Sharp estimates of unimodular Fourier multipliers on Wiener amalgam spaces

We study the boundedness on the Wiener amalgam spaces $W^{p,q}_s$ of Fourier multipliers with symbols of the type $e^{i\mu(\xi)}$, for some real-valued functions $\mu(\xi)$ whose prototype is $|\xi|^{\beta}$ with $\beta\in (0,2]$. Under some suitable assumptions on $\mu$, we give the characterization of $W^{p,q}_s\rightarrow W^{p,q}$ boundedness of $e^{i\mu(D)}$, for arbitrary pairs of $0< p,q\leq \infty$. Our results are an essential improvement of the previous known results, for both sides of sufficiency and necessity, even for the special case $\mu(\xi)=|\xi|^{\beta}$ with $1<\beta<2$

Matrix dilation and Hausdorff operators on modulation spaces

In this paper, we establish the asymptotic estimates for the norms of the matrix dilation operators on modulation spaces. As an application, we study the boundedness on modulation spaces of Hausdorff operators. The definition of Hausdorff operators are also revisited for fitting our study