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arXiv math/02070952002-07-11math.CA

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

E. Ournycheva

备注：: LaTex, 21 pages

英文摘要：

$(L_p, L_q)$ estimates are obtained for oscillatory potentials $(K^\alphaf)(x)=\int\limits_{R^n}\frac{\exp(i|y|)}{|y|^{n-α}}f(x-y)dy$, $0<α<n$, $n\geq 2$, 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 $α$, determination of the corresponding pairs $(L_p, L_q)$ represents an open problem. The range of $α$ for which the problem is open is just the same as for the Bochner-Riesz means.

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