$L_p-L_q$-estimates for potential-type operators with oscillating kernels

M. N. Gurov ; V. A. Nogin

Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 4, pp. 35-42

Résumé

We consider a class of multidimensional potential-type operators whose kernels are oscillating at infinity. The characteristics of these operators are from a wide class of functions including the product of a homogeneous function infinitely differentiable in $\Bbb R^n\setminus\{0\}$ and any function from $C^{m,\gamma}(\dot{R}^1_{+})$. 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, the accuracy of the estimates obtained is proved. In particular, necessary and sufficient conditions for the boundedness of the operators under considered in $ L_p $ are obtained. Currently, there is a number of papers on $L_p-L_q$-estimates for convolution operators with oscillating kernels, in particular, for the Bochner–Riesz operators and acoustic potentials arising in various problems of analysis and mathematical physics. These papers cover kernels containing only the radial characteristic $b(r)$, which stabilized at infinity as a Hëlder function. Due to this property, the derivation of estimates for the indicated operators was reduced to the case of an operator with the characteristic $b(r)\equiv1$. Such a reduction is impossible when the Riesz potential kernel contains a homogeneous characteristic $a(t')$. To receive the results we use new method which based on special representation of the symbols 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 $\mathbb{R}^n$.