Fourier multipliers on weighted $L^p$ spaces

The paper provides a complement to the classical results on Fourier multipliers on $L^p$ spaces. In particular, we prove that if $q\in (1,2)$ and a function $m:\mathbb{R} \rightarrow \mathbb{C}$ is of bounded $q$-variation uniformly on the dyadic intervals in $\mathbb{R}$, i.e. $m\in V_q(\mathcal{D})$, then $m$ is a Fourier multiplier on $L^p(\mathbb{R}, wdx)$ for every $p\geq q$ and every weight $w$ satisfying Muckenhoupt's $A_{p/q}$-condition. We also obtain a higher dimensional counterpart of this result as well as of a result by E. Berkson and T.A. Gillespie including the case of the $V_q(\mathcal{D})$ spaces with $q>2$. New weighted estimates for modified Littlewood-Paley functions are also provided.