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Fiorenza, A.; Gogatishvili, A.; Kopaliani, T.

Estimates for imaginary powers of Laplace operator in variable Lebesgue spaces and applications. (English) Zbl 1319.42012

J. Contemp. Math. Anal., Armen. Acad. Sci. 49, No. 5, 232-240 (2014) and Izv. Nats. Akad. Nauk Armen., Mat. 49, No. 5, 11-22 (2014).
Summary: In this paper we study some estimates of norms in variable exponent Lebesgue spaces for singular integral operators that are imaginary powers of the Laplace operator in \(\mathbb{R}^n\). Using the Mellin transform argument, from these estimates we obtain the boundedness of a family of maximal operators in variable exponent Lebesgue spaces, which are closely related to the (weak) solution of the wave equation.

Cited in 2 Documents

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42B37 Harmonic analysis and PDEs
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
35L05 Wave equation
44A10 Laplace transform

Keywords:

singular integral operators; Laplace operator; imaginary powers; spherical maximal function; variable exponent Lebesgue spaces; Mellin transform; wave equation
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References:

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