Dilation estimates for Orlicz amalgam spaces on the affine group

1 Introduction

An amalgam space consists of functions whose norm distinguishes between local and global properties. The first appearance of amalgam spaces was due to Wiener in his studies of generalized harmonic analysis [23, 24, 25]. Amalgam spaces of Lebesgue spaces were investigated by many authors [4, 5, 7, 19]. The most general definition of Wiener amalgam spaces was introduced by Feichtinger in 1980s [11, 13, 12, 14, 10].

Amalgam spaces have proven to be very fruitful within pure and applied mathematics, for example, in sampling theory [16] and in time-frequency analysis [15]. It turned out that continuity properties of certain operators can be conveniently described in the context of Wiener amalgam spaces [8, 9] and are mostly considered for the Lebesgue spaces on the real line. On the other hand, for 1 ≤ p < ∞ , Heil and Kutyniok studied amalgam spaces W ⁢ ( L ∞ ⁢ ( 𝔸 ) , L p ⁢ ( 𝔸 ) ) on the affine group 𝔸 [17, 18], which is not abelian unlike the real line. They proved a useful convolution relation for the amalgam space W ⁢ ( L ∞ ⁢ ( 𝔸 ) , L 1 ⁢ ( 𝔸 ) ) .

It is well known that the affine group plays a prominent role in harmonic analysis, especially in wavelet theory. It is not an IN group, i.e., a locally compact group with a compact and invariant neighborhood of identity, and it includes all abelian groups as well as some non-abelian groups such as the reduced Heisenberg group which is important for time-frequency analysis. A key feature of the affine group is that the left Haar measure and the right Haar measure are not equal.

An Orlicz space is a type of function space which significantly generalizes the Lebesgue spaces L p . Besides the L p spaces, a variety of function spaces naturally arise in analysis such as L ⁢ log + ⁡ L , which is a Banach space related to Hardy–Littlewood maximal functions. Orlicz spaces contain certain Sobolev spaces as subspaces.

In [2], Arıs and Öztop considered Wiener amalgam spaces with respect to Orlicz spaces W ⁢ ( L Φ ⁢ ( 𝔸 ) , L 1 ⁢ ( 𝔸 ) ) and W ⁢ ( L ∞ ⁢ ( 𝔸 ) , L Φ ⁢ ( 𝔸 ) ) on the affine group 𝔸 . They obtained some properties of Wiener amalgam spaces of Orlicz type and proved convolution relations for W ⁢ ( L Φ ⁢ ( 𝔸 ) , L 1 ⁢ ( 𝔸 ) ) and W ⁢ ( L ∞ ⁢ ( 𝔸 ) , L Φ ⁢ ( 𝔸 ) ) . In [1], the results of [2] were extended to the more general Orlicz amalgam space W ⁢ ( L Φ 1 ⁢ ( 𝔸 ) , L Φ 2 ⁢ ( 𝔸 ) ) by using the equivalent discrete norm. Meanwhile, in [3], the amalgam spaces W ⁢ ( L Φ 1 ⁢ ( ℝ d ) , L Φ 2 ⁢ ( ℝ d ) ) were considered and dilation estimates were studied for these spaces.

The aim of this paper is to give dilation properties of the Orlicz amalgam spaces W ⁢ ( L Φ 1 ⁢ ( 𝔸 ) , L Φ 2 ⁢ ( 𝔸 ) ) on the affine group 𝔸 . In order to do this, we are motivated to study the equivalent discrete norm on W ⁢ ( L Φ 1 ⁢ ( 𝔸 ) , L Φ 2 ⁢ ( 𝔸 ) ) by using a specific partition of unity of the affine group. We also improve the estimates of dilation operator by using interpolation arguments of Orlicz spaces. To the best of our knowledge, dilation properties of Orlicz amalgam spaces on the affine group have not been studied before.

The paper is organized as follows. In Section 2, we review the necessary definitions and some basic results of the Orlicz spaces L Φ ⁢ ( 𝔸 ) on the affine group. In Section 3, we present the basic structure of Orlicz amalgam spaces on the affine group 𝔸 which we denote by W ⁢ ( L Φ 1 ⁢ ( 𝔸 ) , L Φ 2 ⁢ ( 𝔸 ) ) . Finally, in Section 4, we study the dilation properties of Orlicz amalgam spaces (Theorem 4.2 and Theorem 4.5). These results are also new for the Lebesgue spaces and the standard Orlicz spaces.

2 Preliminaries

Throughout the paper, we consider the affine group 𝔸 = ℝ + × ℝ with the multiplication

where ℝ + denotes the multiplicative group of positive real numbers. The identity element and inverses in 𝔸 are given by

for ( a , b ) ∈ 𝔸 , respectively. It is easy to see that 𝔸 is a non-abelian group under its multiplication.

One can see that the left Haar measure on 𝔸 is d ⁢ μ = d ⁢ x x ⁢ d ⁢ y . The affine group 𝔸 is not unimodular.

We consider Orlicz spaces on the affine group 𝔸 . An Orlicz space is determined by a Young function. A function Φ : [ 0 , ∞ ) → [ 0 , ∞ ] is called a Young function if Φ is convex, Φ ⁢ ( 0 ) = 0 , and lim x → ∞ ⁡ Φ ⁢ ( x ) = ∞ . For a Young function Φ, Φ - 1 is defined by

where inf ⁡ ∅ = ∞ , and we have

For a Young function Φ, the complementary function Ψ of Φ is given by

and Ψ is also a Young function. So ( Φ , Ψ ) is called a complementary Young pair. We have the Young inequality

for complementary functions Φ and Ψ.

By our definition, a Young function can take the value ∞ at a certain point, and hence be discontinuous at such a point. However, we always consider the pair of complementary Young functions ( Φ , Ψ ) with Φ being real-valued and continuous on [ 0 , ∞ ) and increasing on ( 0 , ∞ ) . Note that even though Φ is continuous, it may happen that Ψ is not continuous.

Let ( Φ 1 , Ψ 1 ) and ( Φ 1 , Ψ 2 ) be complementary Young pairs. If Φ 1 ⁢ ( x ) ≤ Φ 2 ⁢ ( x ) for all x ≥ x 0 ≥ 0 , then we have Ψ 2 ⁢ ( y ) ≤ Ψ 1 ⁢ ( y ) for all y ≥ y 0 = Φ 1 ⁢ ( x 0 ) ≥ 0 .

Let Φ 1 , Φ 2 be two Young functions. If there exist c > 0 and x 0 ≥ 0 (depending on c) such that Φ 1 ⁢ ( x ) ≤ Φ 2 ⁢ ( c ⁢ x ) for all x ≥ x 0 , then we say that Φ 2 is stronger than Φ 1 and denote this by Φ 1 ≺ Φ 2 . If, in addition, c = 1 and x 0 = 0 , we say that Φ 2 is strictly stronger than Φ 1 . If Φ 1 ≺ Φ 2 and Φ 2 ≺ Φ 1 , then we write Φ 1 ≍ Φ 2 . Also, Φ 1 ≺ Φ 2 if and only if Φ 2 - 1 ⁢ ( y ) ≤ c ⁢ Φ 1 - 1 ⁢ ( y ) for all y ≥ y 0 = Φ 1 ⁢ ( x 0 ) .

A Young function Φ satisfies the Δ 2 condition if there exist a constant K > 0 and an x 0 ≥ 0 such that Φ ⁢ ( 2 ⁢ x ) ≤ K ⁢ Φ ⁢ ( x ) for all x ≥ x 0 . In this case, we write Φ ∈ Δ 2 .

Let 𝔸 be equipped with the left Haar measure d ⁢ μ = d ⁢ x x ⁢ d ⁢ y . Given a Young function Φ, the Orlicz space on 𝔸 is defined by

Then the Orlicz space is a Banach space under the Orlicz norm ∥ ⋅ ∥ L Φ ⁢ ( 𝔸 ) defined for f ∈ L Φ ⁢ ( 𝔸 ) by

where Ψ is the complementary Young function of Φ.

One can also define the Luxemburg norm ∥ ⋅ ∥ L Φ ⁢ ( 𝔸 ) o on L Φ ⁢ ( 𝔸 ) by

It is known that these norms are equivalent, that is,

and

If ( Φ , Ψ ) is a complementary Young pair and Φ ∈ Δ 2 , then the dual space L Φ ⁢ ( 𝔸 ) ∗ is L Ψ ⁢ ( 𝔸 ) . If, in addition, Ψ ∈ Δ 2 , then the Orlicz space L Φ ⁢ ( 𝔸 ) is a reflexive Banach space [21].

We denote the norm equivalence of Banach spaces ( X , ∥ ⋅ ∥ X ) and ( Y , ∥ ⋅ ∥ Y ) by ∥ ⋅ ∥ X ≍ ∥ ⋅ ∥ Y or X ≍ Y . Although the same notation is used for the equivalence of Young functions, the meaning will be clear from the context.

We also have Hölder’s inequality, which states that if f ∈ L Φ ⁢ ( 𝔸 ) and g ∈ L Ψ ⁢ ( 𝔸 ) , then f ⁢ g ∈ L 1 ⁢ ( 𝔸 ) and

A normed space ( Y , ∥ ⋅ ∥ Y ) consisting of measurable of complex-valued functions on a measurable space X is called solid if for each measurable function f : X → ℂ satisfying | f | ≤ | g | almost everywhere for some g ∈ Y , we have f ∈ Y and ∥ f ∥ Y ≤ ∥ g ∥ Y . Since the Young function Φ is increasing, the Orlicz space L Φ ⁢ ( 𝔸 ) is a solid space (see [22]).

For 1 ≤ p < ∞ and the Young function Φ ⁢ ( x ) = x p , the space L Φ ⁢ ( 𝔸 ) is the classical Lebesgue space L p ⁢ ( 𝔸 ) , and the norm ∥ ⋅ ∥ L Φ ⁢ ( 𝔸 ) is equivalent to the norm of the usual Lebesgue spaces ∥ ⋅ ∥ L p ⁢ ( 𝔸 ) .

If p = 1 , then we obtain the space L 1 ⁢ ( 𝔸 ) . In this case, the complementary Young function of Φ ⁢ ( x ) = x is

and ∥ f ∥ L Φ ⁢ ( 𝔸 ) = ∥ f ∥ L 1 ⁢ ( 𝔸 ) for all f ∈ L 1 ⁢ ( 𝔸 ) . For the Young function Ψ given in (2.2), the space L Ψ ⁢ ( 𝔸 ) coincides with the space L ∞ ⁢ ( 𝔸 ) , and we have ∥ f ∥ L Ψ ⁢ ( 𝔸 ) = ∥ f ∥ L ∞ ⁢ ( 𝔸 ) for all f ∈ L ∞ ⁢ ( 𝔸 ) .

In addition, for the Young function

the space L Φ s ⁢ ( 𝔸 ) becomes L 1 ⁢ ( 𝔸 ) ∩ L ∞ ⁢ ( 𝔸 ) with the norm

for all f ∈ L Φ s ⁢ ( 𝔸 ) . Then we have

For the Young function

we obtain the space L 1 ⁢ ( 𝔸 ) + L ∞ ⁢ ( 𝔸 ) with the norm

We have Φ b ⁢ ( x ) - 1 = x + 1 , x ≥ 0 . In addition, Φ s and Φ b are complementary Young functions [20, p. 52] which satisfy the Δ 2 condition.

Since

for any Young function Φ, we may consider L 1 ⁢ ( 𝔸 ) ∩ L ∞ ⁢ ( 𝔸 ) as the smallest Orlicz space, and L 1 ⁢ ( 𝔸 ) + L ∞ ⁢ ( 𝔸 ) as the largest one [20, p. 100]. We denote the spaces L 1 ⁢ ( 𝔸 ) ∩ L ∞ ⁢ ( 𝔸 ) and L 1 ⁢ ( 𝔸 ) + L ∞ ⁢ ( 𝔸 ) by L Φ s ⁢ ( 𝔸 ) and L Φ b ⁢ ( 𝔸 ) , respectively. Also, we have L Φ s ⁢ ( K ) ≍ L ∞ ⁢ ( K ) for any compact subset K ⊂ 𝔸 .

We denote the translation operator by L ( a , b ) ⁢ f ⁢ ( x , y ) = f ⁢ ( ( a , b ) - 1 ⁢ ( x , y ) ) and the dilation operator by D λ ⁢ f ⁢ ( x , y ) = f ⁢ ( λ ⁢ x , λ ⁢ y ) for ( x , y ) , ( a , b ) ∈ 𝔸 and λ > 0 . Both operators are well-defined, linear and bounded operators on Orlicz spaces [22].

Properties of the dilation operator D λ when acting on Orlicz spaces were recently studied by Blasco and Osançlıol in [6]. Let us modify their results for our aim.

Given λ > 0 , another norm on the Orlicz space L Φ ⁢ ( 𝔸 ) is defined by

When λ = 1 , we obtain ∥ f ∥ L Φ ⁢ ( 𝔸 ) ∘ , 1 = ∥ f ∥ L Φ ⁢ ( 𝔸 ) ∘ , and for λ > 0 we have

By convexity, it can be easily seen that

and

3 Orlicz amalgam spaces on the affine group

Let 𝔸 be the affine group and let Φ 1 , Φ 2 be Young functions. In this section, we give the definition and basic properties of the Orlicz amalgam spaces W ⁢ ( L Φ 1 ⁢ ( 𝔸 ) , L Φ 2 ⁢ ( 𝔸 ) ) consisting of functions that are locally in L Φ 1 ⁢ ( 𝔸 ) and globally in L Φ 2 ⁢ ( 𝔸 ) . In our theorems, the translation invariance and solidity of the Orlicz spaces L Φ 1 ⁢ ( 𝔸 ) and L Φ 2 ⁢ ( 𝔸 ) play important roles.

Next, we summarize some technical results from [1] that will be used in the next section.

By the equivalence of the Orlicz norm and the Luxemburg norm in Orlicz spaces [22], we have

Throughout the paper, we consider the Luxemburg norm on W ⁢ ( L Φ 1 ⁢ ( 𝔸 ) , L Φ 2 ⁢ ( 𝔸 ) ) .

Note that the Orlicz amalgam space W ⁢ ( L Φ 1 ⁢ ( 𝔸 ) , L Φ 2 ⁢ ( 𝔸 ) ) is a Banach space and its definition is independent of the choice of the compact subset Q ⊂ 𝔸 , in the sense that different compact subsets yield equivalent Orlicz amalgam space norms. Moreover, since the Orlicz spaces L Φ 1 ⁢ ( 𝔸 ) and L Φ 2 ⁢ ( 𝔸 ) are solid spaces, W ( L Φ 1 ( 𝔸 ) , L Φ 2 ( 𝔸 ) is also a solid space.

Let us now recall Hölder’s inequality for Orlicz amalgam spaces. If we have f ∈ W ⁢ ( L Φ 1 ⁢ ( 𝔸 ) , L Φ 2 ⁢ ( 𝔸 ) ) and g ∈ W ⁢ ( L Ψ 1 ⁢ ( 𝔸 ) , L Ψ 2 ⁢ ( 𝔸 ) ) , then f ⁢ g ∈ L 1 ⁢ ( 𝔸 ) and

Wiener amalgam spaces can be considered as spaces with a discrete norm. In fact, the norm which is given in Definition 3.1 and discrete norms are equivalent (see [1]).

Let { Q h } h > 0 denote a fixed family of increasing, exhaustive neighborhoods of identity in 𝔸 . In particular, we take Q h = [ e - h , e h ) × [ - h , h ) . The Haar measure of the translated set ( x , y ) ⁢ Q h is

Given h > 0 , for k , j ∈ ℤ , we define particular translates of Q h and Q 2 ⁢ h as

Note that B j ⁢ k ⊆ B j ⁢ k ′ .

To obtain an equivalent discrete norm on these spaces, the following lemma is a key observation.

Hence the set X = { ( e 2 ⁢ j ⁢ h , 2 ⁢ k ⁢ h ⁢ e - h ) : j , k ∈ ℤ } for h > 0 becomes a well-spread family [13, 14].

By Urysohn’s lemma, there exist continuous functions ϕ j ⁢ k : 𝔸 → ℝ such that 0 ≤ ϕ j ⁢ k ⁢ ( x , y ) ≤ 1 , supp ⁡ ( ϕ j ⁢ k ) ⊆ B j ⁢ k ′ and ϕ j ⁢ k ⁢ ( x , y ) = 1 for ( x , y ) ∈ B j ⁢ k . Define

Thus { ψ j ⁢ k } j , k ∈ ℤ is a bounded uniform partition of unity (BUPU). Then, by [11, Theorem 2], we have the following equivalence in W ⁢ ( L Φ 1 ⁢ ( 𝔸 ) , L Φ 2 ⁢ ( 𝔸 ) ) :

To illustrate the usefulness of the discrete norm, we present a duality theorem and some inclusion relations for W ⁢ ( L Φ 1 ⁢ ( 𝔸 ) , L Φ 2 ⁢ ( 𝔸 ) ) given in [1].

4 Dilation properties

In this section, we study dilation properties of Wiener amalgams of the Orlicz spaces W ⁢ ( L Φ 1 ⁢ ( 𝔸 ) , L Φ 2 ⁢ ( 𝔸 ) ) . By using the constructions related to the affine group, we obtain norm estimates of the dilation operator on the Orlicz amalgam space W ⁢ ( L Φ 1 ⁢ ( 𝔸 ) , L Φ 2 ⁢ ( 𝔸 ) ) , differently from those in [3]. In particular, using the dilation properties of Orlicz spaces given in [6] and the interpolation of Orlicz spaces in [20], we improve the norm estimates.

Let f λ ⁢ ( x , y ) = f ⁢ ( λ ⁢ x , λ ⁢ y ) , λ > 0 , ( x , y ) ∈ 𝔸 . The following lemma provides an estimate for the norm of the dilation operator D λ in the Orlicz space L Φ ⁢ ( 𝔸 ) . It will be used when proving subsequent theorems related to dilation properties of the Orlicz amalgam spaces W ⁢ ( L Φ 1 ⁢ ( 𝔸 ) , L Φ 2 ⁢ ( 𝔸 ) ) .

Note that the following result for L Φ ⁢ ( ℝ ) is given by [6, Proposition 2.2]. If we adapt it to the space L Φ ⁢ ( 𝔸 ) , we obtain the following lemma.

Note that, by (2.4) and Lemma 4.1, we have

The first estimates for dilations on Orlicz amalgam spaces W ⁢ ( L Φ 1 ⁢ ( 𝔸 ) , L Φ 2 ⁢ ( 𝔸 ) ) are given in the following theorem.