Approximation properties on Herz spaces

Jhean E. Pérez-López^∗

Universidad Industrial de Santander, Escuela de Matemáticas, Bucaramanga, Colombia.

Abstract

In this paper we consider the Herz spaces K_p,q^α , which are a natural generalization of the Lebesgue spaces L^p. We prove some approximation properties such as density of the space C_c^∞ (Rⁿ), continuity of the translation, continuity of the mollification, global behavior of the convolution with smooth functions, among others.

Keywords: Herz spaces, Mollifiers, Convolution, Functional spaces. MSC2010: 26B05, 26B35, 26B99.

Propiedades de aproximación en espacios de Herz

Resumen

En este artículo consideramos los espacios de Herz, los cuales son una generalización natural de los espacios de Lebesgue L^p. Demostramos algunas propiedades de aproximación tal es como densidad del espacio

, continuidad de la traslación, continuidad de la molificación, comportamiento global de la convolución con funciones suaves, entre otras.

Palabras clave: Espacios de Herz, Molificadores, Convolución, Espacios funcionales.

1. Introduction

The Herz space K_p,q^α was introduced by Herz [7] as a suitable environment for the image of the Fourier transform acting on a class of Lipschitz spaces, in order to obtain a Bernstein type theorem. A characterization of the K_p,q^α -norm in terms of L^p-norms over annuli was given in [8] (see Definition 2.1). Recently, some versions of classical spaces based on Herz spaces K_p,q^α , such as Hardy-Herz, Sobolev-Herz and Triebel-Lizorkin-Herz spaces, have presented an increasing interest in the literature of function spaces and turned out to be a useful tool in harmonic analysis (see [1],[4],[5],[10] and references therein). We also refer the reader to [9] for results of well-posedness of the Navier-Stokes equations in weak-Herz spaces, and to [3] for results of well-posedness of the Euler equations on Besov-Herz spaces.

In this paper we consider the Herz spaces , which are a natural generalization of the Lebesgue spaces L^p. We prove some approximation properties such as density of the spaces, continuity of the translation, continuity of the mollification, global behavior of the convolution with smooth functions, among others. This results generalize the corresponding ones in L^p. They seem necessary for the study of the transport equations as done in [2] in the frame work of the L^p spaces.

This paper is organized as follows. In Section 2 we recall the definition of the Herz spaces K_p,q^α and some basic properties. Section 3 is devoted to the new results about approximation and global behavior of functions in Herz spaces and his convolution with smooth functions.

2. Herz spaces

This section is devoted to recall the definition of Herz spaces and establish some of their properties, which will be useful in the remainder of the paper. For k integer and k ≥−1, let A_k be defined as

where B(x₀,R) = {x ∈ Rⁿ;|x − x₀| < R}. Then we have the disjoint decomposition Rn = ∪k≥−1Ak.

Definition 2.1. Let 1 ≤ p,q ≤∞ and α ∈R. The Herz space K_p,q^α = K_p,q^α (Rⁿ) is defined by

where

,

(3)

In the case p = 1, we consider as a space of signed measures, with kfk_L1_(Ak) denoting the total variation of f on A_k. The pair (K_p,q^α ,k·k_Kp,qα ) is a Banach space for α ∈R,

1 ≤ p < ∞ and 1 ≤ q ≤∞ (see e.g. [6]). Also, note that K_p,⁰₁ ֒→ L^p = K_p,p⁰ ֒→ K_p,⁰ _∞. So, all the results presented here are a generalization of the corresponding results in L^p.

There is a version of Hölder’s inequality in Herz spaces (see [9]). In fact, let 1 ≤ p,p₁,p₂,q,q₁,q₂ ≤∞ and α,α₁,α₂ ∈R be such that , and α = α₁ + α₂. Then,

The following result provides a control in K_p,q^α -spaces for the action of a volume preserving diffeomorphism. Recall that a diffeomorphism X is volume preserving if for all measurable set Ω, we have µ(Ω) = µ(X(Ω)), where µ represent the Lebesgue measure.

Lemma 2.2 ([3]). Let 1 ≤ p,q ≤∞ and α ≥ 0. Let X : Rⁿ →Rⁿ be a volume-preserving diffeomorphism such that, for some fixed ω > 0,

where |·| stands for the Euclidean norm in . Then, there exists C > 0 such that

for all

The next lemma is key in order to prove the new results in Section 3. It provides an estimate for convolution operators in Herz spaces depending on certain weighted norms of the kernel θ.

Lemma 2.3 ([3]). (Convolution). Let α ∈R and 1 ≤ p,q ≤∞. Let θ ∈ L¹ be such that

M_θ < ∞ for some β > 0, where

,

Then, there is a constant C > 0 (independent of θ) such that

To finish this section we present a first result of density in Herz spaces.

Lemma 2.4 ([3]). Let 1 ≤ p,q ≤∞ and α ∈R . Then, Schwartz space S is continuously included in K_p,q^α . Moreover, the inclusion is dense provided that 1 ≤ p,q < ∞.

3. New results

Now we present some new results about approximation of functions in Herz spaces and global behavior of their convolution with smooth functions. This results can be useful in order to analyze the well-posedness of some PDEs in the framework of Herz spaces.

In what follows, let ρ ∈S such that we define

additionally, for a mensurable function f ∈Rⁿ →R we also define f^ǫ (x) = (ρ_ǫ ∗ f)(x).

We first show that not only S but also are dense in Herz spaces.

Theorem 3.1. Let 1 ≤ p,q < ∞ and α ∈R. Then, the space C_c^∞ (Rⁿ) is dense in K_p,q^α .

Proof. From Lemma 2.4 it follows that S is dense in and φ ∈S such that . Now, let N ∈N such that

,

and let be such that in

. Defining ψ = θφ, we have that and

. Moreover, it follows that

Thus,

Since ǫ > 0 is arbitrary, we conclude the proof.

Now we prove the continuity of the translation mapping in Herz spaces.

Theorem 3.2. Let 1 ≤ p,q < ∞ and α ∈R. Then, τ_zf −→ f in K_p,q^α as z −→ 0 for all f ∈ K_p,q^α . Here τ_zf denotes the mapping such that τ_zf(x) = f(x − z).

Proof. Note that, for |z| < 1, we have that the volume preserving diffeomorphism X_z(x) = x − z trivially verifies kX_z − Idk_∞ ≤ 1. Now, let f ∈ K_p,q^α , ǫ > 0 and g ∈ C_c^∞ (Rⁿ) such that kf − gk_Kp,qα < ǫ. Using Lemma 2.2, it follows that

Now we prove that kg − τ_zgk_Kp,qα −→ 0 when z −→ 0. In fact, if |z| < 1, then suppg and suppτ_zg are contained in a common compact set K. Choosing M ∈N such that

k=−1

Thus, taking limsup as z −→ 0 in (9) and using (10) we obtain

where suppρ = K. Since K is compact, we have that |z| < C₂ for all z ∈ K; moreover, for 0 < ǫ ≪ 1 we also have that |ǫz| < 1 for all z ∈ K. Thus, using Lemma 2.2 we have that kτǫzf^k_Kp,qα = kf ◦ Xǫz^k_Kp,qα ≤ C kf^k_Kp,qα and it follows that kτǫzf − f^k_Kp,qα ≤ (1 + C)kfk_Kp,qα . Then, using (11), Proposition 3.2 and the Dominate Convergence Theorem we obtain the result. ^X

The following is an auxiliary lemma.

Proposition 3.5. Let φ ∈ C_c^∞ (Rⁿ), 1 ≤ p,q ≤∞, α ∈R and ρ ∈S. Then, φ^ǫ −→ φ in K_p,q^α as ǫ −→ 0.

Proof. Let N ∈N such that suppφ ⊂ B(0,2^N), and let M > N + 2. For k ≥ M and x ∈ A_k we have that

Thus, for any s > 0 we get

On the other hand,

.

Is clear that S₁ −→ 0 as ǫ −→ 0, because ρ_ǫ ∗ φ −→ φ in L^p(A_k). For S₂, using the Hölder inequality, (12) and taking s > 0 large enough, we have

⁼

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