$L_p-L_q$ Estimates for the Bochner-Riesz Operator of Complex Order

Abstract

We describe convex sets on the $(\frac{1}{p},\frac{1}{q}$)-plane for which the well-known Bochner-Riesz operator with the symbol $(1–|\xi {|}^2){+}^{–\alpha }(0<\mathrm{Re}\alpha <\frac{n+1}{2})$ is bounded from $L_p$ into $L_q$.