$L^{p}$ multipliers with weight $x^{kp-1}$

Abstract:

Let $k$ be a positive integer and $1 < p < \infty$. It is shown that if $T$ is a multiplier operator on ${L^p}$ of the line with weight $|x{|^{kp-1}}$, then $Tf$ equals a constant times $f$ almost everywhere. This does not extend to the periodic case since $m(j) = 1/j, j \ne 0$, is a multiplier sequence for ${L^p}$ of the circle with weight $|x{|^{kp-1}}$. A necessary and sufficient condition is derived for a sequence $m(j)$ to be a multiplier on ${L^2}$ of the circle with weight $|x{|^{2k - 1}}$.