Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications

Ruimin Wu; Songbai Wang

doi:10.1515/math-2022-0595

Article Open Access

Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications

Ruimin Wu and Songbai Wang

Published/Copyright: July 29, 2023

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$Open Mathematics$

From the journal Open Mathematics Volume 21 Issue 1

Abstract

Let m ∈ N and be a generalized Orlicz function. We obtained some interpolation inequalities for derivatives in generalized Orlicz-Sobolev spaces W m , φ ( R n ) . As applications, we established a compact Sobolev embedding on domain and a Landau-Kolmogorov-type inequality in generalized Orlicz spaces. And we introduced the Sobolev φ -capacity and studied some of its properties.

Keywords: interpolation inequality; generalized Orlicz-Sobolev space; maximal function operator

MSC 2010: 26D10; 46E30; 46E35

1 Introduction

The theory of Sobolev spaces as one of the most important tools in analysis has been developed a great deal for more than nine decades now. Scholars have studied various variants of Sobolev space. Recently, Harjulehto et al. [1] studied generalized Sobolev-Orlicz spaces and established the existence of solutions of the φ -Laplace equation with zero and non-zero right-hand side. In the classical Sobolev spaces, it is well known that, for any u ∈ W m , p ( R n ) with m ∈ N and 1 ≤ p < ∞ ,

∥ ∇ j u ∥ L p ( R n ) ≤ C [ ε − j ∕ ( m − j ) ‖ u ‖ L p ( R n ) + ε ∥ ∇ m u ∥ L p ( R n ) ] ,

where 0 ≤ j ≤ m − 1 and ε ∈ ( 0 , ∞ ) . This type of inequality and its applications played an important role in the partial differential equations (simply denoted by PDEs) and variation problems. The purpose of this note is to study these estimates of intermediate derivative of functions in generalized Orlicz-Sobolev spaces.

The classical Orlicz spaces are well known and have been studied for a long period (see, for instance, the monograph [2] and related references). The generalized Orlicz spaces are also called Musielak-Orlicz spaces and Nakano spaces and are a class of Banach function spaces, which include some spaces of interest in harmonic analysis and PDEs as special cases (see, for instance, [3–6]). Some authers in [1,7–9] have systematically studied the operators of classical harmonic analysis and generalized Sobolev spaces and established a very broad theory that unites and extends the previous work. In this article, we studied the generalized Orlicz spaces that adapt some assumptions presented in the recent book [10], where there are more details on this topic.

Different from the classical Lebesgue spaces, the translations are not invariant in generalized Orlicz spaces. In general, the translation and the convolution are not continuous on L φ ( R n ) (for example, see [11,12]). However, there need some other important tools still for us to study the generalized Orlicz spaces and generalized Orlicz-Sobolev spaces. One of the most important and elementary results is that the Hardy-Littlewood maximal function is bounded on L φ ( Ω ) . When the generalized Orlicz function satisfies some minimal assumptions, see [7,8]. By extrapolation, Cruz-Uribe and Hästö [13] discussed some embedding theorems for the generalized Orlicz-Sobolev spaces. Some more details for the generalized Orlicz-Sobolev spaces are collected in [10].

Invoking the boundedness of the Hardy-Littlewood maximal function on the generalized Orlicz spaces and the Sobolev integral representation, we establish some interpolation inequalities in the generalized Orlicz-Sobolev spaces both on R n and on domains Ω . Using these inequalities, we consider a compact Sobolev embedding theorem of the generalized Orlicz-Sobolev spaces. As an application, we obtain a Landau-Kolmogorov-type inequality in generalized Orlicz spaces. In the last section, we introduce the Sobolev φ -capacity and study the properties of the Sobolev capacity. Our idea is strongly motivated by [14].

2 Preliminaries

Hereafter, we say that a function f is almost increasing if there exists L ≥ 1 such that for all s ≤ t and f ( s ) ≤ L f ( t ) . Almost decreasing is defined analogously. We say that f is increasing/decreasing for L = 1 .

Definition 2.1

Let φ : [ 0 , ∞ ) → [ 0 , ∞ ] be an increasing function such that φ ( 0 ) = lim t → 0 + φ ( t ) = 0 and lim t → ∞ φ ( t ) = ∞ . The function φ is called a Φ -prefunction. If the function φ satisfies that t → φ ( t ) t is almost increasing on ( 0 , ∞ ) , we call that φ is a weak Φ -function and denoted by φ ∈ Φ w .

Definition 2.2

Given φ ∈ Φ w , by φ − 1 ,we denote the left-inverse of φ

φ ( τ ) ≔ inf { t ≥ 0 : φ ( t ) ≥ τ } .

By the definition of the left inverse, we know that φ − 1 is increasing and φ − 1 ∘ φ ( t ) ≤ t , and if φ is left-continuous, then φ ∘ φ − 1 ( τ ) ≤ τ .

Definition 2.3

We say that two functions φ and ψ are equivalent, denoted by φ ≃ ψ , if there exists L > 1 such that φ x , t L ≤ ψ ( x , t ) ≤ φ ( x , L t ) for all x and t > 0 .

Remark 2.4

Short calculations show that ≃ is an equivalence relation. Then, two functions φ ≃ ψ if and only if φ − 1 ≈ ψ − 1 (see [10, Theorem 2.3.6]).

To define generalized Orlicz spaces, we first recall the definition of generalized Φ -functions.

Definition 2.5

Given Ω ⊂ R n , a function φ : Ω × [ 0 , ∞ ) → [ 0 , ∞ ) is called a generalized Φ -function if φ ( x , t ) is a Φ w -function for every x ∈ Ω and φ ( x , ∣ f ( x ) ∣ ) is measurable for every measurable function f on Ω . It is denoted by φ ( x , t ) ∈ Φ w ( Ω ) .

Now, we recall generalized Orlicz spaces.

Definition 2.6

Let φ ∈ Φ w ( Ω ) . For any measurable function f on Ω , we define the semimodular ϱ φ ( ⋅ ) by:

ϱ φ ( ⋅ ) ( f ) ≔ ∫ Ω φ ( x , ∣ f ( x ) ∣ ) d x .

The generalized Orlicz space is defined as the set L φ ( ⋅ ) ( Ω ) :

L φ ( ⋅ ) ( Ω ) ≔ f measurable : ϱ φ ( ⋅ ) f λ < ∞ , for some λ > 0 ,

which is equipped with the (Luxemburg) norm by:

‖ f ‖ L φ ( ⋅ ) ( Ω ) ≔ inf λ > 0 : ϱ φ ( ⋅ ) f λ ≤ 1 .

Remark 2.7

If φ ≃ ψ , then L φ ( Ω ) = L ψ ( Ω ) (see [10, Theorem 3.2.6]).

For any x ∈ R n and any locally integrable function f , the Hardy-Littlewood maximal function f is defined by:

M ( f ) ( x ) ≔ sup Q cube : Q ∋ x 1 ∣ Q ∣ ∫ Q ∣ f ( y ) ∣ d y .

We also recall a family of hypotheses that are closely related to the boundedness of the Hardy-Littlewood maximal function on generalized Orilcz spaces.

Definition 2.8

Given φ ∈ Φ w ( Ω ) and 0 < p < ∞ , we say that φ satisfies:

(A0), if there exists α ∈ ( 0 , 1 ] such that α ≤ φ − 1 ( x , 1 ) ≤ α − 1 for almost x ∈ R n , here and hereafter, φ − 1 ( x , t ) ≔ inf { τ ∈ [ 0 , ∞ ) : φ ( x , τ ) ≥ t } .
(A1), if there exists β ∈ ( 0 , 1 ) such that β φ − 1 ( x , t ) ≤ φ − 1 ( y , t ) for every t ∈ 1 , 1 ∣ B ∣ , almost every x , y ∈ Ω ∩ B and every ball B with ∣ x − y ∣ ≤ 1 .
(A2), if for every s > 0 , there exist β ∈ ( 0 , 1 ] and h ∈ L 1 ( Ω ) ∩ L ∞ ( Ω ) such that β φ − 1 ( x , t ) ≤ φ − 1 ( y , t ) for almost every x , y ∈ Ω and every t ∈ [ h ( x ) + h ( y ) , s ] .
(Inc)_p, if t → t − p φ ( x , t ) is increasing for almost x ∈ Ω .
(aInc)_p, if t → t − p φ ( x , t ) is almost increasing for almost x ∈ Ω .
(Dec)_p, if t → t − p φ ( x , t ) is decreasing for almost x ∈ Ω .
(aDec)_p, if t → t − p φ ( x , t ) is almost decreasing for almost in x ∈ Ω .

We say that φ satisfies (aInc) if there exists p > 1 such that φ satisfies (aInc)_p and (aDec) for some p < ∞ .

Remark 2.9

If φ satisfies (aInc)_p and (aDec)_q with p , q ∈ ( 0 , ∞ ) , then p ≤ q .

Lemma 2.10

[7, Theorem 4.7] Let φ ∈ Φ w ( Ω ) satisfy (A0), (A1), (A2), and (aInc). Then, the maximal operator is bounded on L φ ( Ω ) .

Definition 2.11

Let φ ∈ Φ w ( Ω ) . The function u ∈ L φ ( Ω ) ∩ L loc 1 ( Ω ) belongs to the Sobolev space W m , φ ( Ω ) , where m ∈ N , if its weak partial derivatives ∂ α u exist and belong to L φ ( Ω ) for all ∣ α ∣ ≤ m . We define a semimodular on W m , φ ( Ω ) by:

ϱ W m , φ ( Ω ) ( u ) ≔ ∑ 0 ≤ ∣ α ∣ ≤ m ϱ φ ( ⋅ ) ( ∂ α u ) < ∞ .

And it induces a (quasi-)norm by:

‖ u ‖ W m , φ ( Ω ) ≔ inf λ > 0 : ϱ W m , φ ( Ω ) u λ ≤ 1 .

Remark 2.12

If we take φ ( x , t ) = t p , p ∈ ( 0 , ∞ ) , then W m , φ ( Ω ) goes back to the classical Sobolev spaces. From [10, Lemma 6.1.5], we know that

‖ u ‖ W m , φ ( Ω ) ≈ ∑ k = 0 m ∥ ∇ k u ∥ L φ ( Ω ) .

From now on, we let ∣ u ∣ φ ≔ ‖ u ‖ L φ ( Ω ) , for any m ∈ Z and ∣ u ∣ m , φ ≔ ∑ ∣ α ∣ = j ∣ ∂ α u ∣ L φ ( Ω ) , where α is the multi-index, α = ( α 1 , … , α n ) ∈ Z + n and ∣ α ∣ = α 1 + ⋯ + α n .

Lemma 2.13

[10, Theorem 6.1.4 and Theorem 6.2.8] Let φ ∈ Φ w ( Ω ) and L φ ( Ω ) ⊂ L loc 1 ( Ω ) .

If φ satisfies (aInc) and (aDec), then W k , φ ( Ω ) is uniformly convex and reflexive.
If Ω is a bounded domain and φ satisfies (A0) and (A1), then the Pooinvaré inequality holds, that is, for every u ∈ W 0 1 , φ ( Ω ) ,
∣ u ∣ φ ≤ C diam ( Ω ) ∣ ∇ u ∣ φ ( Ω ) .

Definition 2.14

Let φ ∈ Φ w ( Ω ) be such that m ∈ N . The Sobolev space W 0 m , φ ( Ω ) with zero boundary values is the closure of C 0 ∞ ( Ω ) ∩ W m , φ ( Ω ) in W m , φ ( Ω ) .

Lemma 2.15

[10, Lemma 6.1.6] Let φ ∈ Φ w ( Ω ) satisfy (A0) and (aInc) p and p ∈ [ 1 , ∞ ) , Then, W m , φ ( Ω ) ↪ W loc m , 1 ( Ω ) . Moreover, if ∣ Ω ∣ < ∞ , W m , φ ( Ω ) ↪ W m , p ( Ω ) .

Lemma 2.16

[10, Theorem 6.4.7] Let φ ∈ Φ w ( Ω ) satisfy (A0), (A1), (A2), and (aDec). Then, C ∞ ( Ω ) ∩ W 1 , φ ( Ω ) is dense in W 1 , φ ( Ω ) .

Lemma 2.17

[10, Corollary 6.3.3] Let φ ∈ Φ w ( R n ) satisfy (A0), (A1), (A2), and (aDec) p + for p + < n . Define ψ − 1 ( x , t ) ≔ t − 1 n φ − 1 ( x , t ) , Then,

W 1 , φ ( Ω ) ↪ L ψ ( Ω ) .

Lemma 2.18

[13, Theorem 6.15] or [10, Theorem 6.3.7] Let φ ∈ Φ w ( R n ) satisfy (A0), (A1), (A2), and (aDec) p + for p + < n . Define ψ − 1 ( x , t ) ≔ t − α φ − 1 ( x , t ) with α ∈ 0 , 1 n . Then, the embedding

W 0 1 , φ ( Ω ) ↪ L ψ ( Ω )

is compact.

The following technique lemma is also needed.

Lemma 2.19

[15, p. 17, Theorem 1.32] Let f ∈ L loc 1 ( R n ) . If 0 < α < n , β > 0 , and r > 0 , then for any x ∈ R n ,

∫ B r ( x ) ∣ f ( y ) ∣ ∣ x − y ∣ n − α d y ≤ C α r α M ( f ) ( x ) .

3 Interpolation inequalities and their applications

We first state the interpolation inequalities on R n for the generalized Orlicz-Sobolev spaces.

Theorem 3.1

Let 0 ≤ j ≤ m and let φ ∈ Φ w ( R n ) satisfy (A0), (A1), (A2), and (aInc). Then, there exists finite constants K and K ′ depending only on n , m , and φ such that for every u ∈ W m , φ ( Ω ) and all ε ∈ ( 0 , ∞ ) , we have

(3.1) ∣ u ∣ j , φ ≤ K ε ∣ u ∣ m , φ + ε − j m − j ∣ u ∣ φ ;

(3.2) ∣ u ∣ j , φ ≤ 2 K ∣ u ∣ m , φ j m + ∣ u ∣ φ n − j m ;

(3.3) ‖ u ‖ j , φ ≤ K ′ ε ‖ u ‖ m , φ + ε − j m − j ∣ u ∣ φ ;

(3.4) ‖ u ‖ j , φ ≤ 2 K ′ ‖ u ‖ m , φ j m ∣ u ∣ φ m − j m .

Proof

We first prove (3.1). Let j ∈ { 1 , … , m } and u ∈ W m , φ ( R n ) . Fix a ball B r ( x ) centered at x ∈ R n with the radius r > 0 . By the fact that W m , φ ( R n ) ↪ W loc m , 1 ( R n ) (see Lemma 2.15) and the Sobolev’s integral representation [16, p. 113 (3.58)], we conclude that, for any β ∈ Z + n with ∣ β ∣ = j ,

∣ D β u ( x ) ∣ ≤ C 1 r − j − m ∫ B r ( x ) ∣ u ( y ) ∣ d y + ∑ ∣ α ∣ = m ∫ B r ( x ) ∣ D α u ∣ ∣ x − y ∣ n − m + j d y ≤ C r − j 1 ∣ B r ( x ) ∣ ∫ B r ( x ) ∣ u ( y ) ∣ d y + ∑ ∣ α ∣ = m ∫ B r ( x ) ∣ D α u ∣ ∣ x − y ∣ n − m + j d y ≤ C r − j M u ( x ) + ∑ ∣ α ∣ = m ∫ B r ( x ) ∣ D α u ∣ ∣ x − y ∣ n − m + j d y ,

where C is only dependent on n and m .

If m − j < n , from Lemma 2.19, there exists a positive constant C such that for any δ ∈ ( 0 , ∞ ) , have

∫ B r ( x ) ∣ D α u ( y ) ∣ ∣ x − y ∣ n − m + j d y ≤ C r m − j M ( D α u ) ( x ) ,

which implies that

∣ D β u ( x ) ∣ ≤ C r − j M u ( x ) + r m − j ∑ ∣ α ∣ = m M ( D α u ) ( x ) .

If m − j ≥ n , then

∣ D β u ( x ) ∣ ≤ C r − j M u ( x ) + ∑ ∣ α ∣ = m r m − j − n ∫ B r ( x ) ∣ D α u ∣ d y ≤ C r − j M u ( x ) + ∑ ∣ α ∣ = m r m − j M ( D α u ) ( x ) .

Since L φ ( R n ) is a quasi-Banach space and M is bounded on L φ ( R n ) (see Lemma 2.10), we have

(3.5) ∣ D β u ( x ) ∣ φ ≤ C r − j ∣ M u ∣ φ + r m − j ∑ ∣ α ∣ = m ∣ M ( D α u ) ∣ φ ≤ C r − j ∣ u ∣ φ + r m − j ∑ ∣ α ∣ = m ∣ D α u ∣ φ .

So we obtain (3.1) by setting ε = r m − j .

The remainder of inequalities are applications of (3.1). In fact, (3.3) follows from (3.1) directly. We obtain (3.2) and (3.4) by choosing ε = [ ∣ u ∣ m , φ − 1 ∣ u ∣ φ ] m − j m as in (3.1) and (3.3). Thus, we finish the proof of Theorem 3.1.□

To obtain the similar interpolation inequalities for the generalized Orlicz-Sobolev spaces on domains Ω , we need to strengthen the assumptions of the generalized Orlicz function φ .

Definition 3.2

Let Ω ⊂ R n . We say that φ ∈ Φ w ( Ω ) and satisfies (A1) Ω , if there exists a constant β ∈ ( 0 , 1 ] such that β ∣ x − y ∣ t 1 ∕ n + 1 φ − 1 ( y , t ) ≤ φ − 1 ( x , t ) for all x , y ∈ Ω and t > 1 .

Definition 3.3

We say that ψ ∈ Φ w ( R n ) is an extension of φ ∈ Φ w ( Ω ) if ψ ∣ Ω ≃ φ .

Lemma 3.4

[17, Theorem 3.5] Suppose that Ω ⊂ R n and φ ∈ Φ w ( Ω ) . Then, there exists an extension ψ ∈ Φ w ( R n ) of φ , when φ satisfies (A0), (A1), and (A2), and where, if and only if φ satisfies (A0), (A1) Ω , and (A2).

If φ satisfies (aInc) p and/or (aDec) q , then the extension ψ can be taken to satisfy it/them as well.

Definition 3.5

Let x ∈ R n , B 1 be an open ball centered at x and B 2 be an open ball with x ∉ B 2 . A finite cone with vertex at x , which is associated with B 1 and B 2 and denoted by C x , is a set of the form:

C x = B 1 ∩ { x + λ ( y − x ) : y ∈ B 2 , λ > 0 } .

Definition 3.6

An open set Ω ⊂ R n has the uniform cone property if there exists a finite collection of open sets { U j } (not necessarily bounded) and an associated collection { C j } of finite cones such that the following hold:

there exists δ > 0 such that Ω δ = { x ∈ Ω : dist ( x , ∂ Ω ) < δ } ⊂ ∪ j U j ;
for every index j and every x ∈ Ω ∩ U j , x + C j ⊂ Ω .

Theorem 3.7

Given an open set Ω ⊂ R n having the uniform cone property, and provided φ ∈ Φ w ( Ω ) satisfying (A0), (A1) Ω , and (A2), then for any k ∈ N , there exists an extension operator

E k : W k , φ ( Ω ) → W k , φ ( R n ) ,

such that E k u ( x ) = u ( x ) , a.e. x ∈ Ω , and

‖ E k u ‖ W k , φ ( R n ) ≤ C φ , k , Ω ‖ u ‖ W k , φ ( Ω ) .

Proof

The proof of Theorem 3.7 in generalized Orlicz-Sobolev spaces is nearly identical to that in the classical setting. We can follow the proof of [18, Theorem 5.28] step by step, since the following facts are true under our hypotheses.

By Lemma 3.4, φ immediately extends to a generalized Orlicz function on R n .
By Lemma 2.16, functions in C ∞ ( Ω ) are dense in W k , φ ( Ω ) .
If ϕ is a mollifier, then ϕ ε ∗ f converges to f in the sense of L φ ( Ω ) as ε → 0 + .
Singular integral operators with kernels of the form:
K ( x ) = G ( x ) ∣ x ∣ n ,
where G is bounded on R n \ { 0 } , has compact support, is homogeneous of degree zero on B R ( 0 ) \ { 0 } for some R > 0 , and has ∫ ∣ x ∣ = R G ( x ) d x = 0 , which is bounded on L φ ( Ω ) . By [13, Corollary 6.2], kernels are bounded. We remark here that [13, Corollary 6.2] holds true for the standard Calderón-Zygmund operators; however, it is also true for the rough singular integral operators with these kernels.

We omit the details and finish the proof of Theorem 3.7.□

Theorem 3.8

Suppose that Ω ⊂ R n and φ ∈ Φ w ( Ω ) are as in Theorem 3.7. Moreover, if φ satisfies (aInc), then there exists a positive constant K depending only on n , m , φ , and Ω , such that for all u ∈ W m , φ ( Ω ) , 0 ≤ j < m , and all ε ∈ ( 0 , ∞ ) , have

(3.6) ‖ u ‖ j , φ ≤ K ε ‖ u ‖ m , φ + ε − j n − j ∣ u ∣ φ

(3.7) ‖ u ‖ j , φ ≤ 2 K ‖ u ‖ m , φ j m ∣ u ∣ φ m − j m .

Proof

By Lemma 3.4, there exists a generalized Orlicz function φ ∈ Φ w ( R n ) , which satisfies (A0), (A1), (A2), and (aInc). From Theorems 3.7 and 3.1, we deduce that

‖ u ‖ j , φ ≤ ‖ E u ‖ j , φ ˜ , R n ≤ C ε ‖ E u ‖ m , φ ˜ , R n + ε − j m − j ∣ E u ∣ φ ˜ , R n ≤ C ε ‖ E u ‖ m , φ + ε − j m − j ∣ E u ∣ φ

and

‖ u ‖ j , φ ≤ ‖ E u ‖ j , φ ˜ , R n ≤ 2 C ‖ E u ‖ m , φ ˜ , R n j m ∣ E u ∣ φ ˜ , R n m − j m ≤ 2 C ‖ E u ‖ m , φ j m ∣ E u ∣ φ m − j m .

It completes the proof of Theorem 3.8.□

Theorem 3.9

If Ω ⊂ R n and φ are as in Theorem 3.8, then for any ε > 0 , Inequalities (3.1)–(3.4) hold for any u ∈ W 0 m , φ ( Ω ) .

Proof

By Lemma 3.4 again, there exists a generalized Orlicz function φ ˜ satisfying (A0), (A1), (A2), and (aInc). Let u ˜ denote the zero extension of u to R n \ Ω . Similar to the proof of [18, Lemma 3.27], we can prove that the mapping u → u ˜ maps W 0 m , φ ( Ω ) isometrically into W m , φ ˜ ( R n ) . Thus, we obtain the conclusion immediately by Theorem 3.1.□

Now, we turn to apply the interpolation inequalities to study a Sobolev embedding theorem for the generalized Orlicz-Sobolev spaces.

Lemma 3.10

Let m ∈ N , j ∈ Z + , j < m , and Ω ⊂ R n be an open set. Let φ , ψ ∈ Φ c ( Ω ) .

If the embedding
(3.8) W m , φ ( Ω ) ↪ W j , ψ ( Ω )
is compact, then for any ε ∈ ( 0 , ∞ ) there exists a positive constant C ε such that for all f ∈ W m , φ ( Ω ) , have
(3.9) ‖ f ‖ j , ψ ≤ C ε ∣ f ∣ φ + ε ‖ f ‖ m , φ .
If (3.9) holds and the embedding W m , φ ( Ω ) ↪ L φ ( Ω ) is compact for any ε > 0 , then embedding (3.8) is also compact.

Proof

Suppose that (3.9) does not hold for all ε ∈ ( 0 , ∞ ) . Then, there exist ε 0 ∈ ( 0 , ∞ ) and a sequence of functions f k ∈ W m , φ ( Ω ) , k ∈ N , such that ‖ f k ‖ m , φ = 1 and
(3.10) ‖ f k ‖ j , ψ > k ∣ f k ∣ φ + ε 0 ‖ f k ‖ m , φ .
By the compact embedding (3.8), there exists a positive constant M such that ‖ f k ‖ j , ψ ≤ M for all k ∈ N . From this and (3.10), it follows that ∣ f k ∣ φ < M ∕ k , which further implies that
(3.11) lim k → ∞ ∣ f k ∣ φ = 0 .
Letting k → ∞ on both sides of (3.10), the assumption that ‖ f k ‖ m , φ = 1 yields
(3.12) liminf k → ∞ ‖ f k ‖ m , φ ≥ ε 0 .
By the compactness of the set { f k } k ∈ N in W j , ψ ( Ω ) , there exists a subsequence of { f k } k ∈ N , which is still denoted by { f k } k , converging to a function in W j , ψ ( Ω ) . Obviously, { f k } k ∈ N also converges to f in L ψ ( Ω ) to almost x ∈ Ω . And from (3.11) yields f = 0 to almost everywhere in Ω . This contradicts Inequality (3.12). Thus, (3.9) holds true.
For any M > 0 and any bounded set S ≔ { f ∈ W m , φ ( Ω ) : ‖ f ‖ m , φ ≤ M } , since the embedding W m , φ ( Ω ) ↪ L φ ( Ω ) is compact, there exists a sequence of function { f k } k ∈ N ⊂ S such that { f k } is a Cauchy sequence in L φ ( Ω ) . Then, using Inequality (3.9), for any ε > 0 , we have
‖ f k − f l ‖ W j , ψ ( Ω ) ≤ C ε ∣ f k − f l ∣ φ + ε ‖ f k − f l ‖ m , φ
for all k , l ∈ N . From this, it is easy to show that { f k } k ∈ N is also a Cauchy sequence in W j , ψ ( Ω ) . The completeness of W j , ψ ( Ω ) completes the proof of Lemma 3.10(2).

We finish the proof of Lemma 3.10.□

Then, we have the following compact embedding theorem.

Theorem 3.11

Let m , j ∈ N with j < m , and Ω be a bounded domain with Lipschitz boundary. Let φ ∈ Φ w ( Ω ) satisfy (A0), (A1), (A2), and (aDec) p + for 1 < p + < n ∕ m . Define ψ − 1 ( x , t ) ≔ t − α φ − 1 ( x , t ) , with α ∈ [ 0 , 1 n ) . Then, the embedding

W m , φ ( Ω ) ↪ W j , ψ ( Ω )

is compact.

Proof

By Lemma 3.4, there exists a generalized Orlicz function φ ˜ satisfying (A0), (A1), (A2), and (aDec) p + for 1 < p + < n ∕ m . We first consider the case m = 1 . Let f k , f ∈ W 1 , φ ( Ω ) with f k ⇀ f in W 1 , φ ( Ω ) as k → ∞ (weak limit). By the property of the compact operator, we need to prove that f k converges to f in L ψ ( Ω ) . To end this, let ψ ∗ ∈ Φ w ( Ω ) satisfy ψ ∗ − 1 = t − 1 n φ − 1 ( x , t ) . Note that ( ψ θ ) − 1 ( x , t ) = [ ψ − 1 ( x , t ) ] θ . Then, ψ − 1 = ( ψ θ ) − 1 ( x , t ) ( φ 1 − θ ) − 1 ( x , t ) with α = θ ∕ n and θ ∈ [ 0 , 1 ) . From Hölder’s inequality [19, Theorem 2.12], we deduce that

(3.13) ‖ f k − f ‖ L ψ ( Ω ) ≤ C ‖ f k − f ‖ ( ψ ∗ ) θ ‖ f k − f ‖ φ 1 − θ = C ‖ f k − f ‖ ψ ∗ θ ‖ f k − f ‖ φ 1 − θ .

By Lemma 2.17, we have f k ⇀ f in L ψ ∗ ( Ω ) as k → ∞ , and hence, { f k − f } k ∈ N is bounded in L ψ ∗ ( Ω ) . Then, by Lemma 2.18, we know that ‖ f k − f ‖ φ → 0 as k → ∞ . Thus, it follows that f k converges to f in L ψ ( Ω ) by (3.13).

Now, we prove this theorem for the case m ≥ 2 . By the aforementioned statement, we know that W m , φ ( Ω ) ↪ L φ ( Ω ) is compact. Then, by Hölder’s inequality [19, Theorem 2.12] and Theorem 3.8, for any f ∈ W m , φ ( Ω ) , we have

‖ f ‖ j , ψ ≤ ‖ 1 Ω ‖ L 1 ∕ α ‖ f ‖ j , φ ≤ K ‖ 1 Ω ‖ L 1 ∕ α ε ‖ u ‖ m , φ + ε − j m − j ∣ u ∣ φ ,

which shows that (3.9) holds true for any ε ∈ ( 0 , ∞ ) . Again, Lemma 3.10 yields our result. Thus, we finish the proof of Theorem 3.11.□

For any u ∈ W 0 1 , φ ( Ω ) ∩ W 2 , φ ( Ω ) , define the norm

‖ u ‖ = ‖ u ‖ W 0 1 , φ ( Ω ) ∩ W 2 , φ ( Ω ) ≔ ‖ u ‖ 1 , φ + ‖ u ‖ 2 , φ = ∣ u ∣ φ + ∣ ∇ u ∣ φ + ∑ ∣ α ∣ = 2 ∣ D α u ∣ φ .

Theorem 3.12

Assume that Ω is a bounded domain with Lipschitz boundary and φ ∈ Φ w ( Ω ) satisfies (A0), (A1), (A2), (aInc), and (aDec). The norms ‖ ⋅ ‖ and ∣ Δ ⋅ ∣ φ are equivalent in the space W 0 1 , φ ( Ω ) ∩ W 2 , φ ( Ω ) .

Proof

Let V = { u ∈ C ∞ ( Ω ) : u ∣ ∂ Ω = 0 } . By the definition of W 0 1 , φ ( Ω ) ∩ W 2 , φ ( Ω ) , we find that the set V is dense in W 0 1 , φ ( Ω ) ∩ W 2 , φ ( Ω ) . To prove our result, we only need to show that for any u ∈ V , there exists a positive constant C such that

∑ ∣ α ∣ = 2 ∣ D α ∣ φ ≤ C ∣ Δ u ∣ φ .

From [20, p. 59, (11)], we know that

∂ 2 u ∂ x i ∂ x j = − R i R j ( Δ u ) ,

where R i , i = 1 , … , n are the Riesz transforms. It is well known that the Riesz transforms are the standard Calderón-Zygmund singular integrals, and hence, R i is bounded on L φ ( Ω ) for each i ∈ { 1 , … , n } by [13, Corollary 6.2]. Therefore, since u ∈ V ⊂ L φ ( Ω ) (see, for instance, [10, Theorem 3.7.15]), we have

∂ 2 u ∂ x i ∂ x j φ = ∣ − R i R j ( Δ u ) ∣ φ ≤ C ∣ Δ ∣ φ

for all i , j = 1 , … , n . Then,

(3.14) ∣ Δ u ∣ φ ≤ ∑ ∣ α ∣ = 2 ∣ D α u ∣ φ ≤ C ∣ Δ u ∣ φ .

Let Γ ∗ ( Δ u ) be the Newtonian potential of ( Δ u ) . By virtues of [21, (2.17), (4.8), and (2.14)], we have

∣ D u ∣ = ∣ D Γ ∗ ( Δ u ) ∣ ≤ C I 1 ( Δ u ) ,

where I 1 denotes the Riesz potential, I 1 ( f ) ( x ) = ∣ ⋅ ∣ 1 − n ∗ f ( x ) ; see also [10, Lemma 6.2.3(1)]. Applying the fact that Ω is bounded domain and Lemma 2.19, we obtain ∣ I 1 ( Δ u ) ∣ ≤ C M ( Δ u ) and so, by Lemma 2.10, ∣ D u ∣ φ ≤ C ∣ Δ u ∣ φ , where C depends only on Ω and n . From the Poincaré inequality and Lemma 2.13(2), we have ∣ u ∣ φ ≤ C ∣ ∇ u ∣ φ ≤ C ∣ D u ∣ φ ≤ C ∣ Δ u ∣ φ . We sum all the estimates

(3.15) ∣ Δ u ∣ φ ≤ ‖ u ‖ W 0 1 , φ ( Ω ) ∩ W 2 , φ ( Ω ) ≤ C ∣ Δ u ∣ φ .

The density of V in W 0 1 , φ ( Ω ) ∩ W 2 , φ ( Ω ) implies that (3.15) holds true for all u ∈ W 0 1 , φ ( Ω ) ∩ W 2 , φ ( Ω ) . Thus, we finish the proof of Theorem 3.12.□

The following proposition was proved in [10, Theorem 3.7.15 and Theorem 6.4.4] for the case 0 ≤ m ≤ 1 . For other cases m ≥ 2 , the proof has no essential difference, the details being omitted.

Proposition 3.13

Let φ ∈ Φ w ( R n ) satisfy (A0)–(A2) and (aDec). Then, C 0 ∞ ( R n ) is dense in W m , φ ( R n ) and L φ ( R n ) .

Theorem 3.14

Let φ ∈ Φ w ( R n ) . If f ∈ L φ ( R n ) and Δ f ∈ L φ ( R n ) , then ∣ ∇ f ∣ ∈ L φ ( R n ) and

∣ ∇ f ∣ φ ≤ C ∣ f ∣ φ 1 2 ∣ Δ f ∣ φ 1 2 ,

where C is independent of f.

Proof

We first prove that f ∈ W 2 , φ ( R n ) . Let X ≔ W 2 , φ ( Ω ) and Y ≔ { f ∈ L φ ( R n ) : Δ f ∈ L φ ( R n ) } equipped with the norms ‖ f ‖ X ≔ ‖ f ‖ 2 , φ and ‖ f ‖ Y ≔ ∣ f ∣ φ + ∣ Δ u ∣ φ , respectively. Easily, X and Y are Banach spaces and C 0 ∞ ( R n ) is dense in both X and Y by Proposition 3.13. From Theorem 3.1, ‖ ∇ u ‖ φ ≤ K ‖ Y ‖ , which together with (3.14), further implies that ‖ u ‖ X ≤ C ‖ u ‖ Y for all u ∈ C 0 ∞ ( R n ) . By the density, we have ‖ u ‖ X ≤ ‖ u ‖ Y for all u ∈ Y . This shows that Y ⊂ X . Then, by Theorem 3.1(3.2) in which we take j = 1 , m = 2 , and together with (3.14), we immediately obtain

∣ ∇ f ∣ φ ≤ C ∣ f ∣ φ 1 2 ∣ f ∣ m , φ 1 2 ≤ C ∣ f ∣ φ 1 2 ∣ Δ f ∣ φ 1 2 .

Thus, we finish the proof of Theorem 3.14.□

4 Sobolev φ -capacity

In this section, we study the Sobolev φ -capacity.

Definition 4.1

Let φ satisfy (aInc) and (aDec). For E ⊂ R n , we denote

S φ ( E ) ≔ { u ∈ W 1 , φ ( R n ) : u ≥ 1 in an open set containing E and u ≥ 0 } .

Functions u ∈ S φ ( E ) are said to be φ -admissible for the capacity of the set E . The Sobolev φ -capacity of E is defined by:

C φ ( E ) ≔ inf u ∈ S φ ( E ) ϱ W 1 , φ ( R n ) ( u ) ≔ inf u ∈ S φ ( E ) ∫ R n φ ( u ) + φ ( ∇ u ) d x .

Remark 4.2

When S φ ( E ) = ∅ , we set C φ ( E ) = ∞ .
If u ∈ S φ ( E ) , then min { 1 , u } ∈ S φ ( E ) and ϱ W 1 , φ ( R n ) ( min { 1 , u } ) ≤ ϱ W 1 , φ ( R n ) ( u ) . Thus, it is enough to test the Sobolev capacity by u ∈ S φ ( E ) with 0 ≤ u ≤ 1 .

Lemma 4.3

[22] Every uniformly convex space has the Banach-Saks property, that is, if u i ⇀ u as i → ∞ then 1 m ∑ i = 1 m u i → u as m → ∞ .

We first give some properties for the Sobolev φ -capacity.

Theorem 4.4

Let φ satisfy (aInc) and (aDec). The set function E → C φ ( E ) has the following properties:

C φ ( ∅ ) = 0 .
If E 1 ⊂ E 2 , then C φ ( E 1 ) ≤ C φ ( E 2 ) .
If E is a subset of R n , then
C φ ( E ) = inf E ⊂ U U open C φ ( U ) .
If E 1 and E 2 are subsets of R n , then
C φ ( E 1 ∪ E 2 ) + C φ ( E 1 ∩ E 2 ) ≤ C φ ( E 1 ) + C φ ( E 2 ) .
If K 1 ⊃ K 2 ⊃ ⋯ are compact sets, then
lim i → ∞ C φ ( K i ) = C φ ⋂ i = 1 ∞ K i .
If E 1 ⊂ E 2 ⊂ ⋯ are subsets of R n , then
lim i → ∞ C φ ( K i ) = C φ ⋃ i = 1 ∞ K i .
For E i ⊂ R n , i ∈ N , we have
C φ ⋃ i = 1 ∞ E i ≤ ∑ i = 1 ∞ C φ ( E i ) .

Proof

Property (1) follows from the fact that zero is in S φ ( ∅ ) .

To prove Property (2), let E 1 ⊂ E 2 . By Definition 4.1, we have S φ ( E 1 ) ⊃ S φ ( E 2 ) and C φ ( E 1 ) ≤ C φ ( E 2 ) .

Now, we prove Property (3). By Property (2), it is easy to see that

C φ ( E ) ≤ inf E ⊂ U U open C φ ( U ) .

To prove the opposite inequality, For any given ε > 0 . By Definition 4.1, there exists a function u ∈ S φ ( E ) so that ϱ W 1 , φ ( R n ) ( u ) ≤ C φ ( E ) + ε . Denote U ≔ int { u ≥ 1 } , where int { u ≥ 1 } denotes the interior of the set { u ≥ 1 } , then E ⊂ U , and C φ ( U ) ≤ ϱ W 1 , φ ( R n ) ( u ) ≤ C ε ( E ) + ε , and thus, the claim follows as ε → 0 + .

Next, we prove Property (4). For any given ε > 0 , we can choose two functions u 1 ∈ S φ ( E 1 ) such that ϱ W 1 , φ ( R n ) ( u 1 ) ≤ C φ ( E 1 ) + ε , and u 2 ∈ S φ ( E 2 ) such that ϱ W 1 , φ ( R n ) ( u 2 ) ≤ C φ ( E 2 ) + ε . We have max { u 1 , u 2 } ∈ S φ ( E 1 ∪ E 2 ) and min { u 1 , u 2 } ∈ S φ ( E 1 ∩ E 2 ) , and, by [23, Theorem 1.20] (see also [10, Lemma 6.1.7]), have

∫ R n φ ( x , ∣ ∇ max { u 1 , u 2 } ∣ ) d x + ∫ R n φ ( x , ∣ ∇ min { u 1 , u 2 } ∣ ) d x = ∫ R n φ ( x , ∣ ∇ u 1 ∣ ) d x + ∫ R n φ ( x , ∣ ∇ u 2 ∣ ) d x .

By this and Definition 4.1, we have

C φ ( E 1 ∪ E 2 ) + C φ ( E 1 ∩ E 2 ) ≤ ϱ W 1 , φ ( R n ) ( max { u 1 , u 2 } ) + ϱ W 1 , φ ( R n ) ( min { u 1 , u 2 } ) = ϱ W 1 , φ ( R n ) ( u 1 ) + ϱ W 1 , φ ( R n ) ( u 2 ) ≤ C φ ( E 1 ) + C φ ( E 2 ) + 2 ε ,

from which Property (3) follows as ε tends to zero.

To prove Property (5), let K 1 ⊃ K 2 ⊃ ⋯ be compact sets. Since ∩ i = 1 ∞ K i ⊂ K j for each j = 1 , 2 , … , Property (2) gives

C φ ⋂ i = 1 ∞ E i ≤ lim j → ∞ C φ ( E j ) .

Now, we choose an open set U such that ∩ i = 1 ∞ K i ⊂ U . Since K j is compact for every j ∈ N , then the set ∩ j = 1 ∞ E j is also compact. The compact and the monotonicity of { E i } i ∈ N , there is a positive integer K such that K j ⊂ U for all j ≥ k . Thus,

lim j → ∞ C φ ( K j ) ≤ C φ ( U ) ,

and by Property (3)

lim j → ∞ C φ ( K j ) ≤ C φ ( ∩ i = 1 ∞ E i ) .

To prove Property (6), set E ≔ ∪ i = 1 ∞ E i . The monotonicity of { E i } i ∈ N and Property (2) yield

lim i → ∞ C φ ( E i ) ≤ C φ ( E ) .

To prove the opposite inequality, we may assume that lim i → ∞ C φ ( E i ) < ∞ . Let u i ∈ S φ ( E i ) and ϱ W 1 , φ ( R n ) ( u i ) < C φ ( E i ) + 2 − 1 for every i ∈ N . Lemma 2.13 shows that W 1 , φ ( R n ) is reflexive. Note that the sequence of { u i } i ∈ N is bounded in W 1 , φ ( R n ) . There exists a subsequence of { u i } i ∈ N , which converges weakly to a function u ∈ W 1 , φ ( R n ) . Then, Lemma 4.3 yields 1 m ∑ i = 1 m u i → u in the Sobolev space W 1 , φ ( R n ) as m → ∞ . Set v j ≔ 1 j ( j − 1 ) ∑ i = j + 1 j 2 u i . We have

1 j 2 ∑ i = 1 j 2 u i − v j 1 , φ ≤ 1 j ( j − 1 ) ∑ i = 1 j u i 1 , φ + 1 j 2 ( j − 1 ) ∑ i = 1 j 2 u i 1 , φ ≤ 1 j − 1 1 j ∑ i = 1 j u i 1 , φ + 1 j − 1 1 j 2 ∑ i = 1 j 2 u i 1 , φ ,

which converges to zero as j goes to infinity. Thus, v j → u in W 1 , φ ( R n ) . Since E j is an increasing sequence, it follows that u i ≥ 1 in an open set containing E j for every j ≤ i . Hence, E j ⊂ int { v j ≥ 1 } . By the convexity of the modular and Property (2), we obtain

ϱ W 1 , φ ( R n ) ( v j ) ≤ 1 j ( j − 1 ) ∑ i = j + 1 j 2 ∫ R n φ ( x , u i ) + φ ( x , ∣ ∇ u i ∣ ) d x ≤ sup i ≥ j ϱ W 1 , φ ( R n ) ( u i ) ≤ sup i ≥ j ( C φ ( E i ) + 2 − i ) ≤ lim i → ∞ C φ ( E i ) + 2 − j .

Note that { v j } converges strongly to u in W 1 , φ ( R n ) . By considering a subsequence, if necessary, we may assume that ‖ v j + 1 − v j ‖ 1 , φ ≤ 2 − j for all j ∈ N . Then, we let

w j ≔ v j + ∑ i = j ∞ ∣ v i + 1 − v i ∣ .

It is obvious that w j ∈ W 1 , φ ( R n ) . Since w j ≥ sup i ≥ j v i , we see that w j ≥ 1 in the open set

⋃ i = j ∞ int { v i ≥ 1 } ⊃ E ,

so w j ∈ S φ ( E ) . By Definition 4.1, we have C φ ( E ) ≤ ϱ ( w j ) for all j = 1 , 2 , … . This also yields

‖ w j − v j ‖ 1 , φ ≤ ∑ i = j ∞ ‖ v i + 1 − v i ‖ 1 , φ ≤ ∑ i = j ∞ 2 − j = 2 − j + 1 ,

which further implies that

ϱ W 1 , φ ( R n ) ( w j − v j ) → 0 as j → ∞ .

Therefore,

C φ ( E ) ≤ lim j → ∞ ϱ W 1 , φ ( R n ) ( w j ) = lim j → ∞ ϱ W 1 , φ ( R n ) ( v j ) ≤ lim j → ∞ lim i → ∞ C φ ( E i ) + 2 − j = lim i → ∞ C φ ( E i ) ,

which completes the proof of Property (6).

It remains to Property (7). From Property (4) it follows by induction that

C φ ( ∪ i = 1 k E i ) ≤ ∑ i = 1 k C φ ( u i ) ≤ ∑ i = 1 ∞ C φ ( u i ) ,

for any finite family of subsets E 1 , E 2 , … , E k in R n . Since { E i } i = 1 ∞ increases to ∪ i = 1 ∞ E i , Property (6) implies Property (7). Hence, it completes the proof of Theorem 4.4.□

By the definition of outer measure [24, p. 42], Properties (1), (2), and (7) of Sobolev φ -capacity yield:

Corollary 4.5

Let φ satisfy (aInc) and (aDec), then the Sobolev φ -capacity is an outer measure.

A set function that satisfies Properties (1), (2), (5), and (6) is called a Choquet capacity. A set E ⊂ R n is capacitable if

sup K ⊂ E K compact C φ ( K ) = C φ ( E ) = inf E ⊂ U U open C φ ( U ) .

Let N be the set of all sequences of positive integers. Consider the projection

p : R n × N → R n , p ( x , n ) = x whenever x ∈ R n , n ∈ N .

By a Suslin set, we mean the p image of some closed subset of R n × N . Every Borel set is a Suslin set in metric spaces (see, for instance, [25, p. 66]). For a Choquet capacity, every Suslin set is capacitable [26]. Thus, we obtain the following result:

Corollary 4.6

Let φ satisfy (aInc) and (aDec), then the set function E → C φ ( E ) , E ⊂ R n , is a Choquet capacity. In particular, all Borel sets E ⊂ R n are capacitable.

5 Two examples of generalized Orlicz spaces

In this section, we give two classical examples of generalized Orlicz spaces. Besides the classical Lebesgue space, the variable exponent Lebesgue spaces with φ ( x , t ) = t p ( x ) and the function spaces generated by the double-phase functionals φ ( x , t ) = t p ( x ) + a ( x ) t q ( x ) are illustrative examples of the generalized Orlicz spaces. Under some hypotheses, our main results could be applied to the Sobolev spaces associated with these function spaces.

5.1 Variable Lebesgue spaces

Let p : Ω → [ 1 , ∞ ] be a measurable function. Define

φ ( x , t ) ≔ t p ( x ) ,

where we interpret t ∞ = ∞ 1 ( 1 . ∞ ] ( t ) . In this case, we call the special generalized Orlicz space L p ( ⋅ ) ( Ω ) by the variable exponent Lebesgue spaces. There are some books to obtain more information about them (see, for instance [3,27]).

We define p − ≔ ess inf x ∈ Ω p ( x ) and p + ≔ ess sup x ∈ Ω p ( x ) . By [10, Lemma 7.1.1], we know that the Φ -function φ ( x , t ) = t p ( x ) satisfies (A0), (aInc) p − and (aDec) p + . Moreover, in [10, Propositions 7.1.2 and 7.1.3], Harjulehto and Hästö reveal the relationship between (A1) and (A2) for φ ( x , t ) = t p ( x ) and the log -Hölder continuous, and Nekvinda’s decay condition for the variable exponent function p ( x ) , respectively. Precisely,

Lemma 5.1

The Φ -function φ ( x , t ) = t p ( x ) satisfies (A1) if and only if p − 1 is log -Hölder continuous, that is, there exists a positive constant c such that, for every distinct x , y ∈ Ω ,

1 p ( x ) − 1 p ( y ) ≤ c log ( e + ∣ x − y ∣ − 1 ) .

Lemma 5.2

The Φ -function φ ( x , t ) = t p ( x ) satisfies (A2) if and only if p satisfies Nekvinda’s decay condition that is, there exists a positive constant c such that

∫ { p ( x ) ≠ p ∞ } c 1 ∣ 1 p ( x ) − 1 p ∞ ∣ < ∞ ,

where p ∞ = lim ∣ x ∣ → ∞ p ( x ) .

5.2 Double-phase functionals with variable exponents

The double-phase functional, namely, t p + a ( x ) t q , was initiated by Zhikov [28] in 1986, and then has been studied by many researchers (see, for instance, [29]). Let p ( x ) and q ( x ) be measurable functions on Ω with 1 ≤ p ( x ) < q ( x ) < ∞ . Suppose that a ( x ) is a positive and bounded measurable function on Ω . Next, we turn to double-phase functionals with variable exponents:

φ ( x , t ) ≔ t p ( x ) + a ( x ) t q ( x ) .

We also define p − = ess inf x ∈ Ω p ( x ) , p + ≔ ess sup x ∈ Ω q ( x ) , q − = ess inf x ∈ Ω q ( x ) and q + ≔ ess sup x ∈ Ω q ( x ) . Then, we can conclude that φ satisfies (aInc) p − and (aDec) q + . Note that 1 ≤ φ ( x , 2 ) ≤ 2 p + + 2 q + ‖ a ‖ L ∞ ( Ω ) , from [10, Corollary 3.7.5], we know that φ satisfies (A0). For any given s > 0 , when t ∈ [ 0 , s ] , we have

t p ( x ) + a ( x ) t q ( x ) ≤ ( 1 + ‖ a ‖ L ∞ ( Ω ) s q + − p − ) t p ( x ) ≤ ( 1 + ‖ a ‖ L ∞ ( Ω ) s q + − p − ) ( t p ( x ) + a ( x ) t q ( x ) ) .

Thus, if p ( x ) satisfies Nekvinda’s decay condition, then φ satisfies (A2).

By a similar statement of the proof of [10, Proposition 7.2.2], we have

Lemma 5.3

The Φ -function φ ( x , t ) = t p ( x ) + a ( x ) t q ( x ) satisfies (A1) if and only if there exists a positive constant β such that, for all ball B ⊂ R n with ∣ B ∣ ≤ 1 , x , y ∈ B ∩ Ω , and t ∈ [ 1 , ∣ B ∣ − 1 ] , such that

β min t 1 p ( x ) , t a ( x ) 1 q ( x ) ≤ min t 1 p ( y ) , t a ( y ) 1 q ( y ) .

Acknowledgments

We are greatly indebted to the anonymous referee for helpful comments and stimulating hints.

Funding information: This project was supported by Higher Education Innovation Fund project of Gansu Province (Grant No.2021A-193) and the Natural Science Foundation Project of Chongqing, China (Grant No. cstc2021jcyj-msxmX0705).
Author contributions: Ruimin Wu deduced and verified the conclusion; Songbai Wang put forward many valuable opinions and suggestions, and checked and corrected the manuscript. All authors read and approved the final manuscript; we confirmed that all the aforementioned information is true and valid.
Conflict of interest: The authors declare that they have no conflicts of interest.
Data availability statement: No data and material were used to support this study.

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Received: 2022-07-15

Revised: 2023-02-18

Accepted: 2023-05-29

Published Online: 2023-07-29

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2022-0595

Keywords for this article

interpolation inequality; generalized Orlicz-Sobolev space; maximal function operator

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