We establish a characterization for the homogeneous Weinstein-Besov spaces via the Weinstein heat semigroup. Next, we obtain the generalized Sobolev embedding theorems.
We consider the Weinstein operator defined on
The harmonic analysis associated with the Weinstein operator is studied by Ben Nahia and Ben Salem (cf. [
In the present paper, we intend to continue our study of generalized spaces of type Sobolev associated with the Weinstein operator that started in [
In this paper we consider the Weinstein heat equation
Characterize the homogeneous Weinstein-Besov spaces via the Weinstein heat semigroup. Prove the imbedding Sobolev theorems.
I have studied the generalized Sobolev spaces in the context of differential-differences operators (cf. [
The remaining part of the paper is organized as follows. Section
In order to confirm the basic and standard notations, we briefly overview the Weinstein operator and the related harmonic analysis. Main references are [
In this subsection, we collect some notations and results on the Weinstein kernel, the Weinstein intertwining operator and its dual, the Weinstein transform, and the Weinstein convolution.
In the following,
We consider the Weinstein operator
The Weinstein kernel For each For all For all
where
The Weinstein intertwining operator is the operator
We denote by
The Weinstein transform is given for For For For all For
(i) The Weinstein transform
(ii) In particular, the renormalized Weinstein transform
In the Fourier analysis, the translation operator is given by
In harmonic analysis associated for the operator
By using the Weinstein kernel, we can also define a generalized translation. For functions For all Let
We define the tempered distribution
The Weinstein transform
In particular, for
The Weinstein transform
The generalized convolution product of a distribution
Let
For each
One of the main tools in this paper is the homogeneous Littlewood-Paley decomposition of distribution associated with the Weinstein operators into dyadic blocs of frequencies.
One defines by
Throughout this paper, we define
When dealing with the Littlewood-Paley decomposition, it is convenient to introduce the functions
We remark that
We put
One denotes by
For all
Using Remark
In the following, we define analogues of the homogeneous Besov, Triebel-Lizorkin, and Riesz potential spaces associated with the Weinstein operators on
From now, we make the convention that for all nonnegative sequence
Let
Let
For
The nonhomogeneous Besov space
We give now another definition equivalent to the nonhomogeneous Besov space
Let
Let
For
For
Let
The operator
We obtain these results by the similar ideas used in the nonhomogeneous case (cf. [
Let
We obtain these results by the similar ideas used in the nonhomogeneous case (cf. [
As in the Euclidean case (cf. [
If
One assumes that
In order to prove the inclusion, we use the estimate
(1) If
(2) If
(3) If
(1) is obvious from the Hölder’s inequality. As for
Let
We obtain these results by the similar ideas used in the nonhomogeneous case (cf. [
Let
Let
Let
By choosing
The Weinstein heat equation reads
The function
Let
It follows from the relations (
In this section, we prove estimates for the Weinstein heat semigroup. These estimates are based on the following result.
Let
We again consider a function
For any interval
Let
It suffices to use the fact that
Let
To prove this result we need the following lemma.
There exist two positive constants
The result follows immediately by applying Lemma
Using Lemma
In order to prove the other inequality, let us observe that for any
We now prove that
Let
By density, we can suppose that
On the other hand, by Theorem
This leads to
Let
It suffices to prove that
We decompose the integral in two parts as follows:
We proceed as in [
For any interval
For any interval
Let
Since
Let
Since
The author gratefully acknowledges the Deanship of Scientific Research at the University of Taibah. The author is deeply indebted to the referee for providing constructive comments and help in improving the contents of this paper.