\(L_p-L_q\)-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.

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