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The Rellich-Kondrachov Theorem for Gelfand Pairs Over Hypergroups

Ky T. Bataka1, Yaogan Mensah1,2,∗
1Department of Mathematics, University of Lomé, POB 1515 Lomé 1,Togo

btaka.2005@gmail.com
2International Chair in Mathematical Physics and Applications (ICMPA)-Unesco Chair, University of

Abomey-Calavi, Benin
mensahyaogan2@gmailcom

∗Correspondence: mensahyaogan2@gmailcom

Abstract. Embedding results play important rôles in mathematical analysis. This paper addressessome embedding theorems in the context of Sobolev spaces theory on Gelfand pairs over hypergroups.Mainly, the analogue of the Rellich-Kondrachov theorem is proved.

1. Introduction
Sobolev spaces are well studied on subsets of Rn [1, 4] and on other classical spaces such asRiemannian manifolds [13, 14], metric measure spaces [12], etc. More recently, these studies areextended to other topological algebraic structures such as topological abelian groups, locally com-pact groups, Gelfand pairs over locally compact groups, locally compact commutative hypergroups,etc. More precisely, in [10, 11], Górka et al. constructed a class of Sobolev spaces on Hausdorfflocally compact abelian groups by the means of the Fourier transform. This construction is gener-alized to Gelfand pairs over locally compact groups by Krukowski [16], to compact groups by Kumarand Kumar [17], to noncommutative locally compact groups by Mensah [18] and to noncommutativehypergroups by Bataka et al. [2].In Sobolev spaces theory, embedding theorems are among the useful results that one may ex-pect. They appear as support points in the analysis of partial differential equations and integralequations. Among such embedding theorems is the Rellich-Kondrachov theorem. It is a com-pact embedding theorem in Sobolev spaces theory which intervenes for instance in the proof ofthe Poincaré inequality. The Rellich-Kondrachov theorem took it origin in a special result by
Received: 3 Jul 2024.
Key words and phrases. hypergroup; Sobolev space; Sobolev embedding theorem; Rellich-Kondrachov theorem.1

https://adac.ee
https://doi.org/10.28924/ada/ma.5.3


Eur. J. Math. Anal. 10.28924/ada/ma.5.3 2Rellich [19] and the general case was obtained by Kondrachov [15]. Such compact embedding the-orem has many important applications in analysis for instance in linear elliptic partial differentialequations defined over bounded domains [8, 9], in engineering applications [20], etc.In this paper, we are mainly concerned with a generalization of the Rellich-Kondrachov theoremto a class of Sobolev spaces on Gelfand pairs associated with compact hypergroups. We paved theway with some results which amount to the proof of the Rellich-Kondrachov theorem in the presentframework.The paper is organized as follows. In Section 2, we recall some results which we may need. InSection 3, we present the main results, the culmination of which is the analogue of the Rellich-Kondrachov theorem.
2. Preliminaries

The important ingredients which constitute this section are borrowed from [3, 5, 6]. Let G be alocally compact space. Denote by
• C(G), the set of complex-valued continuous functions on G,
• M(G), the set of Radon measures on G,
• Mb(G), the subset of M(G) consisting of bounded measures,
• M1(G), the subset of Mb(G) consisting of probability measures,
• C(G), the set of compact subspaces of G,
• δx , the point measure at the element x .The setM(G) is endowed with the cône topology while C(G) is endowed with the Michael topology.

Definition 2.1. A locally compact space G is called a hypergroup if the following properties hold.(1) There exists a binary operation ∗ (the convolution) on Mb(G) which turns it into an asso-
ciative algebra such that(a) the mapping (µ, ν) 7→ µ ∗ ν is continuous from Mb(G)×Mb(G) into Mb(G),(b) ∀x, y ∈ G, δx ∗ δy is a probability measure such that supp(δx ∗ δy ) is compact.(c) The mapping (x, y)→ supp(δx ∗ δy ) is continuous from G × G into C(G).(2) There exists a unique element e in G (the neutral element) such that

∀x ∈ G, δx ∗ δe = δe ∗ δx = δx .

(3) There exists an involutive homeomorphism � : G → G such that for all x, y ∈ G,

(δx ∗ δy )� = δy� ∗ δx� .

(4) ∀x, y , z ∈ G, z ∈ supp(δx ∗ δy )⇐⇒ x ∈ supp(δz ∗ δy�).

Definition 2.2. A closed nonempty subset H of a hypergroup G is called a subhypergroup of G if(1) ∀x ∈ H, x� ∈ H,

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Eur. J. Math. Anal. 10.28924/ada/ma.5.3 3(2) ∀x, y ∈ H, supp(δx ∗ δy ) ⊂ H.

Let G be a hypergroup and let K be a compact subhypergroup of G. For x, y ∈ G, x ∗ y standsfor the support of δx ∗ δy . The double coset of x with respect to K is
KxK = {k1 ∗ x ∗ k2 : k1, k2 ∈ K} =

⋃
k1,k2∈K

supp(δk1 ∗ δx ∗ δk2).

For f ∈ C(G), we set f (x ∗ y) =

∫
G

f (z)d(δx ∗ δy )(z) and f �(x) = f (x�).A function f ∈ C(G) is said to be K-bi-invariant if
∀k1, k2 ∈ K,∀x ∈ G, f (k1 ∗ x ∗ k2) = f (x).

Denote by K(G) the set of continuous functions on G with compact support and by K\(G) thesubset of K(G) consisting of K-bi-invariant functions. Now, assume that the hypergroup G isprovided with a left Haar measure and that K is equipped with a normalized Haar measure. For
f ∈ K(G), put

f \(x) =

∫
K

∫
K

f (k1 ∗ x ∗ k2)dk1dk2.

For a measure µ ∈ M(G), set µ\(f ) = µ(f \), f ∈ K(G). The measure µ is called K-bi-invariant if
µ\ = µ. Denote by M\

c(G) the set of complex Radon measures with compact support that are also
K-bi-invariant. For µ, ν ∈ M(G), we define µ ∗ ν by

µ ∗ ν(f ) =

∫∫
G

f (x ∗ y)dµ(x)dν(y), f ∈ C(G).

Also, for f , g ∈ K(G), the convolution product of f and g is the function f ∗ g defined by
(f ∗ g)(x) =

∫
G

f (y)g(y� ∗ x)dy =

∫
G

f (x ∗ y)g(y�)dy.

Provided with this convolution product, K(G) is an algebra and K\(G) is a subalgebra of K(G).
Definition 2.3. Let G be a hypergroup and let K be a compact subhypergroup of G. The pair
(G,K) is called a Gelfand pair if the space (M\

c(G), ∗) is commutative.

We may refer to this Gelfand pair as a hypergroup Gelfand pair. If (G,K) is a hypergroupGelfand pair and if G has a Haar measure then G is unimodular [6].In the rest of the paper, (G,K) is assumed to be a hypergroup Gelfand pair. We denote by Ĝ\the set of bounded continuous functions φ : G −→ C such that(1) φ is K-bi-invariant,(2) φ(e) = 1,(3) ∀x, y ∈ G, ∫
K

φ(x ∗ k ∗ y)dk = φ(x)φ(y),(4) ∀x ∈ G, φ(x�) = φ(x), where φ(x) is the complex conjugate of φ(x).

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Eur. J. Math. Anal. 10.28924/ada/ma.5.3 4

The set Ĝ\ is called the dual set of the hypergroup G [5]. When equipped with the topology ofuniform convergence on compact sets, the space Ĝ\ is a locally compact Hausdorff space.
Definition 2.4 ( [5]). Let (G,K) be a hypergroup Gelfand pair. Let f ∈ K\(G). The Fourier
transform of f is the map f̂ : Ĝ\ −→ C defined by

f̂ (φ) =

∫
G

φ(x�)f (x)dx.

By a classical argument, the inverse Fourier transform is given by
f (x) =

∫
Ĝ\
φ(x)f̂ (φ)dπ(φ)

where the existence of the measure π is ensured by the following theorem (Theorem 2.5).
Theorem 2.5 ( [5]). Let (G,K) be a hypergroup Gelfand pair. There exists a unique nonnegative
measure π on Ĝ\ such that∫

G

|f (x)|2dx =

∫
Ĝ\
|f̂ (φ)|2dπ(φ), ∀f ∈ L1(G) ∩ L2(G).

Hereafter are the analogue of the Hausdorff-Young inequality and its inverse inequality.
Theorem 2.6. [7] Let p, q be such that 1 ≤ p ≤ 2 and 1

p
+

1

p′
= 1. Then, the following inequalities

hold.(1) ‖f̂ ‖p′ ≤ ‖f ‖p , for all f ∈ Lp(G).(2) ‖f ‖p′ ≤ ‖f̂ ‖p , for all f ∈ Lp′(G).

3. Sobolev spaces and embedding results
Definition 3.1. [2] Let (G,K) be a hypergroup Gelfand pair. Let γ : Ĝ\ −→ R+ be a positive
measurable function and let s ∈ (0,+∞). The set

Hs,\γ (G) =

{
f ∈ L2,\(G) :

∫
Ĝ\

(1 + γ(φ)2)s |f̂ (φ)|2dπ(φ) <∞
}

provided with the norm

‖f ‖
Hs,\γ

=

(∫
Ĝ\

(1 + γ(φ)2)s |f̂ (φ)|2dπ(φ)

) 1
2

will be called a Sobolev space.

In the sequel, the symbol ↪→ denotes the continuous embedding.
Theorem 3.2. Let (G,K) be a hypergroup Gelfand pair. Let α > s > 0 and let p =

2α

α+ s
. Let p′

be such that 1

p
+

1

p′
= 1. If (1 + γ2)−1 ∈ Lα(Ĝ\), then Hs,\γ (G) ↪→ Lp

′,\(G).

https://doi.org/10.28924/ada/ma.5.3


Eur. J. Math. Anal. 10.28924/ada/ma.5.3 5

Proof. The conditions about α and s imply 1 < p < 2. Then, by the inverse Hausdorff-Younginequality in Theorem 2.6, we have ‖f ‖p′ ≤ ‖f̂ ‖p .
‖f̂ ‖pp =

∫
Ĝ\
|f̂ (φ)|pdπ(φ)

=

∫
Ĝ\
|f̂ (φ)|p ·

(1 + γ(φ)2)
sp
2

(1 + γ(φ)2)
sp
2

dπ(φ)

=

∫
Ĝ\
|f̂ (φ)|

2p
2 (1 + γ(φ)2)

sp
2 (1 + γ(φ)2)

− sp(2−p)
2(2−p) dπ(φ).

Since p
2

+
2− p

2
= 1, then by the Hölder’s inequality, we have

‖f̂ ‖pp ≤
(∫

Ĝ\
(1 + γ(φ)2)s |f̂ (φ)|2dπ(φ)

) p
2
(∫

Ĝ\
(1 + γ(φ)2)−

sp
2−p dπ(φ)

) 2−p
2

‖f̂ ‖p ≤
(∫

Ĝ\
(1 + γ(φ)2)s |f̂ (φ)|2dπ(φ)

) 1
2
(∫

Ĝ\
(1 + γ(φ)2)−

sp
2−p dπ(φ)

) 2−p
2p

≤ ‖f ‖
Hs,\γ

(∫
Ĝ\

(1 + γ(φ)2)−
sp
2−p dπ(φ)

) 2−p
2p

≤ ‖f ‖
Hs,\γ
‖(1 + γ2)−1‖

s
2
α since α =

sp

2− p .

Finally, ‖f ‖p′ ≤ ‖f̂ ‖p ≤ ‖f ‖Hs,\γ ‖(1 + γ2)−1‖
s
2
α. Thus, Hs,\γ (G) ↪→ Lp

′,\(G). �

Lemma 3.3. Let (G,K) be a hypergroup Gelfand pair. If φ ∈ Ĝ\, then ∀g ∈ K\(G), g ∗φ = ĝ(φ)φ.

Proof. Let f , g ∈ K\(G). Consider Φ(g) = ĝ(φ). Set A =

∫
G

f (x)Φ(g)φ(x�)dx . We have
A = Φ(f )Φ(g) = Φ(f ∗ g) (the convolution Theorem)

=

∫
G

f ∗ g(x)φ(x�)dx

=

∫
G

φ(x�)

(∫
G

f (x ∗ y)g(y�)dy

)
dx

=

∫
G

g(y�)

(∫
G

f (x ∗ y)φ(x�)dx

)
dy (the Fubini’s Theorem)

=

∫
G

g(y�)

(∫
G

f (x)φ(y ∗ x�)dx
)
dy

=

∫
G

f (x)

(∫
G

g(y�)φ(y ∗ x�)dy
)
dx (again the Fubini’s Theorem)

=

∫
G

f (x)

(∫
G

g(y)φ(y� ∗ x�)dy
)
dx (change of variable y → y�)

=

∫
G

f (x)(g ∗ φ)(x�)dx.

https://doi.org/10.28924/ada/ma.5.3


Eur. J. Math. Anal. 10.28924/ada/ma.5.3 6

Since ∫
G

f (x)Φ(g)φ(x�)dx =

∫
G

f (x)(g ∗ φ)(x�)dx for all f ∈ K\(G), then Φ(g)φ(x�) = (g ∗

φ)(x�). Therefore, g ∗ φ = Φ(g)φ = ĝ(φ)φ. �

Theorem 3.4. Let (G,K) be a hypergroup Gelfand pair. Let f ∈ Hs,\γ (G). If y ∈ G, then∫
G

|f (x ∗ y�)− f (x)|2dx ≤

(
sup
φ∈Ĝ\

|φ(y)− 1|2

(1 + γ(φ)2)s

)
· ‖f ‖2

Hs,\γ
.

Proof. Fix y ∈ G. Let f ∈ K\(G). Set fy (x) = f (x ∗ y�), x ∈ G. We have,
f̂y (φ) =

∫
G

φ(x�)f (x ∗ y�)dx

=

∫
G

φ(y� ∗ x�)f (x)dx (change of variable x → x ∗ y�)

=

∫
G

φ(y� ∗ x)f (x�)dx (change of variable x → x�)

= (φ ∗ f )(y�) = (f ∗ φ)(y�)

= f̂ (φ)φ(y�)(Lemma 3.3).∫
G

|fy (x)− f (x)|2dx =

∫
Ĝ\
|f̂y (φ)− f̂ (φ)|2dπ(φ)(Theorem 2.5)

=

∫
Ĝ\
|f̂ (φ)φ(y�)− f̂ (φ)|2dπ(φ)

=

∫
Ĝ\
|f̂ (φ)(φ(y�)− 1)|2dπ(φ)

=

∫
Ĝ\
|f̂ (φ)|2|φ(y�)− 1|2dπ(φ)

=

∫
Ĝ\
|f̂ (φ)|2|φ(y)− 1|2

(1 + γ(φ)2)s

(1 + γ(φ)2)s
dπ(φ)

≤

(
sup
φ∈Ĝ\

|φ(y)− 1|2

(1 + γ(φ)2)s

)
· ‖f ‖2

Hs,\γ
.

Since K\(G) is dense in Hs,\γ (G), the result holds for all f ∈ Hs,\γ (G). �

Theorem 3.5. Let (G,K) be a hypergroup Gelfand pair. If f ∈ Hs,\γ (G), then there exists η ∈ K\(G)

such that
‖f ∗ η − f ‖2 ≤ sup

y∈supp(η)
sup
φ∈Ĝ\

|φ(y)− 1|
(1 + γ(φ)2)

s
2

· ‖f ‖
Hs,\γ
.

Proof. Since G is a locally compact compact Hausdorff space, then it is a Tychonoff space. Therefore,there exists η ∈ K\(G) such that η(e) 6= 0, η ≥ 0 and ∫
G

η(x)dx = 1. Then, we have
‖f ∗ η − f ‖2 =

(∫
G

|f ∗ η(x)− f (x)|2dx
) 1
2

https://doi.org/10.28924/ada/ma.5.3


Eur. J. Math. Anal. 10.28924/ada/ma.5.3 7

=

(∫
G

∣∣∣∣∫
G

f (x ∗ y�)η(y)dy − f (x)

∣∣∣∣2 dx
) 1
2

=

(∫
G

∣∣∣∣∫
G

f (x ∗ y�)η(y)dy − f (x)

∫
G

η(y)dy

∣∣∣∣2 dx
) 1
2

=

(∫
G

∣∣∣∣∫
G

(f (x ∗ y�)− f (x))η(y)dy

∣∣∣∣2 dx
) 1
2

≤
∫
G

(∫
G

|f (x ∗ y�)− f (x)|2|η(y)|2dx
) 1
2

dy

≤
∫
G

|η(y)|
(∫

G

|f (x ∗ y�)− f (x)|2dx
) 1
2

dy

≤ sup
y∈supp(η)

sup
φ∈Ĝ\

|φ(y)− 1|
(1 + γ(φ)2)

s
2

· ‖f ‖
Hs,\γ

(use Theorem 3.4).

�

In the rest of the paper, we assume that G is compact.
Theorem 3.6. Let G be a compact hypergroup. Let (G,K) be a hypergroup Gelfand pair. Let
p, q ∈ (1,∞). If a sequence (fn) ⊂ Lp,\(G) converges weakly to a function f , then for every
η ∈ K\(G) the sequence (fn ∗ η) converges strongly to f ∗ η in Lq,\(G).

Proof. Since the sequence (fn) converges weakly to f , then by [4, Proposition 3.5] there exists apositive real M such that
‖fn‖p ≤ M and ‖f ‖p ≤ M.

We have
|fn ∗ η(x)| =

∣∣∣∣∫
G

fn(y)η(y� ∗ x)dy

∣∣∣∣
≤
∫
G

|fn(y)η(y� ∗ x)| dy

≤ ‖fn‖p
(∫

G

|η(y� ∗ x)|p′dy
) 1

p′

≤ M‖η‖p′

where p′ is such that 1

p
+

1

p′
= 1. Since G is compact, the constant function x 7−→ M‖η‖p′ isintegrable. Therefore, by the Dominated Convergence Theorem, we have

fn ∗ η(x) =

∫
G

fn(y)η(y� ∗ x)dy
n→∞−−−→

∫
G

f (y)η(y� ∗ x)dy = f ∗ η(x).

https://doi.org/10.28924/ada/ma.5.3


Eur. J. Math. Anal. 10.28924/ada/ma.5.3 8Then,
|fn ∗ η(x)− f ∗ η(x)| =

∣∣∣∣∫
G

fn(y)η(y� ∗ x)dy −
∫
G

f (y)η(y� ∗ x)dy

∣∣∣∣
=

∣∣∣∣∫
G

(fn(y)− f (y))η(y� ∗ x)dy

∣∣∣∣
≤ ‖fn − f ‖p

(∫
G

|η(y� ∗ x)|p′dy
) 1

p′

≤ 2M

(∫
G

|η(z)|p′d(δy� ∗ δx)(z)

) 1
p′

≤ 2M‖η‖p′ .Again, by the Dominated Convergence Theorem, we obtain
lim
n→∞

‖fn ∗ η − f ∗ η‖q = 0.

�

Hereafter is the analogue of the Rellich-Kondrachov theorem for hypergroup Gelfand pairs.
Theorem 3.7. Assume that G is compact. Let (G,K) be a hypergroup Gelfand pair. Let α > s > 0.
Let p := 2α

α+s and let p′ be such that 1

p
+

1

p′
= 1. If (1 + γ2)−1 ∈ Lα(Ĝ\) and

lim
y→e

(
sup
φ∈Ĝ\

|φ(y)− 1|
(1 + γ(φ)2)

s
2

)
= 0,

then Hs,\γ (G) embeds compactly in Lq,\(G) for every q ∈ [1, p′].

Proof. In Theorem 3.2, we obtained that Hs,\γ (G) ↪→ Lp
′,\(G); since G is compact and p′ > q, then

Lp
′,\(G) ↪→ Lq,\(G). Therefore, we have that Hs,\γ (G) ↪→ Lq,\(G). Now, let (fn) be a boundedsequence in Hs,\γ (G). Then, (fn) is a bounded sequence in Lp′,\(G). There exists M > 0 such that

∀n ∈ N, ‖fn‖p′ 6 M.For g ∈ Lp,\(G), we have
|〈fn, g〉| 6 ‖fn‖p′‖g‖p 6 M‖g‖p.Therefore, (fn) is weakly bounded. It admits a subsequence (hn) which converges weakly to

h ∈ Lp′,\(G). Take ε > 0 and η ∈ K\(G) such that ‖h ∗ η − h‖2 < ε.By Theorem 3.5 and Theorem 3.6, we have
‖hn − h‖2 ≤ ‖hn − hn ∗ η‖2 + ‖hn ∗ η − h ∗ η‖2 + ‖h ∗ η − h‖2

≤ sup
y∈supp(η)

(
sup
φ∈Ĝ\

|φ(y)− 1|
(1 + γ(φ)2)

s
2

)
· ‖hn‖Hs,\γ + ‖hn ∗ η − h ∗ η‖2 + ε

≤ 2ε+ ‖hn ∗ η − h ∗ η‖2.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.3 9As the above inequality is realized for an arbitrary ε, then
‖hn − h‖2 ≤ ‖hn ∗ η − h ∗ η‖2.

Therefore,
lim
n→∞

‖hn − h‖2 = lim
n→∞

‖hn ∗ η − h ∗ η‖2 = 0.

Thus, (hn) converges to h in L2,\(G). Since G is compact, we apply the Vitali’s convergence theoremto conclude that (hn) converges to h in Lq,\(G). �

References
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.[2] K. T. Bataka, E. M. Egwe and Y. Mensah, Sobolev spaces on hypergroup Gelfand pairs, Far East J. Math. Sci.,142(1) (2024), 57–69, https://doi.org/10.17654/097208712505.[3] W. R. Bloom and H. Heyer, Harmonic analysis of probability measures on hypergroups, de Gruyter Studies inMathematics 20, Berlin, 1995.[4] H. Brezis, Sobolev spaces and partial differential equations, Springer, 2011.[5] K. G. Brou, I. Toure and K. Kangni, A Plancherel theorem on a noncommutative hypergroup, Int. J. Anal. Appl., 20(2022), 20–32, https://doi.org/10.28924/2291-8639-20-2022-32.[6] K. G. Brou and K. Kangni, On Gelfand pair over hypergroups, Far East J. Math. Sci., 132(1) (2021) 63-76, http:

//dx.doi.org/10.17654/MS132010063.[7] K. G. Brou, P. Coulibaly and K. Kangni, Uncertainly principles for a non commutative hypergroup, Adv. in Math.Sci. J., 12 (2), (2023) 381-392, https://doi.org/10.37418/amsj.12.2.6.[8] S. Chua, S. Rodney and R. Wheeden, A compact embedding theorem for generalized Sobolev spaces, Pacific J.Math., 265(1), (2013) 17-57, 10.2140/pjm.2013.265.17.[9] D. E. Edmunds and W. D. Evans, Spectral theory and differential operators, Oxford Mathematical Monograph, 2ndedition, Oxford Science Publications, Oxford University Press, 2018.[10] P. Górka, T. Kostrzewa and E. G. Reyes, Sobolev spaces on locally compact abelian groups: compact embeddingsand local spaces, Journal of Function Spaces, (2014), Article ID 404738, 6 pages, http://dx.doi.org/10.1155/
2014/404738.[11] P. Górka, T. Kostrzewa and E. G. Reyes, Sobolev spaces on locally compact abelian groups and the bosonic stringequation, J. Aust. Math. Soc., 98 (2015), 39-53, 10.1017/S1446788714000433.[12] P. Hajlasz, Sobolev spaces on an arbitrary metric space, Potential Anal., 5(4) (1996), 403-415, https://doi.org/
10.1007/BF00275475.[13] P. Hajlasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc., 145(688) (2000).[14] E. Hebey, Sobolev Spaces on Riemannian Manifolds, Lecture Notes in Mathematics, 1635, Springer, Berlin, 1996.[15] V. I. Kondrachov, On certain properties of functions in the spaces Lp , Dokl. Akad. Nauk SSSR, 48 (8) (1945) 563-566.[16] M. Krukowski, Sobolev spaces on Gelfand pairs, (2020), https://arxiv.org/abs/2003.08519v1.[17] M. Kumar and N. S. Kumar, Sobolev spaces on compact groups, Forum Math. 35(4) (2023), 901- 911, https:
//doi.org/10.1515/forum-2022-0076.[18] Y. Mensah, Sobolev spaces arising from a generalized spherical Fourier transform, Adv. Math. Sci. J., 10(7) (2021),2947-2955, https://doi.org/10.37418/amsj.10.7.3.[19] F. Rellich, Ein Satz über mittlere Konvergenz, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl., 141, (1930) 30-35.

https://doi.org/10.28924/ada/ma.5.3
 https://doi.org/10.17654/097208712505
https://doi.org/10.28924/2291-8639-20-2022-32
http://dx.doi.org/10.17654/MS132010063
http://dx.doi.org/10.17654/MS132010063
https://doi.org/10.37418/amsj.12.2.6
10.2140/pjm.2013.265.17
http://dx.doi.org/10.1155/2014/404738
http://dx.doi.org/10.1155/2014/404738
10.1017/S1446788714000433
https://doi.org/10.1007/BF00275475
https://doi.org/10.1007/BF00275475
https://arxiv.org/abs/2003.08519v1
https://doi.org/10.1515/forum-2022-0076
https://doi.org/10.1515/forum-2022-0076
https://doi.org/10.37418/amsj.10.7.3


Eur. J. Math. Anal. 10.28924/ada/ma.5.3 10

[20] A. Rozanova-Pierrat, Generalization of Rellich-Kondrachov theorem and trace compactness for fractal boundaries,In : M. Rosaria Lancia, A. Rozanova-Pierrat (Eds.), Fractals in engineering: Theoretical aspects and Numericalapproximations, Springer, (2021) pp. 155-173.

https://doi.org/10.28924/ada/ma.5.3

	1. Introduction
	2. Preliminaries
	3. Sobolev spaces and embedding results
	References

