arXiv subjects
Publications and source records attributed to Joachim Toft.
We discuss continuity of the twisted convolution on (weighted) Fourier modulation spaces. We use these results to establish continuity results for the twisted convolution on Lebesgue spaces. For example we prove that if $ω$ is an appropriate weight and $1\le p\le 2$, then $L^p_{(ω)}$ is an algebra under the twisted convolution.
Let $\Op_t(a)$, for $t\in \mathbf R$, be the pseudo-differential operator $$ f(x) \mapsto (2π)^{-n}\iint a((1-t)x+ty,ξ)f(y)e^{i\scal {x-y}ξ} dydξ$$ and let $\mathscr I_p$ be the set of Schatten-von Neumann operators of order $p\in [1,\infty ]$ on $L^2$. We are especially concerned with the Weyl case (i.{}e. when $t=1/2$). We prove that if $m$ and $g$ are appropriate metrics and weight functions respectively, $h_g$ is the Planck's function, $h_g^{k/2}m\in L^p$ for some $k\ge 0$ and $a\in S(m,g)$, then $\Op_t(a)\in \mathscr I_p$, iff $a\in L^p$. Consequently, if $0\le δ<ρ\le 1$ and $a\in S^r_{ρ,δ}$, then $\Op_t(a)$ is bounded on $L^2$, iff $a\in L^\infty$.
In this paper various properties of global and local changes of variables as well as properties of canonical transforms are investigated on modulation and Wiener amalgam spaces. We establish several relations among localisations of modulation and Wiener amalgam spaces and, as a consequence, we obtain several versions of local and global Beurling-Helson type theorems. We also establish a number of positive results such as local boundedness of canonical transforms on modulation spaces, properties of homogeneous changes of variables, and local continuity of Fourier integral operators on Fourier Lebesgue spaces. Finally, counterparts of these results are discussed for spaces on the torus as well as for weighted spaces.
We consider Fourier integral operators with symbols in modulation spaces and non-smooth phase functions whose second orders of derivatives belong to certain types of modulation space. We establish continuity and Schatten-von Neumann properties of such operators when acting on modulation spaces.
We consider Fourier integral operators with symbols in modulation spaces and non-smooth phase functions whose second orders of derivatives belong to certain types of modulation space. We establish continuity and Schatten-von Neumann properties of such operators when acting on modulation spaces.