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 Vladikavkaz. Mat. Zh., 2017, Volume 19, Number 2, Pages 3–10 (Mi vmj611)

$L_p-L_q$-estimates for generalized Riss potentials with oscillating

M. N. Gurova, V. A. Noginba

b Southern Federal University, Rostov-on-Don

Abstract: We consider a class of multidimensional potential-type operators whose kernels are oscillating at infinity. The characteristics of these operators are infinitely differentiable homogeneous functions. We describe convex sets in the $(1/p;1/q)$-plane for which these operators are bounded from $L_p$ into $L_q$ and indicate the domains where they are not bounded. In some cases we describe their $\mathcal{L}$-characteristics. To obtain these results we use a new method based on special representation of the symbols of multidimensional potential-type operators. To these representations of the symbols we apply the technique of Fourier-multipliers, which degenerate or have singularities on the unit sphere in $\mathbb{R}^n$.

Key words: potential-type operators, oscillating kernel, method of Fourier multipliers, $L_p-L_q$-estimates, $\mathcal L$-characteristic.

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UDC: 517.983.2

Citation: M. N. Gurov, V. A. Nogin, “$L_p-L_q$-estimates for generalized Riss potentials with oscillating”, Vladikavkaz. Mat. Zh., 19:2 (2017), 3–10

Citation in format AMSBIB
\Bibitem{GurNog17} \by M.~N.~Gurov, V.~A.~Nogin \paper $L_p-L_q$-estimates for generalized Riss potentials with oscillating \jour Vladikavkaz. Mat. Zh. \yr 2017 \vol 19 \issue 2 \pages 3--10 \mathnet{http://mi.mathnet.ru/vmj611}