A generalization of parabolic potentials associated to Laplace–Bessel differential operator and its behavior in the weighted Lebesque spaces

In this work we introduce some generalizations of the singular parabolic Riesz and parabolic Bessel potentials. Namely, ∆ν being the Laplace–Bessel singular differential operator, we define the families of operators H α β,ν = ( ∂ ∂t + (−∆ν) β/2 )−α/β and H α β,ν = ( I + ∂ ∂t + (−∆ν) β/2 )−α/β , (α, β > 0), and investigate their properties in the special weighted Lp,ν -spaces.

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[1] Aliev IA. The properties and inversions of B-parabolic potentials. In: Special Problems of Mathematics and Mechanics. Baku, Azerbaijan: Bilik, 1992, pp. 56-75 (in Russian).
[2] Aliev IA, Bayrakci S. On inversion of B-elliptic potentials by the method of Balakrishman-Rubin. Fractional Calculus and Applied Analysis 1998; 1(4): 365-384.
[3] Aliev IA, Rubin B. Parabolic potentials and wavelet transforms with the generalized translation. Studia Mathematica 2001; 145: 1-16.
[4] Aliev IA, Rubin B. Parabolic wavelet transforms and Lebesque spaces of parabolic potentials. Rocky Mountain Journal of Mathematics 2002; 32 (2): 391-408.
[5] Aliev IA, Rubin B, Sezer S, Uyhan SB. Composite wavelet transforms: applications and perspectives. AMS Contemporary Mathematics 2008; 464: 1-25.
[6] Aliev IA, Sağlık E. Generalized Riesz potentials spaces and their characterization via wavelet-type transforms. Filomat 2016; 30 (10): 2809-2823.
[7] Aliev IA , Sekin Ç. A generalization of parabolic Riesz and parabolic Bessel potentials. Rocky Mountain Journal of Mathematics 2020; 50 (3): 815-824.
[8] Bagby R. Lebesgue spaces of parabolic potentials. Illinois Journal of Mathematics 1971; 15 (4): 610-634.
[9] Chanillo S. Hypersingular integrals and parabolic potentials. Transactions of the American Mathematical Society 1981; 267: 531-547.
[10] Gadjiev AD, Aliev IA. Riesz and Bessel potentials generated by a generalized translation and their inverses. In: Proc. IV All-Union Winter Conference, Theory of functions and approximation; Saratov, Russia; 1988. pp. 7-53.
[11] Gopala VR. A characterization of parabolic function spaces. American Journal of Mathematics 1977; 99: 985-993.
[12] Gopala VR. Parabolic function spaces with mixed norm. Transactions of the American Mathematical Society 1978; 246: 451-461.
[13] Jones BF. Lipschitz spaces and heat equation. Journal of Applied Mathematics and Mechanics 1968; 18: 379-410.
[14] Kipriyanov IA. Singular Eliptic Boundary Value Problems. Moscow, Russia: Nauka Fizmatlit, 1997 (in Russian).
[15] Nogin VA, Rubin BS. Inversion and description of parabolic potentials with Lp -densities. Proceedings of the USSR Academy of Sciences 1985; 284: 535-538.
[16] Nogin VA, Rubin BS. Inversion of parabolic potentials with Lp -densities. Matematicheskie Zametki 1986; 38: 831- 840.
[17] Nogin VA, Rubin BS. The spaces L α p,r(R n+1) of parabolic potentials. Analysis Mathematica 1987; 13: 321-338 (in Russian).
[18] Samko SG, Kilbas AA, Marichev OI. Fractional Integrals and Derivatives: Theory and applications. New York, NY, USA: Cordon and Breach Science Publishers Inc., 1993. [19] Samko SG. Hypersingular Integrals and Their Applications. Oxfordshire, UK: Taylor& Francis, 2002.
[20] Samko SG. Fractional powers of operators via hypersingular integrals. In: Balakrishnan AV (editor). Semigroups of Operators: Theory and Applications. Progress in Nonlinear Differential Equations and Their Applications, Vol. 42. Basel, Switzerland: Birkhäuser Verlag AG, 2000, pp. 259-273.
[21] Sampson CH. A characterization of parabolic Lebesgue spaces. PhD, Rice University, Houston, TX, USA, 1968.
[22] Sezer S, Aliev IA. On space of parabolic potentials associated with the singular heat operator. Turkish Journal of Mathematics 2005; 29: 299-319.
[23] Stempak K. La theorie de Littlewood-Paley pour la transformation de Fourier-Bessel. Comptes rendus de l’Académie des Sciences Series I 1986; 303: 15-18 (in French).
[24] Uyhan S, Gadjiev AD, Aliev IA. On approximation properties of the parabolic potentials. Bulletin of the Australian Mathematical Society 2006; 74 (3): 449-460.