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

E. Ournycheva
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Abstract

$(L_p, L_q)$ 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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BibTeXRIS

E. Ournycheva. 2002-07-11. $(L_p, L_q)$ estimates of potentials with oscillating kernel. https://arxiv.org/abs/math/0207095

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