Approximation
properties on Herz spaces
Jhean E.
Pérez-López∗
Universidad Industrial de Santander, Escuela de Matemáticas, Bucaramanga, Colombia.
In
this paper we consider the Herz spaces Kp,qα , which
are a natural generalization of the Lebesgue spaces Lp. We prove some approximation
properties such as density of the space Cc∞ (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.
En este artículo consideramos los
espacios de Herz,
los cuales son una generalización natural de los espacios de Lebesgue Lp.
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.
The Herz
space Kp,qα 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 Kp,qα -norm
in terms of Lp-norms over annuli was given in [8]
(see Definition 2.1). Recently, some versions of classical spaces based on Herz
spaces Kp,qα ,
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 Lp. 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 Lp.
They seem necessary for the study of the transport equations as done in [2] in
the frame work of the Lp spaces.
This paper is organized as follows. In Section
2 we recall the definition of the Herz spaces Kp,qα 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.
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 Ak 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 Kp,qα = Kp,qα (Rn) is defined by
where
,
(3)
In the case p = 1, we consider as a space of signed measures, with kfkL1(Ak) denoting the total variation of f on Ak.
The pair (Kp,qα ,k·kKp,qα ) is a Banach
space for α ∈R,
1 ≤ p < ∞ and 1 ≤ q ≤∞ (see
e.g. [6]). Also, note that Kp,01 ֒→ Lp =
Kp,p0 ֒→ Kp,0
∞. So,
all the results presented here are a generalization of the corresponding
results in Lp.
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 Kp,qα -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
Mθ < ∞ 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 Kp,qα . Moreover, the inclusion is dense
provided that 1 ≤ p,q < ∞.
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 fǫ (x) = (ρǫ ∗ 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 Cc∞ (Rn) is dense in Kp,qα .
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, τzf −→ f in Kp,qα as z −→ 0 for all f ∈ Kp,qα .
Here τzf denotes the mapping such that τzf(x) = f(x − z).
Proof. Note
that, for |z| < 1, we have
that the volume preserving diffeomorphism Xz(x) = x − z trivially verifies kXz − Idk∞
≤ 1. Now, let f ∈ Kp,qα , ǫ > 0 and g ∈ Cc∞ (Rn) such that kf − gkKp,qα < ǫ. Using Lemma 2.2, it follows that
Now we
prove that kg − τzgkKp,qα −→ 0 when z −→ 0. In fact,
if |z| <
1, then suppg and suppτzg 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 kτǫzfkKp,qα = kf ◦ XǫzkKp,qα ≤ C kfkKp,qα and it follows that kτǫzf − fkKp,qα ≤ (1 + C)kfkKp,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 ∈ Cc∞ (Rn), 1 ≤ p,q ≤∞, α ∈R and ρ ∈S. Then, φǫ −→ φ in Kp,qα as ǫ −→ 0.
Proof. Let N ∈N such that
suppφ ⊂ B(0,2N), and let M > N + 2. For k ≥ M and x ∈ Ak we have that
Thus, for any s > 0 we get |
|
|
|
On the
other hand,
.
Is clear that S1 −→ 0 as ǫ −→ 0, because ρǫ ∗ φ −→ φ in Lp(Ak). For S2, using the Hölder inequality, (12)
and taking s > 0 large enough, we have
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