Convolution operators with fractional measures associated to holomorphic functions (Q1603230)

Let \(\Omega\) be an open subset of the complex plane \(C\) and \(K\) be a compact subset of \(\Omega\) with non-empty interior such that \(0\notin \partial K\). Consider a holomorphic function \(\phi\) on \(\Omega\) such that \(\phi''\) never vanishes on \(\partial K\). Define the Borel measure \(\mu\) on \(R^4\sim C^2\) by \[ \mu(E)=\int_K 1_E(z,\phi(z))\left|z\right|^{\gamma-2}d\sigma(z), \] with \(0<\gamma\leq 2\) and \(d\sigma\) stands for the Lebesgue measure on \(R^2\sim C\). Let \(T_{\mu}\) be the convolution operator given by \(T_{\mu}(f)=\mu\ast f\). The aim of this paper is to give a complete description of the type set \(E_{\mu}\) of \(T_{\mu}\), defined by \[ E_{\mu}=\left\{\left(\frac 1p, \frac 1q\right);\;\left\|T_{\mu}\right\|_{p,q}<+\infty,\;1\leq p,q\leq +\infty\right\}. \] The authors show that it is a closed polygonal region with vertices depending on \(\gamma\) and on the function \(\phi\) in an explicit way, unless \(\phi\) is a polynomial of degree \(1\) (in this case, \(E_{\mu}\) is the principal diagonal, as was shown by \textit{F. Ricci} [Boll. Unione Mat. Ital., VII. Ser., A 11, No. 2, 237-252 (1997; Zbl 0946.42006)]. The proof relies on results by \textit{M. Christ} [UCLA Preprint (1988)] and \textit{S. W. Drury} and \textit{K. Guo} [Math. Proc. Camb. Philos. Soc. 110, No. 1, 151-159 (1991; Zbl 0739.42011)].