Convolution estimates for some distributions with singularities on the light cone (Q810826)

Let \(K^ z\) be the family of distributions on \(R^{n+1}\) defined by \textit{I. M. Gelfand} and \textit{G. E. Shilov} [Generalized functions, Vol. I (1964; Zbl 0115.331)]. The main result of this paper is the following theorem: Theorem 1. Suppose \(-n/2\leq z<0\). If \(1/p-1/q=1+2z/(n+1)\) and \(1+z/n<1/p<1+(z(n-1))/(n(n+1))\) then the inequality \[ \| K^ z*f\|_ q\leq C(p)\| f\|_ p \] holds for measurable f on \(R^{n+1}.\) The proof of the theorem and its consequences used extensions of techniques developed by the author [Ill. J. Math. 33, 143-152 (1989; Zbl 0688.42018)]. The results obtained extend a theorem of the author to the case \(n>2\), settles a point left open by Ricci and Stein (Preprint) and generalizes a result of \textit{R. S. Strichartz} [J. Funct. Anal. 5, 218- 235 (1970; Zbl 0189.407)].