On the Boundedness and Compactness of a Class of Integral Operators

Abstract

Let α > 0. The operator of the form Tαf(x)=υ(x)xα∫0xf(y)dy(x‐y)1‐α,x>0, is considered, where the real weight function v(x) is locally integrable on R+ := (0, ∞). In case v(x) = 1 the operator coincides with the Riemann–Liouville fractional integral, Lp → Lq estimates of which with power weights are well known. This work gives Lp → Lqboundedness and compactness criteria for the operator Tα in the case 0 < p, q < ∞, p > max(1/α, 1).

Boundedness and compactness of positive integral operators on cones of homogeneous groups

“…Ashraf et al Positivity from L p (R + ) to L q v (R + ), 1 < p, q < ∞, 1/p < α < 1 have been obtained in [19] and [28]. This result was generalized in [20] (see also [6], Ch.…”

Kernel operators on the upper half-space: boundedness andcompactness criteria

“…were found in [18] (see also [24]). That result was generalized in [20] for kernel operators involving, for example, Riemann-Liouville, power-logarithmic, Erdelyi-Köber, and Hadamard kernels (see also monograph [5], Ch.2).…”

Редукционные Теоремы Для Весовых Интегральных Неравенств На Конусе Монотонных Функций

“…для частных случаев параметров суммирования и весовых функций [69], [61], [65], [79], [86]. Наиболее общий результат получен в [61] при 1 < p q < ∞, но он имеет труднопроверяемый характер.…”