Fractional Sobolev Spaces with Kernel Function on Compact Riemannian Manifolds

In this paper, a new class of Sobolev spaces with kernel function satisfying a Lévy-integrability-type condition on compact Riemannian manifolds is presented. We establish the properties of separability, reflexivity, and completeness. An embedding result is also proved. As an application, we prove the existence of solutions for a nonlocal elliptic problem involving the fractional p(·,·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\cdot , \cdot )$$\end{document}-Laplacian operator. As one of the main tools, topological degree theory is applied.

For more results on the functional framework, we refer to Bahrouni and Rȃdulescu [9] who proved the solvability of the following problems Lw(y) + |w(y)| q(y)−2 w(y) = λ|w(y)| r(y)−2 w(y) in U, by using Ekeland's variational method, where U is an open bounded subset of R N , λ > 0, r(y) < p − = min (y,z)∈U ×U p(y, z), and Lw is the fractional p(y, •) Laplacian operator.
Bahrouni [8] continued to study the space W s,q(y,p(y,z) (U).More specially, he proved the strong comparison principle for (−∆) s p(y,.) and by using the sub-supersolution method, he showed the solvability of the following nonlocal equation where U is an open bounded domain, s ∈ (0, 1), p is a continuous function, and h satisfies the following growth |h(y, z)| ≤ A 1 |z| r(y)−1 + A 2 , for every (y, z) ∈ R N +1 , where r ∈ C(R N , R), 1 < r(y) < p ⋆ s (y), for every y ∈ R N .The generalized fractional Sobolev space was studied in [8,9,27] and further developed in [26].They proved a fundamental compact embedding for this space and investigated the multiplicity and boundedness of solutions to the following problem where f : R N × R → R is a Carathéodory function, and −∆ p(y) s is an operator defined by where The approaches for ensuring the existence of weak solutions for a class of nonlocal fractional problems with variable exponents were addressed in greater depth in [1,2,6,8,9,10,11,17,18,26,27,29,31,33] and the references therein.In the non-Euclidean case, classical Sobolev spaces on Riemannian manifolds have been investigated for more than seventy years [5,25,30].The theory of these spaces has been applied to isoperimetrical inequalities [25] and the Yamabe problem [35].In [22] the authors investigated the theory of generalized Sobolev spaces on compact Riemannian manifolds.Moreover, they proved the compact embeddability of these spaces into the Hölder space.They also studied a PDE problem involving p(•)-Laplacian operator.
In addition, the authors in [21] studied variable exponent function spaces on complete non-compact Riemannian manifolds.They used classical assumptions on the geometry to establish compact embeddings between Sobolev spaces and the Hölder function space.Finally, they also showed the existence of solutions to the p(•)-Laplacian problem.The authors in [24] introduced the fractional Sobolev spaces on Riemannian manifolds.As a consequence, they investigated fundamental properties, such as compact embeddings, completeness, density, separability, and reflexivity.They also investigated the existence of solutions to the following equation: where Aberqi, Benslimane, Ouaziz, and Repovš [2] introduced the space W s,p(y,z) (M) and proved some important properties of this space and studied the following problem Fractional Sobolev spaces and problems involving the p(•, •)-Laplacian operator have attracted significant attention in recent decades.This class of operators appears rather naturally in a variety of applications, including optimization and financial mathematics, we cite the well-known example by Carbotti, Dipierro, and Valdinoci [16] who obtained the following equation: where A := a∂ 2 − b(−∆) s with a, b 0 and r ∈ R. Here, S t is the price at time t and V the value of option.They are also useful in optimal control, engineering, quantum mechanics, obstacle problems, elasticity, image processing, minimal surfaces, stabilization of Lévy processes, game theory, population dynamics, fluid filtration, and stochastics, see for example [4,7,15,17,19,20,23,32,33] and the references therein.Our work's novelty is extending general Sobolev spaces to Sobolev spaces W q(y),p(y,z) K (U) with kernel function K on M. We shall prove important properties of this new class of spaces.In particular, we shall investigate the existence of solutions to problem (1.1) using the topological degree method.This work generalizes previous results [1,2,8,9,12,24,26,27].However, the main difficulty is presented by the fact that the p(•, •)-Laplacian operator has a more complicated nonlinearity than the p-Laplacian operator.For example, it is non-homogeneous.Other complications are due to the non-Euclidean framework of our problem.Also checking for example the density of the space C ∞ (M) in W q(y),p(y,z) K (U), because the notion the translation in Riemannian manifolds is not defined.To the best of our knowledge, there were no such results prior to this work.
Our first major result is the following theorem.Then the space W q(y),p(y,z) K (U) is continuously embeddable in L ℓ(y) (U) and there exists a positive constant C = C(N, s, p, q, U) such that |w| L l(y) (U ) ≤ ||w|| W q(y),p(y,z) K (U ) , for every w ∈ W q(y),p(y,z) K
Our second main result is related to the investigation of the following fractional p(y, •)-Laplacian problem with a general kernel K Using Berkovits' topological degree, we study the existence of solutions and prove the following theorem.
Theorem 1.2.Suppose that (M, g) is a compact N -dimensional Riemannian manifold, U is a smooth open subset of M, K : U × U → (0, +∞) is a symmetric function satisfying Lévy-integrability and coercivity conditions.Assume that assumption (B 1 ) holds.Then problem (1.1) has at least one weak solution w ∈ W q(y),p(y,z) K

(U).
The paper is organized as follows: In Section 2, we collect the main definitions and properties of generalized Lebesgue spaces and generalized Sobolev spaces on compact manifolds and provide crucial background on recent Berkovits degree theory.In Section 3, we establish completeness, separability, and reflexivity properties of our spaces (Lemmas 3.2, 3.3, and 3.5).In Section 4, we prove our first main result (Theorem 1.1).In Section 5 we prove our second main result (Theorem 1.2).Finally, in Section 6, we prove some lemmas needed for the proofs of our main results.

Preliminaries 2.1. Generalized Lebesgue Spaces on Compact Manifolds
Throughout this section, (M, g) will be a compact Riemannian manifold of dimension N .To start, we briefly review some fundamental Riemannian geometry concepts that will be needed.For more details see [5,24,25].
A local chart on M is a pair (U, ϕ), where U is an open subset of M and ϕ is a homeomorphism of U onto an open subset of R N .Furthermore, a collection (U i , ϕ i ) i∈J of local charts such that M = j∈J U j , is called an atlas of manifold M. For some atlas (U j , ϕ j ) j∈J of M, we say that a family (U j , ϕ j , β j ) j∈J is a partition of unity subordinate to the covering (U j , ϕ j ) j∈J if the following holds:

1)
j∈J β j = 1, 2) (U j , ϕ j ) j∈J is an atlas of M, 3) supp(β j ) ⊂ U j , for every j ∈ J. Definition 2.1.(see [25]) Suppose that w : M → R is a continuous function with compact support, (U j , ϕ j ) j∈J is an atlas of M, and (U j , ϕ j , β j ) j∈J is a partition of unity subordinate to (U j , ϕ j ) j∈J .We define the Riemannian measure of w in M as follows: where dy is the Riemannian volume element on (M, g), g ij are the components of the metric g in the local chart (U j , ϕ j ) j∈J , and dy is the Lebesgue volume of R N .Definition 2.2.(see [5] Then the length of γ is given by Definition 2.3.(see [5]) For any (y, z) ∈ M 2 , we define the distance d g (y, z) between y and z as follows Next, we recall basic definitions and preliminary facts on the generalized Lebegue spaces L q(x) (U) on compact manifolds, where U is an open subset of manifold M. For more background, we refer to [2,5,25].We need to recall the notion of the covariant derivative.Definition 2.5.(see [25]) Let ∇ be the Levi-Civita connection.For w ∈ C ∞ (M ), ∇ k w denotes the k-th covariant derivative of w.In local coordinates, the pointwise norm of ∇ k w is given by When k = 1, the components of ∇w in local coordinates are given by (∇w We consider the set: for every y ∈ M, q − = min y∈M q(y), q + = max y∈M q(y).Definition 2.6.(see [21]) Let q ∈ C + (M) and k ∈ N. We define the Sobolev space L q(y) k (M) as the completion of C q(y) k (M) with respect to the norm |w| L q(y) k (M), where where |∇ j w| is the k-the covariant derivative of w.

Topological Degree Theory
Let E be a real separable Banach space and E * its dual.Given a non-empty set U ⊂ E, denote by Ū and by ∂U its closure and boundary, respectively.
Definition 2.11.(see [13]) Let f : U ⊂ E → E * be an operator.1) We say that f is an (S + )-map if for {{z n } n∈N , z} ⊂ U, we have z n weakly ⇀ z weakly and lim sup We say that f is a quasi-monotone operator if for every {{z n } n∈N , z} ⊂ U, we have Definition 2.12.(Condition (S + ) B , see [28]) Suppose that 1) We say that f satisfies condition (S + ) B if for every {{z n } n∈N , z} ⊂ U, the following combined properties 2) (Property (QM ) B ).We say that f satisfies condition (QM ) B if for every {{z n } n∈N , z} ⊂ U, we have We consider the following sets , and is bounded}.
Let U ⊂ D f and B ∈ F ⋆ 0 (U ), where D f denotes the domain of f .We denote by N the collection of all bounded open sets in E. The following operators will be considered  Theorem 2.17.(see [11]) There exists a unique degree function satisfying the following properties: 2) If G : [0, 1] × B → F is a bounded admissible affine homotopy with a common continuous essential inner map and b :

Fractional Sobolev Spaces with a General Kernel on Compact Riemannian Manifolds
In this section, we shall introduce fractional Sobolev spaces with a general kernel and prove several qualitative lemmas.3).We define fractional Sobolev space W q(y),p(y,z) K (U) with general kernel K(y, z) on compact manifold M as the set of all measurable functions w ∈ L q(y) (U) such that λ p(y,z) K(y, z)dv g (y)dv g (z) < ∞, for some λ > 0 and endow it with the natural norm w q(y),p(y,z) K where is the Gagliardo seminorm of u and (L q(y) (U), |.| q(y) ) is a variable exponent Lebesgue space.Proof.Let {w n } n∈N be a Cauchy sequence in W q(y),p(y,z) K (U).For any ε > 0, there exists N ε ≥ 0, such that for every n, m ∈ N, n, m ≥ N ε , Since (L q(y) (U), |.| q(y) ) is a Banach space, there exists w ∈ L q(y) (U) such that w n → w strongly in L q(y) (U) as n → +∞.Thanks to the converse of the Dominated Convergence Theorem, it follows that for a subsequence still denoted {w n }, we have that w n → w as n → +∞ a.e on U.

Proof of Theorem 1.1
In this section we shall prove Theorem 1.1, establishing an embedding of W q(y),p(y,z) K (U) into L ℓ(y) (U).
So, we have the following , for every y ∈ U j .(4.3) ≤ q(y), for every y ∈ U j .(4.4) By [24,Lemma 2.4 ], there exists a constant C = C(N, t, ε, p j , U j ) such that (see [24] for more details) for every w ∈ W s,pj (U j ).(4.5) Now, we shall prove the following three inequalities.a) There exists a constant c 1 > 0, such as: b) There exists a constant c 2 > 0, such as: c) There exists a constant c 3 > 0, such as: We shall first prove (a).We have that where χ Uj is a characteristic function.Hence, we have Combining the statement (4.3) with the Hölder inequality, we obtain , , for every y ∈ U j .
Similarly, by using the fact that q(y) > p j for every y ∈ U j , we get (b).Now, we show (c).Put We use the Hölder inequality and the definition of p j , to get where and dµ g (y, z) = dv g (y)dv g (z) We show that this embedding is compact.Let {w n } be a bounded sequence in W q(y),p(y,z) K (U), we need to prove that there exists w ∈ L ℓ(y) (M) such that for every ℓ(y) ∈ (1, p * s ), Since M is a compact Riemannian N-manifold, we can cover M by a finite number of charts (U j , ϕ j ) j=1,...,m satisfying 1 where g s ij are bilinear forms and Q > 1.Let η j be a smooth partition of unity subordinate to the chart (U j , ϕ j ) j=1,...,m .Let w n ∈ W q(y),p(y,z) K (M).Then where B 0 (1) is an open unit ball of R N .By [27, Theorem 1.1], there exists
Proof.Let w ∈ W q(y),p(y,z) K (U).Then w is a weak solution of problem (1.1) if and only if where L, S are the operators defined in Lemma 6.1 and Lemma 6.To solve (5.2), we shall use the Berkovits topological degree introduced in Section 2. To this end, we first show that the set is bounded.Let h ∈ D and take w = Gh.Using the growth condition (B 1 ), the Hölder inequality, the Young inequality, and continuous embedding W q(y),p(y,z) K (U) ֒→ L q(y) (U), we get Since S is bounded, it follows that D is bounded in (W q(y),p(y,z) K (U)) * .As a result, there exists a positive constant η > 0 such that ||h|| W q(y),p(y,z) K (U ) * < η, for every h ∈ D.
By the statements (1)-(2) in Theorem 2.17, we can deduce by applying the homotopy invariance and normalization properties of the degree d from Theorem 2.17.Therefore there exists w ∈ B η (0) such that h + S • Gh = 0. We can now deduce that w = Gh is a weak solution to problem (1.1) in W q(y),p(y,z) K (U).This completes the proof.

Appendix
Lemma 6.1.Suppose that (M, g) is a compact N -dimensional Riemannian manifold, U is a smooth open subset of M, K : U × U → (0, +∞) is a symmetric function satisfying Lévy-integrability and coercivity conditions.Assume that assumption (B 1 ) holds.Then the operator L : W q(y),p(y,z) K (U) → (W q(y),p(y,z) K (U)) * is continuous, bounded and strictly monotone, and i) L is an operator of type (S + ), ii) L : W q(y),p(y,z) K (U) → (W q(y),p(y,z) K (U)) * is a homeomorphism.
Proof.It is obvious that L is bounded.We show that L is continuous.Assume that w n → w in W q(y),p(y,z) K (U) and we show that L(w n ) → L(w) in (W q(y),p(y,z) K (U)) * .Indeed, where, 1 p(y, z) + 1 p ′ (y, z) = 1.Thanks to the Hölder inequality, we have

and
V (y, z) = (w(y) − w(z))K(y, z) Since w n → w in W q(y),p(y,z) K (U), we have V n → V in L p(y,z) (U × U).So, there exists a subsequence of {V n } n∈N and h(y, z) ∈ L p(y,z) (U × U) such that V n → V a.e in U × U and |V n | ≤ h(y, z).Therefore we have G n → G a.e in U × U and We use the Dominated Convergence Theorem to get By Lemma 2.10, L is strictly monotone.Now, we show that L is mapping of type (S + ).Let {w n } n∈N ⊂ W q(y),p(y,z) K (U) be a sequence with w n ⇀ w in W q(y),p(y,z) K According to (6.1) -( 6.3), we get As a consequence of the Brezis-Lieb Lemma [13], (6.1), and (6.4), L is of type (S + )• We show that L is a homeomorphism.It is easy to see that L is coercive and injective.Thanks to the Minty-Browder Theorem [36, Theorem 26 A], L is surjective.So, L is a bijection.There exists a map G : (W q(y),p(y,z) K (U)) * → W q(y),p(y,z) K (U) such that G • L = id W q(y),p(y,z) K (U ) and L • G = id (W q(y),p(y,z) K (U )) * .We show that G is continuous.Let g n , g ∈ W q(y),p(y,z) K (U) be such that g n → g in W q(y),p(y,z) K (U).Let t n = G(g n ), w = G(g).Then L(w n ) = g n and L(w) = g.Since {t n } n∈N is bounded in W q(y),p(y,z) K (U), we have t n ⇀ w in W q(y),p(y,z) K (U).It follows that Since L is of type (S + ), we get t n → w in W q(y),p(y,z) K (U).This completes the proof.We shall show that S 1 and S 2 are both bounded and continuous.For every w ∈ W q(y),p(y,z) K (U), |S 1 w| q ′ (y) = λ M |β(y)|w(y)| r(y)−2 w(y)| q ′ (y) dv g (y) ≤ λ||β|| ∞ M ||w(y)| r(y)−1 | q ′ (y) dv g (y) ≤ λC||β|| ∞ M ||w(y)| q(y)−1 | q ′ (y dv g (y) ≤ λC||β|| ∞ M |w(y)| q(y) dv g (y).
The Dominated Convergence Theorem implies that S 2 w n → S 2 w in L q ′ (y) (U), so S 2 is continuous in W q(y),p(y,z) K (U).Because the canonical embedding i : W q(y),p(y,z) K (U) ֒→ L q(y) (U) is compact, its adjoint operator i * : L q ′ (y) (U) → (W q(y),p(y,z) K (U)) * is also compact.As a result, compositions i * •S 2 and S 2 •i * are compact, so we come to the conclusion that the operator S is compact and this completes the proof.

Theorem 2 . 4 .
(Stine's theorem[5]) For any (a, b) ∈ M 2 , d g (a, b) defines a distance on (M, g), and the topology determined by d g (a, b) is equivalent to the topology of M as a manifold.

1 ) 2 ) 3 )
13.  (see[28]) Let ω be a bounded open set in uniformly convex Banach space E, B : ω → E * a bounded operator, and f :ω → E. Then we have If f is locally bounded and satisfies condition (S + ) B and B is continuous, then f has the property (QM ) B .The operator f has the property (QM ) B , if for all {{z n } n∈N , z} ⊂ Uz n ⇀ z and a n = Bz n ⇀ a ⇒ lim inf f z n , a n − a ≥ 0. If operators f 1 , f 2 : ω → E satisfy (QM ) B condition, then f 1 + f 2and αf 1 also satisfy (QM ) B condition, for every positive numbers α. 4) Let f 1 : ω → E be an operator of the type (S + ) B and f 2 : ω → E an operator satisfying the property (QM ) B .Then f 1 +f 2 satisfies condition (S + ) B .Lemma 2.14.(see [11]) Let B be a bounded open set in E, B ∈ F ⋆ 0 (B) continuous, and g : D g ⊂ E * → E a demi-continuous operator such that B( B) ⊂ D g .Then the following properties hold: a) If g is quasi-monotone operator, then I + g • B ∈ F B ( B), where I denotes the identity operator.b) If g is an operator of type (S + ), then g • B ∈ F B (B).Definition 2.15.(see [14]) Let B ⊂ E be a bounded open set, B ∈ F ⋆ 0 (B) continuous, and f, g ∈ F B (E).Then the map H : [0, 1] × E → E given by H(s, w) = (1 − s)f w + sgw, for every (s, w) ∈ [0, 1] × B, is called an admissible affine homotopy.