Nonlocal operators of small order

Abstract

In this work we study nonlocal operators and corresponding spaces with a focus on operators of order near zero. We investigate the interior regularity of eigenfunctions and of weak solutions to the associated Poisson problem depending on the regularity of the right-hand side. Our method exploits the variational structure of the problem and we prove that eigenfunctions are of class $C^{\infty }$ if the kernel satisfies this property away from its singularity. Similarly in this case, if in the Poisson problem the right-hand is of class $C^{\infty }$, then also any weak solution is of class $C^{\infty }$.

1 Introduction and main results

A crucial role in the investigation of differential operators is the study of eigenfunctions and corresponding eigenvalues, if they exist. In the classical case of the Laplacian $-\Delta$ in a bounded domain $\Omega$ in $\mathbb {R}^N$, it is well known that there exists a sequence of functions $u_n\in H^1_0(\Omega )$, $n\in \mathbb {N}$ and corresponding values $\lambda _{n}>0$ such that

$$\begin{aligned} -\Delta u_n=\lambda _n u_n \quad \text { in} \,\,\Omega ~\text { and}~ u_n=0~\text { in}\,\, \partial \Omega . \end{aligned}$$

Here, $H^1_0(\Omega )$ is as usual the closure of $C^{\infty }_c(\Omega )$ with respect to the norm $u\mapsto \Big (\Vert u\Vert _{L^2(\Omega )}^2+\Vert |\nabla u|\Vert _{L^2(\Omega )}^2\Big )^{\frac{1}{2}}$.

With the well-known De Giorgi iteration in combination with the Sobolev embedding, it follows that $u_n$ must be bounded and by a boot strapping argument using the regularity theory of the Laplacian it follows that $u_n$ is smooth in $\Omega$. In the model case of a nonlocal problem, one usually studies the fractional Laplacian $(-\Delta )^s$ with $s\in (0,1)$. This operator can be defined via its Fourier symbol $|\cdot |^{2s}$ and it can be shown that for $\phi \in C^{\infty }_c(\mathbb {R}^N)$ we have

$$\begin{aligned} (-\Delta )^s\phi (x)=c_{N,s}p.v.\int _{\mathbb {R}^N}\frac{u(x)-u(y)}{|x-y|^{N+2s}}\ \textrm{d}y=c_{N,s}\lim _{\epsilon \rightarrow 0^+}\int _{\mathbb {R}^N{\setminus } B_{\epsilon }(x)}\frac{u(x)-u(y)}{|x-y|^{N+2s}}\ \textrm{d}y, \quad x\in \mathbb {R}^N \end{aligned}$$

with a suitable normalization constant $c_{N,s}>0$. As in the above classical case, it can be shown that there exists a sequence of functions $u_n\in {\mathscr {H}}^s_0(\Omega )$, $n\in \mathbb {N}$ and corresponding values $\lambda _{s,n}>0$ such that

$$\begin{aligned} (-\Delta )^s u_n=\lambda _{s,n} u_n ~\text {in}~ \Omega ~ \text {and}~u_n=0~\text {in}~ \mathbb {R}^N{\setminus }\Omega . \end{aligned}$$

Here, the space ${\mathscr {H}}^s_0(\Omega )$ is given by the closure of $C^{\infty }_c(\Omega )$—understood as functions on $\mathbb {R}^N$—with respect to the norm $u\mapsto \Big (\Vert u\Vert _{L^2(\Omega )}^2+{\mathscr {E}}_s(u,u)\Big )^{\frac{1}{2}}$, where for $u,v\in C^{\infty }_c(\mathbb {R}^N)$ we set

$$\begin{aligned} {\mathscr {E}}_s(u,v):=\frac{c_{N,s}}{2}\int _{\mathbb {R}^N}\int _{\mathbb {R}^N} \frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+2s}}\ \textrm{d}x\textrm{d}y. \end{aligned}$$

With similar methods as in the classical case, it follows that $u_n$ is smooth in the interior of $\Omega$.

In the following we investigate the above discussion to the case where the kernel function $z\mapsto c_{N,s}|z|^{-N-2s}$ is replaced by a measurable function $j:\mathbb {R}^N\rightarrow [0,\infty ]$ such that for some $\sigma \in (0,2]$ we have

$$\begin{aligned} j(z)=j(-z)~\text { for all}~ z\in \mathbb {R}^N ~{ and}~\int _{\mathbb {R}^N}\min \{1,|z|^{\sigma }\}j(z)\ \textrm{d}z<\infty . \end{aligned}$$

(1.1)

With $\sigma =2$, the above yields that $j\,\textrm{d}z$ is a Lévy measure and the associated operator to this choice of kernel is of order below 2. In the following we focus on the case where the singularity of j is not too large, that is on the case $\sigma <1$ so that the associated operator is of order strictly below one. We call the operator in this case also of small order.

Motivated by applications using nonlocal models, where a small order of the operator captures the optimal efficiency of the model [1, 24], nonlocal operators with possibly order near zero, i.e. if (1.1) is satisfied for all $\sigma >0$, have been studied in linear and nonlinear integro-differential equations, see [5, 6, 11,12,13, 17, 18, 25] and the references in there. From a stochastic point of view, general classes of nonlocal operators appear as the generator of jump processes, where the jump behavior is modeled through types of Lévy measures and properties of associated harmonic functions have been studied, see [14, 16, 21, 23] and the references in there. In particular, operators of the form $\phi (-\Delta )$ for a certain class of functions $\phi$ are of interest from a stochastic and analytic point of view, see e.g. [2, 3] and the references in there.

In the following, we aim at investigating properties of bilinear forms and operators associated to a kernel j satisfying (1.1) for some $\sigma \in (0,2]$ from a variational point of view. For this, some further assumptions on j are needed in our method and we present certain explicit examples at the end of this introduction, where our results apply.

Let $\Omega \subset \mathbb {R}^N$ open, $u,v\in C^{0,1}_c(\Omega )$ understood as functions defined on $\mathbb {R}^N$, and consider the bilinear form

$$\begin{aligned} b_{j,\Omega }(u,v):=\frac{1}{2}\int _{\Omega }\int _{\Omega }(u(x)-u(y))(v(x)-v(y))j(x-y)\ \textrm{d}x\textrm{d}y, \end{aligned}$$

(1.2)

where we also write $b_{j}(u,v):=b_{j,\mathbb {R}^N}(u,v)$ and $b_{j,\Omega }(u):=b_{j,\Omega }(u,u)$, $b_j(u)=b_{j}(u,u)$ resp. We denote

$$\begin{aligned} D^j(\Omega ):=\{u\in L^2(\Omega )\;:\; b_{j,\Omega }(u)<\infty \}. \end{aligned}$$

(1.3)

Associated to $b_j$ there is a nonlocal self-adjoint operator $I_j$ which for $u,v\in C^{0,1}_c(\Omega )$ satisfies

$$\begin{aligned} b_{j}(u,v)= & {} \int _{\mathbb {R}^N}I_{j}u(x)v(x)\ \textrm{d}x~\text { and is represented by}\nonumber \\{} & {} I_{j}u(x)=p.v.\int _{\mathbb {R}^N}(u(x)-u(y))j(x-y)\ \textrm{d}y,\quad x\in \Omega . \end{aligned}$$

(1.4)

To investigate the eigenvalue problem for $I_j$, we further need the space

$$\begin{aligned} {\mathscr {D}}^j(\Omega ):=\{u\in D^j(\mathbb {R}^N)\;:\; 1_{\mathbb {R}^N{\setminus } \Omega } u\equiv 0\}. \end{aligned}$$

Note that clearly ${\mathscr {D}}^j(\mathbb {R}^N)=D^j(\mathbb {R}^N)$ and both $D^j(\Omega )$ and ${\mathscr {D}}^j(\Omega )$ are Hilbert spaces with scalar products

$$\begin{aligned} \langle \cdot ,\cdot \rangle _{D^j(\Omega )}:=\langle \cdot ,\cdot \rangle _{L^2(\Omega )}+b_{j,\Omega }(\cdot ,\cdot ) ~ \text {and}~ \langle \cdot ,\cdot \rangle _{{\mathscr {D}}^j(\Omega )}:=\langle \cdot ,\cdot \rangle _{L^2(\Omega )}+b_j(\cdot ,\cdot )~\text {resp.}~ \end{aligned}$$

Our first result concerns the eigenfunctions for the operator $I_j$.

Theorem 1.1

Let (1.1) hold with $\sigma =2$ and assume j satisfies additionally

$$\begin{aligned} \int _{\mathbb {R}^N}j(z)\ \textrm{d}z=\infty . \end{aligned}$$

(1.5)

Let $\Omega \subset \mathbb {R}^N$ be a bounded open set. Then there exists a sequence $u_n\in {\mathscr {D}}^j(\Omega )$, $n\in \mathbb {N}$ and values $\lambda _n$ such that

$$\begin{aligned} b_j(u_n,v)= & {} \lambda _n\int _{\Omega } u_n(x)v(x)\ \textrm{d}x ~\text { for all}~ v\in {\mathscr {D}}^j(\Omega ), n\in \mathbb {N}~\text {and}~\\{} & {} 0<\lambda _1<\lambda _2\le \ldots \le \lambda _n\rightarrow \infty ~\text { for}~ n\rightarrow \infty . \end{aligned}$$

Here, we have

$$\begin{aligned} \lambda _1=\Lambda _1(\Omega ):=\inf _{\begin{array}{c} u\in {\mathscr {D}}^j(\Omega )\\ u\ne 0 \end{array}}\frac{b_j(u)}{\Vert u\Vert _{L^2(\Omega )}^2} \end{aligned}$$

(1.6)

and $u_1$ is unique up to a multiplicative constant—that is, $\lambda _1$ is simple. Moreover, $u_1$ can be chosen to be positive in $\Omega$. Furthermore, the following statements hold.

• (1) If in addition j satisfies

  $$\begin{aligned} \int _{B_R(0){\setminus } B_r(0)} j^2(z)\ \textrm{d}z<\infty ~\text {for all}~ 0<r<R, \end{aligned}$$

  (1.7)

  then $u_n\in L^{\infty }(\Omega )$ for every $n\in \mathbb {N}$ and there is $C=C(\Omega ,j,n)>0$ such that

  $$\begin{aligned} \Vert u_n\Vert _{L^{\infty }(\Omega )}\le C\Vert u_n\Vert _{L^2(\Omega )}. \end{aligned}$$

• (2) Assume (1.1) holds with $\sigma <\frac{1}{2}$, (1.7) holds, and there is $m\in \mathbb {N}\cup \{\infty \}$ such that the following holds: We have $j\in W^{l,1}(\mathbb {R}^N{\setminus } B_{\epsilon }(0)$ for every $l\in \mathbb {N}$ with $l\le 2m$, and there is some constant $C_j>0$ such that

  $$\begin{aligned} |\nabla j(z)|\le C_j|z|^{-1-\sigma -N}~\text {for all}~ 0<|z|\le 3. \end{aligned}$$

  Then $u_n\in H^m_{loc}(\Omega )$. In particular, $u_n\in C^{\infty }(\Omega )$ if $m=\infty$.

For the definition of the Sobolev spaces $W^{l,1}$ and $H^m$ we refer to Sect. 2.1. The first part of Theorem 1.1 indeed follows immediately from the results of [19]. To show the boundedness, we emphasize that in our setting, there are no Sobolev embeddings available and thus it is not clear how to implement the approach via the De Giorgi iteration. We circumvent this, by generalizing the $\delta$-decomposition introduced in [13]. The proof of the regularity statement is inspired by the approach used in [7], where the author studies regularity of solutions to equations involving nonlocal operators which are in some sense comparable to the fractional Laplacian and uses Nikol’skii spaces. We emphasize that some of our methods generalize to the situation where the operator is not translation invariant and maybe perturbed by a convolution type operator. We treat these in the present work, too, see e.g. Theorem 4.3 below. Using a probabilistic and potential theoretic approach, a local smoothness of bounded harmonic solutions solving in a certain very weak sense $I_ju=0$ in $\Omega$, have been obtained in [16, Theorem 1.7] for radial kernel functions using the same regularity as we impose in statement (2) of Theorem 1.1 (see also [14, 23]). See also [15] for related regularity properties of solutions.

To present our generalization of the above mentioned $\delta$-decomposition, let us first note that the first equality in (1.4) can be extended, see Sect. 2. For this, let ${\mathscr {V}}^j(\Omega )$ denote the space of those functions $u:\mathbb {R}^N\rightarrow \mathbb {R}$ such that $u|_{\Omega }\in D^j(\Omega )$ and

$$\begin{aligned} \sup _{x\in \mathbb {R}^N}\int _{\mathbb {R}^N{\setminus } B_r(x)} |u(y)|j(x-y)\ \textrm{d}y<\infty ~\text {for all}~ r>0. \end{aligned}$$

(1.8)

Given $f\in L^2_{loc}(\Omega )$, we then call $u\in {\mathscr {V}}^j(\Omega )$ a (weak) supersolution of $I_ju=f$ in $\Omega$, if

$$\begin{aligned} b_j(u,v)\ge \int _{\Omega } f(x)v(x)\ \textrm{d}x~\text {for all}~ v\in C^{\infty }_c(\Omega ). \end{aligned}$$

(1.9)

In this situation, we also say that u satisfies in weak sense $I_ju\ge f$ in $\Omega$. Similarly, we define subsolutions and solutions.

We emphasize that this definition of supersolution is larger than the one considered in [19]. In the case where $\sigma <1$ in (1.1) this allows via a density result to extend the weak maximum principles presented in [19] as follows.

Proposition 1.2

(Weak maximum principle) Assume (1.1) is satisfied with $\sigma <1$ and assume that

$$\begin{aligned} j ~\text { does not vanish identically on} ~B_r(0) \text { for any} r>0. \end{aligned}$$

(1.10)

Let $\Omega \subset \mathbb {R}^N$ open and with Lipschitz boundary, $c\in L^{\infty }_{loc}(\Omega )$, and assume either

• (1) $c\le 0$ or

• (2) $\Omega$ and c are such that $\Vert c^{+}\Vert _{L^{\infty }(\Omega )}<\inf _{x\in \Omega } \int _{\mathbb {R}^N{\setminus } \Omega }j(x-y)\ \textrm{d}y$.

If $u\in {\mathscr {V}}^j(\Omega )$ satisfies in weak sense

$$\begin{aligned} I_ju\ge c(x)u~\text { in}~ \Omega , u\ge 0 ~\text { almost everywhere in}~ \mathbb {R}^{N}{\setminus } \Omega , ~\text { and}~ \liminf \limits _{|x|\rightarrow \infty }u(x)\ge 0, \end{aligned}$$

then $u\ge 0$ almost everywhere in $\mathbb {R}^N$.

Remark 1.3

• (1) The Lipschitz boundary assumption on $\Omega$ is a technical assumption for the approximation argument.

• (2) Note that Assumption (1.10) readily implies the positivity $\Lambda _1(\Omega )$ defined in (1.6), whenever $\Omega$ is an open set in $\mathbb {R}^N$ which is bounded in one direction, that is $\Omega$ is contained (after a rotation) in a strip $(-a,a)\times \mathbb {R}^{N-1}$ for some $a>0$ (see [11, 20]).

Up to our knowledge Proposition 1.2 is even new in the case of $j=|\cdot |^{-N-2s}$, that is, the case of the fractional Laplacian (up to a multiplicative constant). In this situation, it holds ${\mathscr {V}}^j(\Omega )=H^s(\Omega )\cap {\mathscr {L}}^1_s$, where

$$\begin{aligned} {\mathscr {L}}^1_s=\left\{ u\in L^1_{loc}(\mathbb {R}^N)\;:\; \int _{\mathbb {R}^N}\frac{|u(x)|}{1+|x|^{N+2s}}\ \textrm{d}y<\infty \right\} , \end{aligned}$$

and the proposition can be reformulated as follows.

Corollary 1.4

Let $s\in (0,\frac{1}{2})$, $\Omega \subset \mathbb {R}^N$ open and with Lipschitz boundary, $c\in L^{\infty }_{loc}(\Omega )$, and assume either

• (1) $c\le 0$ or

• (2) $\Omega$ and c are such that $\Vert c^{+}\Vert _{L^{\infty }(\Omega )}<\Lambda _1(\Omega )$.

If $u\in H^s(\Omega )\cap {\mathscr {L}}^1_s$ satisfies in weak sense

$$\begin{aligned} (-\Delta )^su\ge c(x)u~\text { in}~ \Omega , u\ge 0 ~\text {almost everywhere in}~ \mathbb {R}^{N}{\setminus } \Omega ,~\text { and}~ \liminf \limits _{|x|\rightarrow \infty }u(x)\ge 0, \end{aligned}$$

then $u\ge 0$ almost everywhere in $\mathbb {R}^N$.

Similarly to the extension of the weak maximum principle, we have also the following extension of the strong maximum principle presented in [19] in the case $\sigma <1$.

Proposition 1.5

(Strong maximum principle) Assume (1.1) is satisfied with $\sigma <1$ and assume that j satisfies additionally (1.5). Let $\Omega \subset \mathbb {R}^N$ open and $c\in L^{\infty }_{loc}(\Omega )$ with $\Vert c^+\Vert _{L^{\infty }(\Omega )}<\infty$. Moreover, let $u\in {\mathscr {V}}^j(\Omega )$, $u\ge 0$ satisfy in weak sense $I_ju\ge c(x) u$ in $\Omega$. Then the following holds.

• (1) If $\Omega$ is connected, then either $u\equiv 0$ in $\Omega$ or ${{\,\textrm{essinf}\,}}_Ku>0$ for any $K\subset \subset \Omega$.

• (2) j given in (2.2) satisfies ${{\,\textrm{essinf}\,}}_{B_r(0)}j>0$ for any $r>0$, then either $u\equiv 0$ in $\mathbb {R}^N$ or ${{\,\textrm{essinf}\,}}_Ku>0$ for any $K\subset \subset \Omega$.

Our last results concern the Poisson problem associated to the operator $I_j$.

Theorem 1.6

Let (1.1) hold with $\sigma =2$ and assume j satisfies additionally (1.10). Let $\Omega \subset \mathbb {R}^N$ be a bounded open set. Then for any $f\in L^2(\Omega )$ there is a unique solution $u\in {\mathscr {D}}^j(\Omega )$ of $I_ju=f$. Moreover, if j satisfies (1.7) and $f\in L^{\infty }(\Omega )$, then also $u\in L^{\infty }(\Omega )$ and there is $C=C(\Omega ,j)>0$ such that

$$\begin{aligned} \Vert u\Vert _{L^{\infty }(\Omega )}\le C\Vert f\Vert _{L^\infty (\Omega )}. \end{aligned}$$

Additionally, if (1.1) holds with $\sigma <\frac{1}{2}$, there is $m\in \mathbb {N}\cup \{\infty \}$ such that j satisfies the assumptions in Theorem 1.1(2), and it holds $f\in C^{2m}(\overline{\Omega })$, then $u\in H^m_{loc}(\Omega )$. More precisely in this case, for every $\beta \in \mathbb {N}_0^N$, $|\beta |\le m$ and $\Omega '\subset \subset \Omega$ there is $C=C(\Omega ,\Omega ',j,\beta )>0$ such that

$$\begin{aligned} \Vert \partial ^{\beta }u\Vert _{L^2(\Omega ')}\le C\Vert f\Vert _{C^{2m}(\Omega )}. \end{aligned}$$

In particular, if $m=\infty$ and $f\in C^{\infty }(\Omega )$, then $u\in C^{\infty }(\Omega )$.

Our approach to prove Theorems 1.1 and 1.6 is uniformly by considering an equation of the form

$$\begin{aligned} I_ju=h*u+\lambda u+f ~\text {in}~ \Omega \end{aligned}$$

for $f\in L^2(\Omega )$, $h\in L^{1}(\mathbb {R}^N)\cap L^2(\mathbb {R}^N)$, and $\lambda \in \mathbb {R}$. Moreover, several of our results need $I_j$ not to be translation invariant and this setup is discussed in Sect. 2.

1.1 Examples

We close this introduction with some classes of operators covered by our results.

• (1) As introduced in [5, 13, 18] the logarithmic Laplacian

  $$\begin{aligned} L_{{ \Delta \,}}\phi (x)=c_NP.V.\int _{B_1(0)}\frac{\phi (x)-\phi (x+y)}{|y|^{N}}\ \textrm{d}y - c_N\int _{\mathbb {R}^N{\setminus } B_1(0)}\frac{\phi (x+y)}{|y|^{N}}\ \textrm{d}y+\rho _N\phi (x),\nonumber \\ \end{aligned}$$

  (1.11)

  appears as the operator with Fourier-symbol $-2\ln (|\cdot |)$ and can be seen as the formal derivative in s of $(-\Delta )^s$ at $s=0$. Here

  $$\begin{aligned} c_N=\frac{\Gamma (\frac{N}{2})}{\pi ^{N/2}}= \frac{2}{|S^{N-1}|} ~ \text {and}~ \rho _N :=2\ln (2)+\psi \left( \frac{N}{2}\right) - \gamma \end{aligned}$$

  (1.12)

  where $\psi := \frac{\Gamma '}{\Gamma }$ denotes the digamma function and $\gamma := -\psi (1)=-\Gamma '(1)$ is the Euler-Mascheroni constant. With $j(z)=c_N1_{B_1(0)}(z)|z|^{-N}$ the operator $L_{{ \Delta \,}}$ can be seen as a bounded perturbation of the operator class discussed in the introduction. The following sections cover in particular this operator.

• (2) The logarithmic Schrödinger operator $(I-\Delta )^{\log }$ as in [12] is an integro-differential operator with Fourier-symbol $\log (1+|\cdot |^2)$ and also appears as the formal derivative in s of the relativistic Schrödinger operator $(I-\Delta )^s$ at $s=0$,

  $$\begin{aligned} (I-\Delta )^{\log }u(x)= d_{N}P.V.\int _{\mathbb {R}^N}\frac{u(x)-u(x+y)}{|y|^{N}}\omega (|y|)\ \textrm{d}y, \end{aligned}$$

  where $d_N=\pi ^{-\frac{N}{2}}$, $\omega (r)=2^{1-\frac{N}{2}}r^{\frac{N}{2}}K_{\frac{N}{2}}(r)$ and $K_{\nu }$ is the modified Bessel function of the second kind with index $\nu$. More generally, operators with symbol $\log (1+|\cdot |^{\beta })$ for some $\beta \in (0,2]$ are studied in [22].

• (3) Finally, also nonradial kernels of the type considered in [19] satisfy in particular the assumptions (2.1) and (4.1). See also also [14, 22, 23] and references in there.

The paper is organized as follows. In Sect. 2 we collect some general results concerning the spaces used in this paper and resulting definitions of weak sub- and supersolutions. Section 3 is devoted to show several density results, which then are used to show the Propositions 1.2 and 1.5. In Sect. 4 we present a general approach to show boundedness of solutions and in Sect. 5 we give the proof of an interior $H^1$-regularity estimate for solutions from which we then deduce the interior regularity statement as claimed in Theorems 1.1(2) and 1.6.

Notation In the remainder of the paper, we use the following notation. Let $U,V\subset \mathbb {R}^N$ be nonempty measurable sets, $x \in \mathbb {R}^N$ and $r>0$. We denote by $1_U: \mathbb {R}^N \rightarrow \mathbb {R}$ the characteristic function, |U| the Lebesgue measure, and $\text {diam}(U)$ the diameter of U. The notation $V \subset \subset U$ means that ${\overline{V}}$ is compact and contained in the interior of U. The distance between V and U is given by $\text {dist}(V,U):= \inf \{|x-y|\,:\, x \in V,\, y \in U\}$. Note that this notation does not stand for the usual Hausdorff distance. If $V= \{x\}$ we simply write $\text {dist}(x,U)$. We let $B_r(U):=\{x\in \mathbb {R}^N\;:\; \text {dist}(x,U)<r\}$, so that $B_r(x):=B_r(\{x\})$ is the open ball centered at x with radius r. We also put $B:=B_1(0)$ and $\omega _N:=|B|$. Finally, given a function $u: U\rightarrow \mathbb {R}$, $U \subset \mathbb {R}^N$, we let $u^+:= \max \{u,0\}$ and $u^-:=-\min \{u,0\}$ denote the positive and negative part of u, and we write $\text {supp}\ u$ for the support of u given as the closure in $\mathbb {R}^N$ of the set ${\{ x\in U\;:\; u(x)\ne 0\}}$.

2 Preliminaries

In the following, we generalize the translation invariant setting of the introduction. For this and from now on, let $k:\mathbb {R}^N\times \mathbb {R}^N\rightarrow [0,\infty ]$ be a measurable function satisfying for some $\sigma \in (0,2]$

$$\begin{aligned} \begin{aligned} k(x,y)=k(y,x)&~\text {for all}~ x,y\in \mathbb {R}^N ~\text { and }~ \sup _{x\in \mathbb {R}^N}\int _{\mathbb {R}^N}\min \{1,|x-y|^{\sigma }\}k(x,y)\ \textrm{d}y<\infty . \end{aligned}\nonumber \\ \end{aligned}$$

(2.1)

We let $j:\mathbb {R}^N\rightarrow [0,\infty ]$ be the symmetric lower bound of k given by

$$\begin{aligned} j(z):={{\,\textrm{essinf}\,}}\{k(x,x\pm z)\;:\; x\in \mathbb {R}^N\}~\text {for}~ z\in \mathbb {R}^N. \end{aligned}$$

(2.2)

For $\Omega \subset \mathbb {R}^N$ open let

$$\begin{aligned} b_{k,\Omega }(u,v)&:=\frac{1}{2}\int _{\Omega }\int _{\Omega }(u(x)-u(y))(v(x)-v(y))k(x,y)\ \textrm{d}x\textrm{d}y,\\ \kappa _{k,\Omega }(x)&:=\int _{\mathbb {R}^N{\setminus } \Omega }k(x,y)\ \textrm{d}y\in [0,\infty ] ~\text {for}~ x\in \mathbb {R}^N,~\text { and}~\\ K_{k,\Omega }(u,v)&:=\int _{\Omega } u(x)v(x)\kappa _{k,\Omega }(x)\ \textrm{d}x, \end{aligned}$$

where, if $u=v$ we put

$$\begin{aligned} b_{k,\Omega }(u):=b_{k,\Omega }(u,u)~\text {and}~ K_{k,\Omega }(u):=K_{k,\Omega }(u,u) \end{aligned}$$

and we drop the index $\Omega$, if $\Omega =\mathbb {R}^N$. Note that we have for any fixed $x\in \Omega$ that $\kappa _{k,\Omega }(x)<\infty$ by (2.1). We consider the function spaces

$$\begin{aligned} D^k(\Omega )&:=\Big \{u\in L^2(\Omega )\;:\; b_{k,\Omega }(u)<\infty \Big \},\\ {\mathscr {D}}^k(\Omega )&:=\Big \{u\in D^k(\mathbb {R}^N)\;:\; u=0 ~\text {on}~ \mathbb {R}^{N}{\setminus } \Omega \Big \},\\ {\mathscr {V}}^k(\Omega )&:=\Big \{u:\mathbb {R}^N\rightarrow \mathbb {R}\;:\; u|_{\Omega }\in D^k(\Omega )~\text { and, for all}~ r>0,\\&\sup _{x\in \mathbb {R}^N}\int _{\mathbb {R}^N{\setminus } B_r(x)}|u(y)|k(x,y)\ \textrm{d}y<\infty \Big \}~\text {and}~\\ {\mathscr {V}}_{loc}^k(\Omega )&:=\Big \{u:\mathbb {R}^N\rightarrow \mathbb {R}\;:\; u|_{\Omega '}\in {\mathscr {V}}^k(\Omega ') ~\text { for all} ~\Omega '\subset \subset \Omega \Big \}. \end{aligned}$$

Lemma 2.1

Let $U\subset \Omega \subset \mathbb {R}^N$ open and $u:\mathbb {R}^N\rightarrow \mathbb {R}$. Then the following hold:

• (1) $u\in {\mathscr {D}}^k(\Omega )\ \Rightarrow \ u|_{\Omega }\in D^k(\Omega )$.

• (2) ${\mathscr {D}}^k(U)\subset {\mathscr {D}}^k(\Omega )\subset {\mathscr {V}}^k(\Omega )\subset {\mathscr {V}}^k(U)\subset {\mathscr {V}}^k_{loc}(U)$.

Proof

This follows immediately from the definitions (see also [19, Section 3]). $\square$

Lemma 2.2

(See Proposition 3.3 in [19] or Proposition 1.7 in [20]) For $\Omega \subset \mathbb {R}^N$ open, let $\Lambda _1(\Omega )$ be given by (c.f. (1.6))

$$\begin{aligned} \Lambda _1(\Omega ):=\inf _{\begin{array}{c} u\in {\mathscr {D}}^j(\Omega )\\ u\ne 0 \end{array}}\frac{b_j(u)}{\Vert u\Vert _{L^2(\Omega )}^2} \end{aligned}$$

and let

$$\begin{aligned} \lambda (r)=\inf \{\Lambda _1(\Omega )\;:\;\Omega \subset \mathbb {R}^N ~\text { open with}~ |\Omega |=r \}. \end{aligned}$$

Then $\lim \limits _{r\rightarrow \infty }\lambda (r)\ge \int _{\mathbb {R}^N}j(z)\ \textrm{d}z$ with $j(z):={{\,\textrm{essinf}\,}}\{k(x,x\pm z)\;:\; z\in \mathbb {R}^N\}$ for $z\in \mathbb {R}^N$ as in (2.2).

Lemma 2.3

Let $\Omega \subset \mathbb {R}^N$ open and let X be any of the above function spaces. Then the following hold:

• (1) $b_{k,\Omega }$ is a bilinear form and in particular we have $b_{k,\Omega }(u,v)\le b_{k,\Omega }^{1/2}(u)b_{k,\Omega }^{1/2}(v)$. Moreover, $D^k(\Omega )$ and ${\mathscr {D}}^k(\Omega )$ are Hilbert spaces with scalar products

  $$\begin{aligned} \langle u,v\rangle _{D^k(\Omega )}&=\langle u,v\rangle _{L^2(\Omega )}+b_{k,\Omega }(u,v),\\ \langle u,v\rangle _{{\mathscr {D}}^k(\Omega )}&=\langle u,v\rangle _{L^2(\Omega )}+b_{k,\mathbb {R}^N}(u,v). \end{aligned}$$

• (2) If $u\in X$, then $u^{\pm },|u|\in X$ and we have $b_{k,\Omega '}(u^+,u^-)\le 0$ for all $\Omega '\subset \Omega$ with $b_{k,\Omega '}(u)<\infty$.

• (3) If $g\in C^{0,1}(\mathbb {R}^N)$, $u\in X$, then $g\circ u\in X$.

• (4) $C^{0,1}_c(\Omega )\subset X$.

• (5) $\phi \in C^{0,1}_c(\Omega )$, $u\in X$, then $\phi u\in {\mathscr {D}}^k(\Omega )$, where if necessary we extend u trivially to a function on $\mathbb {R}^N$. Moreover, there is $C=C(N,k,\Vert \phi \Vert _{C^{0,1}(\Omega )})>0$ such that

  $$\begin{aligned} b_{k,\mathbb {R}^N}(\phi u)\le C\Big (\Vert u\Vert _{L^2(\Omega ')}^2+b_{k,\Omega '}(u)\Big ) \end{aligned}$$

  for any $\Omega '\subset \Omega$ with $\text {supp}\,\phi \subset \subset \Omega '$.

Proof

Theses statements follow directly from the definition (c.f. [19, Section 3]). To be precise in the last part, let $\phi \in C^{0,1}_c(\Omega )$ and fix $L:=\Vert \phi \Vert _{C^{0,1}(\Omega )}$. That is, we have

$$\begin{aligned} |\phi (x)|\le L~\text {and}~ |\phi (x)-\phi (y)|\le L|x-y|. \end{aligned}$$

Then using the inequality for $x,y\in \mathbb {R}^N$

$$\begin{aligned} |\phi (x)u(x)-\phi (y)u(y)|^2\le 2|\phi (x)-\phi (y)|^2|u(x)|^2+2|\phi (y)|^2|u(x)-u(y)|^2 \end{aligned}$$

we find by the assumptions (2.1)

$$\begin{aligned} b_{k,\mathbb {R}^N}(\phi u)&\le b_{k,\Omega '}(\phi u)+L^2\int _{\text {supp}\,\phi }|u(x)|^2\kappa _{k,\Omega '}(x)\ \textrm{d}x\\&\le 2L^2\int _{\Omega '}\int _{\Omega '}|u(x)|^2|x-y|^2k(x,y)\ \textrm{d}y\ \textrm{d}x+2L^2b_{k,\Omega '}(u)\\&\quad +L^2\sup _{x\in \text {supp}\,\phi }\kappa _{k,\Omega '}(x)\Vert u\Vert _{L^2(\Omega ')}^2\\&\le 2L^2\Big (\sup _{x\in \Omega '}\int _{\Omega '} |x-y|^2k(x,y)\ \textrm{d}y+\sup _{x\in \text {supp}\,\phi }\kappa _{k,\Omega '}(x)\Big )\Vert u\Vert _{L^2(\Omega ')}^2\\&\quad +2L^2b_{k,\Omega '}(u)<\infty . \end{aligned}$$

$\square$

Remark 2.4

• (1) Note that for $u,v\in {\mathscr {D}}^k(\Omega )$ we have

  $$\begin{aligned} b_{k}(u,v)=b_{k,\mathbb {R}^N}(u,v)=b_{k,\Omega }(u,v)+K_{k,\Omega }(u,v). \end{aligned}$$

• (2) It follows in particular that there is a nonnegative self-adjoint operator $I_k$ associated to $b_{k,\mathbb {R}^{N}}=b_k$ as mentioned in the introduction.

Lemma 2.5

Let $\Omega \subset \mathbb {R}^N$ open and $u\in {\mathscr {V}}_{loc}^k(\Omega )$. Then $b_{k}(u,\phi )$ is well-defined for any $\phi \in C^{\infty }_c(\Omega )$.

Proof

Let $\phi \in C^{\infty }_c(\Omega )$ and fix $U\subset \subset \Omega$ such that $\text {supp}\ \phi \subset \subset U$. Then with the symmetry of k

$$\begin{aligned} |b_{k}(u,\phi )|&\le |b_{k,U}(u,\phi )|+\int _{U} |\phi (x)|\int _{\mathbb {R}^N{\setminus } U} |u(x)-u(y)|k(x,y)\ \textrm{d}y\textrm{d}x\\&\le b_{k,U}^{1/2}(u)b_{k,U}^{1/2}(\phi )+\int _{\text {supp}\,\phi }|\phi (x)| \ \textrm{d}x\sup _{x\in \mathbb {R}^N}\int _{\mathbb {R}^N{\setminus } B_{\epsilon }(x)}|u(y)|k(x,y)\ \textrm{d}y\\&\qquad +\int _{\text {supp}\,\phi }|\phi (x)u(x)| \ \textrm{d}x \sup _{x\in \mathbb {R}^N}\int _{\mathbb {R}^N{\setminus } B_{\epsilon }(x)}k(x,y)\ \textrm{d}y<\infty , \end{aligned}$$

where $\epsilon =\text {dist}(\text {supp}\ \phi ,\mathbb {R}^N{\setminus } U)>0$. $\square$

Definition 2.6

Let $\Omega \subset \mathbb {R}^N$ open and $f\in L^1_{loc}(\Omega )$. Then $u\in {\mathscr {V}}_{loc}^k(\Omega )$ is called a weak supersolution of $I_ku=f$ in $\Omega$, if

$$\begin{aligned} b_{k}(u,\phi )\ge \int _{\Omega } f(x)\phi (x)\ \textrm{d}x~\text {for all nonnegative}~ \phi \in C^{\infty }_c(\Omega ). \end{aligned}$$

We also say that u satisfies $I_ku\ge f$ weakly in $\Omega$.

Similarly, we define weak subsolutions and solutions.

Remark 2.7

• (1) We note that by Assumption 2.1, it follows that for any function $u\in {\mathscr {V}}^k_{loc}(\Omega )$ with $u|_{\Omega }\in C^{0,1}(\Omega )$ for $\Omega \subset \mathbb {R}^N$ open, we have $I_ku|_U\in L^{\infty }(U)$ for any $U\subset \subset \Omega$ and

  $$\begin{aligned} I_ku(x)=\int _{\mathbb {R}^N}(u(x)-u(y))k(x,y)\ \textrm{d}y~\text {for} ~x\in \Omega . \end{aligned}$$

  This follows similarly to the proof of the statements in Lemma 2.3.

• (2) If $u\in {\mathscr {V}}^k(\Omega )$, then indeed also $b_k(u,\phi )$ is well-defined for all $\phi \in {\mathscr {D}}^k(\Omega )$. Hence also $b_k$ is well defined on ${\mathscr {V}}^k_{loc}(\Omega )\times {\mathscr {D}}^k(U)$ for all $U\subset \subset \Omega$. In some of our results the statements need a Lipschitz-boundary of $\Omega$, which comes into play due to approximation with $C^{\infty }_c(\Omega )$-functions (see Section 3 below). However, this can be weakened, if $u\in {\mathscr {V}}^k(\Omega )$ and the space of test-functions is adjusted.

Lemma 2.8

Let $\Omega \subset \mathbb {R}^N$ open. Let $\Omega _1\subset \subset \Omega _2\subset \subset \Omega _3\subset \subset \Omega$. Let $\eta \in C^{0,1}_c(\mathbb {R}^N)$ such that $0\le \eta \le 1$ in $\mathbb {R}^N$ and we have

$$\begin{aligned} \eta =1~\text {in}~ \Omega _{2}~\text {and }~ \eta =0~\text {in}~ \mathbb {R}^N{\setminus } \Omega _3. \end{aligned}$$

Let $f\in L^1_{loc}(\Omega )$ and let $u\in {\mathscr {V}}^k_{loc}(\Omega )$ satisfy in weak sense $Iu\ge f$ in $\Omega$. Then the function $v=\eta u\in {\mathscr {D}}^k(\Omega _3)$ satisfies in weak sense $Iv\ge f+g_{\eta ,u}(x)$ in $\Omega _1$, where

$$\begin{aligned} g_{\eta ,u}(x)=\int _{\mathbb {R}^N{\setminus } \Omega _2}(1-\eta (y))u(y)k(x,y)\ \textrm{d}y~\text {for}~ x\in \Omega _1. \end{aligned}$$

Proof

The fact, that $v\in {\mathscr {D}}^k(\Omega _3)$ follows from Lemma 2.3. Let $\phi \in C^{\infty }_c(\Omega _1)$, then

$$\begin{aligned} \int _{\mathbb {R}^N}v(x)I_k\phi (x)\ \textrm{d}x\ge \int _{\mathbb {R}^N}f(x)\phi (x)\ \textrm{d}x-\int _{\mathbb {R}^N}(1-\eta (x))u(x)I_k\phi (x)\ \textrm{d}x. \end{aligned}$$

Here, since $(1-\eta )u\equiv 0$ on $\Omega _2$, we have

$$\begin{aligned} \int _{\mathbb {R}^N}(1-\eta (x))u(x)I_k\phi (x)\ \textrm{d}x&=\int _{\mathbb {R}^N}\phi (x) [I_k(1-\eta )u](x)\ \textrm{d}x\\&=-\int _{\Omega _1}\phi (x) \int _{\mathbb {R}^N{\setminus } \Omega _2}(1-\eta (y))u(y)k(x,y)\ \textrm{d}y\ \textrm{d}x. \end{aligned}$$

Thus the claim follows. $\square$

Remark 2.9

The same result as in Lemma 2.8 also holds if “$\ge$” in the solution type is replaced by “$\le$” or “$=$”.

In the following, it is useful to understand functions $u\in D^k(\Omega )$ satisfying $b_{k,\Omega }(u)=0$.

Proposition 2.10

Assume that the symmetric lower bound j of k defined in (2.2) satisfies $\int _{\mathbb {R}^N} j(z)\ \textrm{d}z=\infty$. Let $\Omega \subset \mathbb {R}^N$ open and bounded and let $u\in D^k(\Omega )$ such that $b_{k,\Omega }(u)=0$. Then u is constant.

Proof

Let $x_0\in \Omega$ and fix $r>0$ such that $B_{2r}(x_0)\subset \Omega$. Denote $q(z):=\min \{c,j(z)\}1_{B_r(0)}(z)$, where we may fix $c>0$ such that $|\{q>0\}|>0$ due to the assumption on j. Then by Lemma A.1 we have

$$\begin{aligned} 0=2b_{k,\Omega }(u)\ge 2b_{q,\Omega }(u)\ge \frac{1}{2\Vert q\Vert _{L^1(\mathbb {R}^N)}} b_{q*q,B_{r}(x_0)}(u), \end{aligned}$$

where $a*b=\int _{\mathbb {R}^N}a(\cdot -y)b(y)\ \textrm{d}y$ denotes as usual the convolution. Note that since $q\in L^1(\mathbb {R}^N)\cap L^{\infty }(\mathbb {R}^N)$ with $q=0$ on $\mathbb {R}^N{\setminus } B_r(0)$, it follows that $q*q\in C(\mathbb {R}^N)$ with support in $B_{2r}(0)$ and we have

$$\begin{aligned} q*q(0)=\int _{\mathbb {R}^N}q(z)^2\ \textrm{d}z>0 \end{aligned}$$

by the assumption on j. Hence there is $R>0$ with $q*q\ge \epsilon$ for some $\epsilon >0$ and thus we have

$$\begin{aligned} 0=b_{q*q,B_{r}(x_0)}(u)\ge b_{q*q,B_{\rho }(x_0)}(u)\ge \frac{\epsilon }{2}\int _{B_{\rho }(x_0)}\int _{B_{\rho }(x_0)}(u(x)-u(y))^2\ \textrm{d}x\textrm{d}y. \end{aligned}$$

for any $\rho \in (0,\frac{R}{2}]$. But then $u(x)=u(y)$ for almost every $x,y\in B_{R/2}(x_0)$ so that u is constant a.e. in $B_{\rho }(x_0)$. Since $b_{k,\Omega }(u)=b_{k,\Omega }(u-m)$ for any $m\in \mathbb {R}$, we may next assume that $u=0$ in $B_{R/2}(x_0)$ and show that indeed we have $u=0$ a.e. in $\Omega$. Denote by W the set of points $x\in \Omega$ such that there is $r>0$ with $u=0$ a.e. in $B_r(x)$. By definition W is open and the above shows that W is nonempty. Next, let $(x_n)_n\subset W$ be a sequence with $x_n\rightarrow x\in \Omega$ for $n\rightarrow \infty$. Then there is $r_x>0$ such that $B_{4r_x}(x)\subset \Omega$ and we can find $n_0\in \mathbb {N}$ such that $x\in B_{r_x}(x_n)\subset B_{2r_x}(x_n)\subset \Omega$ for $n\ge n_0$. Repeating the above argument, it follows that u must be zero in $B_{r_x}(x_n)$ and thus $x\in W$. Hence, W is relatively open and closed in $\Omega$ and since W is nonempty, we have $W=\Omega$. That is $u=0$ in $\Omega$. $\square$

2.1 On Sobolev and Nikol’skii spaces

We recall here the notations and properties of Sobolev and Nikol’skii spaces as introduced in [7, 26]. In the following, let $p\in [1,\infty )$ and $\Omega \subset \mathbb {R}^N$ open.

2.1.1 Sobolev spaces

If $k\in \mathbb {N}_0$, we set as usual

$$\begin{aligned} W^{k,p}(\Omega ):=\Big \{u\in L^p(\Omega )\;:\; \partial ^{\alpha }u ~\text { exists for all}~ \alpha \in \mathbb {N}_0^{n}, |\alpha |\le k ~\text { and belongs to}~ L^p(\Omega )\Big \} \end{aligned}$$

for the Banach space of k-times (weakly) differentialable functions in $L^p(\Omega )$. Moreover, as usual, for $\sigma \in (0,1)$, $p\in [1,\infty )$ we set

$$\begin{aligned} W^{\sigma ,p}(\Omega ):=\Big \{u\in L^p(\Omega )\;:\; \frac{u(x)-u(y)}{|x-y|^{\frac{n}{p}+\sigma }}\in L^{p}(\Omega \times \Omega )\Big \}. \end{aligned}$$

With the norm

$$\begin{aligned}{} & {} \Vert u\Vert _{W^{\sigma ,p}(\Omega )}=\Vert u\Vert _{L^p(\Omega )}^p+[u]_{W^{\sigma ,p}(\Omega )},~\text { where}\\{} & {} ~ [u]_{W^{\sigma ,p}(\Omega )}=\Bigg (\iint _{\Omega \times \Omega }\frac{|u(x)-u(y)|^p}{|x-y|^{n+\sigma p}}\ \textrm{d}x\textrm{d}y\Bigg )^{1/p} \end{aligned}$$

the space $W^{\sigma ,p}(\Omega )$ is a Banach space. For general $s=k+\sigma$, $k\in \mathbb {N}_0$, $\sigma \in [0,1)$ the Sobolev space is defined as

$$\begin{aligned} W^{s,p}(\Omega ):=\Big \{u\in W^{k,p}(\Omega )\;:\; \partial ^{\alpha }u\in W^{\sigma ,p}(\Omega )~ \text {for all}~ \alpha \in \mathbb {N}_0^{n} ~\text {with}~ |\alpha |= k\Big \}. \end{aligned}$$

Finally, in the particular case $p=2$ the space $H^{s}(\Omega ):=W^{s,2}(\Omega )$ is a Hilbert space.

2.1.2 Nikol’skii spaces

For $u:\Omega \rightarrow \mathbb {R}$ and $h\in \mathbb {R}$, let $\Omega _h:=\{x\in \Omega \;:\; \text {dist}(x,\partial \Omega )>h\}$ and, with $e\in \partial B_1(0)$, we let

$$\begin{aligned} \delta _hu(x)=\delta _{h,e}u(x):=u(x+he)-u(x). \end{aligned}$$

Moreover, for $l\in \mathbb {N}$, $l>1$ let

$$\begin{aligned} \delta _{h}^{l}u(x)=\delta _h(\delta _h^{l-1}u)(x). \end{aligned}$$

For $s=k+\sigma >0$ with $k\in \mathbb {N}_0$ and $\sigma \in (0,1]$ define

$$\begin{aligned} N^{s,p}(\Omega ):=\Big \{u\in W^{k,p}(\Omega )\;:\; [\partial ^{\alpha }u]_{N^{\sigma ,p}(\Omega )}<\infty ~\text {for all}~ \alpha \in \mathbb {N}_0^{n} ~\text {with}~|\alpha |= k\Big \}, \end{aligned}$$

where

$$\begin{aligned}{}[u]_{N^{\sigma ,p}(\Omega )}=\sup _{\begin{array}{c} e\in \partial B_1(0)\\ h>0 \end{array}}h^{-\sigma }\Vert \delta _{h,e}^2u\Vert _{L^p(\Omega _{2h})}. \end{aligned}$$

It follows that $N^{s,p}(\Omega )$ is a Banach space with norm $\Vert u\Vert _{N^{s,p}(\Omega )}:=\Vert u\Vert _{W^{k,p}(\Omega )}+\sum _{|\alpha |=k}[\partial ^{\alpha }u]_{N^{\sigma ,p}(\Omega )}$. It can be shown that this norm is equivalent to

$$\begin{aligned} \Vert u\Vert _{L^p(\Omega )}+\sum _{|\alpha |=k}\sup _{\begin{array}{c} e\in \partial B_1(0)\\ h>0 \end{array}}h^{m-\sigma }\Vert \delta _{h,e}^lu\Vert _{L^p(\Omega _{lh})} \end{aligned}$$

for any fixed $m,l\in \mathbb {N}_0$ with $m<\sigma$ and $l>\sigma -m$ (see [26, Theorem 4.4.2.1]).

Proposition 2.11

(See e.g. Propositions 3 and 4 in [7]) Let $\Omega \subset \mathbb {R}^N$ open and with $C^{\infty }$ boundary. Moreover, let $t>s>0$ and $1\le p<\infty$. Then

$$\begin{aligned} N^{t,p}(\Omega )\subset W^{s,p}(\Omega )\subset N^{s,p}(\Omega ). \end{aligned}$$

3 Density results and maximum principles

The main goal of this section is to show the following.

Theorem 3.1

Let either $\Omega =\mathbb {R}^N$ or $\Omega \subset \mathbb {R}^N$ open and bounded with Lipschitz boundary. In the following, let $X(\Omega ):={\mathscr {D}}^k(\Omega )$ or $D^k(\Omega )$. Then $C^{\infty }_c(\Omega )$ is dense in $X(\Omega )$. Moreover, if $u\in X(\Omega )$ is nonnegative, then we have

• (1) There exists a sequence $(u_n)_n\subset X(\Omega )\cap L^{\infty }(\Omega )$ with $\lim \limits _{n\rightarrow \infty }u_n= u$ in $X(\Omega )$ satisfying that for every $n\in \mathbb {N}$ there is $\Omega '_n\subset \subset \Omega$ with $u_n=0$ on $\Omega {\setminus } \Omega '_n$ and $0\le u_n\le u_{n+1}\le u$.

• (2) There exists a sequence $(u_n)_n\subset C^{\infty }_c(\Omega )$ with $u_n\ge 0$ for every $n\in \mathbb {N}$ and $\lim \limits _{n\rightarrow \infty }u_n= u$ in $X(\Omega )$.

Remark 3.2

To put Theorem 3.1 into perspective, we consider the following examples.

• (1) In the case $k(x,y)=|x-y|^{-2s-N}$ for some $s\in (0,\frac{1}{2})$, the above Theorem is well-known and leads to the interesting property that for any open, bounded Lipschitz set $\Omega \subset \mathbb {R}^N$ we have

  $$\begin{aligned} D^k(\Omega )=H^s(\Omega )=H^s_0(\Omega ). \end{aligned}$$

  We emphasize that the above equality also holds for $s=\frac{1}{2}$. Moreover, if $s<\frac{1}{2}$, it also holds $H^s(\Omega )=\{u\in H^s(\mathbb {R}^N)\;:\; 1_{\mathbb {R}^N{\setminus }\Omega }u\equiv 0\}$.

• (2) If $k(x,y)=1_{B_1(0)}(x-y)|x-y|^{-N}$, $D^k(\Omega )$ is associated to the function space of the localized logarithmic Laplacian (see [5]).

The proof is split into several smaller steps. Recall that ${\mathscr {D}}^k(\mathbb {R}^N)=D^k(\mathbb {R}^N)$ by definition.

Lemma 3.3

Let $u\in D^k(\mathbb {R}^N)$. Then there is a sequence $(u_n)_n\subset D^k(\mathbb {R}^N)$ with $\lim \limits _{n\rightarrow \infty }u_n= u$ in $D^k(\mathbb {R}^N)$ satisfying that for every $n\in \mathbb {N}$ there is $\Omega _n\subset \subset \mathbb {R}^N$ with $u_n=0$ on $\mathbb {R}^N{\setminus } \Omega _n$. Moreover, if $u\ge 0$, then $(u_n)_n$ can be chosen to satisfy in addition $0\le u_n\le u_{n+1}\le u$.

Proof

For $n\in \mathbb {N}$ let $\phi _n\in C^{0,1}_c(\mathbb {R}^N)$ be radially symmetric and such that $\phi _n\equiv 1$ on $B_n(0)$, $\phi _n\equiv 0$ on $B_{n+1}(0)^c$. Clearly, we may assume that $[\phi _n]_{C^{0,1}(\mathbb {R}^N)}=1$. By Lemma 2.3 there is hence some $C=C(N,k)>0$ with $b_{k,\mathbb {R}^N}(\phi _n u)\le C\Vert u\Vert _{D^k(\mathbb {R}^N)}$ for all $n\in \mathbb {N}$. In the following, let $u_n:=\phi _nu$ and without loss of generality we may assume $u\ge 0$. Since then $0\le u-u_n\le u$ on $\mathbb {R}^N$ and $u-u_n=0$ on $B_n$, by dominated convergence we have $\lim \limits _{n\rightarrow \infty }\Vert u-u_n\Vert _2=0$. Moreover, by choice of $\phi _n$ we have for $x,y\in \mathbb {R}^N$

$$\begin{aligned} |u(x)(1-\phi _n(x))-u(y)(1-\phi _n(y))|&\le |u(x)-u(y)|(1-\phi _n(x)) +|u(y)||\phi _n(x)-\phi _n(y)|\\&\le |u(x)-u(y)| +|u(y)|\min \{1,|x-y|\}=:U(x,y). \end{aligned}$$

Here, $U(x,y)\in L^2(\mathbb {R}^N\times \mathbb {R}^N,k(x,y)\ d(x,y))$, since

$$\begin{aligned} \iint _{\mathbb {R}^N\times \mathbb {R}^N} U(x,y)k(x,y)\ \textrm{d}x\textrm{d}y&=b_{k,\mathbb {R}^N}(u)+\int _{\mathbb {R}^N}|u(y)|^2\int _{\mathbb {R}^N} \min \{1,|x-y|^{2}\}k(x,y)\ \textrm{d}x\textrm{d}y\\&\le b_{k,\mathbb {R}^N}(u)+\int _{\mathbb {R}^N}|u(y)|^2\ \textrm{d}y\,\\&\text {supp}_{x\in \mathbb {R}^N}\int _{\mathbb {R}^N} \min \{1,|x-y|^{2}\}k(x,y)\textrm{d}y<\infty . \end{aligned}$$

Thus $\lim \limits _{n\rightarrow \infty }b_{k,\mathbb {R}^N}(u-u_n)=0$ by the dominated convergence Theorem. $\square$

Proposition 3.4

We have that $C^{\infty }_c(\mathbb {R}^N)$ is dense in $D^k(\mathbb {R}^N)$. Moreover, if $u\in D^k(\mathbb {R}^N)$ is nonnegative, then there exists $(\phi _n)_n\subset C^{\infty }_c(\mathbb {R}^N)$ with $\phi _n\ge 0$ for every $n\in \mathbb {N}$ and $\lim \limits _{n\rightarrow \infty }\phi _n= u$ in $D^k(\mathbb {R}^N)$.

Proof

Let $u\in D^k(\mathbb {R}^N)$. Moreover, let $\phi _n\in C^{0,1}_c(\mathbb {R}^N)$ for $n\in \mathbb {N}$ be given by Lemma 3.3 such that $\Vert u-\phi _n u\Vert _{s,p}<\frac{1}{n}$. Then $v_n:=\phi _n u\in D^k(\mathbb {R}^N)$ and there is $R_n>0$ with $v_n\equiv 0$ on $\mathbb {R}^N{\setminus } B_{R_n}(0)$. Next, let $(\rho _\epsilon )_{\epsilon \in (0,1]}$ by a Dirac sequence and denote $v_{n,\epsilon }:=\rho _\epsilon *v_n$. Then $v_{n\epsilon }\in C^{\infty }_c(\mathbb {R}^N)$ for all $n\in \mathbb {N}$, $\epsilon \in (0,1]$ and

$$\begin{aligned} b_{k,\mathbb {R}^N}(u-v_{n,\epsilon })\le b_{k,\mathbb {R}^N}(u-v_n)+ b_{k,\mathbb {R}^N}(v_n-v_{n,\epsilon })\le \frac{1}{n}+ b_{k,\mathbb {R}^N}(v_n-v_{n,\epsilon }). \end{aligned}$$

It is hence enough to show that $v_{n,\epsilon }\rightarrow v_n$ in $D^k(\mathbb {R}^N)$ for $\epsilon \rightarrow 0$. In the following, we write v in place of $v_n$ and $v_\epsilon =\rho _\epsilon *v$ in place of $v_{n,\epsilon }$ for $\epsilon \in (0,1]$. Moreover, let $R=R_n>0$ with $v=v_n=0$ on $\mathbb {R}^N{\setminus } B_R(0)$. Clearly, $v_{\epsilon }\rightarrow v$ in $L^2(\mathbb {R}^N)$ for $\epsilon \rightarrow 0$ and this convergence is also pointwise almost everywhere. Hence it is enough to analyze the convergence of $b_{k,\mathbb {R}^N}(v-v_{\epsilon })$ as $\epsilon \rightarrow 0$. From here, the proof follows along the lines of [19, Proposition 4.1] noting that there it is not used that k only depends on the difference of x and y. Note here, that if u is nonnegative then the above constructed sequence is also nonnegative. $\square$

Lemma 3.5

Let $\Omega \subset \mathbb {R}^N$ open and such that $\partial \Omega$ is bounded. Denote $\delta (x):=\text {dist}(x,\mathbb {R}^N{\setminus } \Omega )$. Then the following is true.

• (1) There is $C=C(N,\Omega ,k)>0$ such that $\kappa _{k,\Omega }(x)\le C\delta ^{-\sigma }(x)\$ for $x\in \Omega$.

• (2) If $\Omega$ is bounded, then $1_{\Omega }\in D^k(\mathbb {R}^N)$.

Proof

Let $C=C(N,\Omega ,k)>0$ be constants varying from line to line and denote $U:=\{x\in \mathbb {R}^N\;:\; \text {dist}(x,\Omega )\le 1\}$. To see item 1., let $x\in \Omega$ and fix $p\in \partial \Omega$ such that $\delta (x)=|x-p|$. Then

$$\begin{aligned} \kappa _{k,\Omega }(x)\le C+\int _{U{\setminus } \Omega }\frac{|x-p|^{\sigma }}{|x-p|^{\sigma }}k(x,y)\ \textrm{d}y\le C+\delta (x)^{-\sigma }\int _{U{\setminus } \Omega } |x-y|^{\sigma }k(x,y)\ \textrm{d}y\le C\delta ^{-\sigma }(x), \end{aligned}$$

where we have used that $|x-p|\le |x-y|$ for $y\in \mathbb {R}^N{\setminus } \Omega$. Now 2. follows immediately from 1., since we have

$$\begin{aligned} b_{k,\mathbb {R}^N}(1_{\Omega })=\int _{\Omega }\int _{\mathbb {R}^N{\setminus }\Omega }k(x,y)\ \textrm{d}y\textrm{d}x\le C \int _{\Omega } \delta ^{-\sigma }(x)\ \textrm{d}x <\infty . \end{aligned}$$

$\square$

Theorem 3.6

(See Theorem 3.1) Let $\Omega \subset \mathbb {R}^N$ be an open bounded set with Lipschitz boundary. Then $C^{\infty }_c(\Omega )$ is dense in ${\mathscr {D}}^k(\Omega )$. Moreover, if $u\in {\mathscr {D}}^k(\Omega )$ is nonnegative, then we have

• (1) There exists a sequence $(u_n)_n\subset {\mathscr {D}}^k(\Omega )$ with $\lim \limits _{n\rightarrow \infty }u_n= u$ in ${\mathscr {D}}^k(\Omega )$ satisfying that for every $n\in \mathbb {N}$ there is $\Omega '_n\subset \subset \Omega$ with $u_n=0$ on $\mathbb {R}^N{\setminus } \Omega '_n$ and $0\le u_n\le u_{n+1}\le u$.

• (2) There exists a sequence $(u_n)_n\subset C^{\infty }_c(\Omega )$ with $u_n\ge 0$ for every $n\in \mathbb {N}$ and $\lim \limits _{n\rightarrow \infty }u_n= u$ in ${\mathscr {D}}^k(\Omega )$.

Proof

Note that the second claim follows immediately from the first one using [19, Proposition 4.1] as in the proof of Proposition 3.4. Then also the main claim follows by considering $u^{\pm }$ separately. Hence it is enough to show 1. We proceed similar to [5, Theorem 3.1]. Denote $\delta (x):=\text {dist}(x,\mathbb {R}^N{\setminus } \Omega )$. For $r>0$, define the Lipschitz map

$$\begin{aligned} \phi _r:\mathbb {R}^N\rightarrow \mathbb {R},\quad \phi _r(x)=\left\{ \begin{aligned}&0{} & {} \delta (x)\ge 2r,\\&2-\frac{\delta (x)}{r}{} & {} r\le \delta (x)\le 2r,\\&1{} & {} \delta (x)\le r. \end{aligned}\right. \end{aligned}$$

Note that we have $\phi _s\le \phi _r$ for $0<s\le r$. We show

$$\begin{aligned} u\phi _r\in {\mathscr {D}}^k(\Omega )~ \text { for}~ r>0 ~\text {sufficiently small and }~ b_{k,\mathbb {R}^N}(u\phi _r)\rightarrow 0 ~\text { for}~ r\rightarrow 0. \end{aligned}$$

(3.1)

Note that once this is shown, we have $u(1-\phi _r)\in {\mathscr {D}}^k(\Omega )$ for $r>0$ sufficiently small and $u(1-\phi _r)\rightarrow u$ for $r\rightarrow 0$. Since also $0\le u(1-\phi _r)\le u(1-\phi _s)$ for $0<s\le r$ and $u(1-\phi _r)=0$ for $x\in \mathbb {R}^N$ with $\delta (x)\le r$, it follows that (3.1) implies 1.

The remainder of the proof is to show (3.1). For this, let $C=C(N,\Omega ,k)>0$ be a constant which may vary from line to line. Let $A_t:=\{x\in \Omega \;:\; \delta (x)\le t\}$. Note that $u\phi _r$ vanishes on $\mathbb {R}^N{\setminus } A_{2r}$, we have $0\le \phi _r\le 1$ and, moreover,

$$\begin{aligned} |\phi _r(x)-\phi _r(y)|\le \min \Big \{C\frac{|x-y|}{r},1\Big \}~\text {for}~ x,y\in \mathbb {R}^N. \end{aligned}$$

Then proceeding similarly to the proof of Lemma 2.3.(5) we find for r small enough

$$\begin{aligned} b_{k,\mathbb {R}^N}(u\phi _r)&=\frac{1}{2}\int _{A_{4r}}\int _{A_{4r}}(u(x)\phi _r(x)-u(y)\phi _r(y))^2k(x,y)\ \textrm{d}x\textrm{d}y\\&\quad +\int _{A_{2r}}u^2(x)\phi _r^2(x)\kappa _{k,A_{4r}}(x)\ \textrm{d}x\\&\le \int _{A_{4r}}\int _{A_{4r}}\Big (u(x)^2(\phi _r(x)-\phi _r(y))^2 +(u(x)-u(y))^2\phi _r^2(y)^2\Big )k(x,y)\ \textrm{d}x\textrm{d}y\\&\quad +\int _{A_{2r}}u^2(x)\kappa _{k,A_{4r}}(x)\ \textrm{d}x\\&\le \frac{C}{r^2}\int _{A_{4r}}u(x)^2\int _{B_r(x)}|x-y|^2k(x,y)\ \textrm{d}y\textrm{d}x+\int _{A_{4r}}u(x)^2\int _{\mathbb {R}^N{\setminus } B_r(x)}k(x,y)\ \textrm{d}y\textrm{d}x\\&\quad + \int _{A_{4r}}\int _{A_{4r}}(u(x)-u(y))^2k(x,y)\ \textrm{d}x\textrm{d}y+\int _{A_{2r}}u^2(x)\kappa _{k,A_{4r}}(x)\ \textrm{d}x\\&\le \frac{C}{r^{\epsilon }}\int _{A_{4r}}u(x)^2\int _{B_1(x)}|x-y|^{\epsilon }k(x,y)\ \textrm{d}y\textrm{d}x\\&\quad +C\int _{A_{4r}}u(x)^2 \textrm{d}x +b_{k,A_{4r}}(u)+\int _{A_{2r}}u^2(x)\kappa _{k,A_{4r}}(x)\ \textrm{d}x\\&\le \frac{C}{r^{\epsilon }}\int _{A_{4r}}u(x)^2\ \textrm{d}x+C\int _{A_{4r}}u(x)^2 \textrm{d}x +b_{k,A_{4r}}(u)+\int _{A_{2r}}u^2(x)\kappa _{k,A_{4r}}(x)\ \textrm{d}x. \end{aligned}$$

Note here, since $u\in D^k(\mathbb {R}^N)$, we have $\int _{A_{4r}}u(x)^2 \textrm{d}x +b_{k,A_{4r}}(u)\rightarrow 0$ for $r\rightarrow 0$. Moreover, we have by Lebesgue’s differentiation theorem

$$\begin{aligned} \frac{C}{r^{\epsilon }}\int _{A_{4r}}u(x)^2 \textrm{d}x&\le \frac{C|B_{4r}|}{r^{\epsilon }}\int _{\partial \Omega }\frac{1}{|B_{4r}|}\int _{B_{4r}(\theta )}u(x)^2\ \textrm{d}x\sigma (d\theta )\\&\le Cr^{1-\epsilon }\int _{\partial \Omega }\frac{1}{|B_{4r}|}\int _{B_{4r}(\theta )}u(x)^2\ \textrm{d}x\sigma (d\theta )\rightarrow 0~\text {for}~ r\rightarrow 0^+. \end{aligned}$$

Finally, since

$$\begin{aligned} K_{k,\Omega }(u)=\int _{\Omega }u^2(x)\kappa _{k,\Omega }(x)\ \textrm{d}x<\infty \end{aligned}$$

(3.2)

and, by Lemma 3.5, we have

$$\begin{aligned} \kappa _{k,A_{4r}}(x)\le \int _{\mathbb {R}^N{\setminus } \Omega }k(x,y)\ \textrm{d}y+\int _{\Omega {\setminus } A_{4r}}k(x,y)\ \textrm{d}y\le C \kappa _{k,\Omega }(x)+Cr^{-\epsilon } \end{aligned}$$

for $x\in A_{2r}$, so that also $\int _{A_{2r}}u^2(x)\kappa _{k,A_{4r}}(x)\ \textrm{d}x\rightarrow 0$ for $r\rightarrow 0$ with a similar argument. $\square$

Proof of Theorem 3.1

for $X(\Omega )={\mathscr {D}}^k(\Omega )$ This statement now follows from Theorem 3.6, Lemma 3.3, and Proposition 3.4. $\square$

Theorem 3.7

Let $\Omega \subset \mathbb {R}^N$ be an open bounded set with Lipschitz boundary. Then $C^{\infty }_c(\Omega )$ is dense in $D^k(\Omega )$. Moreover, if $u\in D^k(\Omega )$ is nonnegative, then we have

• (1) There exists a sequence $(u_n)_n\subset D^k(\Omega )\cap L^{\infty }(\Omega )$ with $\lim \limits _{n\rightarrow \infty }u_n= u$ in $D^k(\Omega )$ satisfying that for every $n\in \mathbb {N}$ there is $\Omega '_n\subset \subset \Omega$ with $u_n=0$ on $\Omega {\setminus } \Omega '_n$ and $0\le u_n\le u_{n+1}\le u$.

• (2) There exists a sequence $(u_n)_n\subset C^{\infty }_c(\Omega )$ with $u_n\ge 0$ for every $n\in \mathbb {N}$ and $\lim \limits _{n\rightarrow \infty }u_n= u$ in $D^k(\Omega )$.

Proof

Consider the Lipschitz map

$$\begin{aligned} g_n:\mathbb {R}\rightarrow \mathbb {R},\qquad g_n(t)=\left\{ \begin{aligned}&0{} & {} t\le 0\\&t{} & {} 0<t<n\\&n{} & {} t\ge n.\end{aligned}\right. \end{aligned}$$

Then $v_n:=g_n(u)\in D^k(\Omega )\cap L^{\infty }(\Omega )$ and we have with $\phi _r$ as in the proof of Proposition 3.6

$$\begin{aligned} b_{k,\Omega }(u-(1-\phi _r)v_n)\le b_{k,\Omega }(u-v_n)+b_{k,\Omega }( \phi _rv_n). \end{aligned}$$

Clearly, $b_{k,\Omega }(u-v_n)\rightarrow 0$ for $n\rightarrow \infty$ by dominated convergence and $b_{k,\Omega }( \phi _rv_n)\rightarrow 0$ for $r\rightarrow 0$ analogously to the proof of Proposition 3.6, noting that the term in (3.2) reads in this case

$$\begin{aligned} K_{k,\Omega }(v_n)\le n^2\int _{\Omega }\kappa _{k,\Omega }(x)\ \textrm{d}x<\infty ~\text {for every}~ n\in \mathbb {N}. \end{aligned}$$

In particular, statement (1) follows. Now statement (2) and the density statement follow analogously, again, to the proof of Proposition 3.6. $\square$

Proof of Theorem 3.1

for $X(\Omega )=D^k(\Omega )$ This statement now follows from Theorem 3.7, Lemma 3.3, and Proposition 3.4. $\square$

Remark 3.8

It is tempting to conjecture the following type of Hardy inequality: There is $C>0$ such that

$$\begin{aligned} K_{k,\Omega }(\phi )\le C\Big (\Vert \phi \Vert _{L^2(\Omega )}^2+b_{k,\Omega }(\phi )\Big )~\text {for all}~ \phi \in C^{\infty }_c(\Omega ) \end{aligned}$$

if $\Omega$ is a bounded Lipschitz set and k is such that its symmetric lower bound j is not in $L^1(\mathbb {R}^N)$. Let us mention that for $k(x,y)=|x-y|^{-2s-N}$ this holds for $s\in (0,1)$, $s\ne \frac{1}{2}$, see [4, 8]. Moreover, for $k(x,y)=1_{B_{1}(0)}(x-y)|x-y|^{-N}$, this has been shown in [5]. In the general framework presented here, however, it is not clear if this is true.

Remark 3.9

With the above density results, we can now note that our definition of weak supersolutions (and similarly of weak subsolutions and solutions), see Definition 2.6, can be extended slightly:

Let $u\in {\mathscr {V}}^k_{loc}(\Omega )$ satisfy weakly $I_ku\ge f$ in $\Omega$ for some $f\in L^1_{loc}(\Omega )$ and $\Omega \subset \mathbb {R}^N$ open and bounded with Lipschitz boundary.

• (1) If $f\in L^2_{loc}(\Omega )$, then by density it also holds

  $$\begin{aligned} b_k(u,v)\ge \int _{U}f(x)v(x)\ \textrm{d}x~\text {for all nonnegative} ~v\in {\mathscr {D}}^k(U), U\subset \subset \Omega . \end{aligned}$$

  (3.3)

• (2) If $u\in {\mathscr {V}}^k(\Omega )\cap L^{\infty }(\mathbb {R}^N)$ and $f\in L^2(\Omega )$, then by density it also holds

  $$\begin{aligned} b_k(u,v)\ge \int _{\Omega }f(x)v(x)\ \textrm{d}x~\text {for all nonnegative}~ v\in {\mathscr {D}}^k(\Omega ). \end{aligned}$$

  (3.4)

Finally note that if $u:\mathbb {R}^N\rightarrow \mathbb {R}$ satisfies $u1_{U}\in D^k(U)$ for some $U\subset \subset \mathbb {R}^N$ and $u\in L^{\infty }(\mathbb {R}^N{\setminus } U)$, then $u\in {\mathscr {V}}^k_{loc}(U)$.

Proposition 3.10

(Weak maximum principle) Assume that the symmetric lower bound j of k defined in (2.2) satisfies

$$\begin{aligned} j ~\text { does not vanish identically on}~ B_r(0) ~\text { for any} ~r>0. \end{aligned}$$

(3.5)

Let $\Omega \subset \mathbb {R}^N$ open and with Lipschitz boundary, $c\in L^{\infty }_{loc}(\Omega )$, and assume either

• (1) $c\le 0$ or

• (2) $\Omega$ and c are such that $\Vert c^{+}\Vert _{L^{\infty }(\Omega )}<\inf _{x\in \Omega } \int _{\mathbb {R}^N{\setminus } \Omega }k(x,y)\ \textrm{d}y$.

If $u\in {\mathscr {V}}^k(\Omega )$ satisfies in weak sense

$$\begin{aligned} I_ku\ge c(x)u~\text { in}~ \Omega , u\ge 0~\text {almost everywhere in}~ \mathbb {R}^{N}{\setminus } \Omega ,~\text { and}~ \liminf \limits _{|x|\rightarrow \infty }u(x)\ge 0, \end{aligned}$$

then $u\ge 0$ almost everywhere in $\mathbb {R}^N$.

Proof

Note that also $u^-\in {\mathscr {V}}^k(\Omega )$ and in particular $u^-\in D^k(\Omega )$. Hence, we can find $(v_n)_n\subset C^{\infty }_c(\Omega )$ with $v_n\rightarrow u^-$ in $D^k(\Omega )$ for $n\rightarrow \infty$ with $0\le v_n\le v_{n+1}\le u^-$ by Proposition 3.7. Then

$$\begin{aligned} b_{k,\mathbb {R}^N}(u,v_n)\ge \int _{\Omega } c(x) u(x)v_n(x)\ \textrm{d}x\ge -\Vert c^+\Vert _{L^{\infty }(\Omega )}\int _{\Omega } u^-(x)v_n(x)\ \textrm{d}x. \end{aligned}$$

On the other hand, since $u^+v_n=0$ for all $n\in \mathbb {N}$ and $u\ge 0$ almost everywhere in $\mathbb {R}^{N}{\setminus } \Omega$, we find

$$\begin{aligned} b_{k,\mathbb {R}^N}(u,v_n)&=b_{k,\Omega }(u,v_n)+\int _{\Omega }v_n(x)\int _{\mathbb {R}^N{\setminus } \Omega }(u(x)-u(y))k(x,y)\ \textrm{d}y\textrm{d}x\\&\le b_{k,\Omega }(u^+,v_n)-b_{k,\Omega }(u^-,v_n)+\int _{\Omega }v_n(x)u(x)\int _{\mathbb {R}^N{\setminus } \Omega } k(x,y)\ \textrm{d}y\textrm{d}x\\&\le - b_{k,\Omega }(u^-,v_n)-K_{k,\Omega }(u^-,v_n). \end{aligned}$$

Hence

$$\begin{aligned} 0\le \int _{\Omega } u^-(x)v_n(x)\Big (\Vert c^+\Vert _{L^{\infty }(\Omega )}-\kappa _{k,\Omega }(x)\Big )\ \textrm{d}x-b_{k,\Omega }(u^-,v_n)\le -b_{k,\Omega }(u^-,v_n). \end{aligned}$$

Since $v_n\rightarrow u^-$ in $D^k(\Omega )$, it follows that $b_{k,\Omega }(u^-,u^-)=0$, but then $u^-$ is constant by Proposition 2.10 in $\Omega$. Assume by contradiction that $u^-=m>0$. Then the above calculation gives

$$\begin{aligned} 0\le m \int _{\Omega } v_n(x)\Big (\Vert c^+\Vert _{L^{\infty }(\Omega )}-\kappa _{k,\Omega }(x)\Big )\ \textrm{d}x, \end{aligned}$$

(3.6)

which is in both cases a contradiction: If in case 1. $c\le 0$, then since $\kappa _{k,\Omega }(x)\not \equiv 0$ and since $v_n\rightarrow m$ in $D^k(\Omega )$ the right-hand side of (3.6) is negative.

In case 2. this contradiction is immediate in a similar way. $\square$

Proof of Proposition 1.2

The statement follows immediately from Proposition 3.10. $\square$

Remark 3.11

Usually, the weak maximum principle is stated with an assumption on the first eigenvalue $\Lambda _1(\Omega )$ in place of $\inf _{x\in \Omega }\kappa _{k,\Omega }(x)$. This can be done once the Hardy inequality in Remark 3.8 is shown. Indeed, following the proof of Proposition 3.10 gives

$$\begin{aligned}{} & {} -\Vert c^+\Vert _{L^{\infty }(\Omega )}\int _{\Omega } u^-(x)v_n(x)\ \textrm{d}x\le b_{k,\mathbb {R}^N}(u,v_n)\le - b_{k,\Omega }(u^-,v_n)\\{} & {} \quad -K_{k,\Omega }(u^-,v_n)=-b_{k,\mathbb {R}^N}(u^-,v_n). \end{aligned}$$

With $n\rightarrow \infty$ and using that by the Hardy inequality it holds ${\mathscr {D}}^k(\Omega )=D^k(\Omega )$, it follows that

$$\begin{aligned} -\Lambda _1(\Omega )\Vert u^-\Vert _{L^2(\Omega )}^2 \ge -b_{k,\mathbb {R}^N}(u^-,u^-)\ge -\Vert c^+\Vert _{L^{\infty }(\Omega )}\Vert u^-\Vert _{L^2(\Omega )}^2 \end{aligned}$$

and the conclusion follows similarly.

Proof of Corollary 1.4

This statement now follows from Remark 3.11 using Remark 3.8. $\square$

Proposition 3.12

(Strong maximum principle) Assume k satisfies additionally (4.1). Let $\Omega \subset \mathbb {R}^N$ open and $c\in L^{\infty }_{loc}(\Omega )$ with $\Vert c^+\Vert _{L^{\infty }(\Omega )}<\infty$. Moreover, let $u\in {\mathscr {V}}^k(\Omega )$, $u\ge 0$ satisfy in weak sense $I_ku\ge c(x) u$ in $\Omega$.

• (1) If $\Omega$ is connected, then either $u\equiv 0$ in $\Omega$ or ${{\,\textrm{essinf}\,}}_Ku>0$ for any $K\subset \subset \Omega$.

• (2) If j given in (2.2) satisfies ${{\,\textrm{essinf}\,}}_{B_r(0)}j>0$ for any $r>0$, then either $u\equiv 0$ in $\mathbb {R}^N$ or ${{\,\textrm{essinf}\,}}_Ku>0$ for any $K\subset \subset \Omega$.

Proof

This statement follows by approximation from [19, Theorem 2.5 and 2.6]. Here, the statement $j\notin L^1(\mathbb {R}^N)$ comes into play since we need

$$\begin{aligned} \inf _{x\in B_r(x_0)}\kappa _{k,B_r(x_0)}(x)\rightarrow \infty ~\text {for}~ r\rightarrow 0 \end{aligned}$$

to conclude the statement for arbitrary c as stated. $\square$

Proof of Proposition 1.5

The statement follows immediately from Proposition 3.12. $\square$

4 On boundedness

In the following, let $h*u(x)=\int _{\mathbb {R}^N} h(x-y)u(y)\ \textrm{d}y$ as usual denote the convolution of two functions.

Theorem 4.1

Assume k is such that

$$\begin{aligned} \text {the symmetric lower bound}~ j ~\text { defined in}~ (2.2) ~\text {satisfies}~\int _{\mathbb {R}^N}j(z)\ \textrm{d}z=\infty \end{aligned}$$

(4.1)

and it holds

$$\begin{aligned} \sup _{x\in \mathbb {R}^N}\int _{K{\setminus } B_{\epsilon }(x)}k(x,y)^2\ \textrm{d}y<\infty ~\text {for all}~ K\subset \subset \mathbb {R}^N~\text { and}~ \epsilon >0. \end{aligned}$$

(4.2)

Let $\Omega \subset \mathbb {R}^N$ be an open set. Let $f\in L^{\infty }(\Omega )$, $h\in L^1(\mathbb {R}^N)\cap L^{2}(\mathbb {R}^N)$, and let $u\in {\mathscr {V}}^k_{loc}(\Omega )$ satisfy in weak sense

$$\begin{aligned} I_ku\le \lambda u+h*u+f ~\text {in}~ \Omega ~\text { for some}~ \lambda >0. \end{aligned}$$

If $u^+\in L^{\infty }(\mathbb {R}^N{\setminus } \Omega ')$ for some $\Omega '\subset \subset \Omega$, then $u^+\in L^{\infty }(\mathbb {R}^N)$ and there is $C=C(\Omega ,\Omega ',k,h,\lambda )>0$ such that

$$\begin{aligned} \Vert u^+\Vert _{L^{\infty }(\Omega ')}\le C\Big (\Vert f\Vert _{L^{\infty }(\Omega )}+\Vert u\Vert _{L^{2}(\Omega ')}+\Vert u^+\Vert _{L^{\infty }(\mathbb {R}^N{\setminus } \Omega ')}\Big ). \end{aligned}$$

Proof

Let $\Omega _1,\Omega _2,\Omega _3\subset \mathbb {R}^N$ be with Lipschitz boundary and such that

$$\begin{aligned} \Omega '\subset \subset \Omega _1\subset \subset \Omega _2\subset \subset \Omega _3\subset \subset \Omega . \end{aligned}$$

Let $\eta \in C^{0,1}_c(\Omega _3)$ such that $0\le \eta \le 1$ and $\eta =1$ on $\Omega _2$. Put $v=\eta u$ and, for $\delta >0$, denote $J_{\delta }(x,y):=1_{B_{\delta }(0)}(x-y)k(x,y)$ and $k_{\delta }(x,y)=k(x,y)-J_{\delta }(x,y)$. Note that by Assumption (2.1) it follows that $y\mapsto k_{\delta }(x,y)\in L^1(\mathbb {R}^N)$ for all $x\in \mathbb {R}^N$. Moreover, by Assumption (4.1)

$$\begin{aligned} c_{\delta }:=\inf _{x\in \mathbb {R}^N}\int _{\mathbb {R}^N}k_{\delta }(x,y)\ \textrm{d}y\ge \int _{\mathbb {R}^N{\setminus } B_{\delta }(0)}j(z)\ \textrm{d}z\rightarrow \infty ~\text {for}~ \delta \rightarrow 0. \end{aligned}$$

Hence, we may fix $\delta >0$ such that

$$\begin{aligned} c_{\delta }>\lambda . \end{aligned}$$

In the following, $C_i>0$, $i=1,\ldots$ denote constants depending on $\Omega '$, $\Omega _i$, $\lambda$, $\delta$, $\Omega$, $\eta$, k, and h but may vary from line to line—clearly, by the choices of these dependencies are actually only through $\lambda$, $\Omega$, $\Omega '$, $\eta$, k, and h. First note that by Lemma 2.8 we have in weak sense

$$\begin{aligned} I_kv\le \lambda u+h*u+\tilde{f}~\text {in}~ \Omega _1~\text {with}~ \tilde{f}(x)=f(x)+\int _{\mathbb {R}^N{\setminus } \Omega _2}(1-\eta (y))u(y)k(x,y)\ \textrm{d}y. \end{aligned}$$

In the following, put

$$\begin{aligned} A:=\Vert f\Vert _{L^{\infty }(\Omega )}+\Vert u\Vert _{L^{2}(\Omega ')}+\Vert u^+\Vert _{L^{\infty }(\mathbb {R}^N{\setminus }\Omega ')}. \end{aligned}$$

Then note that for $x\in \mathbb {R}^N$ we have

$$\begin{aligned} |h*u(x)|\le \Vert h\Vert _{L^{2}(\mathbb {R}^N)}\Vert u\Vert _{L^{2}(\Omega ')}+\Vert u^+\Vert _{L^{\infty }(\mathbb {R}^N{\setminus }\Omega ')}\Vert h\Vert _{L^1(\mathbb {R}^N)}\le C_1A \end{aligned}$$

and, since $\sup \limits _{x\in \Omega _1}\int _{\mathbb {R}^N{\setminus } \Omega _2}(1-\eta (y))k(x,y)\ \textrm{d}y\le C_2\sup \limits _{x\in \Omega _1}\int _{\mathbb {R}^N{\setminus } \Omega _2}\min \{1,|x-y|^{\sigma }\}k(x,y)\ \textrm{d}y<\infty$, it also holds that

$$\begin{aligned} \Vert \tilde{f}\Vert _{L^{\infty }(\Omega _1)}\le \Vert f\Vert _{L^{\infty }(\Omega )}+\Vert u^+\Vert _{L^{\infty }(\mathbb {R}^N{\setminus } \Omega _1)}C_2\le C_3A. \end{aligned}$$

Whence, since $u=v$ in $\Omega _1$, we have in weak sense

$$\begin{aligned} I_kv\le \lambda v+C_4A~\text {in}~ \Omega _1. \end{aligned}$$

Next, let $\mu \in C^{\infty }_c(\Omega '')$ for some $\Omega '\subset \subset \Omega ''\subset \subset \Omega _1$ such that $0\le \mu \le 1$, $\mu =1$ on $\Omega '$, and $\mu =0$ on $\mathbb {R}^N{\setminus } \Omega ''$. Let $\phi _t=\mu ^2(v-t)^+\in {\mathscr {D}}^k(\Omega '')$ for $t>0$ and note that

$$\begin{aligned} b_k(v,\phi _t)\le \int _{\Omega ''}(\lambda v(x)+C_4A)\phi _t(x)\ \textrm{d}x. \end{aligned}$$

(4.3)

Fix $t>0$ such that

$$\begin{aligned} t\ge \Vert u^+\Vert _{L^{\infty }(\mathbb {R}^N{\setminus } \Omega ')}&~\text {and}~ C_6A+(\lambda -c_{\delta })t\le 0,~\text {where}\\ C_6&=C_4+C_5~\text {with}~ C_5=\sup _{x\in \Omega '}\int _{\Omega '}k_{\delta }^2(x,y)\ \textrm{d}y+\sup _{x\in \Omega '}\int _{\mathbb {R}^N{\setminus }\Omega '}k_{\delta }(x,y)\ \textrm{d}y. \end{aligned}$$

That is, we fix

$$\begin{aligned} t= A\Big (1+\frac{C_6}{c_{\delta }-\lambda }\Big ). \end{aligned}$$

Then with (4.3)

$$\begin{aligned} b_{J_{\delta }}(v,\phi _t)&=b_k(v,\phi _t)-b_{k_{\delta }}(v,\phi _t)\\&\le \int _{\Omega ''}\lambda v(x)\phi _t(x)+C_4A\phi _t(x)\ \textrm{d}x-\int _{\mathbb {R}^N}v(x)\phi _t(x)\int _{\mathbb {R}^N}k_{\delta }(x,y)\ \textrm{d}y\textrm{d}x\\&+\int _{\mathbb {R}^N}\phi _t(x)\int _{\mathbb {R}^N}v(y)k_{\delta }(x,y)\ \textrm{d}y\ \textrm{d}x. \end{aligned}$$

Note here, that for $x\in \mathbb {R}^N$ we have by the integrability assumptions on $k_{\delta }$ and k

$$\begin{aligned} \int _{\mathbb {R}^N}v(y)k_{\delta }(x,y)\ \textrm{d}y&\le \int _{\mathbb {R}^N}u(y)k_{\delta }(x,y)\ \textrm{d}y\le C_5(\Vert u\Vert _{L^2(\Omega ')}+\Vert u^+\Vert _{L^{\infty }(\mathbb {R}^N{\setminus } \Omega ')})\le C_5A \end{aligned}$$

so that using that $v\ge t$ in $\text {supp}\,\phi _t$ we have

$$\begin{aligned} b_{J_{\delta }}(v,\phi _t)\le \int _{\Omega ''}(C_6A+(\lambda -c_{\delta })v(x))\phi _t(x)\ \textrm{d}x\le (C_6A+(\lambda -c_{\delta })t)\int _{\Omega ''}\phi _t(x)\ \textrm{d}x.\nonumber \\ \end{aligned}$$

(4.4)

On the other hand, with $v_t(x)=v(x)-t$, we have

$$\begin{aligned}&(v(x)-v(y))(\phi _t(x)-\phi _t(y))-(\mu (x)v_t^+(x)-\mu (y)v_t^+(y))^2\\&\quad =2\mu (x)\mu (y)v_t^+(x)v_t^+(y)-v_t(y)\mu ^2(x)v_t^+(x)-\mu ^2(y)v_t^+(y)v_t(x)\\&\quad =-v_t^+(x)v_t^+(y)(\mu (x)-\mu (y))^2+v_t^-(y)\mu ^2(x)v_t^+(x)+\mu ^2(y)v_t^+(y)v_t^-(x)\\&\quad \ge -v_t^+(x)v_t^+(y)(\mu (x)-\mu (y))^2. \end{aligned}$$

Whence with Poincaré’s inequality, using that by Assumption 4.1 there is for any $K\subset \mathbb {R}^N$ open and bounded, some $C>0$ such that $b_{J_{\delta }}(u)\ge C\Vert u\Vert _{L^2(\mathbb {R}^N)}^2$ for $u\in {\mathscr {D}}^{J_{\delta }}(K)$, we find for some constant $C_7$

$$\begin{aligned}&b_{J_{\delta }}(v,\phi _t)\ge b_{J_{\delta }}(\mu v_t^+)-\frac{1}{2}\int _{\mathbb {R}^N}\int _{\mathbb {R}^N} v_t^+(x)v_t^+(y)(\mu (x)-\mu (y))^2J_{\delta }(x,y)\ \textrm{d}x\textrm{d}y\nonumber \\&\quad \ge C_7\int _{\mathbb {R}^N}\mu ^2(x)(v_t^+(x))^2\ \textrm{d}x-\frac{1}{2}\int _{\mathbb {R}^N}\int _{\mathbb {R}^N} v_t^+(x)v_t^+(y)(\mu (x)-\mu (y))^2J_{\delta }(x,y)\ \textrm{d}x\textrm{d}y\end{aligned}$$

(4.5)

$$\begin{aligned}&\quad = C_7\int _{\mathbb {R}^N}\mu ^2(x)(v_t^+(x))^2\ \textrm{d}x-\frac{1}{2}\int _{\Omega '}\int _{\Omega '} v_t^+(x)v_t^+(y)(\mu (x)-\mu (y))^2J_{\delta }(x,y)\ \textrm{d}x\textrm{d}y\end{aligned}$$

(4.6)

$$\begin{aligned}&\quad = C_7\int _{\mathbb {R}^N}\mu ^2(x)(v_t^+(x))^2\ \textrm{d}x\ge C_7\int _{\Omega '}(v_t^+(x))^2\ \textrm{d}x. \end{aligned}$$

(4.7)

Combining (4.7) and (4.4) we have

$$\begin{aligned} C_7\int _{\Omega '}&(v_t^+(x))^2\ \textrm{d}x\le \Big (C_6A+(\lambda -c_{\delta })t\Big )\int _{\Omega ''} \phi _t(x)\ \textrm{d}x\le 0. \end{aligned}$$

Whence $v_t^+=0$ in $\Omega '$ and thus $u=v\le t=A\cdot C_{8}$ in $\Omega '$ as claimed. $\square$

Corollary 4.2

If in the situation of Theorem 4.1 we have in weak sense $I_ku=\lambda u+h*u+f$ in $\Omega$, then we have $u\in L^{\infty }(\Omega ')$ and there is $C=C(\Omega ,\Omega ',k,\lambda ,h)>0$ such that

$$\begin{aligned} \Vert u\Vert _{L^{\infty }(\Omega ')}\le C\Big (\Vert f\Vert _{L^{\infty }(\Omega )}+\Vert u\Vert _{L^{2}(\Omega ')}+\Vert u\Vert _{L^{\infty }(\mathbb {R}^N{\setminus } \Omega ')}\Big ). \end{aligned}$$

Proof

This follows by replacing u with $-u$ (and f with $-f$) in the statement of Theorem 4.1. $\square$

Theorem 4.3

If in the situation of Theorem 4.1 we have in weak sense $I_ku=\lambda u+h*u+f$ in $\Omega$ and $u\in {\mathscr {D}}^k(\Omega )$, then we have $u\in L^{\infty }(\Omega )$ and there is $C=C(\Omega ,k,\lambda ,h)>0$ such that

$$\begin{aligned} \Vert u\Vert _{L^{\infty }(\Omega )}\le C\Big (\Vert f\Vert _{L^{\infty }(\Omega )}+\Vert u\Vert _{L^{2}(\Omega )}\Big ). \end{aligned}$$

Proof

Using in the proof of Theorem 4.1 the test-function $u_t^+$ instead of $\phi _t$ (and similarly for Corollary 4.2), we find

$$\begin{aligned} \Vert u\Vert _{L^{\infty }(\Omega )}\le C\Big (\Vert f\Vert _{L^{\infty }(\Omega )}+\Vert u\Vert _{L^{2}(\Omega )}\Big ). \end{aligned}$$

as claimed. $\square$

5 On differentiability of solutions

In the following, $\Omega \subset \mathbb {R}^N$ is an open bounded set and k satisfies through out the assumptions (2.1) with some $\sigma <\frac{1}{2}$, (4.1), and (4.2). Moreover, we assume that there is $j:\mathbb {R}^N\rightarrow [0,\infty ]$ such that $k(x,y)=j(x-y)$ for $x,y\in \mathbb {R}^N$ and that for some $m\in \mathbb {N}\cup \{\infty \}$ the following holds: We have $j\in W^{l,1}(\mathbb {R}^N{\setminus } B_{\epsilon }(0)$ for every $l\in \mathbb {N}$ with $l\le 2m$, and there is some constant $C_j>0$ such that

$$\begin{aligned} |\nabla j(z)|\le C_j|z|^{-1-\sigma -N}~\text {for all} ~0<|z|\le 3. \end{aligned}$$

For simplicity, we write j in place of k and fix

$$\begin{aligned} \alpha :=1-\sigma \in \left( \frac{1}{2},1\right) . \end{aligned}$$

Theorem 5.1

Let $f\in H^1(\Omega )$, $\lambda \in \mathbb {R}$ and $u\in {\mathscr {V}}^j_{loc}(\Omega )\cap L^{\infty }(\mathbb {R}^N)$ satisfy in weak sense $I_ju=f+\lambda u$ in $\Omega$. Then for any $\Omega '\subset \subset \Omega$ there is $C=C(N,\Omega ,\Omega ',j,\lambda )>0$ such that

$$\begin{aligned} \Vert \delta _{h,e} u\Vert _{L^2(\Omega ')}\le h^{\alpha }C\Big (\Vert f\Vert _{H^1(\Omega )}^2+\Vert u\Vert _{L^\infty (\mathbb {R}^N)}^2\Big )^{\frac{1}{2}}~\text {for all} ~h>0, e\in \partial B_1(0). \end{aligned}$$

(5.1)

Proof

Let $\Omega '\subset \subset \Omega$ and fix $r\in (0,\frac{1}{8})$ small such that $8r\le \text {dist}(\Omega ',\mathbb {R}^N{\setminus }\Omega )$. Moreover, fix $x_0\in \Omega '$ and denote $B_n:=B_{nr}(x_0)$. Note that by using assumption (4.1) with Lemma 2.2 we achieve, by making $r>0$ small enough,

$$\begin{aligned} \lambda <\lambda _1=\min _{\begin{array}{c} w\in {\mathscr {D}}^j(B_4)\\ w\ne 0 \end{array}}\frac{b_j(w)}{\Vert w\Vert _{L^2(\mathbb {R}^N)^2}}. \end{aligned}$$

Let $\eta \in C^{0,1}_c(B_4)$ with $0\le \eta \le 1$, $\eta \equiv 1$ on $B_2$. Note that it holds

$$\begin{aligned} |\eta (x)-\eta (y)|\le 2\Vert \eta \Vert _{C^{0,1}(\mathbb {R}^N)}\min \{1,|x-y|\}, \end{aligned}$$

where we put as usual

$$\begin{aligned} \Vert \eta \Vert _{C^{0,1}(\mathbb {R}^N)}:=\sup _{x\in \mathbb {R}^N}|\eta (x)|+\sup _{\begin{array}{c} x,y\in \mathbb {R}^N\\ x\ne y \end{array}}\frac{|\eta (x)-\eta (y)|}{|x-y|}. \end{aligned}$$

Note that by choice we have $\Vert \eta \Vert _{C^{0,1}(\mathbb {R}^N)}\le 1+\frac{1}{r}\le \frac{2}{r}$, so that for all $x,y\in \mathbb {R}^N$

$$\begin{aligned} |\eta (x)-\eta (y)|\le \frac{4}{r}\min \{1,|x-y|\}. \end{aligned}$$

(5.2)

Fix $e\in \partial B_1(0)$ and $h\in (0,r)$. Let

$$\begin{aligned} A:=\Vert u\Vert _{L^{\infty }(\mathbb {R}^N)}. \end{aligned}$$

Let $\psi =\eta ^2\delta _{h}u\in {\mathscr {D}}^j(B_4)$, where in the following $\delta _hu:=\delta _{h,e}u$. Note that

$$\begin{aligned} (\delta _{h}u(x)-\delta _{h}u(y))(\psi (x)-\psi (y))&=(\eta (x)\delta _{h}u(x)-\eta (y)\delta _{h}u(y))^2\\&\quad -\delta _{h}u(x)\partial _{h}u(y)(\eta (x)-\eta (y))^2. \end{aligned}$$

Hence, we have

$$\begin{aligned} b_j(\delta _{h}u,\psi )=b_j(\eta \delta _{h}u)-\frac{1}{2}\iint _{\mathbb {R}^{N}\times \mathbb {R}^N} \delta _{h}u(x)\delta _{h}u(y)(\eta (x)-\eta (y))^2j(x-y)\ \textrm{d}x\textrm{d}y. \end{aligned}$$

and using the translation invariance, we also have

$$\begin{aligned} b_j(\delta _h u,\psi )=\int _{\Omega } [\delta _hf(x)+\lambda \delta _hu]\psi (x)\ \textrm{d}x. \end{aligned}$$

In the following, for simplicity, we put $v(x)=\eta (x)\delta _hu(x)$, $x\in \mathbb {R}^N$. Note that by Definition, $v\in {\mathscr {D}}^j(B_4)$. Then with the help of Young’s inequality for some $\mu \in (0,1)$ such that

$$\begin{aligned} 2\mu < \lambda _1-\lambda \end{aligned}$$

(5.3)

we find

$$\begin{aligned}&\lambda _1\Vert v\Vert _{L^2(\Omega '')}^2\le b_j(v)=b_j(\delta _{h}u,\psi )+\frac{1}{2}\iint _{\mathbb {R}^{n}\times \mathbb {R}^N}\delta _{h}u(x)\delta _{h}u(y)(\eta (x)\nonumber \\&\qquad -\eta (y))^2j(x-y)\ \textrm{d}x\textrm{d}y\nonumber \\&\quad =\int _{\Omega ''}[\delta _{h}f(x)+\lambda \delta _hu]\eta ^2(x)\delta _{h}u(x)\ \textrm{d}x+\frac{1}{2}\iint _{\mathbb {R}^{n}\times \mathbb {R}^N}\delta _{h}u(x)\delta _{h}u(y)(\eta (x)\nonumber \\&\qquad -\eta (y))^2j(x-y)\ \textrm{d}x\textrm{d}y\nonumber \\&\quad \le (\mu +\lambda )\Vert v\Vert _{L^{2}(\Omega '')}^2+\mu ^{-1}h^2\Vert \frac{\delta _hf}{h}\Vert _{L^2(\Omega '')}^2 +\frac{1}{2}\iint _{\mathbb {R}^N\times \mathbb {R}^N}\delta _{h}u(x)\delta _{h}u(y)(\eta (x)\nonumber \\&\qquad -\eta (y))^2j(x-y)\ \textrm{d}x\textrm{d}y. \end{aligned}$$

(5.4)

By a rearrangement of the double integral with Young’s inequality for the same $\mu \in (0,1)$ as above we have

$$\begin{aligned}&\frac{1}{2}\iint _{\mathbb {R}^{n}\times \mathbb {R}^N}\delta _{h}u(x)\delta _{h}u(y)(\eta (x)-\eta (y))^2j(x-y)\ \textrm{d}x\textrm{d}y\nonumber \\&\quad =\iint _{\mathbb {R}^N\times \mathbb {R}^N} \delta _{h}u(x)\delta _{h}u(y)\eta (x)(\eta (x)-\eta (y))j(x-y)\ \textrm{d}x\textrm{d}y\nonumber \\&\quad =\int _{\mathbb {R}^N} \eta (x)\delta _{h}u(x)\int _{\mathbb {R}^N}u(y)\delta _{-h,y}\Big ((\eta (x)-\eta (y))j(x-y)\Big )\ \textrm{d}y\textrm{d}x\nonumber \\&\quad \le \mu \Vert v\Vert _{L^2(B_4)}^2+\mu ^{-1}\int _{B_4}\Bigg (\ \int _{\mathbb {R}^N}|u(y)|\Big |\delta _{-h,y}\Big ((\eta (x)-\eta (y))j(x-y)\Big )\Big |\ \textrm{d}y\Bigg )^2 \textrm{d}x\nonumber \\&\quad \le \mu \Vert v\Vert _{L^2(B_4)}^2+\mu ^{-1}A^2\int _{B_4}\Bigg (\ \int _{\mathbb {R}^N}\Big |\delta _{-h,y}\Big ((\eta (x)-\eta (y))j(y-x)\Big )\Big |\ \textrm{d}y\Bigg )^2 \textrm{d}x\nonumber \\&\quad \le \mu \Vert v\Vert _{L^2(B_4)}^2+\mu ^{-1}A^2\int _{B_4}\Bigg (\ \int _{\mathbb {R}^N}\Big |\delta _{-h,z}\Big ((\eta (x)-\eta (z+x))j(z)\Big )\Big |\ \textrm{d}z\Bigg )^2 \textrm{d}x. \end{aligned}$$

(5.5)

Here, we indicate with $\delta _{-h,y}$ (resp. $\delta _{-h,z}$) that $\delta _{-h}$ acts on the y (resp. z) variable. Note that

$$\begin{aligned}&\delta _{-h,z}\Big ((\eta (x)-\eta (z+x))j(z)\Big )\nonumber \\&\quad =\delta _{-h,z}(\eta (x)-\eta (z+x))j(z)+ (\eta (x)-\eta (z+x-he))\delta _{-h}j(z)\nonumber \\&\quad =\Big (\eta (z+x)-\eta (z+x-he)\Big ) j(z)+(\eta (x)-\eta (z+x-he))\Big (j(z-he)-j(z)\Big )\end{aligned}$$

(5.6)

$$\begin{aligned}&\quad = \Big (\eta (z+x)-\eta (z)\Big ) j(z)+ \Big (\eta (x)-\eta (z+x-he)\Big )j(z-he). \end{aligned}$$

(5.7)

Note here, that (5.6) satisfies

$$\begin{aligned} \begin{aligned} \Bigg |\Big (\eta (z+x)-\eta (z+x-he)\Big )&j(z)+(\eta (x)-\eta (z+x-he))\Big (j(z-he)-j(z)\Big )\Bigg |\\&\le \frac{4h}{r}J(z)+\frac{4h}{r}\min \{1,|z-he|\}\int _0^1|\nabla j(z-\tau he)|\ d\tau \end{aligned} \end{aligned}$$

(5.8)

and (5.7) can be written as

$$\begin{aligned} \begin{aligned} \Bigg | \Big (\eta (z+x)-\eta (z)\Big )&j(z)+ \Big (\eta (x)-\eta (z+x-he)\Big )j(z-he)\Bigg |\\&\le \frac{4}{r}\min \{1,|z|\}j(z)+\frac{4}{r}\min \{1,|z-he|\}j(z-he). \end{aligned} \end{aligned}$$

(5.9)

For $h\in (0,r)$, $z\in \mathbb {R}^N{\setminus }\{0\}$ put

$$\begin{aligned} k_h(z)=\min&\Bigg \{h\Big (j(z)+\min \{1,|z-he|\}\int _0^1|\nabla j(z-\tau he)|\ d\tau \Big ),\\&\qquad \qquad \min \{1,|z|\}j(z)+\min \{1,|z-he|\}j(z-he)\Bigg \}. \end{aligned}$$

Then, by combining (5.4) and (5.5), we find

$$\begin{aligned} \begin{aligned} \Vert \delta _h u\Vert _{L^2(B_2)}^2&\le \Vert v\Vert _{L^2(B_4)}^2\\&\le \frac{\mu ^{-1}|B_4|}{\lambda _1-\lambda -2\mu }\Bigg (h^2\Vert \frac{\delta _hf}{h}\Vert _{L^{2}(B_4)}^2+ \frac{16}{r^2}\Vert u\Vert _{L^\infty (\mathbb {R}^N)}^2 \Big (\ \int _{\mathbb {R}^N}k_h(z)\ \textrm{d}z\Big )^2\Bigg ). \end{aligned}\nonumber \\ \end{aligned}$$

(5.10)

Next we show that we have $\int _{\mathbb {R}^N}k_h(z)\ \textrm{d}z \le C h^{\alpha }$ for some $C>0$. Clearly, we can bound

$$\begin{aligned} \int _{\mathbb {R}^N{\setminus } B_2(0)}k_h(z)\ \textrm{d}z\le C_1h \end{aligned}$$

(5.11)

for some $C_1=C_1(N,j)>0$, using that $B_1(0)\cup B_1(he)\subset B_2(0)$ and the properties of j. In the following, by making $C_j$ larger if necessary, we may also assume that assumption (2.1) reads

$$\begin{aligned} \sup _{x\in \mathbb {R}^N}\int _{\mathbb {R}^N}\min \{1,|x-y|^{\sigma }\}j(x-y)\ \textrm{d}y=\int _{\mathbb {R}^N}\min \{1,|z|^{\sigma }\}j(z)\ \textrm{d}z\le C_j. \end{aligned}$$

Then note that $B_{2h}(he)\subset B_{3h}(0)$ and we have

$$\begin{aligned}&\int _{B_{2h}(0)} \min \{1,|z|\}j(z)+\min \{1,|z-he|\}j(z-he)\ \textrm{d}z\nonumber \\&\quad \le C_j\int _{B_{3h}(0)} |z|^{1-\sigma -n}\ \textrm{d}z + C_j\int _{B_{3h}(he)}|z-he|^{1-\sigma -n}\ \textrm{d}z\nonumber \\&\quad =\frac{2|B_{1}(0)|C_j}{n} \int _0^{3h}\rho ^{-\sigma }\ d\rho =\frac{2|B_{1}(0)|C_j}{n(1-\sigma )}(3h)^{1-\sigma }. \end{aligned}$$

(5.12)

While with $b_{\sigma }(t)=\frac{1}{\sigma }t^{-\sigma }$ we have

$$\begin{aligned}&h\int _{B_2(0){\setminus } B_{2h}(0)}j(z)+\min \{1,|z-he|\}\int _0^1|\nabla j(z-\tau he)|\ d\tau \ \textrm{d}z\nonumber \\&\quad \le \frac{h|B_{1}(0)|C_j}{n}\int _{2h}^2\rho ^{-\sigma -1}\ d\rho +hC_j\int _0^1 \int _{B_{3}(\tau he){\setminus } B_{h}(\tau he)}|z||z-\tau he|^{-1-\sigma -n}\ \textrm{d}z\ d\tau \nonumber \\&\quad \le \frac{h|B_{1}(0)|C_j}{n}b_{\sigma }(2h)+hC_j\int _0^1 \int _{B_{3}(0){\setminus } B_{h}(0)}|z+\tau h e||z|^{-1-\sigma -n}\ \textrm{d}z\ d\tau \nonumber \\&\quad \le \frac{h|B_{1}(0)|C_j}{n} b_{\sigma }(2h)+hC_J \int _{B_{3}(0){\setminus } B_{h}(0)}|z|^{-\sigma -n}\ \textrm{d}z +h^2C_j \int _{B_{3}(0){\setminus } B_{h}(0)}|z|^{-1-\sigma -n}\ \textrm{d}z\nonumber \\&\quad \le \frac{2h|B_{1}(0)|C_j}{n} b_{\sigma }(h)+\frac{h^2|B_{1}(0)|C_j}{n} \int _h^3\rho ^{-2-\sigma }\ d\rho \nonumber \\&\quad \le \frac{2|B_{1}(0)|C_j}{n} hb_{\sigma }(h)+\frac{|B_{1}(0)|C_j}{n(1+\sigma )}h^{1-\sigma }. \end{aligned}$$

(5.13)

Combining (5.11) with (5.12) and (5.13) and the choice $\alpha =1-\sigma \in (0,1)$ we find $C_2=C_2(N, j,\alpha )>0$ such that

$$\begin{aligned} \int _{\mathbb {R}^N}k_h(z)\ \textrm{d}z\le C_2h^{\alpha }. \end{aligned}$$

(5.14)

Whence, from (5.10) with (5.14) we have

$$\begin{aligned} \Vert \delta _h u\Vert _{L^2(B_2)}^2\le \Vert v\Vert _{L^2(B_4)}^2\le h^{2\alpha }C_4\Big (\Vert \frac{\delta _hf}{h}\Vert _{L^2(B_4)}^2+ \Vert u\Vert _{L^\infty (\mathbb {R}^N)}^2\Big ), \end{aligned}$$

(5.15)

for a constant $C_4=C_4(N,j,r,\alpha ,\lambda )>0$. By a standard covering argument, we then also find with a constant $C_5=C_5(N,j,\Omega ,\Omega ',\alpha ,\lambda )>0$ and $\Omega ''=\{x\in \Omega \;:\; \text {dist}(x,\mathbb {R}^N{\setminus }\Omega )>4r\}$

$$\begin{aligned} \Vert \delta _h u\Vert _{L^2(\Omega ')}^2\le h^{2\alpha }C_4\Big (\Vert \frac{\delta _hf}{h}\Vert _{L^2(\Omega '')}^2+ \Vert u\Vert _{L^\infty (\mathbb {R}^N)}^2\Big ), \end{aligned}$$

(5.16)

The claim (5.1) then follows since $f\in H^1(\Omega )$. $\square$

Remark 5.2

If additionally $f\in L^{\infty }(\Omega )$, combining Theorem 5.1 with Corollary 4.2 it follows that we have in the situation of Theorem 5.1 for every $\Omega '\subset \subset \Omega$

$$\begin{aligned} \Vert \delta _{h,e} u\Vert _{L^2(\Omega ')}\le h^{\alpha }C\Big (\Vert f\Vert _{C^1(\Omega )}^2+\Vert u\Vert _{L^2(\Omega ')}^2+\Vert u\Vert _{L^\infty (\mathbb {R}^N{\setminus } \Omega ')}^2\Big )^{\frac{1}{2}}~\text {for all}~ h>0, e\in \partial B_1(0).\nonumber \\ \end{aligned}$$

(5.17)

Corollary 5.3

Assume $m=1$. Let $f\in C^2(\overline{\Omega })$, $\lambda \in \mathbb {R}$, and let $u\in {\mathscr {V}}^j_{loc}(\Omega )\cap L^{\infty }(\mathbb {R}^N)$ satisfy in weak sense $I_ju=\lambda u+ f$ in $\Omega$. Then $u\in H^{1}(\Omega ')$ and $\partial _iu\in D^j(\Omega ')$ for any $\Omega '\subset \subset \Omega$. More precisely, with $\alpha$ as above there is for any $\Omega '\subset \subset \Omega$ a constant $C=C(N,\Omega ,\Omega ', j,\lambda )>0$ such that

$$\begin{aligned} \sup _{\begin{array}{c} e\in \partial B_1(0)\\ h>0 \end{array}}h^{-2\alpha }\Vert \delta _{h,e}^2 u\Vert _{L^2(\Omega ')}\le C\Big (\Vert f\Vert _{C^2(\Omega )}^2+\Vert u\Vert _{L^2(\Omega ')}+\Vert u\Vert ^2_{L^\infty (\mathbb {R}^N{\setminus } \Omega ')} \Big )^{\frac{1}{2}}, \end{aligned}$$

(5.18)

so that $u\in N^{2\alpha ,2}(\Omega ')\subset H^1(\Omega ')$, that is, there is also $C'=C'(N, j,\Omega ,\Omega ',\alpha ,\lambda )>0$ such that

$$\begin{aligned} \Vert \nabla u\Vert _{L^2(\Omega ')}\le C'\Big (\Vert f\Vert _{C^2(\Omega )}^2+\Vert u\Vert _{L^2(\Omega ')}+\Vert u\Vert ^2_{L^\infty (\mathbb {R}^N{\setminus } \Omega ')} \Big )^{\frac{1}{2}} \end{aligned}$$

(5.19)

and, moreover,

$$\begin{aligned} b_{j,\Omega '}(\partial _iu)\le C'~\text {for}~ i=1,\ldots ,N. \end{aligned}$$

Proof

Let $\Omega _i\subset \subset \Omega$, $i=1,\ldots ,7$ such that

$$\begin{aligned} \Omega '\subset \subset \Omega _i\subset \subset \Omega _j ~\text {for}~ 1\le i<k\le 7. \end{aligned}$$

Let $\eta \in C^{\infty }_c(\Omega _{7})$ with $\eta =1$ on $\Omega _6$ and $0\le \eta \le 1$. Fix $e\in \partial B_1(0)$ and $h\in (0,\frac{1}{2}r)$, where $r=\min \{\text {dist}(\Omega _i,\Omega {\setminus } \Omega _{i+1})\;:\; i=1,\ldots ,6\}$. Then by Lemma 2.8 the function $v=\eta \delta _h u$, where we write $\delta _h$ instead of $\delta _{h,e}$, satisfies $I_jv=\lambda v+\tilde{f}$ in $\Omega _5$, where $\tilde{f}=\delta _hf+g_{\eta ,\delta _hu}$. Following the proof of Theorem 5.1 to (5.16) it follows with Theorem 4.1 that there is $C=C(N, j, r,\alpha ,\lambda )>0$ (changing from line to line) such that

$$\begin{aligned} \Vert \delta _{h}^2 u\Vert ^2_{L^2(\Omega ')}&= \Vert \delta _hv\Vert ^2_{L^2(\Omega ')}\le h^{2\alpha }C\Big (\Vert \frac{\delta _{h}\tilde{f}}{h}\Vert _{L^{2}(\Omega _1)}^2+ \Vert v\Vert ^2_{L^\infty (\mathbb {R}^N)}\Big )\\&\le h^{2\alpha }C\Big (\Vert \frac{\delta _{h}\tilde{f}}{h}\Vert _{L^{2}(\Omega _1)}^2+ \Vert \tilde{f}\Vert _{L^{\infty }(\Omega _4)}^2+\Vert v\Vert _{L^{2}(\Omega _3)}^2+\Vert v\Vert _{L^{\infty }(\mathbb {R}^N{\setminus } \Omega _3)}^2\Big )\\&\le h^{2\alpha }C\Big (\Vert \frac{\delta _{h}\tilde{f}}{h}\Vert _{L^{2}(\Omega _1)}^2+ \Vert \tilde{f}\Vert _{L^{\infty }(\Omega _4)}^2+\Vert \delta _hu\Vert _{L^{2}(\Omega _3)}^2\Big )\\&\le h^{2\alpha }C\Big (\Vert \frac{\delta _{h}\tilde{f}}{h}\Vert _{L^{2}(\Omega _1)}^2+ \Vert \tilde{f}\Vert _{L^{\infty }(\Omega _4)}^2+h^{2\alpha }\Big (\Vert f\Vert _{C^1(\Omega )}^2+\Vert u\Vert _{L^{\infty }(\mathbb {R}^N)}^2\Big )\Big ), \end{aligned}$$

where we applied once more Theorem 5.1. Here, for $x\in \Omega _4$ using the assumptions on the differentiability of j it follows that there is $C=C(j)>0$ such that

$$\begin{aligned} |\tilde{f}(x)|&\le |\delta _hf(x)|+\Bigg |\int _{\mathbb {R}^N{\setminus } \Omega _6}(1-\eta (y))\delta _hu(y)j(x-y)\ \textrm{d}y\Bigg |\\&= |\delta _hf(x)|+\Bigg |\int _{\mathbb {R}^N{\setminus } \Omega _5}|u(y)|\delta _h[(1-\eta (y))j(x-y)]\ \textrm{d}y\\&\le hC\Big (\Vert \nabla f\Vert _{L^{\infty }(\Omega )} +\Vert u\Vert _{L^{\infty }(\mathbb {R}^N{\setminus } \Omega ')}\Big ). \end{aligned}$$

Moreover, for $x\in \Omega _1$ in a similar way, there is $C=C(j)>0$ such that

$$\begin{aligned} |\delta _h\tilde{f}(x)|&\le |\delta _h^2f(x)|+\Bigg |\int _{\mathbb {R}^N{\setminus } \Omega _6}(1-\eta (y))\delta _hu(y)\delta _hj(x-y)\ \textrm{d}y\Bigg |\\&\le h^2\Vert f\Vert _{C^2(\Omega )}+ \Vert u\Vert _{L^{\infty }(\mathbb {R}^N{\setminus } \Omega ')}\Bigg |\int _{\mathbb {R}^N{\setminus } \Omega _5}\delta _h[(1-\eta (y))\delta _hj(x-y)]\ \textrm{d}y\Bigg |\\&\le h^2C\Bigg (\Vert f\Vert _{C^2(\Omega )}+ \Vert u\Vert _{L^{\infty }(\mathbb {R}^N{\setminus } \Omega ')}\Bigg ). \end{aligned}$$

Thus we have

$$\begin{aligned} \Vert \delta _{h}^2 u\Vert ^2_{L^2(\Omega ')}\le Ch^{4\alpha }\Bigg (\Vert f\Vert _{C^{2}(\Omega )}^2+\Vert u\Vert _{L^2(\Omega ')}^2+\Vert u\Vert _{L^{\infty }(\mathbb {R}^N{\setminus } \Omega ')}^2\Bigg ). \end{aligned}$$

The proof of the first part then is finished with Proposition 2.11 since $2\alpha >1$. Next, write $D_hp(x)=\frac{p(x+he)-p(x)}{h}$ for any function $p:\mathbb {R}^N\rightarrow \mathbb {R}$, with $e\in \partial B_1(0)$ fixed and $h\in \mathbb {R}{\setminus }\{0\}$. Then with Lemma 2.8 for some $\eta \in C^{\infty }_c(\Omega )$ such that $0\le \eta \le 1$ and $\eta \equiv 1$ on $\Omega _2\subset \subset \Omega$ with $\Omega '\subset \subset \Omega _1\subset \subset \Omega _2$ we have with $v=\eta u$,

$$\begin{aligned} I_jv=f+\lambda v+g_{\eta ,u}~\text {in}~ \Omega _1,~\text { where}~ g_{\eta ,u}=\int _{\mathbb {R}^N{\setminus }\Omega _2}(1-\eta (y))u(y)j(x-y)\ \textrm{d}y. \end{aligned}$$

Next, let $\mu \in C^{\infty }_c(\Omega _1)$ with $0\le \mu \le 1$ and $\mu \equiv 1$ on $\Omega '$. Then with $\phi =D_{-h}[\mu ^2D_hv]\in {\mathscr {D}}^j(\Omega _1)$ for h small enough we have for some $C>0$ (which may change from line to line independently of h)

$$\begin{aligned} |b_j(v,\phi )|=\Bigg |\int _{\Omega _1}D_hf\mu ^2D_hv+ \lambda (\mu D_hv)^2+D_hg_{\eta ,u}\mu ^2 D_hv\ \textrm{d}x\Bigg |\le C, \end{aligned}$$

(5.20)

since

$$\begin{aligned}{} & {} \int _{\Omega _1}|D_hf\mu ^2D_hv|\ \textrm{d}x\le C\Vert f\Vert _{C^1(\Omega )}\Vert \nabla u\Vert _{L^2(\Omega _2)}<\infty ,\\{} & {} \quad \int _{\Omega _1}|\lambda (\mu D_hv)^2|\ \textrm{d}x\le 2|\lambda |\Vert \nabla u\Vert _{L^2(\Omega _2)}^2<\infty , \end{aligned}$$

and

$$\begin{aligned}{} & {} \int _{\Omega _1}|D_hg_{\eta ,u}\mu ^2 D_hv|\ \textrm{d}x\le C\Big (\int _{\Omega _1}\int _{\mathbb {R}^N{\setminus } \Omega _2}|(1-\eta (y))u(y)|[D_hj](x-y)|\ \textrm{d}y\ \textrm{d}x\Big )^{1/2}\\{} & {} \quad \Vert \nabla u\Vert _{L^2(\Omega _2)}<\infty \end{aligned}$$

due to assumptions on the differentiability of j. Moreover, with a similar calculation as in the proof of Theorem 5.1 we have

$$\begin{aligned} b_j(v,\phi )&=b_j(\mu D_hv,\mu D_hv)-\frac{1}{2}\int _{\mathbb {R}^N}\int _{\mathbb {R}^N} D_hv(x)D_hv(y)(\mu (x)-\mu (y))^2j(x-y)\ \textrm{d}x\textrm{d}y, \end{aligned}$$

where for some $\Omega _2\subset \subset \Omega _3\subset \subset \Omega _4\subset \subset \Omega$  with h small enough

$$\begin{aligned}&\int _{\mathbb {R}^N}\int _{\mathbb {R}^N} |D_hv(x)D_hv(y)(\mu (x)-\mu (y))^2 j(x-y)|\ \textrm{d}x\textrm{d}y\\&\quad \le C\int _{\Omega _3} \int _{\Omega _3}|D_h(\eta u)(x)D_h(\eta u)(y)||x-y|^2j(x-y)\ \textrm{d}x\textrm{d}y\\&\quad \le C\int _{\Omega _3} |D_h(\eta u)(x)|^2\int _{\Omega _3} |x-y|^2j(x-y)\ \textrm{d}y\textrm{d}x\\&\quad \le C\Vert \nabla u\Vert _{L^2(\Omega _4)}\int _{\mathbb {R}^N}\min \{1,|z|^2\}j(z)\ \textrm{d}z<\infty . \end{aligned}$$

Combining this with (5.20) we find

$$\begin{aligned} b_j(\mu D_hv,\mu D_hv)\le C~\text {for all}~ h>0~\text { small enough.} \end{aligned}$$

Since also $\mu D_hv\in {\mathscr {D}}^j(\Omega _2)$ for all $h>0$ small enough (see Lemma 2.3) and since $D^j(\Omega _2)$ is a Hilbert space, we conclude that $\mu \partial _ev\in {\mathscr {D}}^j(\Omega _2)$ with

$$\begin{aligned} b_j(\mu \partial _ev)\le C \end{aligned}$$

for $h\rightarrow 0$. This finishes the proof. $\square$

Corollary 5.4

Let $f\in C^{2m}(\overline{\Omega })$, $\lambda \in \mathbb {R}$, and let $u\in {\mathscr {V}}^j_{loc}(\Omega )\cap L^{\infty }(\mathbb {R}^N)$ satisfy in weak sense $I_ju=\lambda u+f$ in $\Omega$. Then $u\in H^{m}(\Omega ')$ for any $\Omega '\subset \subset \Omega$ and there is $C=C(N, j,\Omega ,\Omega ',m)>0$ such that

$$\begin{aligned} \Vert u\Vert _{H^m(\Omega ')}\le C\Big (\Vert f\Vert _{C^{2m}(\Omega )}^2+\Vert u\Vert ^2_{L^2(\Omega ')}+\Vert u\Vert ^2_{L^\infty (\mathbb {R}^N{\setminus } \Omega ')} \Big )^{\frac{1}{2}}. \end{aligned}$$

(5.21)

In particular, if $m=\infty$, then $u\in C^{\infty }(\Omega )$.

Proof

By Corollary 5.3 the claim holds for $m=1$ in particular with $u|_{\Omega '}\in D^j(\Omega ')$ for all $\Omega '\subset \subset \Omega$. Assume next, the claim holds for $m-1$ with $m\in \mathbb {N}$, $m\ge 2$ in the following way: We have $u\in H^{m-1}(\Omega ')$ and $\partial ^{\beta }u|_{\Omega '}\in D^j(\Omega ')$ for any $\Omega '\subset \subset \Omega$ and $\beta \in \mathbb {N}_0^N$ with $|\beta |\le m-1$, and there is $C=C(N, j,\Omega ,\Omega ',m)>0$ such that

$$\begin{aligned} \Vert u\Vert _{H^{m-1}(\Omega ')}\le C\Big (\Vert f\Vert _{C^{2m-2}(\Omega )}^2+\Vert u\Vert _{L^2(\Omega ')}+\Vert u\Vert ^2_{L^\infty (\mathbb {R}^N{\setminus } \Omega ')} \Big )^{\frac{1}{2}}. \end{aligned}$$

(5.22)

Fix $\Omega '\subset \subset \Omega$ and let $\Omega _i\subset \subset \Omega$, $i=1,\ldots ,7$ and $\eta \in C^{\infty }_c(\Omega _7)$ as in the proof of Corollary 5.3. Put $v=\partial ^{\beta }(\eta u)$ for some $\beta \in \mathbb {N}_0^N$, $|\beta |=m-1$. Then $Iv=\partial ^{\beta }f+\lambda v+\partial ^{\beta }g_{\eta ,u}$ in $\Omega _5$ by Lemma 2.8 and direct computation using the assumptions on J. From here, proceeding as in the proof of Corollary 5.3 by applying Theorem 5.1 the claim follows. $\square$

Proof of Theorem 1.1

By assumption, it follows from [20] that ${\mathscr {D}}^j(\Omega )$ is compactly embedded into $L^2(\Omega )$. This gives the existence of the sequence of eigenfunctions and corresponding eigenvalues. The fact that the first eigenfunction can be chosen to be positive follows from the fact that $b_j(|u|)\le b_j(u)$, Proposition 1.2 and Proposition 1.5 (see also [19]). Now statement (1) follows from Theorem 4.3 (with $h=f=0$) and statement (2) follows directly from Corollary 5.4. $\square$

Proof of Theorem 1.6

The first part follows from the Poincaré inequality, i.e. under the assumptions it holds $\Lambda _1(\Omega )>0$, and Theorem 4.3 with $h=0=\lambda$. The last assertion follows from Corollary 5.4. $\square$

Appendix A: An inequality

The following is a variant of [9, Lemma 10] (see also [20, Lemma 5.1]).

Lemma A.1

Let $q\in L^1(\mathbb {R}^N)$ be a nonnegative even function with $q=0$ on $\mathbb {R}^N{\setminus } B_r(0)$ for some $r>0$. Let $\Omega \subset \mathbb {R}^N$ open and $x_0\in \Omega$ such that $B_{2r}(x_0)\subset \Omega$. Then for all measurable functions $u:\Omega \rightarrow \mathbb {R}$ we have

$$\begin{aligned} b_{q*q,B_r(x_0)}(u)\le 4\Vert q\Vert _{L^1(\mathbb {R}^N)} b_{q,\Omega }(u). \end{aligned}$$

Proof

In the following, we identify u with its trivial extension $\tilde{u}:\mathbb {R}^N\rightarrow \mathbb {R}$, $\tilde{u}(x)=u(x)$ for $x\in \Omega$ and $\tilde{u}(x)=0$ otherwise. Denote $g(x,y)=(u(x)-u(y))^2$ for $x,y\in \mathbb {R}^N$. Note that we have

$$\begin{aligned} 0\le g(x,y)=g(y,x)\le 2g(x,z)+2g(y,z)~\text { for all}~ x,y,z\in \mathbb {R}^N. \end{aligned}$$

By Fubini’s theorem we have

$$\begin{aligned}&\int _{B_r(x_0)}\int _{B_r(x_0)}g(x,y) (q*q)(x-y)\ \textrm{d}x\textrm{d}y=\\&\quad \int _{B_r(x_0)}\int _{B_r(x_0)} \int _{\mathbb {R}^N} g(x,y) q(x-z)q(y-z)\ \textrm{d}z\textrm{d}x\textrm{d}y\\&\quad \le 2\int _{B_r(x_0)}\int _{B_r(x_0)} \int _{\mathbb {R}^N} [g(x,z)+g(y,z)] q(x-z)q(y-z)\ \textrm{d}z\textrm{d}x\textrm{d}y\\&\quad \le 4\int _{B_r(x_0)}\int _{\mathbb {R}^N}g(x,z)q(x-z) \\&\quad \int _{\mathbb {R}^{N}}q(y-z)\ \textrm{d}y\textrm{d}z\textrm{d}x= 4\Vert q\Vert _{L^1(\mathbb {R}^N)}\int _{B_r(x_0)}\int _{\mathbb {R}^N}g(x,z)q(x-z)\ \textrm{d}z\textrm{d}x. \end{aligned}$$

Note that since $q=0$ on $\mathbb {R}^N{\setminus } B_r(0)$, q is even, and $B_r(x)\subset B_{2r}(x_0)\subset \Omega$ for any $x\in B_r(x_0)$, we have

$$\begin{aligned} \int _{B_r(x_0)}\int _{\mathbb {R}^N}g(x,z)q(x-z)\ \textrm{d}z\textrm{d}x&=\int _{B_r(x_0)}\int _{B_r(x)}(u(x)-u(z))^2q(x-z)\ \textrm{d}z\textrm{d}x\le 2b_{q,\Omega }(u). \end{aligned}$$

$\square$