Sufficiency conditions for $L^ p$ multipliers with general weights

Abstract:

Weighted norm inequalities in ${R^1}$ are proved for multiplier operators with the multiplier function satisfying Hörmander type conditions. The operators are initially defined on the space ${\mathcal {S}_{0,0}}$ of Schwartz functions whose Fourier transforms have compact support not including 0. This restriction on the domain of definition makes it possible to use a larger class of weight functions than usually considered; weight functions used here are of the form ${\left | {g(x)} \right |^p}V(x)$ where $g(x)$) is a polynomial of arbitrarily high degree and $V(x)$ is in ${A_p}$. For weight functions in ${A_p}$, the results hold for all Schwartz functions. The periodic case is also considered.