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DOI: 10.23671/VNC.2018.4.23385

\(L_p-L_q\)-Estimates for Potential-Type Operators with Oscillating Kernels

Gurov, M. N. , Nogin, V. A.
Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 4.
Abstract:
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 Helder 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 \(\mathbb{R}^n\).
Keywords: potential-type operators, oscillating kernel, \(L_p-L_q\)-estimates, \(\cal L\)-characteristics.
Language: Russian Download the full text  
For citation: Gurov, M. N. and Nogin, V. A. \(L_p-L_q\)-Estimates for Potential-Type Operators with Oscillating Kernels, Vladikavkaz Math. J., 2018, vol. 20, no. 4, pp. 35-42 (in Russian). DOI 10.23671/VNC.2018.4.23385
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