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On boundedness and compactness of Riemann-Liouville fractional operators

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Abstract

Let α ∈ (0, 1). Consider the Riemann-Liouville fractional operator of the form

$f \to T_\alpha f(x): = v(x)\int\limits_0^x {\frac{{f(y)u(y)dy}} {{(x - y)^{1 - \alpha } }}} ,x > 0, $

with locally integrable weight functions u and v. We find criteria for the L p → L q-boundedness and compactness of T α when 0 < p,q < ∞, p > 1/α under the condition that u monotonely decreases on ℝ+:= [0,∞). The dual versions of this result are given.

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Authors and Affiliations

  1. People’s Friendship University of Russia, Moscow, Russia

    S. M. Farsani

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  1. S. M. Farsani
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Correspondence to S. M. Farsani.

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Original Russian Text Copyright © 2013 Farsani S.M.

__________

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 54, No. 2, pp. 468–479, March–April, 2013.

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Farsani, S.M. On boundedness and compactness of Riemann-Liouville fractional operators. Sib Math J 54, 368–378 (2013). https://doi.org/10.1134/S0037446613020183

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