Piecewise linear and step Fourier multipliers for modulation spaces

Abstract

This note significantly extends various earlier results concerning Fourier multipliers of modulation spaces. It combines not so widely known characterizations of pointwise multipliers of Wiener amalgam spaces with novel geometric ideas and a new approach to piecewise linear functions belonging to the Fourier algebra. Thus the paper provides two original types of results.

On the one hand we establish results for step functions (i.e. piecewise constant, bounded functions), which are multipliers on the modulation spaces $\left(M_{\omega }^{p,q}\right(R^d),{\Vert \cdot \Vert }_{M_{\omega }^{p,q}})$ with $1<p<\infty$, fixed. Instead of regular patterns with a discrete subgroup structure we demonstrate that there is a significant freedom in the choice of the domains of constant values. In particular for higher dimensions (i.e., $d\ge 2$), this widens the scope of possible multipliers very much. Adding some geometric considerations we show that the step functions, which arise as nearest neighborhood interpolation (using the so-called Voronoi cells) from roughly well-spread sets with bounded values define Fourier multipliers in this range, with uniform control for large families of such sets. Parameterized families of lattices are just simple special cases.

In the second part of the paper we aim at sufficient conditions for piecewise linear Fourier multipliers, with uniform estimates for the range $p\in [1,\infty ]$ (and independent from q and s). These results are based on the control on the Fourier algebra norm of (oblique) triangular functions on $R$. This result is of independent interest, as it provides new sufficient conditions for the membership of piecewise linear functions (with irregular nodes) in the modulation space $M^1\left(R^d\right)$, also known as the Segal algebra $S_0\left(R^d\right)$ (see [6] and [25]).