On the symbol calculus for multidimensional Hausdorff operators

The aim of this work is to derive a symbol calculus on $L^2(\mathbb{R}^n)$ for multidimensional Hausdorff operators. Two aspects of this activity result in two almost independent parts. While throughout the perturbation matrices are supposed to be self-adjoint and form a commuting family, in the second part they are additionally assumed to be positive definite. What relates these two parts is the powerful method of diagonalization of a normal Hausdorff operator elaborated earlier by the second author

Weighted Fourier inequalities for radial functions

Weighted Lp(Rn) ! Lq(Rn) Fourier inequalities are studied. We prove Pitt-Boas type results on integrability with power weights of the Fourier transform of a radial function