We establish strong and weak Hardy–Littlewood–Sobolev inequalities for the subadditive operators majorized by operators from a certain class of integral convolutions of the Riesz-potential type with almost monotone kernels generated both by operators of ordinary shift and by operators of generalized shift associated with the differential Laplace–Bessel operator.
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I. A. Kipriyanov, Singular Elliptic Boundary-Value Problems [in Russian], Nauka, Moscow (1997).
I. A. Kipriyanov and L. A. Ivanov, “Determination of the fundamental solutions for homogeneous equations with singularities in several variables,” Tr. Sem. Soboleva, No. 1, 55–77 (1983).
B. M. Levitan, “Expansion in Fourier series and integrals in Bessel functions,” Usp. Mat. Nauk, 6, No. 2, 102–143 (1951).
I. A. Kipriyanov and M. I. Klyuchantsev, “Estimates for the surface potential generated by the operator of generalized shift,” Dokl. Akad. Nauk SSSR, 188, No. 5, 997–1000 (1969).
S. L. Sobolev, “On one theorem of functional analysis,” Mat. Sb., 4, No. 3, 471–497 (1938).
A. D. Gadzhiev and I. A. Aliev, “On the classes of potential-type operators generated by generalized shifts,” Dokl. Rasshir. Sem. Vekua Inst. Prikl. Mat., 5, No. 2, 30–32 (1990).
V. S. Guliev, “Sobolev theorem for the Riesz B-potentials,” Dokl. Ros. Akad. Nauk, 358, No. 4, 450–451 (1998).
V. S. Guliev, “Sobolev theorem for the anisotropic Riesz–Bessel potential in the Morrey–Bessel spaces,” Dokl. Ros. Akad. Nauk, 367, No. 2, 155–156 (1999).
V. S. Guliyev, N. N. Garakhanova, and Y. Zeren, “Pointwise and integral estimates for B-Riesz potentials in terms of B-maximal and B-fractional maximal functions,” Siberian Math. J., 49, No. 6, 1008–1022 (2008). 10. V. S. Guliev, “Some properties of the anisotropic Riesz–Bessel potential,” Anal. Math., 26, No. 2, 99–118 (2000).
S. K. Abdullaev and Z. A. Damirova, Nauch. Ped. Izv. Univ. “Odlar Yurdu,” Ser. Fiz., Tekh., Mat., Estestv. Nauk, No. 13 (2005).
S. K. Abdullaev and B. K. Agarzaev, “Hardy–Littlewood–Sobolev inequality for the generalized Riesz potentials,” in: Proc. of the Mathematical Sci. Conf. Dedicated to 50 Years of the Chair of Numerical Mathematics at the Baku State University (Baku, November 15–16, 2012) (2012), pp. 85–90.
E. Nakai and H. Sumitomo, “On generalized Riesz potentials and spaces of some smooth functions,” Sci. Math. Jap., 54, No. 3, 463–472 (2001).
M. A. Krasnosel’skii and Ya. V. Rutitskii, Convex Functions and Orlicz Spaces [in Russian], Fizmatgiz, Moscow (1958).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 1, pp. 3–19, January, 2020.
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Abdullayev, S.K., Mammadov, E.A. On One Class of Subadditive Operators with Generalized Shift. Ukr Math J 72, 1–20 (2020). https://doi.org/10.1007/s11253-020-01760-7
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DOI: https://doi.org/10.1007/s11253-020-01760-7