$(L_p, L_q)$ estimates of potentials with oscillating kernel

Abstract

$(L_p, L_q)$ estimates are obtained for oscillatory potentials $(K^\alphaf)(x)=\int\limits_{R^n}\frac{\exp(i|y|)}{|y|^{n-\alpha}}f(x-y)dy$, $0<\alpha<n$, $n\geq 2$, whose symbol has a singularity on the unit sphere. These potentials are natural modifications of the celebrated Bochner-Riesz operator and Helmholtz potential arising in Fourier analysis and PDE. For some values of $\alpha$, determination of the corresponding pairs $(L_p, L_q)$ represents an open problem. The range of $\alpha$ for which the problem is open is just the same as for the Bochner-Riesz means.