The Malgrange-Ehrenpreis theorem for nonlocal Schr\"odinger operators with certain potentials

In this paper, we prove the Malgrange-Ehrenpreis theorem for nonlocal Schr\"odinger operators $L_K+V$ with nonnegative potentials $V\in L^q_{\loc}(\BR^n)$ for $q>\f{n}{2s}$ with $0<s<1$ and $n\ge 2$; that is to say, we obtain the existence of a fundamental solution $\fe_V$ for $L_K+V$ satisfying \begin{equation*}\bigl(L_K+V\bigr)\fe_V=\dt_0\,\,\text{ in $\BR^n$ }\end{equation*} in the distribution sense, where $\dt_0$ denotes the Dirac delta mass at the origin. In addition, we obtain a decay of the fundamental solution $\fe_V$.

Set L = {L K : K ∈ K} where K denotes the family of all kernels K satisfying (1.2). In particular, if K(y) = c n,s |y| −n−2s , then L K = (−∆) s is the fractional Laplacian and it is well-known that lim s→1 − (−∆) s = −∆u for any function u in the Schwartz space S (R n ).
In what follows, we consider the nonlocal Schrödinger operators given by where K ∈ K and V ∈ L q loc (R n ), q > n 2s , s ∈ (0, 1), n ≥ 2, is a nonnegative potential. We are interested in the existence of a fundamental solution for the operator L V . Let D ′ (R n ) be the space of all distributions on R n . Given f ∈ D ′ (R n ), we say that a real-valued Lebesgue measurable function u on R n satisfies the equation L V u = f in the sense of D ′ (R n ), if the linear map L V u : C ∞ c (R n ) → R given by is well-defined and a distribution, and also L V u, ϕ = f, ϕ for all ϕ ∈ C ∞ c (R n ). Theorem 1.1. There exists a fundamental solution e V ∈ D ′ (R n ) for the nonlocal Schrödinger operator L V , i.e. it satisfies that where δ 0 is the Dirac delta mass at the origin. Moreover, there exists a universal constant C > 0 depending on n, s, λ and Λ such that (1.5) 0 ≤ e V (x) ≤ C |x| n−2s for any x ∈ R n \ {0}.
Remark. In case that L K = (−∆) s and V = 0, its fundamental solution was obtained in [CS].
The paper is organized as follows. In Section 2, we define several function spaces and give the fractional Sobolev embedding theorem which was proved in [SV, DPV]. In Section 3, we define weak solutions of the nonlocal equation L V = f in a bounded domain Ω ⊂ R n with Lipschitz boundary and obtain a relation between weak solutions (weak subsolutions, weak supersolutions) and minimizers (subminimizers, superminimizers) of the energy functional for the operator L V , respectively. In Section 4, we obtain a Rellich-Kondrachov compactness theorem, a weak maximum principle and a comparison principle for L V . In Section 5, we obtain an extension of the Malgrange-Ehrenpreis theorem to nonlocal Schrödinger operators L V , that is, we furnish the proof of the existence of a fundamental solution e V for L V by using functional analysis stuffs. Moreover we obtain a decay of the fundamental solution e V .

Preliminaries
Let F n be the family of all real-valued Lebesgue measurable functions on R n . Let Ω be a bounded open domain in R n with Lipschitz boundary and let K ∈ K s . Let X s (Ω) be the function space of all u ∈ F n on R n such that u| Ω ∈ L 2 (Ω) and We also denote by (2.1) X s 0 (Ω) = {v ∈ X s (Ω) : u = 0 a.e. in R n \ Ω } Note that X s (Ω) and X s 0 (Ω) are not empty, because C 2 0 (Ω) ⊂ X s 0 (Ω). Then we see that (X s (Ω), · X s (Ω) ) is a normed space, where the norm · X s (Ω) is defined by For p ≥ 1 and s ∈ (0, 1), let W s,p (Ω) be the usual fractional Sobolev spaces with the norm In what follows, we write H s (Ω) = W s,2 (Ω). When Ω = R n in (2.3), we can similarly define the spaces W s,p (R n ) and H s (R n ) = W s,2 (R n ) for s ∈ (0, 1). By [SV], there exists a constant c > 1 depending only on n, s and Ω such that Thus · X s 0 (Ω) is a norm on X s 0 (Ω) which is equivalent to (2.2). Moreover it is known [SV] that (X s 0 (Ω), · X s 0 (Ω) ) is a Hilbert space with inner product Let X s 0 (Ω) * be the dual space of X s 0 (Ω); that is, the family of all bounded linear functionals on X s 0 (Ω). Then we know that (X s 0 (Ω) * , · X s 0 (Ω) * ) is a Hilbert space, where the norm · X s 0 (Ω) * is given by where the Fourier transform of u is defined by Then, by the Plancherel theorem, it is easy to check that H s (R n ) = H s (R n ) and they are norm-equivalent.
We also define the homogeneous fractional Sobolev spacesḢ s (R n ) by the closure of S(R n ) with respect to the norm From a direct calculation [DPV], it turns out that where c n,s is the universal constant given in (1.3).
(a) It easily follows from (2.7) and Lemma 5 in [SV]. (b) It is also very straightforward.
Next we state the fractional Sobolev embedding theorem, which was proved in [SV, DPV].
Proposition 2.2. If 0 ≤ s < n 2 , then the spaceḢ s (R n ) is continuously embedded in L 2n n−2s (R n ), i.e. there is a constant C > 0 depending only on n, s such that

Weak solutions and minimizers for L K + V
In this section, we define weak solutions, weak subsolutions and supersolutions of the nonlocal equation L K u + V u = 0 on Ω. To comprehend them well, we obtain a relation between weak solutions (weak subsolutions and weak supersolutions) and minimizers (subminimizers and superminimizers) of the energy functional for the nonlocal operator L K + V , respectively.
From now on, we always assume that V is a nonnegative potential in L q loc (R n ) for q > n 2s with s ∈ (0, 1) and n ≥ 2. We denote by L 2 V (Ω) the weighted L 2 class of all real-valued measurable functions g on R n satisfying We consider a bilinear form defined by From (b) of Corollary 5.2 below, we see in advance that for u, v ∈ X s 0 (Ω).
Definition 3.1. Let V ∈ L q loc (R n ) for q > n 2s with s ∈ (0, 1) and n ≥ 2. Then we say that a function u ∈ X s 0 (Ω) is a weak solution of the nonlocal equation for any ϕ ∈ X s 0 (Ω). Here · , · denotes the pair between X s 0 (Ω) * and X s 0 (Ω).
In fact, it turns out that the weak solution of the equation (3.2) is the minimizer of the energy functional We consider function spaces Y s 0 (Ω) + and Y s . Then we see that X s 0 (Ω) = Y s 0 (Ω) + ∩ Y s 0 (Ω) − . We now define weak subsolutions and supersolutions of the nonlocal equation (3.2) as follows.
Definition 3.2. We say a function u ∈ Y s 0 (Ω) − ( Y s 0 (Ω) + ) is a weak subsolution (weak supersolution) of the nonlocal equation (3.2), if it satisfies that for every nonnegative ϕ ∈ X s 0 (Ω). Also we say that a function u is a weak solution of the nonlocal equation (3.2), if it is both a weak subsolution and a weak supersolution. So any weak solution u of the equation (3.2) must be in X s 0 (Ω) and satisfies (3.3). In the next, we furnish the definition of subminimizer and superminimizer of the functional (3.4) to get better understanding of weak subsolutions and supersolutions of the nonlocal equation (3.2).
for all nonpositive ϕ ∈ X s 0 (Ω). Also we say that a function u ∈ Y s 0 (Ω) + is a superminimizer of the functional (3.4) over Y s 0 (Ω) + , if it satisfies (3.6) for all nonnegative ϕ ∈ X s 0 (Ω). (b) We say that a function u is a minimizer of the functional (3.4) over X s 0 (Ω), if it is both a subminimizer and a superminimizer. So any minimizer u must be in X s 0 (Ω) and satisfies (3.6) for all ϕ ∈ X s 0 (Ω).
Lemma 3.4. If s ∈ (0, 1), then there is a unique minimizer of the functional (3.4). Proof. Using standard method of calculus of variation, we proceed with our proof. We now take any minimizing sequence {u k } ⊂ X s 0 (Ω). By applying Theorem 4.1 below, we can take a subsequence {u kj } ⊂ X s 0 (Ω) converging strongly to u ∈ L 2 (Ω). So there exists a subsequence {u ki } of {u kj } which converges a.e. in Ω to u ∈ X s 0 (Ω). Thus by applying Fatou's lemma we can show that the energy functional E V is weakly semicontinuous in X s 0 (Ω). This implies that u is a minimizer of (3.4). Its uniqueness also follows from the strict convexity of the functional (3.4).
Next, we show the equivalency only for the weak supersolution case, because the other case can be done in a similar way. First, if u ∈ Y s 0 (Ω) + , then we observe that for all nonnegative ϕ ∈ X s 0 ((Ω). This implies that a weak supersolution u ∈ Y s 0 (Ω) + of the equation (3.2) is a superminimizer of the functional (3.4) over Y s 0 (Ω) + . On the other hand, we suppose that u ∈ Y s 0 (Ω) + is a superminimizer of the functional (3.4). Then it follows from (3.7) that for all nonnegative ϕ ∈ X s 0 (Ω). Since εϕ ∈ X s 0 (Ω) and it is nonnegative for any ε > 0 and ϕ ∈ X s 0 (Ω), we obtain that for any ε > 0. Taking ε → ∞, we can conclude that for any nonnegative ϕ ∈ X s 0 (Ω). Hence u is a weak supersolution of the equation (3.2). Therefore we are done.

Rellich-Kondrachov Compactness Theorem, a Weak Maximum
Principle and a Comparison Principle for L K + V In this section, we obtain the Rellich-Kondrachov compactness theorem, a weak maximum principle and a comparison principle for L K + V which will play a crucial role in proving the existence of a fundamental solution for the nonlocal Schrödinger operators in the next section.
First we get a compactness result Y s 0 (Ω) ֒→ L 2 (Ω) by using the fact that X s 0 (Ω) and Y s 0 (Ω) are norm-equivalent and the precompactness of Y s 0 (Ω) in L 2 (Ω).
Theorem 4.1. Let n ≥ 1, s ∈ (0, 1) and 2s < n. If u ∈ Y s 0 (Ω), then there is a universal constant C > 0 depending on n, s and Ω such that We observe that Y s 0 (Ω) = X s 0 (Ω) and they are norm-equivalent, because it follows from Lemma 2.1 and the fractional Sobolev inequality [DPV] that with a universal constant C > 0 depending on n, s and Ω, and so for any u ∈ X s 0 (Ω). Applying Lemma 2.1 and the fractional Sobolev inequality again, we conclude that . For the proof of the second part, take any bounded sequence {u k } in Y s 0 (Ω). Then it is also a bounded sequence in X s 0 (Ω). Thus by Lemma 8 [SV] there is a subsequence {u kj } of the sequence and u ∈ L 2 (Ω) such that u kj → u in L 2 (Ω) as j → ∞. Hence we complete the proof.
Next, we give a weak maximum principle and a comparison principle for the nonlocal Schrödinger operators L V as follows.
Theorem 4.2. If u is a weak supersolution of the nonlocal equation L V u = 0 in Ω such that and u ≥ 0 in R n \ Ω, then u ≥ 0 in Ω.
Proof. From the assumption, we see that u − = 0 in R n \ Ω and u + ∈ X s 0 (Ω), and thus u − , u ∈ X s 0 (Ω). Since u + u − = 0 in R n and u + (x)u − (y) ≥ 0 for all x, y ∈ R n , we have that This implies that u − = 0 in R n , and hence u ≥ 0 in Ω.
Proof. It immediately follows from Theorem 4.2.

Proof of the Malgrange-Ehrenpreis theorem for L K + V
In this section, we study the existence of a fundamental solution e V for the nonlocal Schrödinger operators L V , where V is a nonnegative potential with V ∈ L q loc (R n ) for q > n 2s and s ∈ (0, 1) and n ≥ 2. Let T be an unbounded densely defined linear operator with domain D(T ) in a Hilbert space H with the inner product ·, · H . We denote by D(T * ) the class of η ∈ H for which there exists a ν ∈ H such that For each η ∈ D(T * ), we define T * (η) = ν. Then we call T * the adjoint of T .
Let Γ(T ) be the graph of such an operator T ; that is, it is the linear subspace The operator T is said to be closed, if Γ(T ) is closed in H × H. Also the operator T is said to be closable, if there is a closed extension T 0 of T ; that is, there is a closed operator T 0 with T ⊂ T 0 ( i.e. Γ(T ) ⊂ Γ(T 0 ) ). We call T the closure of T , i.e. the smallest closed extension of T . Then the following two facts are well-known [RS]: (a) If T is closable, then Γ(T ) = Γ(T ) and T = T * * .
(b) If T 1 , T 2 are densely defined operators with T 1 ⊂ T 2 , then T * 2 ⊂ T * 1 . (5.1) For s ∈ (0, 1), we denote by X s c := X s c (R n ) the class of all u ∈ F n such that u ∈ X s 0 (Ω) for some bounded domain Ω in R n . Similarly, we can define Y s c := Y s c (R n ). Lemma 5.1. If K ∈ K s for s ∈ (0, 1), then the operator L K : L 2 (R n ) → L 2 (R n ) is a densely defined operator with domain D(L K ) = C ∞ c (R n ) and it is positive and symmetric. Moreover, there exists a unique closure L K of L K which is self-adjoint and D(L K ) = H 2s (R n ).
Proof. Note that C ∞ c (R n ) is dense in L 2 (R n ) and C ∞ c (R n ) ⊂ X s c (R n ). By Theorem 4.1, it is easy to check that L K : L 2 (R n ) → L 2 (R n ) is a densely defined operator with domain D(L K ) = C ∞ c (R n ). Also we see that L K is positive and symmetric, because To prove the existence of the closure L K which is self-adjoint, it suffices to show that [L K ± i](C ∞ c (R n )) is dense in L 2 (R n ) by verifying that its orthogonal complement is [L K ± i](C ∞ c (R n )) ⊥ = 0 (refer to p.257, [RS]). Indeed, let us take any ϕ ∈ L 2 (R n ) satisfying is the nonnegative function given by (1 − cos y, ξ )K(y) dy.
Next, we show the uniqueness of the closure L K which is self-adjoint. Indeed, if S is another self-adjoint closed extension of L K , then we see that L K ⊂ S. Conversely, it follows from (5.1) that [L K ] * * ⊂ S, and hence Finally, we consider the multiplication operator M 0 : where m is the function in (5.4). Then, by the Plancherel theorem, we see that L K is unitarily equivalent to M 0 . As in (5.3), we can also prove that there is a unique closure M 0 of M 0 which is self-adjoint and Thus we have that D(M 0 ) = H 2s (R n ), because we easily obtain that λ|ξ| 2s ≤ m(ξ) ≤ Λ|ξ| 2s by (1.2) and (1.3). Since the Fourier transform F is a unitary isomorphism from L 2 (R n ) to L 2 (R n ), the closure L K of L K is unitarily equivalent to M 0 . Therefore we conclude that D(L K ) = H 2s (R n ).
As by-products of Lemma 5.1, we get a very useful nonlocal version of integration by parts and the norm equivalence between the space X s 0 (Ω) and the usual fractional Sobolev space H s (R n ) on a dense subspace C ∞ c (Ω) of X s 0 (Ω).
Corollary 5.2. (a) If K ∈ K for s ∈ (0, 1), then there is a unique positive selfadjoint square root operator Q of L K , i.e. Q • Q = L K . Also, it satisfies that (5.5) L K u, v L 2 (R n ) = Qu, Qv L 2 (R n ) and Qu(ξ) = m(ξ) u(ξ) for any u, v ∈ C ∞ c (R n ), where m(ξ) is the multiplier of L K given in (5.4).
Remark. If K(y) = c n,s |y| −n−2s for s ∈ (0, 1), then we note that L K = (−∆) s and Q = (−∆) s/2 . Proof. (a) It can be obtained by Theorem 13.31 [R]. (b) It can be shown by simple calculation. (c) It easily follows from (5.5) and (b). Therefore we complete the proof. Proof. (a) For any fixed v ∈ L 2 (Ω), it is quite easy to check that the linear map By the Hahn-Banach theorem, we can extend T 0 to a continuous linear functional on L 2 (Ω), and so by Riesz representation theorem there is a unique M * V v ∈ L 2 (Ω) such that Thus M V is self-adjoint and it is also positive, because we have that The existence and uniqueness of a positive self-adjoint square root operator P of M V can be obtained by Theorem 13.31 [R]. Hence we are done.
In what follows, by Lemma 5.1 and Lemma 5.3, for simplicity we may write V 2 (Ω), which is positive and self-adjoint. Thus there is a positive self-adjoint square root operator L 1/2 V of L V .
Lemma 5.4. We have the estimate . Proof. It easily follows from Corollary 5.2, Lemma 5.3 and Theorem 4.1.
Next, we obtain the existence and uniqueness of weak solution of the nonlocal equation L V = h in Ω for the forcing term h ∈ L 2 (Ω) and moreover for h ∈ Y s 0 (Ω) * . Lemma 5.5. For each h ∈ Y s 0 (Ω) * , there is a unique weak solution u ∈ Y s 0 (Ω) of the nonlocal equation

By Corollary 5.2, it is easy to check that
for any u, φ ∈ Y s 0 (Ω). Thus the existence result can be obtained by the Lax-Milgram theorem.
Since u ∈ Y s 0 (Ω) is a weak solution of the equation If h ∈ L 2 (Ω), then it follows from the dual form of Theorem 4.1 that for an open subset D of R n , i.e. the class of all smooth functions ϕ with compact support in D which inherits a topology induced by the convergence ϕ k ⇁ ϕ (as k → ∞) for ϕ k , ϕ ∈ D(D) if and only if there is a compact set Q ⊂ D such that supp(ϕ k ) ⊂ Q for all k ∈ N and ∂ α ϕ k converges uniformly on Q to ∂ α ϕ for all multi-indices α = (α 1 , · · · , α n ), where ∂ α denotes Then an equivalent condition of distribution is known as follows (see [R]); T ∈ D ′ (D) if and only if for each compact set Q ⊂ D, there is an integer N = N (Q) > 0 and a constant C = C(Q) > 0 such that where ϕ N = max{sup D |∂ α ϕ| : |α| := n i=1 α i ≤ N } and D Q denotes the class of all smooth functions in D which is supported in Q. In what follows, we write T, ϕ = T (ϕ).
Lemma 5.6. If u ∈Ḣ s (R n ) for s ∈ (0, 1), then the linear map L V u : D (R n Proof. Since 0 < s < n 2 for n ≥ 2, it is easy to check that (5.9) L V ϕ ∈ L 2n n+2s (R n ) for any ϕ ∈ D(R n ). Thus, by Proposition 2.2 and Hölder's inequality, the linear functional L V u is well-defined on D(R n ).
Now it remains only to show that L V u satisfies the continuity property (5.7). Indeed, taking arbitrary compact set Q ⊂ R n and ϕ ∈ D Q , it follows from (2.7) and the mean value theorem that where r Q = sup y∈Q |y|, C 1 = C 1 (n, s, Q) > 0 is a constant and Therefore we complete the proof.
Lemma 5.7. Let V ∈ L q loc (R n ) be a nonnegative potential with q > n 2s , s ∈ (0, 1). If h ∈ L 2n n+2s (R n ), then there exists a function u ∈Ḣ s (R n ) such that Moreover, there is an increasing sequence {a k } k∈N with lim k→∞ a k = ∞ such that u a k ∈ Y s 0 (B a k ) satisfies the nonlocal equation L V u a k = h in B a k in the weak sense and lim k→∞ u a k = u a.e. in R n .
Proof. We observe that By Theorem 4.2 and Lemma 5.5, for each k ∈ N there exists a nonnegative function . Taking ϕ = u k in (5.11), it follows from Hölder's inequality, Lemma 2.1 and the fractional Sobolev embedding onḢ s (R n where C > 0 is a constant depending only on n, s, but not on k. This implies that . By weak compactness, there are a subsequence {u ki } i∈N and u ∈Ḣ s (R n ) such that (5.14) u ki ⇀ u in L 2n n−2s (R n ) and u ki ⇀ u inḢ s (R n ) Let us take any ϕ ∈ D(R n ). Then there is some m ∈ N such that ϕ ∈ D(B m ). We note that L V u k = h in the sense of D ′ (B m ) for any k ≥ m. Thus, by (5.14), Corollary 5.2 and Lemma 5.6, we conclude that Thus this implies that L V u = h in the sense of D ′ (R n ).
Finally, we see from (5.14) and Theorem 7.1 [DPV] that the sequence {u k } is precompact in L 2 (B) for any ball B in R n . This implies the required almost everywhere convergence.
Next, we shall prove our main theorem which is an extension of the Malgrange-Ehrenpreis theorem to nonlocal Schrödinger operators L V .
For l ∈ N, let f l (x) = l n f lx where f ∈ C c (B 1 ) is a nonnegative function with f L 1 (R n ) = 1. From Lemma 5.7, we see that there is a sequence {u l } l∈N ⊂Ḣ s (R n ) such that As a matter of fact, we see from a weak maximum principle (Theorem 4.2) and the proof of Lemma 5.7 that each u l is nonnegative and, for each l ∈ N, there are a sequence {a l k } k∈N with lim k→∞ a l k = ∞ and a sequence {u k l } k∈N of nonnegative functions u k l ∈ Y s 0 (B a l k ) such that u l = lim k→∞ u k l a.e. in R n , L V u k l = f l in B a l k in the weak sense. (5.17) Then we obtain uniform estimates for the sequence {u l } l∈N in the following lemma in order to prove the main theorem.
Proof. (a) Motivated by [KMS], we consider a nonnegative function ϕ : R n → R defined by ϕ = β 1−α − (β + u k l ) 1−α where β > 0 and α ∈ (1, 2) will be chosen later. Since each u k l is supported in B a l k and a(t) = (β + t) 1−α is Lipschitz continuous on (0, ∞), by Lemma 2.1 and the definition of Y s 0 (B a l k ) we see that ϕ ∈ Y s 0 (B a l k ) ⊂ H s (R n ). Thus we can use ϕ as a testing function. From the weak formulation of the nonlocal equation given in (5.17), we have that By the mean value theorem, we have that This gives that, for each l ∈ N, (5.28) u k l L p (Br(x0)) ≤ C r n p −(n−2s) for any k ∈ N.
(d) Take any x ∈ R n with |x| ≥ 3/l. Then we note that B 2r (x) ∩ B 1/l = φ where r = 1 4 |x|. By Lemma 4.5 [CK], we have the estimate with some universal constant C > 0. Using a standard argument in [HL], for any p ∈ [1, ∞) there is a universal constant C = C(n, s, p) > 0 such that Thus it follows from (5.28) and (5.34) with p = 1 that u k l L ∞ (Br(x)) ≤ C r n u k l L 1 (B2r(x)) ≤ C r n−2s . Therefore this and (5.17) imply (5.21). Hence we complete the proof.
From (5.17), Lemma 5.8 [(5.18), (5.19),(5.20)] and Theorem 7.1 [DPV], the standard diagonalization process yields that there exist a subsequence {u li } i∈N of {u l } l∈N and e V ∈ W γ,p loc (R n ) ⊂ D ′ (R n ) such that (5.37) for any (x 0 , r) ∈ R n × (0, ∞). We write {u l } l∈N instead of {u li } i∈N , for simplicity. First, we show that the linear map L V e V : D(R n ) → R defined by is a distribution, i.e. L V e V ∈ D ′ (R n ). Indeed, take any compact set Q ⊂ R n and ϕ ∈ D Q for this proof. If 1 < p < n n−s , then it follows from Hölder's inequality and Young's inequality that where p ′ is the dual exponent of p, because 1 < p < n n−s < 2 for any s ∈ (0, 1) and n ≥ 2, and n + 2s = n + pγ p + n + pγ p ′ + (2s − pγ). By simple calculation, we have that where r Q = sup y∈Q |y|. By the mean value theorem, we obtain that From (5.39), (5.40) and (5.41), we conclude that | L V e V , ϕ | ≤ C(n, s, p, γ, Q) e V W γ,p (B2r Q ) ϕ 1 .
Hence this implies that L V e V ∈ D ′ (R n ). Take any ϕ ∈ D(R n ). Then there exists an integrer m ∈ N such that ϕ ∈ D(B m ). Using Lemma 5.6, as in (5.15) we have that for each l ∈ N. Thus, for each l ∈ N, we see that f l (y)ϕ(y) dy. (5.42) Taking the limit l → ∞ on (5.42), by (5.38) we then claim that For our aim, we write (5.43) Here we denote by I l the last third integral and J l the last fourth integral in the right side of (5.43). From Lemma 5.8 and Fatou's lemma, we note that where C 1 , C 2 , C 3 > 0 are some constants depending only on p, λ, Λ, n, s, r as in (5.18), (5.19) and (5.20). From the weak convergence V u kj ⇀ V e V in L 1 (B m ) obtained in (5.37), we easily derive that lim l→∞ J l = 0. So it remains only to show that lim l→∞ I l = 0. To execute this, we split I l into where B m ⊂ B R for a sufficiently large R > 0 to be determined later.
We have only to show that I 1 l , I 2 l , I 3 l → 0 as l → ∞. We now prove that I 1 l → 0 as l → ∞. To do this, we shall apply Vitali convergence theorem. So we need only to show that the functions v l (x, y) given by v l (x, y) = (u l (x) − e V (x) − u l (y) + e V (y))(ϕ(x) − ϕ(y))K(x − y) are equibounded in L 1 (B 2 m ; dx dy) and equi-integrable in B 2 m . For this, it is enough to show that the sequence {v l } is equibounded in L 1+η (B 2 m ; dx dy) for some η > 0.