition. In view of the lack of the Fourier transform in this setting, the authors obtain Leibniz-type rules of fractional order on (M,ρ,μ) under the assumptions of the heat kernel satisfying the upper bound estimate, the Hölder estimate and the stochastic completeness property.

Appendix

Besides Examples 1.1 and 1.2, we present another three examples of heat kernels generated from differential operators on ${\mathbb {R}^n}$, which also satisfy the setup in Section 1.2.

Example A.1

On the classical Euclidean space ${\mathbb {R}^n}$, consider

$$L= \text{div} (\mathbf{A} \nabla)$$

with A being a real symmetric matrix of bounded measurable functions satisfying the uniformly elliptic condition, that is,

$$\left\langle \mathbf{A}\xi,\xi \right\rangle\ge \lambda |\xi|^{2}\quad \text{for all}\ \xi\in{\mathbb{R}^n},$$

where λ is a positive constant. From Aronson [1, Theorem 1], it follows that the heat kernel {p_t}_t> 0 of $\{e^{-tL}\}_{t\ge 0}$ is non-negative and satisfies the two-sided Gaussian estimate

$$ p_{t}(x,y)\asymp \frac{C}{t^{n/2}}\exp \left( -c\frac{|x-y|^{2}}{t}\right). $$

The Hölder regularity of the heat kernel {p_t}_t> 0 was due to Nash [57, Part IV].

Example A.2

Let $({\mathbb {R}^n}, \omega )$ be the weighted Euclidean space endowed with the Euclidean metric, where ω is a non-negative, locally integrable function which we assume is either in the Muckenhoupt class A₂. On this weighted manifold $({\mathbb {R}^n}, \omega )$, consider the degenerate operator

$$L_{\omega} =\frac1\omega \text{div} \left( \mathbf{A} \nabla \right),$$

where A is a real symmetric matrix of measurable functions satisfying the degenerate ellipticity condition

$$ \begin{cases} \lambda \omega(x)|\xi|^{2}\le \left\langle \mathbf{A}\xi,\xi \right\rangle; \\ \left|\left\langle \mathbf{A}\xi,\eta \right\rangle\right|\le {\Lambda} \omega(x)|\xi||\eta|, \end{cases} $$

for some constants λ and Λ such that $0<\lambda \le {\Lambda }<\infty$, and for all $\xi ,\eta \in \mathbb C^{n}$. Of course, if ω = 1, then L_ω goes back to the operator L in Example A.1.

According to Cruz-Uribe and Rios [22, Theorem 1], upon setting

$$d\mu(x)=\omega(x) dx,$$

the semigroup $\{e^{-tL_{\omega }}\}_{t\ge 0}$ has a heat kernel {p_t}_t> 0 such that

$$ \begin{array}{@{}rcl@{}} e^{-tL_{\omega}} f(x) ={\int}_{{\mathbb{R}^n}} p_{t}(x,y) f(y) d\mu(y)\quad \text{for all}\ f\in L^{2}({\mathbb{R}^n}). \end{array} $$

Moreover, for all $t\in (0,\infty )$ and $x,y, y^{\prime }\in {\mathbb {R}^n}$, the heat kernel p_t satisfies the weighted Gaussian bound

$$ \begin{array}{@{}rcl@{}} |p_{t}(x,y)|\le C \frac 1{\sqrt{\mu(B(x, t^{1/2})) \mu(B(y, t^{1/2}))}} \exp\left( -c\frac{|x-y|^{2}}{t}\right) \end{array} $$

(A.1)

and the Hölder estimate

$$ \begin{array}{@{}rcl@{}} |p_{t}(x,y)-p_{t}(x,y^{\prime})|&\leq& C\left( \frac{|y-y^{\prime}|}{t^{1/2}+|x-y|}\right)^{\theta} \frac 1{\sqrt{\mu(B(x, t^{1/2})) \mu(B(y, t^{1/2}))}} \\ &&\times\exp\left( -c\frac{|x-y|^{2}}{t}\right) \end{array} $$

(A.2)

whenever $|y-y^{\prime }|\leq (t^{1/2}+|x-y|)/2$, where the Hölder exponent 𝜃 is in (0, 1].

From Eqs. A.1 and A.2, it follows that the heat kernel {p_t}_t> 0 satisfies (UE) and (HE) with d_w = 2 and 𝜃 ∈ (0, 1). The stochastic completeness property (SC) was proved in [21, Theorem 1.7].

Example A.3

Let $m\in {\mathbb N}$. Consider on the classical Euclidean space ${\mathbb {R}^n}$ the homogeneous elliptic operator of order 2m of the form

$$L_{0}=(-1)^{m}{\sum}_{|\alpha|=|\beta|=m}\partial^{\alpha}(a_{\alpha,\beta}\partial^{\beta}),$$

where {a_α,β(x)} is a symmetric matrix of complex-valued bounded measurable functions on ${\mathbb {R}^n}$ satisfying the strong Gårding inequality: for all functions u in the Sobolev space $W^{m,2}({\mathbb {R}^n})$,

$$(L_{0}u,u)\ge \delta_{0} \|\nabla^{m} u\|^{2}_{L^{2}({\mathbb{R}^n})}$$

for some constant δ₀ > 0.

• (i) Under the case 2m ≥ n, according to Auscher and Tchamitchian [4, p. 59, Propositon 28], the semigroup $\{e^{-tL_{0}}\}_{t\ge 0}$ has a heat kernel {p_t}_t> 0 satisfying that for all t > 0 and $x,y,y^{\prime }\in {\mathbb {R}^n}$,

  $$ \begin{array}{@{}rcl@{}} |p_{t}(x,y)|\le C\frac{1}{t^{n/(2m)}} \exp\left\{-c\left( \frac{|x-y|}{t^{1/(2m)}}\right)^{\frac{2m}{2m-1}}\right\} \end{array} $$

  (A.4)

  and, for $|y-y^{\prime }|\le (t^{1/(2m)}+|x-y|)/2$,

  $$ \begin{array}{@{}rcl@{}} |p_{t}(x,y^{\prime}) - p_{t}(x,y)|\!\le\! C\frac{1}{t^{n/(2m)}}\left( \frac{|y^{\prime}-y|}{t^{1/(2m)}+|x-y|}\right)^{\theta} \exp\left\{ - c\left( \frac{|x - y|}{t^{1/(2m)}}\right)^{\frac{2m}{2m-1}}\right\},\\ \end{array} $$

  (A.5)

  where C and c are positive constants, and the Hölder exponent 𝜃 can be any number in (0,1).

  Note that Eq. A.4 is just (UE). Based on [4, p. 55, Proposition 25] (see also [4, p. 69, Remarks 3]), the upper bound estimate Eq. A.4 guarantees the stochastic completeness property (SC). Moreover, observe that Eq. A.5 implies that for all $x,y,y^{\prime }\in {\mathbb {R}^n}$ satisfying $|y-y^{\prime }|\le t^{1/(2m)}$,

  $$ \begin{array}{@{}rcl@{}} |p_{t}(x,y^{\prime})-p_{t}(x,y)|\le C\frac{1}{t^{n/(2m)}}\left( \frac{|y^{\prime}-y|}{t^{1/(2m)}}\right)^{\theta} \exp\left\{-c\left( \frac{|x-y|}{t^{1/(2m)}}\right)^{\frac{2m}{2m-1}}\right\}, \end{array} $$

  (A.6)

  thereby leading to (HE). Indeed, when $t^{1/(2m)}/2 \le |y-y^{\prime }|\le t^{1/(2m)}$, by Eq. A.4, we have

  $$ \begin{array}{@{}rcl@{}} |p_{t}(x,y^{\prime})-p_{t}(x,y)| &&\le |p_{t}(x,y^{\prime})|+|p_{t}(x,y)|\\ &&\lesssim \frac{1}{t^{n/(2m)}} \left( \exp\left\{-c\left( \frac{|x-y^{\prime}|}{t^{1/(2m)}}\right)^{\frac{2m}{2m-1}}\right\} +\exp\left\{-c\left( \frac{|x-y|}{t^{1/(2m)}}\right)^{\frac{2m}{2m-1}}\right\}\right)\\ &&\approx \frac{1}{t^{n/(2m)}}\exp\left\{-c\left( \frac{|x-y|}{t^{1/(2m)}}\right)^{\frac{2m}{2m-1}}\right\}\\ &&\approx C\frac{1}{t^{n/(2m)}}\left( \frac{|y^{\prime}-y|}{t^{1/(2m)}}\right)^{\theta}\exp\left\{-c\left( \frac{|x-y|}{t^{1/(2m)}}\right)^{\frac{2m}{2m-1}}\right\}. \end{array} $$

  Altogether, under n ≤ 2m, the heat kernel {p_t}_t> 0 satisfies (SC), (UE) and (HE) with d_w = 2m and arbitrary 𝜃 ∈ (0, 1).

• (ii) In general, assuming further that L₀ has the Dirichlet property (see [3, Definition 5]), Auscher and Qafsaoui [3, Theorem 12] proved that the heat kernel {p_t}_t> 0 satisfies Eq. A.4 and, for all t > 0 and all $x,y,y^{\prime }\in {\mathbb {R}^n}$,

  $$ \begin{array}{@{}rcl@{}} |p_{t}(x,y^{\prime})- p_{t}(x,y)|\le C\frac{1}{t^{n/(2m)}}\left( \frac{|y^{\prime}-y|}{t^{1/(2m)}}\right)^{\nu}, \end{array} $$

  (A.7)

  where C is a positive constant, and ν can be any number in (0,1). We claim that an application of Eqs. A.4-A.7 and the geometric mean inequality yields that Eq. A.6 holds whenever $|y-y^{\prime }|\le t^{1/(2m)}$. Indeed, when $|y-y^{\prime }|\le t^{1/2m}$, it follows from Eq. A.4 that

  $$ \begin{array}{@{}rcl@{}} |p_{t}(x,y^{\prime})-p_{t}(x,y)| &\le |p_{t}(x,y^{\prime})|+|p_{t}(x,y)|\lesssim \frac{1}{t^{n/(2m)}}\exp\left\{\!-c\!\left( \!\frac{|x-y|}{t^{1/(2m)}}\!\right)^{\frac{2m}{2m-1}}\!\right\}. \end{array} $$

  Given any 𝜖 ∈ (0, 1), taking the geometric mean between the last inequality and Eq. A.7 leads to

  $$ \begin{array}{@{}rcl@{}} |p_{t}(x,y^{\prime})-p_{t}(x,y)| &\le \left( \frac{|y^{\prime}-y|}{t^{1/(2m)}}\right)^{\nu \epsilon} \frac{1}{t^{n/(2m)}}\exp\left\{-c\left( \frac{|x-y|}{t^{1/(2m)}}\right)^{\frac{2m}{2m-1}}\right\}. \end{array} $$

  So, setting 𝜃 = ν𝜖 yields Eq. A.6. Consequently, the heat kernel {p_t}_t> 0 satisfies (SC), (UE) and (HE) with d_w = 2m and arbitrary 𝜃 ∈ (0, 1).

To continue, we present several examples (especially when n ≤ 2m) enjoying the aforementioned Dirichlet property so that the corresponding heat kernel {p_t}_t> 0 satisfies (SC), (UE) and (HE) with d_w = 2m and 𝜃 ∈ (0, 1):

• (a) Assume that the coefficients a_α,β are constants. Under such an assumption, the strong Gårding inequality is equivalent to the following ellipticity condition (see [61, Section 1.4])

  $$ \text{Re}{\sum}_{|\alpha|=|\beta|=m} a_{\alpha,\beta}(x) \zeta^{\beta}\zeta^{\alpha} \ge \delta |\zeta|^{2m}\quad \text{for all}\ \zeta\in {\mathbb{R}^n}, $$

  where δ is a positive constant. In this case, Auther-Qafsaoui [3, Proposition 45] proved that the Dirichlet property holds automatically for L₀, so that (UE) and (HE) hold with d_w = 2m and 𝜃 ∈ (0, 1).

• (b) Let $BMO({\mathbb {R}^n} )$ be the space of locally integrable functions f on ${\mathbb {R}^n}$ such that

  $$ \begin{array}{@{}rcl@{}} \|f\|_{BMO({\mathbb{R}^n})}:=\sup_{\text{balls} B\subset {\mathbb{R}^n}}\frac{1}{B}{\int}_{B}|f(x)-f_{B}| dx<\infty, \end{array} $$

  where f_B denotes the integral average of f over the ball B. Obviously, constant functions are in $BMO({\mathbb {R}^n})$. According to [3, Proposition 47], there exists a small constant $\varepsilon _{0}\in (0,\infty )$ such that if

  $$\sup\left\{\|a_{\alpha,\beta}\|_{BMO({\mathbb{R}^n})}:\ |\alpha|=|\beta|=m\right\}<\varepsilon_{0},$$

  then L₀ has the Dirichlet property, which ensures (UE) and (HE) with d_w = 2m and 𝜃 ∈ (0, 1).

• (c) Assume that the coefficients a_α,β are constants and a is a complex bounded function on ${\mathbb {R}^n}$. Consider the operator $L=L_{0}^{\ast } a L_{0}$ with non-smooth coefficients. According to [3, Proposition 51], the heat kernel $\{{p_{t}^{L}}\}_{t>0}$ of the semigroup $\{e^{-tL}\}_{t\ge 0}$ satisfies (UE) and (HE) with d_w = 4m and 𝜃 ∈ (0, 1).

  In particular, taking L₀ = (−Δ)^m and a ≡ 1, we see that the associated heat kernel of (−Δ)^2m satisfies (UE) and (HE) with d_w = 4m and 𝜃 ∈ (0, 1).