Flett Potential Spaces and a Weighted Wavelet-Like Transform

In this study, the Flett potential spaces are defined and a characterization of these potential spaces is given. Most of the known characterizations of classical potential spaces such as Riesz, Bessel potentials spaces and their generalizations are given in terms of finite differences. Here, by taking wavelet measure instead of finite differences, a weighted wavelet-like transform associated with Poisson semigroup is defined. And, by making use of this weighted wavelet-like transform, a new “truncated" integrals are defined, then using these integrals a characterization of the Flett potential spaces is given.


Introduction
The importance of weak-singular integral operators such as classical Riesz, Bessel and parabolic potentials and their various generalizations in harmonic analysis and its applications is well known. Producing inversion formulas for potentials is one of the important problems in potential theory. A number of approaches to this problem are known. The hypersingular integral technique, a very powerful tool for inversion of potentials, was introduced and studied by E. Stein [1], P. Lizorkin [2], S. Samko [3], [4], B. Rubin [5], [6] and many other. We refer the interested reader also to the papers [7]- [9] for various properties, generalizations and applications.
Continuous wavelet transforms are an alternative approach to find inversion formulas of potentials and this approach has been defined by B. Rubin ([5], [10]) and developed by I. A. Aliev and B. Rubin [11], [12], I. A. Aliev and M. Eryigit [13].

The
Bessel potentials The purpose of this paper is to attempt to study and define space of Flett potentials. The rest of the artical organized as follows: Notations and auxilary lemmas introduced in Section 2. Main results proved in Section 3.
In this work, we define the space of Flett potentials Then using the weighted wavelet-like transform Which is generated by a finite Borel measure on [0, ) and the Poisson integral t f , we obtain a characterization of the Flett potential spaces () p L .

Notation and Auxiliary Lemmas
The inverse Fourier transform is defined by The action of a distribution f as a functional on the test function will be denoted by ( , ). f For a locally integrable function f , ( , ) f is defined by Provided that the last integral is finite for every . The Flett potential of order  is defined by (1), has the following convolution type representation: The kernel () y is as follows It is not difficult to show that the kernel () y has the following properties (see also: [22], [4]).
If we use (5) and the Poisson integral t f , we obtain the following equality for the Flett potential: where is the Poisson integral with the Poisson kernel ( , ) P y t , defined by , . (| | ) n n n n ct n P y t c yt (9) The following lemma gives some properties of the Poisson integral t f which will be used later. Lemma 2.1 [5] Let ; where the limit is interpreted in p L -norm and pointwise a.e.. For 0 fC the convergence is uniform on n .

Definition 2.2 Let is a signed Borel measure on
Then is called a wavelet measure.

Definition 2.3
The weighted wavelet-like transform of p fL is defined by where is a finite Borel measure on [0, ) , ([0, )) 0 and t f is the Poisson integral.
Owing to (15), it is assumed that is the Riemann-Liouville fractional integral of order 1 of the measure . Let further, 0 Re , and let satisfy the following conditions: The following theorem gives an inversion formula for wf, defined in (17). where the constant , k is defined as (21). (24)

Main Results
In this section we define Flett potential spaces () p L and give a characterization of these spaces by using the weighted wavelet-like transform wf.  Proof. We use some ideas from [5]. Suppose that ( ).