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DOI: 10.23671/VNC.2017.2.6503

\(L_p-L_q\)-estimates for Generalized Riss Potentials with Oscillating Kernels

Gurov M. N. , Nogin V. A.
Vladikavkaz Mathematical Journal 2017. Vol. 19. Issue 2.
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 \(\cal{L}\)-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 \(\mathbb{R}^n\).
Keywords: potential-type operators, oscillating kernel, method of Fourier multipliers, \(L_p-L_q\)-estimates, \(\cal L\)-characteristic.
Language: Russian Download the full text  
For citation: Gurov M. N., Nogin V. A. \(L_p-L_q\)-estimates for generalized Riss potentials with oscillating† kernels. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 19, no. 1, pp. 3-10. DOI 10.23671/VNC.2017.2.6503
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