Spaces ofDLptype and a convolution product associated with the spherical mean operator

Spaces of DLp type and a convolution product associated with the spherical mean operator

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 (see [7, 8]), or in the linearized inverse scattering problem in acoustics [6]. In [10], the authors associate ...

The spherical mean value operator with centers on a sphere

Let B represent the ball of radius ρ in Rn and S its boundary; consider the map M : C∞ 0 (B) → C∞(S × [0,∞)) where (Mf)(p, r) = 1 ωn−1 ∫ |θ|=1 f(p+ rθ) dθ represents the mean value of f on a sphere of radius r centered at p. We summarize and discuss the results concerning the injectivity of M, the characterization of the range of M, and the inversion of M. There is a close connection between me...

Product - Convolution Operators and Mlxed - Norm Spaces

Conditions for boundedness and compactness of product-convolution operators g —» PhCß = h ■ (/» g) on spaces L^G) are studied. It is necessary for boundedness to define a class of "mixed-norm" spaces L,p>q){G) interpolating the Lp(G) spaces in a natural way (L^^ = Z^,). It is then natural to study the operators acting between L(/1?)(G) spaces, where G has a compact invariant neighborhood. The t...

Convolution and Homogeneous Spaces

On convolution properties for some classes of meromorphic functions associated with linear operator

In this paper, we defined two classes $S_{p}^{ast }(n,lambda ,A,B)$ and\ $ K_{p}(n,lambda ,A,B)$ of meromorphic $p-$valent functions associated with a new linear operator. We obtained convolution properties for functions in these classes.