Abstract
The author has proposed a new approach to extrapolation of operators from the scale of Lebesgue spaces to the Orlicz spaces beyond this scale. In this article comprising two parts we develop some mathematical method that enables us to prove extrapolation theorems for arbitrary behavior of an operator in the Lebesgue scale (i.e., in the case when the norm of the operator is an arbitrary function of p) and also in the case when the basic scale is an interval of the Lebesgue scale with exponents separated from 1 or +∞. In this event, we face ill-posed problems of inversion of the classical Mellin and Laplace type integral transforms over nonanalytic functions in terms of their asymptotic behavior on the real axis and also the question about the properties of convolution type integral transforms on classes of N-functions. In the first part of the article we study integral representations for N-functions by expansions in power functions with a positive weight and the behavior of convolution type integral transforms on classes of N-functions.
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References
Krasnosel'skii M. A. and Rutitskii Ya. B., Convex Functions and Orlicz Spaces [in Russian], Fizmatgiz, Moscow (1958).
Simonenko I. B., “Interpolation and extrapolation of linear operators in Orlicz spaces,” Mat. Sb., 63, No.4, 536–553 (1964).
Berezhnoi E. I. and Perfil'ev A. A., “An exact extrapolation theorem for operators,” Funktsional. Analiz i Prilozhen., 34, No.3, 66–68 (2000).
Vaigant V. A. and Kazhikhov A. V., “On existence of global solutions to the two-dimensional Navier-Stokes equations for a compressible viscous fluid,” Siberian Math. J., 36, No.6, 1108–1141 (1995).
Mamontov A. E., “Existence of global solutions of multidimensional Burgers's equations of a compressible viscous fluid,” Mat. Sb., 190, No.8, 61–80 (1999).
Mamontov A. E., “Global solvability of the multidimensional Navier-Stokes equations of a compressible fluid with nonlinear viscosity. I,” Siberian Math. J., 40, No.2, 351–362 (1999).
Mamontov A. E., “Global solvability of the multidimensional Navier-Stokes equations of a compressible nonlinearly viscous fluid. II,” Siberian Math. J., 40, No.3, 541–555 (1999).
Mamontov A. E. and Kazhikhov A. V., “On a certain class of convex functions and the exact well-posedness classes of the Cauchy problem for the transport equation in Orlicz spaces,” Siberian Math. J., 39, No.4, 716–734 (1998).
Mamontov A. E., “Extrapolation of linear operators from L p into the Orlicz spaces generated by N-functions of slow or fast growth,” in: Current Problems of Modern Mathematics [in Russian], Novosibirsk Univ., Novosibirsk, 1996, 2, pp. 95–103.
Mamontov A. E., Orlicz Spaces in the Existence Problem of Global Solutions to Viscous Compressible Nonlinear Fluid Equations [Preprint, No. 2-96], Lavrent'ev Inst. of Hydrodynamics (Novosibirsk), Novosibirsk (1996).
Kufner A., Fucik S., and John O., Function Spaces, Academia, Prague (1977).
Bateman H. and Erdelyi A., Tables of Integral Transformations. Vol. 1: Fourier, Laplace, and Mellin Transforms [Russian translation], Nauka, Moscow (1969).
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The author was supported by the Russian Science Support Foundation, the Russian Foundation for Basic Research (Grant 05-01-00131), and the Federal Agency for Education (Grant 8247).
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Translated from Sibirskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 123–145, January–February, 2006.
Original Russian Text Copyright © 2006 Mamontov A. E.
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Mamontov, A.E. Integral Representations and Transforms of N-Functions. I. Sib Math J 47, 97–116 (2006). https://doi.org/10.1007/s11202-006-0012-y
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DOI: https://doi.org/10.1007/s11202-006-0012-y