, Volume 54, Issue 2, pp 368-378
Date: 21 Apr 2013

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.