Asymmetry of twisted convolution operators
Let K be a distribution on \({\mathbb{R}}^ 2\), and define a twisted convolution operator \(T_ K\) by the formula \(T_ K\phi(x,y):=K\times \phi(x,y):=\iint K(x-u,y-v)f(u,v)e^{i(xv-yu)}du dv.\) Such twisted convolutions arise naturally, since the usual \(L^ 1\)-algebra of the Heisenberg group with compact center is the topological sum of algebras, which are either isomorphic to the usual convolution algebra on \({\mathbb{R}}^ 2\) or to a twisted convolution algebra on \({\mathbb{R}}^ 2\). The following main result is proved. There exists a continuous Radon measure K on \({\mathbb{R}}^ 2\) whose support is a union of sets \(\{y=y_ n\}\) where \(\{y_ n\}\) is a discrete set in \({\mathbb{R}}\) such that \(T_ K\) is bounded on \(L^ p({\mathbb{R}}^ 2)\) if and only if \(1<p\leq 2.\) For ordinary convolution on \({\mathbb{R}}^ 2\), such an asymmetry of a convolution operator could not hold. The measure K in the theorem above is given explicitly, and the proof of the most difficult part, namely the \(L^ p\)-boundedness of \(T_ K\) for \(1<p\leq 2\), is based on the introduction of a suitable atomic Hardy space \(H^ 1\) on \({\mathbb{R}}^ 2\), which turns out to be ''locally'' the same as \(L^ 1({\mathbb{R}},h^ 1({\mathbb{R}}))\). Here \(h^ 1({\mathbb{R}})\) denotes Goldberg's local Hardy space on \({\mathbb{R}}\) (see \textit{D. Goldberg} [Duke Math. J. 46, 27-42 (1979; Zbl 0409.46060)]). The \(L^ p\)-boundedness is then obtained by complex interpolation between \(H^ 1\) and \(L^ 2.\) In a subsequent paper [\textit{M. Cowling}, and the author, Lect. Notes Math. 908, 210-216 (1982; Zbl 0487.43006)] the result above has been sharpened as follows: There exist twisted convolution operators which are bounded on \(L^ p({\mathbb{R}}^ 2)\) if and only if p lies in an interval of the form [q,2] or (q,2], where \(1<q<2\).
scientific article; zbMATH DE number 3846191