A Hörmander type criterion for quasiradial Fourier multipliers

Abstract:

We state practicable sufficient conditions on quasi-radial functions $m \circ \rho (\xi ) = m(\rho (\xi ))$ to be Fourier multipliers in ${L^p}({{\mathbf {R}}^n})$. Here $m$ is a bounded function and $\rho$ is a homogeneous distance function. The conditions on $m$ are given in terms of localized Bessel potentials and those on $\rho$ reflect and generalize basic properties of the norm in ${{\mathbf {R}}^n}$. The results are related to those of Madych [7] and Fabes and Rivière [3] and improve their results (specialized to quasi-radial multipliers). The proof utilizes Madych’s approach [7] and interpolation properties of localized Bessel potential spaces [2].