Abstract
In this paper, we first introduce some new Morrey-type spaces containing generalized Morrey space and weighted Morrey space with two weights as special cases. Then we give the weighted strong type and weak type estimates for fractional integral operators I_{alpha} in these new Morrey-type spaces. Furthermore, the weighted strong type estimate and endpoint estimate of linear commutators [b,I_{alpha}] formed by b and I_{alpha} are established. Also we study related problems about two-weight, weak type inequalities for I_{alpha} and [b,I_{alpha}] in the Morrey-type spaces and give partial results.
Highlights
For given α, < α < n, the fractional integral operator Iα of order α is defined by f (y)Iαf (x) := γ (α) Rn |x – y|n–α dy, and γ (α) = π n α ( n–α )It is well known that the Hardy-Littlewood-Sobolev theorem states that the fractional integral operator Iα is bounded from Lp(Rn) to Lq(Rn) for < α < n, < p < n/α and /q = /p – α/n
The fractional integral operator Iα is bounded from Lp(wp) to Lq(wq)
We denote by W Lp,κ (w) the weighted weak Morrey space of all measurable functions f for which f
Summary
Let ≤ p < ∞, < κ < and w be a weight on Rn. We denote by W Lp,κ (w) the weighted weak Morrey space of all measurable functions f for which f We denote by W Mp,θ (v) the generalized weighted weak Morrey space of all measurable functions f for which f To obtain endpoint estimate for the linear commutator [b, Iα], we first need to define the weighted A-average of a function f over a ball B by means of the weighted Luxemburg norm; that is, given a Young function A and w ∈ A∞, we define (see [ , ] for instance)
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