Local cone multipliers and Cauchy-Szego projections in bounded symmetric domains

We show that the cone multiplier satisfies local $L^p$-$L^q$ bounds only in the trivial range $1\le q\le 2\le p\le \mathrm{\infty }$. To do so, we suitably adapt to this setting the proof of Fefferman for the ball multiplier. As a consequence we answer negatively a question by Békollé and Bonami (Colloq. Math. 68, 1995, 81-100), regarding the continuity from $L^p\to L^q$ of the Cauchy-Szegö projections associated with a class of bounded symmetric domains in ${\mathbb{C}}^n$ with rank $r\ge 2$.