Bi-Parametric Potentials, Relevant Function Spaces and Wavelet-Like Transforms

Abstract

We introduce new potential type operators J α β = E+(−∆) β/2 −α/β (α > 0, β > 0), and bi-parametric scale of function spaces H α β,p (R n ) associated with J α β . These potentials generalize the classical Bessel potentials (for β = 2), and Flett potentials (for β = 1). A characterization of the spaces H α β,p (R n ) is given with the aid of a special wavelet-like transform associated with a β-semigroup, which generalizes the well-known Gauss-Weierstrass semigroup (for β = 2) and the Poisson one (for β = 1). Mathematics Subject Classification (2000). Primary 26A33; Secondary 46E35, 42C40.

A new characterization of the Riesz potential spaces with the aid of a composite wavelet transform

“…We show that, if one take β > α, only one vanishing moment ( ∞ 0 dμ(t) = 0) is sufficient. Note that the same effect is also true for the Bessel potentials and some their generalizations (see [1]). …”

“…This semigroup arises in diverse contexts of analysis and integral geometry (see, e.g., [11,15,1,6]). For β = 1 and β = 2,…”

“…However, by taking the "homogenous" property of w (β) (|y|, t) into account, other properties of the kernel w (β) (|y|, t) and semigroup B (β) t f are well determined by the following lemma. [11, p. 44] for n = 1, and [1,6] for any n 1.) Let t > 0, y ∈ R n , and 0 < β < ∞.…”

Square-like Functions Generated by a Composite Wavelet Transform

“…Landkof [9], M.V. Fedorjuk [5], B.Rubin [11], I.A.Aliev, B.Rubin, S.Sezer and S. Uyhan [3], I.A.Aliev [2]. The following lemma contains some properties of kernels v (β) (y, t) and semigroups V [(see [2], [3])] Let β > 0, t > 0 and y ∈ R n .…”

On the rate of $L_p$-convergence of Balakrishnan--Rubin-type hypersingular integrals associated to the Gauss-Weierstrass semigroup

“…In particular, the wavelet approach to inversion of the potentials was developed by Rubin [19,20], Rubin and Aliev [1], and Aliev [2]; see also [3][4][5]27]. …”