Weighted inequalities for the disc multiplier

Abstract:

Charles Fefferman has shown that the disc multiplier is not a bounded operator on ${L^p}({{\mathbf {R}}^n})$, $n > 1$, $p \ne 2$. On the other hand, Carl Herz has shown that when this operator is restricted to radial functions in ${{\mathbf {R}}^n}$, it is bounded in ${L^p}({{\mathbf {R}}^n})$ provided $p$ satisfies $2n/(n + 1) < p < 2n/(n - 1)$. In this paper, sufficient conditions on the weight function $\omega$ are given in order that the disc multiplier restricted to radial functions should be bounded in ${L^p}({{\mathbf {R}}^n},\omega (\left | x \right |))$. When applied to power weights $\omega (r) = {r^\alpha }$ these conditions are also necessary.