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Home , Gaetano Fichera, Sergei Sobolev

arXiv:2003.08519v1 [math.FA] 19 Mar 2020 ewrs: Keywords er fSblvsae.Teesae r ae fe Sergei after named are spaces These spaces. 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Hausdorff-Young the with g p adu fslce xmls mn h most the Among examples. selected of handful tigequation string toa nlss ooe pcsadPartial and Spaces Sobolev Analysis, ctional fpstv-eient ucin,through functions, positive-semidefinite of “ erm n elc-odahvtheorem Rellich-Kondrachov and heorems n oiainfrtersace subject researched the for motivation and ooe (1908 Sobolev csadm n17 atn ihr wrote Fichera Gaetano 1977 In stardom. ic nGladpisadsuisteprop- the studies and pairs Gelfand on s ioFbn h netgtdteDirichlet the investigated who Fubini uido tce n oorps ti uieeffort futile a is It monographs. and rticles n hycm pwt h ae“Beppo name the with up came they and .Wa olw sagnrlotieof outline general a is follows What e. giSblvadters shistory... is rest the and Sobolev rgei ,wr nw ic h eybgnigof beginning very the since known were 2, ei(1875 Levi ffrnileutos h a not has who equations, ifferential ecneto efn pairs. Gelfand of context he lt,Sblvsae,Sobolev spaces, Sobolev ality, ´ ´ 91 isl,s h name the so himself, 1961) 99,atog hywere they although 1989), Differential Equations” (comp. [12]) or Leoni’s “A First Course in Sobolev spaces” (comp. [37]). These are by no means the only worth-reading monographs ´ we should also mention Adams’ and Fournier’s “Sobolev spaces” (comp. [1]), Françoise and Gilbert Demengel’s “ Spaces for the Theory of Elliptic Partial Differential Equations” (comp. [20]), Evans’ “Partial Differential Equations” (comp. [24]), Necas’ “Direct Methods in the Theory of Elliptic Equations” (comp. [43]), Tartar’s “An Introduction to Sobolev Spaces and Interpolation Spaces” (comp. [55]) and many more. Although more than 70 years have passed since the birth of Sobolev spaces, they still remain an active field of research. Mathematicians study Sobolev spaces on time scales (comp. [2, 25, 38, 3]) as well as the Sobolev spaces with exponent (comp. [22, 23, 36, 41, 50]). Plenty of papers is devoted to fractional Sobolev spaces (comp. [4, 40, 44, 53, 54]). Even at the author’s alma mater, Łódź University of Technology, there is a research group studying Sobolev’s legacy (comp. [9, 10, 11]). The current paper is was primarily inspired by the works of Przemysław Górka, Tomasz Kostrzewa and Enrique Reyes, particularly [28], [29] and [30]. In their articles, they defined and studied Sobolev spaces on locally compact abelian groups. It turned out that much of the classical theory, when the domain is r0, 1s or R, can be recovered in the setting of locally compact abelian groups. It is of crucial importance that the tools that Górka, Kostrzewa and Reyes used in their work come from the field of abstract harmonic analysis. This is vital, because as a general rule of thumb “harmonic analysis on locally compact abelian groups can be generalized to harmonic analysis on Gelfand pairs” (naturally there are notable exceptions to that “rule”). This is our starting point and, at the same time, one of the main ideas permeating the whole paper. Chapter 2 serves the purpose of laying out the preliminaries of harmonic analysis on Gelfand pairs. Our exposition mainly follows (but is not restricted to) van Dijk’s “Introduction to Harmonic Analysis and Generalized Gelfand pairs” (comp. [21]) and Wolf’s “Harmonic Analysis on Commutative Spaces” (comp. [59]). We recap the basic properties of positive-semidefinite and spherical functions and summarize the notion of Gelfand pairs. We proceed with a brief description of the counterpart of in the context of Gelfand pairs, namely the spherical transform. We invoke the celebrated Plancherel theorem, which can be viewed as a kind of bridge between the functions on group G and functions on the dual object of G. A significant amount of space is devoted to Hausdorff-Young inequality (which follows from Riesz-Thorin interpolation) and its inverse - we even go to some lengths to provide a not-so-well-known proof of the latter. Chapter 3 aspires to be the main part of the paper. It focuses on the investigation of Sobolev spaces on Gelfand pairs. Theorems 8, 9 and 10 generalize the acclaimed Sobolev embedding theorems. It is safe to say that Theorem 16 is the climax of the paper. It is the counterpart of the eminent Rellich-Kondrachov theorem from the classical theory of Sobolev spaces.

2 Harmonic analysis on Gelfand pairs

The current chapter serves as a rather brief overview of the harmonic analysis on Gelfand pairs. We commence with the concept of positive-semidefinite functions, explaining their basic properties in Lemma 1. Subsequently we introduce the notion of spherical functions and define the spherical transform, whose theory bears a striking resemblance to the theory of the Fourier transform for locally compact abelian groups. We discuss a counterpart of the Plancherel theorem (Theorem 3) as well as the inverse spherical transform. The closing part of the chapter is centred around the Hausdorff-Young inequality (Theorem 6) and its inverse (Theorem 7), which are applied in the sequel.

2 2.1 Positive-semidefinite functions, spherical functions and Gelfand pairs First of all, let us emphasize that from this point onward, we work under the assumption that G is a locally compact (Hausdorff) group. Later on, we will add more assumptions on G, but it will never cease to be at least a locally compact group. Now, without further ado, we introduce the concept of positive-semidefinite functions (in the definition below, and throughout the whole paper : stands for the Hermitian transpose): Definition 1. (comp. Definition 32.1 in [34], p. 253 or Definition 8.4.1 in [59], p. 165) A φ: G ÝÑ C is called positive-semidefinite if

: ´1 ´1 ´1 z1 φ x1 x1 φ x1 x2 ... φ x1 xN z1 ´1 ´1 ´1 z2 φ x2 x1 φ x2 x2 φ x2 xN z2 ¨ . ˛ ¨ ¨ ` . ˘ ` ˘ . ` . ˘ ˛ ¨ ¨ . ˛ ě 0, (1) . ` . ˘ ` ˘ .. ` . ˘ . ˚ ‹ ˚ ´1 ´1 ´1 ‹ ˚ ‹ ˚ zN ‹ ˚ φ x x φ x x ... φ x x ‹ ˚ zN ‹ ˚ ‹ ˚ N 1 N 2 N N ‹ ˚ ‹ ˝ ‚ ˝ ‚ ˝ ‚ N N ` ˘N ` CN ˘ ` ˘ for every N P , pxnqn“1 Ă G and pznqn“1 Ă . N For a fixed of elements pxnqn“1 Ă G, the condition (1) is known as the positive-semidefiniteness of the matrix

´1 ´1 ´1 φ x1 x1 φ x1 x2 ... φ x1 xN ´1 ´1 ´1 φ x2 x1 φ x2 x2 φ x2 xN ¨ ` . ˘ ` ˘ . ` . ˘ ˛ . (2) ` . ˘ ` ˘ .. ` . ˘ ˚ ‹ ˚ φ x´1x φ x´1x ... φ x´1x ‹ ˚ N 1 N 2 N N ‹ ˝ ‚ To reiterate, a function φ is positive-semidefinite` ˘ ` if all˘ such matrices,` ˘ regardless of the size N and the N sequence pxnqn“1 Ă G, are positive-semidefinite. We can also express condition (1) more explicitly: a N N N C function φ satisfies (1) if for every N P , pxnqn“1 Ă G and pznqn“1 Ă we have

N N ´1 φ xn xm ¨ zn ¨ zm ě 0. (3) n“1 m“1 ÿ ÿ ` ˘ It will shortly become apparent that positive-semidefinite functions play a crucial role in our un- derstanding of harmonic analysis on Gelfand pairs. We should therefore become acquainted with the properties of these functions. To this end we invoke the following result: Lemma 1. (comp. Lemma 5.1.8 in [21], p. 54, Theorem 32.4 in [34] or Proposition 8.4.2 in [59], p. 165) If φ: G ÝÑ C is a positive-semidefinite function, then

1. φpeq is a real and nonnegative number, 2. φ x´1 “ φpxq for every x P G, 3. |φ`pxq| ď˘ φpeq for every x P G,

3 4. φ is continuous.

To make further progress we take a compact subgroup K of the locally compact (Hausdorff) group G (we shall repeat this assumption ad nauseam) and consider the double coset space (comp. [8], p. 101):

KzG{K :“ KgK : g P G .

Let π : G ÝÑ KzG{K be the projection defined by πpgq :“ KgK.(If

• CcpGq is the space of all continuous functions on G with compact support, and

• CcpKzG{Kq is the space of all continuous functions on KzG{K with compact support then every function F P CcpKzG{Kq determines a function F ˝ π : G ÝÑ C belonging to the set of all bi´K´invariant continuous functions on G with compact support:

f P CcpGq : @k1,k2PK fpk1xk2q“ fpxq . (4) " xPG *

Furthermore, the map F ÞÑ F ˝ π is a bijection, and thus we identify CcpKzG{Kq with the set (4). An analogous reasoning works for the space CpKzG{Kq of continuous functions, the space C0pKzG{Kq of continuous functions which vanish at infinity and the space L1pKzG{Kq of integrable functions. If it turns out that L1pKzG{Kq is a commutative convolution algebra, the pair pG, Kq is called a Gelfand pair. The simplest instance of a Gelfand pair is pG, teuq where G is a locally compact abelian group. A less trivial example is pEpnq, SOpnqq, where SOpnq is the special orthogonal group and Epnq is the group of Euclidean motions on Rn (Epnq is in fact a semi-direct product of the group T pnq and the orthogonal group Opnq). Other instances of Gelfand pairs include: • pGLpn, Rq, Opnqq, where GLpn, Rq is the general linear group on Rn, • pGLpn, Cq,Upnqq, where Upnq is the unitary group, • pOpn ` kq, Opnqˆ Opkqq, • pSOpn ` kq, SOpnqˆ SOpkqq, • pUpn ` kq,Upnqˆ Upkqq, • pSUpn ` kq,SUpnqˆ SUpkqq, where SUpnq is the special unitary group.

Given a Gelfand pair pG, Kq, every function χ: CcpKzG{Kq ÝÑ C satisfying

@f,gPCcpKzG{Kq χpf ‹ gq“ χpfq ¨ χpgq, where ‹ denotes the convolution in CcpKzG{Kq, is called a character. Furthermore, every continuous, bi´K´invariant function φ: G ÝÑ C for which

@fPCcpKzG{Kq χφpfq :“ fpxq ¨ φpxq dx żG is a nontrivial character, is called a spherical function. These functions admit the following characteriza- tion:

4 Theorem 2. (comp. Propositions 6.1.5 and 6.1.6 in [21], p. 77, Proposition 2.2 in [33], p. 400 or Theorem 8.2.6 in [59], p. 157) The following conditions are equivalent:

• φ: G ÝÑ C is a spherical function. • φ: G ÝÑ C is a nonzero, such that

@x,yPG φpxkyq dk “ φpxq ¨ φpyq, żK where dk is the normalized Haar on K.

• φ: G ÝÑ C is a continuous, bi´K´invariant function with φpeq“ 1 and for every f P CcpKzG{Kq there exists a ´1 λf “ fpxq ¨ φ x dx żG ` ˘ such that f ‹ φ “ λf ¨ φ.

2.2 Spherical transform The previous section concluded with a characterization of spherical functions. Currently, our objective is to employ these functions to describe the spherical transform (and its inverse), which is a counterpart of the prominent Fourier transform on locally compact abelian groups. To this end, we introduce the following notation:

• SpG, Kq denotes the set of all spherical functions on the Gelfand pair pG, Kq, • SbpG, Kq denotes the set of all bounded spherical functions on pG, Kq, • S`pG, Kq denotes the set of all positive-semidefinite spherical functions on pG, Kq. Let us remark that due to Lemma 1, positive-semidefinite functions are automatically bounded, so S`pG, KqĂ SbpG, Kq. Furthermore, since the Gelfand pair pG, Kq is usually understood from the context, we frequently simplify the notation SpG, Kq, SbpG, Kq and S`pG, Kq to S, Sb and S`, respectively. Definition 2. (comp. Definition 6.4.3 in [21], p. 83 or Definition 9.2.1 in [59], p. 184) For every f P L1pKzG{Kq the function f : Sb ÝÑ C given by

fpφpq :“ fpxq ¨ φ x´1 dx żG ` ˘ is called a spherical transform. p As in [59], p. 185 we topologize Sb (which is a maximal ideal space for the commutative Banach algebra L1pKzG{Kq) with the weak from the family of maps f : f P L1pKzG{Kq - this turns Sb into a locally compact Hausdorff space. In the sequel we will focus on the subspace S` of Sb, with the induced ( topology. It is remarkable that the induced weak topology on Sp` coincides with the topology of uniform

5 convergence on compact of G (comp. Proposition 6.4.2 in [21], p. 83 or Theorem 3.31 in [26], p. 82). Let us define the function Γpfq :“ f for every f P L1pKzG{Kq. Since every spherical transform f is a continuous function which vanishes at infinity (comp. Corollary 9.1.14 or Corollary 9.2.10 in [59]), we have p p 1 b Γ: L pKzG{Kq ÝÑ C0pS q. This is a counterpart of the classical Riemann-Lebesgue lemma, a classical result in the theory of Fourier transform (comp. Lemma 3.3.7 in [18], p. 47 or Theorem 1.7 in [35], p. 136). Furthermore, we define

1 X1 :“ BpKzG{KqX L pKzG{Kq, where BpKzG{Kq is the set of all linear combinations of positive-semidefinite and bi´K´invariant func- tions. Theorem 9.4.1 in [59], p. 191 asserts the existence of a measure µ on S` such that for every f P X we have f P L1pS`, µq and p p p @xPG fpxq“ fpφq ¨ φpxq dµpφq. (5) S` ż In other words Γ is a bijection between X Ă L1pKzGp{Kq and thep image ΓpX q Ă L1pS`, µq. This insight is vital while proving the counterpart of the Plancherel formula: Theorem 3. (comp. Theorem 6.4.6 in [21], p. 85 or Theorem 9.5.1 in [59], p. 193) p 1 2 2 ` If f P L pKzG{KqX L pKzG{Kq then f P L pS , µq and }f}2 “}f}2. As in the classical case (comp. Theorem 3.5.2 in [18], p. 53), by Theorem 1.7 in [47], p. 9 we can extend p p Γ (a linear and bounded defined on a densep set L1pKzG{KqX L2pKzG{Kq in L2pKzG{Kq) to an isometric isomorphism between L2pKzG{Kq and L2pS`, µq. In order to simplify the notation, we still denote this extension by Γ. The map Γ´1 is called the inverse spherical transform and we often write ´1 2 S` F :“ Γ pF q for F P L p , µq. p As a corollary to Plancherel theorem (Theorem 3) we have the following result: q Corollary 4. (Corollary 9.5.2p in [59], p. 194) If f,g P L2pKzG{Kq, then

xf|gyL2pKzG{Kq “xf|gyL2pS`,µq.

p p p In the sequel we abbreviate both inner products x¨|¨yL2pKzG{Kq and x¨|¨yL2pS`,µq to x¨|¨y2. The distinction between the two should be obvious from the context. p 2.3 Hausdorff-Young inequality and its inverse The final section of the current chapter opens with a renowned Riesz-Thorin interpolation theorem: Theorem 5. (comp. Theorem 6.27 in [27], p. 200) Let pX, ΣX , µX q and pY, ΣY , µY q be measure spaces and let p0,p1, q0, q1 Pr1, 8s. If q0 “ q1 “8, suppose also that µY is semifinite. For 0 ă t ă 1, define pt and qt by 1 1 ´ t t 1 1 ´ t t :“ ` and :“ ` . pt p0 p1 qt q0 q1

6 p0 p1 q0 q1 If T is a from L pX, µX q` L pX, µX q to L pX, µY q` L pX, µY q such that there exist constants M0,M1 ą 0 with

p0 }Tf}q0 ď M0 ¨}f}p0 for every f P L pX, µX q, and p1 }Tf}q1 ď M1 ¨}f}p1 for every f P L pX, µX q, then

1´t t pt }Tf}qt ď M0 M1 ¨}f}pt for every f P L pX, µX q, t P p0, 1q.

Riesz-Thorin interpolation theorem is a rather advanced tool and one of its corollaries is the aforemen- tioned Hausdorff-Young inequality: Theorem 6. (comp. Theorem 31.20 in [34], p. 226 or [59], p. 200) p 1 If p P r1, 2s then for every f P L pKzG{Kq the inequality }f}p1 ď }f}p holds (p stands for the Hölder conjugate of p). p In the sequel, we will also need the inverse Hausdorff-Young inequality, which we present below. We take the liberty of providing a rather short proof as it is far less known than the proof of the “classical” Hausdorff-Young inequality. Theorem 7. (comp. Theorem 31.24 in [34], p. 229 or [59], p. 200) 2 If p P p1, 2s then for every f P L pKzG{Kq the inequality }f}p1 ď}f}p holds.

Proof. Firstly, if }f}p “ 8 then we are immediately done. Thus withoutp loss of generality, we suppose that f P LppS`, µq. We start off withp an observation that for every function g P L2pKzG{KqX LppKzG{Kq we have p p Corollary 4 Hölder ineq. Theorem 6 gpxq ¨ fpxq dx “ |xg|fy2| “ xg|fy2 ď }g}p1 ¨}f}p ď }g}p ¨}f}p. G ˇż ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ p p ˇ p p 2 p Consequently,ˇ the map gˇ ÞÑ G gpxq ¨ fpxq dx is a linear and bounded functional on L pKzG{KqX LppKzG{Kq. Since L2pKzG{KqXLppKzG{Kq is dense in LppKzG{Kq we can extend this map to a linear pş and bounded functional Φ: L pKzG{Kq ÝÑ C such that }Φ}ď}f}p. By Riesz representation theorem 1 (comp. [12], Theorem 4.11, p. 97) there exists a function h P Lp pKzG{Kq such that p

@gPLppKzG{Kq Φpgq“ gpxq ¨ hpxq dx and }h}p1 “}Φ}. żG It follows that f ” h almost everywhere. Lastly, we have

}f}p1 “}h}p1 “}Φ}ď}f}p, which concludes the proof. p

7 3 Sobolev spaces

After the revision of harmonic analysis on Gelfand pairs in the previous chapter, we are in position to define Sobolev spaces on Gelfand pair pG, Kq. Although Sobolev spaces are frequently introduced via the notion of a weak , the expression “differentiation on a group” may be meaningless, so we must resort to a different approach than suggested in [12], p. 202 or [48], p. 204 (to be fair, the latter discusses the characterization of Sobolev spaces via the Fourier transform a couple of pages further). We pursue a path inspired by the works of Górka, Kostrzewa, Reyes (comp. [28, 29, 30]) as well as Grafakos (comp. [31], p. 13), Malliavin (comp. [39], p. 142) or Triebel (comp. [57], p. 3). ` Definition 3. For a measurable function γ : S ÝÑ R` and s P R`, the set

s 2 2 s 2 HγpKzG{Kq :“ f P L pKzG{Kq : 1 ` γpφq ¨ |fpφq| dµpφqă8 (6) S` " ż * ` ˘ is called a . p p Let us immediately explain the reason why the above definition encompasses the definitions encountered in the literature: • A Sobolev space (with weight γ) on R can be defined as

s R 2 R 2 s 2 Hγp q“ f P L p q : 1 ` γpyq ¨ |fpyq| dy ă8 . (7) R " ż * ` ˘ p p If γpyq “ |y| , p P R` and s “ 1, then (7) is exactly the Definition 7.14 on page 208 in [48]. A similar approach is presented in [56] on page 271. This is perfectly compatible with Definition 3: if G “ R and K “ t0u, then (comp. [21], p. 77) we have S “ x ÞÑ eλx : λ P C . Furthermore, if λx ` φpxq“ e is an element of S then (3) implies (for N “ 2, z1 “ 1, z2 “ i) that ( 2 ` i eλpx2´x1q ´ e´λpx2´x1q ě 0. ´ ¯ This is possible only if Repλq“ 0, which means that S` Ă x ÞÑ eiyx : y P R . Moreover, for every N N y P R and pxnq Ă R, pznq Ă C we have n“1 n“1 ( N N N N iypxm´xnq iyxn iyx e ¨ zn ¨ zm “ e ¨ zn ¨ e n ¨ zn ě 0, n“1 m“1 ˜n“1 ¸ ˜n“1 ¸ ÿ ÿ ÿ ÿ so we conclude that S` “ x ÞÑ eiyx : y P R . In other words, S` “ R – R (comp. [18], p. 101-102) so (6) infallibly reconstructs (7). ( p • The reasoning from the first case can be generalized as follows: if G is a locally compact abelian group and K “teu, then by Theorem 5.3.3, p. 61 and Theorem 6.2.5, p. 81 in [21] we conclude that S` “ G. Consequently, (6) reads

s 2 2 s 2 p Hγ pGq“ f P L pGq : 1 ` γpyq ¨ |fpyq| dµpyqă8 , " żG * ` ˘ which is precisely the definition of Górka andp Kostrzewa (compp. [28, 29,p 30]).

8 Having justified the way we chose to define the Sobolev spaces on a Gelfand pair pG, Kq we take a s 2 moment to dwell on the space HγpKzG{Kq. It is obviously a linear subspace of L pKzG{Kq and for every s function f P HγpKzG{Kq we can define

1 2 2 s 2 s }f}Hγ :“ 1 ` γpφq ¨ |fpφq| dνpφq . S` ˆż ˙ ` ˘ s We briefly argue that it is in fact a norm, which turns HγpKpzG{Kq into a Banach space. s s s Obviously, }0}Hγ “ 0 so let f P Hγ pKzG{Kq be such that }f}Hγ “ 0. This implies that |fpφq| “ 0 for ` every φ P S . By Theorem 3 we conclude that }f}2 “ 0, which means f “ 0 almost everywhere. s s s C p s It is trivial to see that }αf}Hγ “ |α|¨}f}Hγ for every f P HγpKzG{Kq and α P . Lastly, }¨}Hγ obeys the triangle inequality due to the classical Minkowski inequality (comp. Theorem 6.5 in [27], p. 183). In s s H K G K summary, }¨}Hγ is a norm in γp z { q. s s As far as the completeness of pHγ pKzG{Kq, }¨}Hγ q is concerned, we could try to prove it in a tedious manner “from scratch”. However, in order to save time let us resort to a cunning trick which makes the s problem significantly easier: by Theorem 3 we know that the space HγpKzG{Kq is isometrically isomorphic (via the spherical transform) to the space L2pS`, νq of square-integrable functions on S` with respect to s 2 2 2 S` s the measure dν “ p1 ` γ q dµ. Since L p , νq is complete, then so is Hγ pKzG{Kq. We conclude that s Hγ pKzG{Kq is a Banach space. p 3.1 Sobolev embedding theorems Our next big topic is the embedding theorems. In general, we say (comp. Definition 7.15 in [48], p. 209) that a Banach space X is continuously embedded in a Banach space Y if X Ă Y and there exists a constant C ą 0 such that @uPX }u}Y ď C ¨}u}X . We denote this situation by X ãÑ Y . The question that permeates the current section is: “In what Banach spaces can be embed the Sobolev s spaces Hγ pKzG{Kq defined previously?” This is by far not a trivial task, as is confirmed by Chapter 4 in [1], Theorem 8.8 in [12], p. 212, Chapter 2 in [20], Chapter 5.6 in [24], Chapters 7.2.3 and 7.2.4 in [48] or Chapter 2.4 in [60]. s Theorem 8. s For every f P HγpKzG{Kq we have }f}2 ď}f}Hγ . s Proof. For every function f P Hγ pKzG{Kq we have

1 1 2 2 Theorem 3 2 2 s 2 s }f}2 “}f}2 “ |fpφq| dµpφq ď 1 ` γpφq ¨ |fpφq| dµpφq “}f}Hγ , S` S` ˆż ˙ ˆż ˙ ` ˘ which concludes the proof.p p p p p s To put it differently, Theorem 8 says that the Sobolev space Hγ pKzG{Kq is continuously embedded in 2 s ã 2 L pKzG{Kq, i.e. HγpKzG{Kq Ñ L pKzG{Kq. Our next results embeds (under certain circumstances) s the Sobolev space HγpKzG{Kq in the space of continuous functions: Theorem 9. If ´ s 1 ` γ2 2 P L2pS`, µq s ã s then HγpKzG{Kq Ñ CpKzG{Kq, i.e. Hγ`pKzG{˘Kq embeds continuously in CpKzG{Kq. p 9 ` Proof. Fix x˚ P G and ε ą 0. Since S is equipped with the topology of uniform convergence on compact sets, then (by the Arzelà-Ascoli theorem) S` is equicontinuous: there exists an open neighbourhood U of x˚ such that

@ xPU |φpxq´ φpx˚q| ă ε. (8) φPS`

s Consequently, for every function f P Hγ pKzG{Kq and x P U we have

|fpxq´ fpx˚q| “ fpφq ¨ φpxq´ φpx˚q dµpφq ď sup |φpxq´ φpx˚q| ¨ |fpφq| dµpφq S` S` S` ˇż ˇ φP ż ˇ ` ˘ ˇ 1 1 Cauchy-Schwarz ineq. 2 2 ˇ p p 2ˇ s 2 p 2p ´s ď sup ˇ|φpxq´ φpx˚q| ¨ 1 ` γpφq ˇ ¨ |fpφq| dµpφq ¨ 1 ` γpφq dµpφq S` S` S` φP ˆż ˙ ˆż ˙ ` ˘ ` ˘ s p8q s 2 ´ 2 p 2 ´ 2 s p s p ď sup |φpxq´ φpx˚q|¨}f}Hγ ¨} 1 ` γ }2 ď ε ¨}f}Hγ ¨} 1 ` γ }2, φPS` ` ˘ ` ˘ s which proves that Hγ pKzG{KqĂ CpKzG{Kq. Finally, a similar estimate to the one above leads to

s Lemma s s 2 ´ 2 1 2 ´ 2 2 ´ 2 s s s |fpxq| ď sup |φpxq|¨}f}Hγ ¨} 1 ` γ }2 ď sup φpeq¨}f}Hγ ¨} 1 ` γ }2 ď}f}Hγ ¨} 1 ` γ }2, φPS` φPS` ` ˘ ` ˘ ` ˘ where the last inequality stems from the fact that φpeq“ 1 for every spherical function φ. This concludes the proof.

s Finally, we prove that (under certain assumptions) Sobolev space Hγ pKzG{Kq embeds continuously 1 in Lp pKzG{Kq. Let us recall that p1 is the Hölder conjugate of p.

2α Theorem 10. Let α ą s ą 0 and let p :“ α`s . If

´1 1 ` γ2 P LαpS`, dµq,

s ã p1 ` ˘ then HγpKzG{Kq Ñ L pKzG{Kq. p

2 Proof. Since p P p1, 2s then by Theorem 7 we know that for every f P L pKzG{Kq we have }f}p1 ď}f}p. Furthermore, we have p 1 2 sp ´p 2 p 2 γ φ 2 Hölder ineq. sp p p p1 ` p q q 2 ´ 2 s ´p }f}p “ |fpφq| ¨ sp dµpφq ď }f}Hγ ¨ p1 ` γpφq q dµpφq . p1 ` γpφq2q 2 S` ˜żG ¸ ˆż ˙ p sp p p p Since α “ 2´p we finally obtain

s 2 ´1 2 1 s }f}p ď}f}p ď}f}Hγ ¨} 1 ` γ }α , ` ˘ which concludes the proof. p

10 3.2 Rellich-Kondrachov theorem Having discussed the embedding theorems in the previous section, we proceed with the next topic, namely the Rellich-Kondrachov theorem. This is one of the central results in the classical theory of Sobolev spaces and appears in virtually any text on the subject: Theorem 6.3 in [1], p. 168, Theorem 1 in [24], p. 272, Theorem 11.10 in [37], p. 320, Theorem 6.1 in [43], p. 102 or Theorem 2.5.1 in [60], p. 62. The main theorem in this section, the counterpart of Rellich-Kondrachov theorem for Gelfand pairs, is Theorem 16. Before we dig into the proof of this result we demonstrate three auxilary lemmas (Lemma 11, 13 and 14). We should also emphasize the fact that Lemma 14 and (as a consequence) Theorem 16 require the group G to be compact while previous lemmas work without that assumption. s Lemma 11. Let f P HγpKzG{Kq. If y P G then

2 ´1 2 |φpyq´ 1| 2 |f xy ´ fpxq| dx ď sup 2 s ¨}f}Hs . S` p1 ` γpφq q γ żG ˜φP ¸ ` ˘ 2 Proof. We fix y P G and, for a moment, we suppose that f P CcpKzG{Kq. If the map Ry : L pKzG{Kq ÝÑ 2 ´1 L pKzG{Kq is given by Ryfpxq :“ f xy , then

` ˘ ´1 ´1 ´1 @φPS` Ryfpφq“ Ryfpxq ¨ φ x dx “ f xy ¨ φ x dx żG żG xÞÑxy ` ˘ ´`1 ˘ ` ˘ y “ fpxq ¨ φ y´1x´1 dx xÞÑ“x fpxq ¨ φ y´1x´1 dx (9) żG żG ` ˘ Theorem 2 ` ˘ “ φ ‹ f y´1 “ f ‹ φ y´1 “ fpφq ¨ φ y´1 . By Theorem 3 we have ` ˘ ` ˘ ` ˘ p 2 ´1 2 p9q 2 ´1 2 |f xy ´ fpxq| dx “ Ryfpφq´ fpφq dµpφq “ |fpφq| ¨ φ y ´ 1 dµpφq G S` S` ż ż ˇ ˇ ż ` ˘ ˇ ˇ 2 ˇ ` 2 ˘ ˇ Lemma 1 2 ˇ 2ys |φpyq´p 1ˇ| p p |φpyq´ˇ 1| ˇ 2 p “ |fpφq| ¨ 1 ` γpφq ¨ 2 s dµpφqď sup 2 s ¨}f}Hs . S` p1 ` γpφq q S` p1 ` γpφq q γ ż ˜φP ¸ ` ˘ p s To conclude the proof it suffices to note that CcpKzG{Kq isp dense in Hγ pKzG{Kq. Prior to Lemma 13 we revisit a very popular Minkowski’s inequality: Theorem 12. (comp. Theorem 6.19 in [27], p. 194 or [51], p. 271) Let X, Y be σ´finite measure spaces, 1 ď p ă8 and let F : X ˆ Y ÝÑ C be a measurable function. Then

1 1 p p p |F px, yq| dy dx ď |F px, yq|p dx dy. (10) ˆżX ˆżY ˙ ˙ żY ˆżX ˙ In the lemma below, ^ stands for the logical “and”, while _ stands for the logical “or”. s Lemma 13. Let f P HγpKzG{Kq. If η P CcpKzG{Kq then

|φpyq´ 1| }f ‹ η ´ f}2 ď sup sup s ¨}f}Hs . 2 2 γ yPsupppηq ˜φPS` p1 ` γpφq q ¸

11 Proof. At first, we choose fB to be a Borel-measurable function such that f “ fB almost everywhere. Since G is a Tychonoff space (as a locally compact group), then there exists η P CcpKzG{Kq such that • ηpeq‰ 0, η ě 0, and • G ηpyq dy “ 1. For everyş x P G we have 2 2 |f ‹ ηpxq´ fpxq|2 “ f xy´1 ¨ ηpyq dy ´ fpxq ¨ ηpyq dy “ f xy´1 ´ fpxq ¨ ηpyq dy . ˇżG żG ˇ ˇżG ˇ ˇ ` ˘ ˇ ˇ ` ` ˘ ˘ ˇ We define ˇ ˇ ˇ ˇ ˇ ´1 ˇ ˇ ˇ F px, yq :“ fB xy ´ fBpxq ¨ ηpyq, which is a Borel function as a composition of` Borel` functions˘ : ˘ ´1 ´1 ´1 ´1 px, yq ÞÑ px,y,yq ÞÑ px, y ,yq ÞÑ px, xy , ηpyqq ÞÑ pfBpxq,fB xy , ηpyqq ÞÑ fB xy ´fBpxq ¨ηpyq. 2 Since fB and η are both in L pKzG{Kq, then supppfBq and` suppp˘ηq are σ´compact` ` (comp.˘ Corollary˘ 1.3.5 in [18], p. 10). Consequently, also the sets

psupppfBq ¨ supppηqq ˆ supppηq and supppfBqˆ supppηq are σ´compact. Next, we follow a of logical implications:

´1 px, yqPtF ‰ 0u ùñ fB xy ´ fBpxq‰ 0 ^ ηpyq‰ 0 ˆ ˙ ` ˘ ´1 ùñ xy P supppfBq _ x P supppfBq ^ y P supppηq ˆ ˙ ` ˘ ´1 ùñ xy P supppfBq ^ y P supppηq _ px P supppfBq ^ y P supppηqq ˆ ˙ ` ˘ ùñ px, yq P psupppfBq ¨ supppηqq ˆ supppηq _ px, yqP supppfBqˆ supppηq . ˆ ˙ We conclude that tF ‰ 0u is σ´compact. Finally, we are in position to apply Minkowski’s integral inequality:

1 2 2 ´1 }f ‹ η ´ f}2 “}fB ‹ η ´ fB}2 “ fB xy ´ fBpxq ¨ ηpyq dy dx ˜żG ˇ żG ˇ ¸ ˇ ` ` ˘ 1 ˘ ˇ p10q ˇ 2 ˇ ´1 ˇ 2 2 ˇ ď |fB xy ´ fBpxq| ¨ |ηpyq| dx dy G G ż ˆż ˙ 1 ` ˘ 2 2 Lemma 11 |φpyq´ 1| R f f η y dy sup η y dy f s “ } y B ´ B}2 ¨ | p q| ď 2 s ¨ | p q| ¨} B}Hγ ¨ S` p1 ` γpφq q ˛ żG żG ˜φP ¸ ˝ ‚ |φpyq´ 1| |φpyq´ 1| “ sup sup s ¨}fB}Hs “ sup sup s ¨}f}Hs , 2 2 γ 2 2 γ yPsupppηq ˜φPS` p1 ` γpφq q ¸ yPsupppηq ˜φPS` p1 ` γpφq q ¸ which ends the proof.

12 Up to this point, pG, Kq was an arbitrary Gelfand pair with G a locally compact Hausdorff group and K its compact subgroup. Now we impose a further restriction, namely: we assume that G is a compact group.

For brevity we will write that pG, Kq is a compact Gelfand pair, meaning that G is compact. This assumption somewhat simplifies the “dual object” of pG, Kq. In summary, if G is compact then SpG, Kq“ SbpG, Kq“ S`pG, Kq (comp. Theorem 9.10.1 in [59], p. 204) and this is a due to Theorem 9.1.13 in [59], p. 183 and the fact that L1pKzG{Kq has a unit element.

p Lemma 14. Let pG, Kq be a compact Gelfand pair and let p, q P p1, 8q. If pfnqĂ L pKzG{Kq is a weakly convergent sequence (with function f), then for every η P CcpKzG{Kq the sequence pfn ‹ηq converges (strongly) to f ‹ η in LqpKzG{Kq.

Proof. Firstly, we fix a function η P CcpKzG{Kq. The sequence pfnq is a weakly convergent, so it is p bounded in L pKzG{Kq, i.e. there exists M ą 0 such that }fn}p ď M for every n P N (comp. Proposition 3.5 in [12], p. 58). Furthermore, by definition of weak convergence, for every x P G we have

´1 ´1 yÞÑyx ´1 yÞÑy ´1 fn ‹ ηpxq“ fn xy ¨ ηpyq dy “ fn y ¨ ηpyxq dy “ fnpyq ¨ η y x dy G G G ż ´1 ż ´1 ż nÑ8 ´1` ˘yÞÑy ´1 ` ˘ yÞÑyx ´1 ` ˘ ÝÑ fpyq ¨ η y x dy “ f y ¨ ηpyxq dy “ fn xy ¨ ηpyq dy “ f ‹ ηpxq. żG żG żG ` ˘ ` ˘ ` ˘ In other words, the sequence pfn ‹ ηq converges pointwise to f ‹ η. Last but not least, for every x P G we have

´1 ´1 yÞÑyx ´1 ´1 |fn ‹ ηpxq´ f ‹ ηpxq| “ fn xy ´ f xy ¨ ηpyq dy “ fn y ´ f y ¨ ηpyxq dy ˇżG ˇ ˇżG ˇ ˇ ˇ ˇ ˇ 1 ´1 ` ` ˘ ` ˘˘ Hölder ineq. ` ` ˘ ` ˘˘ 1 p1 yÞÑˇy ´1 ˇ ˇ ´1 p ˇ “ˇ pfnpyq´ fpyqq ¨ η y x dy ˇ ďˇ }fn ´ f}p ¨ η y x dyˇ ˇżG ˇ ˆżG ˙ ˇ ` 1 ˘ ˇ ˇ ` ˘ˇ yÞÑxy 1 p1 ´1 ˇ ´1 p yÞÑy ˇ ˇ ˇ ď 2ˇM ¨ η y dy “ ˇ 2M ¨}η}p1 . ˆżG ˙ ˇ ` ˘ˇ ˇ ˇ Since G is compact, then 2M ¨ }η}p1 is an integrable (with arbitrary power) dominating function for |fn ‹ η ´ f ‹ η|. Finally, applying the Lebesgue dominated convergence theorem (comp. Theorem 4.2 in [12], p. 90) we obtain lim }fn ‹ η ´ f ‹ η}q “ 0, nÑ8 which concludes the proof. We are almost ready to prove Rellich-Kondrachov theorem for Gelfand pairs. One last, missing piece of the puzzle is Vitali convergence theorem: Theorem 15. (comp. Theorem 7.13 in [7], p. 76) p Let pfnq be a sequence in L pX, µX q, where 1 ď p ă 8. The sequence pfnq converges to a function p f P L pX, µX q if and only if:

1. pfnq converges to f in measure,

13 2. for every ε ą 0 there exists a measurable set Aε with µX pAεqă8 and such that

p p @nPN |fnpxq| dx ă ε , żGzAε

3. for every ε ą 0 there exists δ ą 0 such that for every measurable set A with µX pAqă δ we have

p p @nPN |fnpxq| dx ă ε . żA

At last, we gathered all the required tools to prove the culminating result of the current section, namely Rellich-Kondrachov theorem for compact Gelfand pairs. One last piece of terminology prior to the theorem itself: we say (comp. Definition 7.25 in [48], p. 211) that a Banach space X is compactly embedded in a Banach space Y (which we denote X ãÑc Y ) if it is continuously embedded in Y and if the embedding is a compact map (it maps bounded sets in X to relatively compact sets in Y ).

2α Theorem 16. Let pG, Kq be a compact Gelfand pair, α ą s ą 0 and let p :“ α`s . If

p1 ` γ2q´1 P LαpS`, µq, and p |φpyq´ 1| , lim sup 2 s “ 0 (11) yÑe ˜φPS` p1 ` γpφq q 2 ¸

s ã c q 1 s then Hγ pKzG{Kq Ñ L pKzG{Kq for every q Pr1,p s. In other words, Hγ pKzG{Kq embeds compactly in LqpKzG{Kq.

1 s Proof. Firstly we fix q P r1,p s. By Theorem 10 we already know that HγpKzG{Kq is continuously em- 1 1 bedded in Lp pKzG{Kq. Since G is compact then Lp pKzG{Kq ãÑ LqpKzG{Kq and, as a consequence, s q Hγ pKzG{Kq embedds continuously in L pKzG{Kq. It remains to prove that the embedding is compact. By definition we need to check that every bounded s q set in HγpKzG{Kq is mapped to a relatively compact set in L pKzG{Kq. Since we are dealing with s metric (even Banach!) spaces, it suffices to prove that every bounded sequence pfnqĂ HγpKzG{Kq has a convergent subsequence in LqpKzG{Kq. In other words, instead of checking topological compactness we check the equivalent (in this case) sequential compactness. s We observe that (due to continuous embedding) a bounded sequence pfnq Ă Hγ pKzG{Kq is also p1 bounded in L pKzG{Kq. Consequently, we can choose a weakly convergent subsequence of pfnq´ for no- p1 tational convenience and clarity we still denote this convergent subsequence by pfnq. Let f P L pKzG{Kq be the weak limit of pfnq. Fix ε ą 0 and let η P CcpKzG{Kq be such that }f ‹ η ´ f}q ă ε (comp. Proposition 2.42 in [26], p. 53) and

|φpyq´ 1| sup sup s ¨ M ă ε, (12) 2 2 yPsupppηq ˜φPS` p1 ` γpφq q ¸

14 s where M ą 0 is a constant Hγ pKzG{Kq´bound of the sequence pfnq. The latter condition can be satisfied due to (11). For such a choice of η we have

}fn ´ f}2 ď}fn ´ fn ‹ η}2 `}fn ‹ η ´ f ‹ η}2 `}f ‹ η ´ f}2 Lemma 13 |φpyq´ 1| ď sup sup s ¨}f } s `}f ‹ η ´ f ‹ η} ` ε 2 n Hγ n 2 yPsupppηq ˜φPS` p1 ` γpφq q 2 ¸ p12q ď 2ε `}fn ‹ η ´ f ‹ η}2,

and since }fn ‹ η ´ f ‹ η}2 ÝÑ 0 by Theorem 14, we conclude that limnÑ8 }fn ´ f}2 ď 2ε. Since ε ą 0 was chosen arbitrarily then limnÑ8 }fn ´ f}2 “ 0. 2 Since fn Ñ f in L pKzG{Kq then the sequence pfnq also converges to f in measure (comp. [49], p. 95). Furthermore, G is compact so the second condition in Vitali convergence theorem is automatically q satisfied. Last but not least, the sequence pfnq is uniformly L pKzG{Kq´integrable, so the third condition q of Vitali convergence theorem is satisfied. We conclude that pfnq converges to f in L pKzG{Kq.

References

[1] Adams R. A., Fournier J. J. F. : Sobolev spaces, Academic press, New York, 2003 [2] Agarwal R. P., Otero-Espinar V., Perera K., Vivero D.R. : Basic properties of Sobolev’s spaces on time scales, Adv. Differ. Equ., Article number: 038121 (2006) [3] Ahmad N., Baig H. A., Rahman G., Saleem M. S. : Sobolev’s embedding on time scales, J. Inequal. Appl., Vol. 1, Article number: 134 (2018) [4] Bahrouni A., Rădulescu V. D. : On a new fractional Sobolev space and applications to nonlocal varia- tional problems with variable exponent, Discrete Cont. Dyn.-S, Vol. 11, No. 3, p. 379-389 (2018)

8 ppxq [5] Bandaliyev R. A., Hasanov S. G. : On denseness of C0 pΩq and compactness in L for 0 ă ppxqă 1, Mosc. Math. J., Vol 1, No. 18, p. 1-13 (2018) [6] Barnaby N., Karman N. : Dynamics with infinitely many : the initial value problem, J. High Energy Phys., Vol. 2008 (2008) [7] Bartle R. G. : The Elements of Integration and Lebesgue Measure, A Wiley-Interscience Publication, New York, 1995 [8] Bechtell H. : The theory of groups, Addison-Wesley Publishing Company, London, 1971

[9] Bełdziński M., Galewski M. : Global diffeomorphism theorem applied to the solvability ofdiscrete and continuous boundary value problems, J. Differ. Equ. Appl., Vol. 24, No. 2 (2017) [10] Bełdziński M., Galewski M. : On unique solvability of a with nonlinearity depending on the derivative, Opusc. Math., Vol. 39, No. 2, p. 131-144 (2019) [11] Bełdziński M., Galewski M. : On the Existence and Uniqueness of Dirichlet Problems on a Positive Half-Line, Minimax Theory its Appl., Vol. 4, No. 1, p. 55-69 (2019)

15 [12] Brezis H. : , Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011 [13] Calcagni G., Montobbio M., Nardelli G. : Localization of nonlocal theories, Phys. Lett. B, Vol. 662, No. 3, p. 285-289 (2008) [14] Carmichael, R. D. : Linear differential equations of infinite order, Bull. AMS, Vol. 42, p. 193-218 (1936) [15] Carmichael, R. D. : On non-homogeneous linear differential equations of infinite order with constant coefficients, Am. J. Math., Vol. 58, No. 3, p. 473-486 (1936) [16] Carleson, L. : On infinite differential equations with constant coefficients, I. Math. Scand., Vol. 1, p. 31-38 (1953) [17] Carlsson M., Prado H., Reyes E. G. : Differential Equations with Infinitely Many Derivatives and the Borel Transform, Ann. Henri Poincar’e, Vol. 17, p. 2049-2074 (2016) [18] Deitmar A. : A First Course in Harmonic Analysis, Springer-Verlag, New York, 2005 [19] Deitmar A., Echterhoff S. : Principles of Harmonic Analysis, Springer, New York, 2009 [20] Demengel F., Demengel G. : Functional Spaces for the Theory of Elliptic Partial Differential Equa- tions, Springer-Verlag, London, 2012 [21] van Dijk G. : Introduction to Harmonic Analysis and Generalized Gelfand Pairs, de Gruyter, Berlin, 2009

[22] Edmunds D. E., Rákosník J. : Density of smooth functions in W k,ppxqpΩq, Proc. Roy. Soc. London Ser. A, Vol. 437, No. 1899, p. 229-236 (1992) [23] Edmunds D. E., Rákosník J. : Sobolev embeddings with variable exponent, Stud. Math., Vol. 143, No. 3, p. 267-293 (2000) [24] Evans L. C. : Partial Differential Equations, American Mathematical Society, Providence, 1998 [25] Feng Z., Su Y. H., Yao J. : Sobolev spaces on time scales and applications to semilinear Dirichlet problem, Dyn. Partial Differ. Equ., Vol. 12, No. 3, p.241-263 (2015)

[26] Folland G. B. : A Course in Abstract Harmonic Analysis, CRC Press, London, 1995 [27] Folland G. B. : Real Analysis: Modern techniques and their applications, John Wiley and Sons, New York, 1999 [28] Górka P., Kostrzewa T., Reyes E. G. : The Rellich lemma on compact abelian groups and equations of infinite order, Int. J. Geom. Methods Mod. Phys., Vol. 10, No. 2 (2013) [29] Górka P., Kostrzewa T., Reyes E. G. : Sobolev Spaces on Locally Compact Abelian Groups: Compact Embeddings and Local Spaces, J. Funct. Space, Vol. 2014, Article number: 404738 (2014) [30] Górka P., Reyes E. G. : Sobolev spaces on locally compact abelian groups and the bosonic string equation, J. Aust. Math. Soc., Vol. 98, No. 1, p. 39-53 (2015)

16 [31] Grafakos L. : Modern Fourier analysis, Springer, New York, 2009 [32] Graffi D. (editor) : , Springer-Verlag, Berlin, 2011 [33] Helgason S. : Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions, American Mathematical Society, Providence, 2002 [34] Hewitt E., Ross K. A. : Abstract Harmonic Analysis II, Springer-Verlag, Berlin, 1970 [35] Katznelson Y. : An introduction to harmonic analysis, Cambridge University Press, Cambridge, 2004 [36] Kováčik, Rákosník J. : On spaces Lppxq and W k,ppxq, Czechoslovak Math. J., Vol. 41, No. 4, p. 592-618 (1991) [37] Leoni G. : A First Course in Sobolev Spaces, American Mathematical Society, Providence, 2009 [38] Li Y., Zhou J. : Sobolev’s spaces on time scales and its applications to a of second order Hamiltonian systems on time scales, Nonlinear Anal., Real World Appl., Vol. 73, No. 5, p. 1375-1388 (2010) [39] Malliavin P. (with Airault H., Kay L., Letac G.) : Integration and Probability, Springer-Verlag, New York, 1995 [40] Maz’ya V., Shaposhnikova T. : On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., Vol. 195, No. 2, p. 230-238 (2002) [41] Mihăilescu M., Rădulescu V. : On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc., Vol. 135, p. 2929-2937 (2007) [42] Natanson I. P. : Theory of functions of a real variable, Dover Publications, New York, 2016 [43] Nečas J. : Direct Methods in the Theory of Elliptic Equations, Springer-Verlag, Berlin, 2012 [44] Nezza E., Palatucci G., Valdinoci E. : Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., Vol. 136, No. 5, p. 521-573 (2012) [45] Pego R. L. : Compactness in L2 and the Fourier transform, Proc. Amer. Math. Soc., Vol. 95, No. 2, p. 252-254 (1985) [46] Rastelli L. : Open string fields and D-branes, Fortschr. Phys., Vol. 52, p. 302-337 (2004) [47] Reed M., Simon B. : Methods of modern mathematical physics, Academic Press, San Diego, 1980 [48] Renardy M., Rogers R. C. : An Introduction to Partial differential equations, Springer-Verlag, New York, 2004 [49] Royden H. : Real analysis, Macmillan Publishing Company, New York, 1988 [50] Samko S., Vakulov B. : Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, J. Math. Anal. Appl., Vol. 310, No. 1, p. 229-246 (2005) [51] Stein E. : Singular and differentiability properties of functions, Princeton University Press, Princeton, 1970

17 [52] Stein E., Shakarchi R. : Real analysis: measure theory, integration and Hilbert spaces, Princeton University Press, Princeton 2005 [53] Strichartz R. S. : Multipliers on fractional Sobolev spaces, J. Math. Mech., Vol. 16, No. 9, p. 1031-1060 (1967) [54] Swanson D. : Pointwise inequalities and approximation in fractional Sobolev spaces, Stud. Math., Vol. 149, p. 147-174 (2002) [55] Tartar L. : An Introduction to Sobolev Spaces and Interpolation Spaces, Springer-Verlag, Berlin, 2007 [56] Taylor M. E. : Partial differential equations, Springer, New York, 1996 [57] Triebel H. : Theory of Functions Spaces III, Birkhäuser Verlag, , 2006 [58] Weil A. : L’intégration dans les groupes topologiques et ses applications, Hermann, Paris, 1965 [59] Wolf J. A. : Harmonic Analysis on Commutative Spaces, American Mathematical Society, Providence, 2007 [60] Ziemer W. P. : Weakly Differentiable Functions: Sobolev Spaces and Functions of , Springer-Verlag, New York, 1989

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