Local boundedness of cone multipliers

We show that the cone multiplier satisfies local Lᵖ-Lq bounds only in the trivial range 1≤ q≤ 2≤ p≤∞. 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 B\'ekoll\'e and Bonami (Colloq. Math. 68, 1995, 81-100), regarding the continuity from Lᵖ→ Lq of the Cauchy-Szeg\"o projections associated with a class of bounded symmetric domains in Cⁿ with rank r≥2.