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

M. N. Gurov ; V. A. Nogin

Vladikavkazskij matematičeskij žurnal, Tome 19 (2017) no. 2, pp. 3-10

Résumé

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 $\mathbb{R}^n$.