This article is cited in 5 papers

On boundedness and compactness of Riemann–Liouville fractional operators

S. M. Farsani

People's Friendship University of Russia, Moscow, Russia

Abstract: Let $\alpha\in(0,1)$. Consider the Riemann–Liouville fractional operator of the form
$$ f\to T_\alpha f(x):=v(x)\int_0^x\frac{f(y)u(y)\,dy}{(x-y)^{1-\alpha}},\qquad x>0, $$
with locally integrable weight functions $u$ and $v$. We find criteria for the $L^p\to L^q$-boundedness and compactness of $T_\alpha$ when $0<p,q<\infty$, $p>1/\alpha$ under the condition that $u$ monotonely decreases on $\mathbb R^+:=[0,1)$. The dual versions of this result are given.

Keywords: Riemann–Liouville fractional operator, Lebesgue space, weighted inequality.

UDC: 517.51

Received: 28.03.2012

 English version: Siberian Mathematical Journal, 2013, 54:2, 368–378

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