Abstract
We define and study the spaces ℳp(ℝ × ℝn), 1 ≤ p ≤ ∞, that are of type. Using the harmonic analysis associated with the spherical mean operator, we give a new characterization of the dual space and describe its bounded subsets. Next, we define a convolution product in , 1 ≤ r ≤ p < ∞, and prove some new results.
Highlights
The spherical mean operator is defined, for a function f on Rn+1, even with respect to the first variable, by( f )(r, x) = fdσn(η, ξ), (r, x) ∈ R × Rn, Sn (1.1)where Sn is the unit sphere {(η, ξ) ∈ R × Rn : η2 + ξ 2 = 1} in Rn+1 and σn is the surface measure on Sn normalized to have total measure one.This operator plays an important role and has many applications, for example, in image processing of so-called synthetic aperture radar (SAR) data, or in the linearized inverse scattering problem in acoustics [6]
In [10], the authors associate to the operator a Fourier transform and a convolution product and have established many results of harmonic analysis
In [11], the authors define and study Weyl transforms related to the mean operator
Summary
The spherical mean operator is defined, for a function f on Rn+1, even with respect to the first variable, by. We will define a convolution product and the Fourier transform associated with the spherical mean operator. The Fourier transform associated with the spherical mean operator is defined on L1(dν) by. We denote by (A) S∗(R × Rn) the space of infinitely differentiable functions on R × Rn, even with respect to the first variable, rapidly decreasing together with all their derivatives;. Since Lk is a continuous operator from S∗(R × Rn) into itself, we deduce that
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