Nonlocal operators of small order

In this work we study nonlocal operators and corresponding spaces of order strictly below one and investigate interior regularity properties 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, in particular, we prove that if the right-hand is of class $C^{\infty}$ and the kernel satisfies similar regularity properties away from its singularity, then any weak solution is of class $C^{\infty}$.


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 −∆ in a bounded domain Ω in R N , it is well known that there exists a sequence of functions u n ∈ H 1 0 (Ω), n ∈ N and corresponding values λ n > 0 such that −∆u n = λ n u n in Ω and u n = 0 in ∂ Ω.
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 Ω. In the model case of a nonlocal problem, one usually studies the fractional Laplacian (−∆) s with s ∈ (0, 1). This operator can be defined via its Fourier symbol | · | 2s and it can be shown that for ϕ ∈ C ∞ c (R N ) we have u(x) − u(y) |x − y| N+2s dy, x ∈ R N 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 ∈ H s 0 (Ω), n ∈ N and corresponding values λ s,n > 0 such that (−∆) s u n = λ s,n u n in Ω and u n = 0 in R N \ Ω.
Here, the space H s 0 (Ω) is given by the closure of C ∞ c (Ω) -understood as functions on R N -with respect to the norm u → u 2 L 2 (Ω) + E s (u, u) 1 2 , where for u, v ∈ C ∞ c (R N ) we set N+2s dxdy.
With similar methods as in the classical case, it follows that u n is smooth in the interior of Ω.
In the following we investigate the above discussion to the case where the kernel function z → c N,s |z| −N−2s is replaced by a measurable function j : R N → [0, ∞] such that for some σ ∈ (0, 2] we have j(z) = j(−z) for all z ∈ R N and R N min{1, |z| σ } j(z) dz < ∞. (1.1) With σ = 2, the above yields that j dz 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 σ < 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 σ > 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 ϕ(−∆) for a certain class of functions ϕ 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 σ ∈ (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 Ω ⊂ R N open, u, v ∈ C 0,1 c (Ω) understood as functions defined on R N , and consider the bilinear form (1.4) To investigate the eigenvalue problem for I j , we further need the space Note that clearly D j (R N ) = D j (R N ) and both D j (Ω) and D j (Ω) are Hilbert spaces with scalar products ·, · D j (Ω) := ·, · L 2 (Ω) + b j,Ω (·, ·) and ·, · D j (Ω) := ·, · L 2 (Ω) + b j (·, ·) resp. Our first result concerns the eigenfunctions for the operator I j .
Theorem 1.1. Let (1.1) hold with σ = 2 and assume j satisfies additionally (1.5) Let Ω ⊂ R N be a bounded open set. Then there exists a sequence u n ∈ D j (Ω), n ∈ N and values λ n such that Here, we have and u 1 is unique up to a multiplicative constant -that is, λ 1 is simple. Moreover, u 1 can be chosen to be positive in Ω. Furthermore, the following statements hold.
(1) If in addition j satisfies then u n ∈ L ∞ (Ω) for every n ∈ N and there is C = C(Ω, j, n) > 0 such that (2) Assume (1.1) holds with σ < 1 2 , (1.7) holds, and there is m ∈ N ∪ {∞} such that the following holds: For the definition of the Sobolev spaces W l,1 and H m we refer to Section 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 δ -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 j u = 0 in Ω, 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 δ -decomposition, let us first note that the first equality in (1.4) can be extended, see Section 2. For this, let V j (Ω) denote the space of those functions u : R N → R such that u| Ω ∈ D j (Ω) and In this situation, we also say that u satisfies in weak sense I j u ≥ f in Ω. 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 σ < 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 σ < 1 and assume that j does not vanish identically on B r (0) for any r > 0. (1.10) Let Ω ⊂ R N open and with Lipschitz boundary, c ∈ L ∞ loc (Ω), and assume either (1) If u ∈ V j (Ω) satisfies in weak sense (1) The Lipschitz boundary assumption on Ω is a technical assumption for the approximation argument.
(2) Note that Assumption (1.10) readily implies the positivity Λ 1 (Ω) defined in (1.6), whenever Ω is an open set in R N which is bounded in one direction, that is Ω is contained (after a rotation) in a strip (−a, a) × 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 = | · | −N−2s , that is, the case of the fractional Laplacian (up to a multiplicative constant). In this situation, it holds V j (Ω) = H s (Ω) ∩ L 1 s , where |u(x)| 1 + |x| N+2s dy < ∞ , and the proposition can be reformulated as follows.
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 σ < 1. Proposition 1.5 (Strong maximum principle). Assume (1.1) is satisfied with σ < 1 and assume that j satisfies additionally (1.5). Let Ω ⊂ R N open and c ∈ L ∞ loc (Ω) with c + L ∞ (Ω) < ∞. Moreover, let u ∈ V j (Ω), u ≥ 0 satisfy in weak sense I j u ≥ c(x)u in Ω. Then the following holds.
Our last results concern the Poisson problem associated to the operator I j .
Our approach to prove Theorem 1.1 and Theorem 1.6 is uniformly by considering an equation of the form , and λ ∈ R. Moreover, several of our results need I j not to be translation invariant and this setup is discussed in Section 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 (1.11) appears as the operator with Fourier-symbol −2 ln(| · |) and can be seen as the formal derivative in s of (−∆) s at s = 0. Here where ψ := Γ ′ Γ denotes the digamma function and γ : the operator L ∆ can be seen as a bounded perturbation of the operator class discussed in the introduction. The following sections cover in particular this operator.
The paper is organized as follows. In Section 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 Section 4 we present a general approach to show boundedness of solutions and in Section 5 we give the proof of an interior H 1regularity estimate for solutions from which we then deduce the interior regularity statement as claimed in Theorem 1.1(2) and Theorem 1.6.
Notation. In the remainder of the paper, we use the following notation. Let U,V ⊂ R N be nonempty measurable sets, x ∈ R N and r > 0. We denote by 1 U : R N → R the characteristic function, |U | the Lebesgue measure, and diam(U ) the diameter of U . The notation V ⊂⊂ U means that V is compact and contained in the interior of U . The distance between V and U is given by dist(V,U ) := inf{|x − y| : x ∈ V, y ∈ U }. Note that this notation does not stand for the usual Hausdorff distance. If V = {x} we simply write dist(x,U ). Acknowledgements. This work is supported by DAAD and BMBF (Germany) within the project 57385104. The authors thank Mouhamed Moustapha Fall and Tobias Weth for helpful discussions. We also thank the anonymous referee for valuable suggestions.

PRELIMINARIES
In the following, we generalize the translation invariant setting of the introduction. For this and from now on, let k : be the symmetric lower bound of k given by and we drop the index Ω, if Ω = R N . Note that we have for any fixed x ∈ Ω that κ k,Ω (x) < ∞ by (2.1). We consider the function spaces and, for all r > 0, sup Then the following hold: Proof. This follows immediately from the definitions (see also [19,Section 3]).
and let

Lemma 2.3. Let Ω ⊂ R N open and let X be any of the above function spaces. Then the following hold:
(1) b k,Ω is a bilinear form and in particular we (2) If u ∈ X , then u ± , |u| ∈ X and we have b k, Proof. Theses statements follow directly from the definition (c.f. [19,Section 3]). To be precise in the last part, let ϕ ∈ C 0,1 c (Ω) and fix L : Then using the inequality for we find by the assumptions (2.1) (2) It follows in particular that there is a nonnegative self-adjoint operator I k associated to b k,R N = b k as mentioned in the introduction.
|u(y)|k(x, y) dy We also say that u satisfies I k u ≥ f weakly in Ω.
Similarly, we define weak subsolutions and solutions.
(1) We note that by Assumption 2.1, it follows that for any function This follows similarly to the proof of the statements in Lemma 2.3.
In some of our results the statements need a Lipschitzboundary of Ω, which comes into play due to approximation with C ∞ c (Ω)-functions (see Section 3 below). However, this can be weakened, if u ∈ V k (Ω) and the space of test-functions is adjusted.
Thus the claim follows.
Remark 2.9. The same result as in Lemma 2.8 also holds if "≥" in the solution type is replaced by "≤" or "=".
In the following, it is useful to understand functions u ∈ D k (Ω) satisfying b k,Ω (u) = 0. Proof. Let x 0 ∈ Ω and fix r > 0 such that B 2r (x 0 ) ⊂ Ω. 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 by the assumption on j. Hence there is R > 0 with q * q ≥ ε for some ε > 0 and thus we have for any m ∈ 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 Ω. Denote by W the set of points x ∈ Ω 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.
⊂ Ω for n ≥ n 0 . Repeating the above argument, it follows that u must be zero in B r x (x n ) and thus x ∈ W . Hence, W is relatively open and closed in Ω and since W is nonempty, we have W = Ω. That is u = 0 in Ω.

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 ∈ [1, ∞) and Ω ⊂ R N open.
|α| ≤ k and belongs to L p (Ω) for the Banach space of k-times (weakly) differentialable functions in L p (Ω). Moreover, as usual, for It follows that N s,p (Ω) is a Banach space with norm u N s,p (Ω) := u W k,p (Ω) + ∑ |α|=k [∂ α u] N σ,p (Ω) . It can be shown that this norm is equivalent to for any fixed m, l ∈ N 0 with m < σ and l > σ − m (see [

DENSITY RESULTS AND MAXIMUM PRINCIPLES
The main goal of this section is to show the following.
is associated to the function space of the localized logarithmic Laplacian (see [5]).
The proof is split into several smaller steps. Recall that D k (R N ) = D k (R N ) by definition.
fying that for every n ∈ N there is Ω n ⊂⊂ R N with u n = 0 on R N \ Ω n . Moreover, if u ≥ 0, then (u n ) n can be chosen to satisfy in addition 0 ≤ u n ≤ u n+1 ≤ u.
Thus lim n→∞ b k,R N (u − u n ) = 0 by the dominated convergence Theorem.
Proof. Let u ∈ D k (R N ). Moreover, let ϕ n ∈ C 0,1 c (R N ) for n ∈ N be given by Lemma 3.3 such that u − ϕ n u s,p < 1 n . Then v n := ϕ n u ∈ D k (R N ) and there is R n > 0 with v n ≡ 0 on R N \ B R n (0). Next, let (ρ ε ) ε∈(0,1] by a Dirac sequence and denote v n,ε := ρ ε * v n .
It is hence enough to show that v n,ε → v n in D k (R N ) for ε → 0. In the following, we write v in place of v n and v ε = ρ ε * v in place of v n,ε for ε ∈ (0, 1]. Moreover, let for ε → 0 and this convergence is also pointwise almost everywhere. Hence it is enough to analyze the convergence of b k,R N (v − v ε ) as ε → 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.
The remainder of the proof is to show (3.1). For this, let C = C(N, Ω, k) > 0 be a constant which may vary from line to line. Let A t := {x ∈ Ω : δ (x) ≤ t}. Note that uϕ r vanishes on R N \ A 2r , we have 0 ≤ ϕ r ≤ 1 and, moreover, Then proceeding similarly to the proof of Lemma 2.3.(5) we find for r small enough Note here, since u ∈ D k (R N ), we have A 4r u(x) 2 dx + b k,A 4r (u) → 0 for r → 0. Moreover, we have by Lebesgue's differentiation theorem and, by Lemma 3.5, we have k(x, y) dy ≤ Cκ k,Ω (x) +Cr −ε for x ∈ A 2r , so that also A 2r u 2 (x)κ k,A 4r (x) dx → 0 for r → 0 with a similar argument.  Proof. Consider the Lipschitz map Then v n := g n (u) ∈ D k (Ω) ∩ L ∞ (Ω) and we have with ϕ r as in the proof of Proposition 3.6 Clearly, b k,Ω (u − v n ) → 0 for n → ∞ by dominated convergence and b k,Ω (ϕ r v n ) → 0 for r → 0 analogously to the proof of Proposition 3.6, noting that the term in (3.2) reads in this case In particular, statement (1) follows. Now statement (2) and the density statement follow analogously, again, to the proof of Proposition 3.6.
Remark 3.8. It is tempting to conjecture the following type of Hardy inequality: There is C > 0 such that if Ω is a bounded Lipschitz set and k is such that its symmetric lower bound j is not in L 1 (R N ). Let us mention that for k(x, y) = |x − y| −2s−N this holds for s ∈ (0, 1), s = 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 ∈ V k loc (Ω) satisfy weakly I k u ≥ f in Ω for some f ∈ L 1 loc (Ω) and Ω ⊂ R N open and bounded with Lipschitz boundary.
(1) If f ∈ L 2 loc (Ω), then by density it also holds Proof. Note that also u − ∈ V k (Ω) and in particular u − ∈ D k (Ω). Hence, we can find On the other hand, since u + v n = 0 for all n ∈ N and u ≥ 0 almost everywhere in R N \ Ω, we find Since v n → u − in D k (Ω), it follows that b k,Ω (u − , u − ) = 0, but then u − is constant by Proposition 2.10 in Ω. Assume by contradiction that u − = m > 0. Then the above calculation gives (3.6) which is in both cases a contradiction: If in case 1. c ≤ 0, then since κ k,Ω (x) ≡ 0 and since v n → m in D k (Ω) the right-hand side of (3.6) is negative. In case 2. this contradiction is immediate in a similar way.
Proof of Proposition 1.2. The statement follows immediately from Proposition 3.10.
Remark 3.11. Usually, the weak maximum principle is stated with an assumption on the first eigenvalue Λ 1 (Ω) in place of inf x∈Ω κ k,Ω (x). This can be done once the Hardy inequality in Remark 3.8 is shown. Indeed, following the proof of Proposition 3.10 gives With n → ∞ and using that by the Hardy inequality it holds D k (Ω) = D k (Ω), it follows that and the conclusion follows similarly.
Proof of Corollary 1.4. This statement now follows from Remark 3.11 using Remark 3.8. (2) If j given in (2.2) satisfies essinf B r (0) j > 0 for any r > 0, then either u ≡ 0 in R N or essinf K u > 0 for any K ⊂⊂ Ω.
Proof. This statement follows by approximation from [19, Theorem 2.5 and 2.6]. Here, the statement j / ∈ L 1 (R N ) comes into play since we need to conclude the statement for arbitrary c as stated.
Proof of Proposition 1.5. The statement follows immediately from Proposition 3.12.
Fix t > 0 such that That is, we fix Note here, that for x ∈ R N we have by the integrability assumptions on k δ and k On the other hand, Combining (4.7) and (4.4) we have Hence v + t = 0 in Ω ′ and thus u = v ≤ t = A ·C 8 in Ω ′ as claimed. Corollary 4.2. If in the situation of Theorem 4.1 we have in weak sense I k u = λ u + h * u + f in Ω, then we have u ∈ L ∞ (Ω ′ ) and there is C = C(Ω, Ω ′ , k, λ , h) > 0 such that Proof. This follows by replacing u with −u (and f with − f ) in the statement of Theorem 4.1.
Proof. Using in the proof of Theorem 4.1 the test-function u + t instead of ϕ t (and similarly for Corollary 4.2), we find as claimed.

ON DIFFERENTIABILITY OF SOLUTIONS
In the following, Ω ⊂ R N is an open bounded set and k satisfies through out the assumptions (2.1) with some σ < 1 2 , (4.1), and (4.2). Moreover, we assume that there is j : R N → [0, ∞] such that k(x, y) = j(x − y) for x, y ∈ R N and that for some m ∈ N ∪ {∞} the following holds: We have j ∈ W l,1 (R N \ B ε (0) for every l ∈ N with l ≤ 2m, and there is some constant C j > 0 such that |∇ j(z)| ≤ C j |z| −1−σ −N for all 0 < |z| ≤ 3.
Next, let µ ∈ C ∞ c (Ω 1 ) with 0 ≤ µ ≤ 1 and µ ≡ 1 on Ω ′ . Then with ϕ = D −h [µ 2 D h v] ∈ D j (Ω 1 ) for h small enough we have for some C > 0 (which may change from line to line independently of h) since and Ω 1 |D h g η,u µ 2 D h v| dx ≤ C where for some Ω 2 ⊂⊂ Ω 3 ⊂⊂ Ω 4 ⊂⊂ Ω with h small enough Combining this with (5.20) we find b j (µD h v, µD h v) ≤ C for all h > 0 small enough.
Since also µD h v ∈ D j (Ω 2 ) for all h > 0 small enough (see Lemma 2.3) and since D j (Ω 2 ) is a Hilbert space, we conclude that µ∂ e v ∈ D j (Ω 2 ) with b j (µ∂ e v) ≤ C for h → 0. This finishes the proof. In particular, if m = ∞, then u ∈ C ∞ (Ω).
Proof of Theorem 1.1. By assumption, it follows from [20] that D j (Ω) is compactly embedded into L 2 (Ω). 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|) ≤ b j (u), Proposition 1.2 and Proposition 1.5 (see also [19]). Now statement (1)  Proof. In the following, we identify u with its trivial extensionũ : R N → R,ũ(x) = u(x) for x ∈ Ω and u(x) = 0 otherwise. Denote g(x, y) = (u(x) − u(y)) 2 for x, y ∈ R N . Note that we have 0 ≤ g(x, y) = g(y, x) ≤ 2g(x, z) + 2g(y, z) for all x, y, z ∈ R N .
By Fubini's theorem we have Note that since q = 0 on R N \ B r (0), q is even, and B r (x) ⊂ B 2r (x 0 ) ⊂ Ω for any x ∈ B r (x 0 ), we have