Operator-valued Fourier multipliers of bounded s-variation

TL;DR

This paper proves a new Fourier multiplier theorem for operator-valued functions in weighted spaces, linking geometric properties of Banach spaces with boundedness conditions via $s$-variation and $ $-boundedness.

Contribution

It introduces a novel operator-valued Fourier multiplier theorem in weighted spaces, utilizing $ $-boundedness and $s$-variation conditions, connecting Banach space geometry with multiplier boundedness.

Findings

Established a weighted vector-valued variational Carleson inequality.

Derived a Littlewood--Paley--Rubio de Francia type estimate.

Linked geometric properties of Banach spaces to Fourier multiplier boundedness.

Abstract

In this paper, we establish an operator-valued Fourier multiplier theorem in weighted Lebesgue spaces, Besov and Triebel--Lizorkin spaces, assuming the multiplier has $\mathcal{R}$-bounded range and satisfies an $\ell^r$-summability condition on its bounded $s$-variation seminorms over dyadic intervals. The exponents $r$ and $s$ reflect the relationship between the geometric properties of the underlying Banach spaces (type and cotype) and the boundedness of Fourier multiplier operators. As our main tool we prove a weighted vector-valued variational Carleson inequality and deduce an estimate of Littlewood--Paley--Rubio de Francia type.