Abstract
In this paper, we introduce a new class of weights, the A_{lambda, infty} weights, which contains the classical A_{infty} weights. We prove a mixed A_{p,q}–A_{lambda,infty} type estimate for fractional integral operators.
Highlights
Introduction and the main resultsFractional integral operators and the associated maximal functions are very useful tools in harmonic analysis and PDE, especially in the study of differentiability or smoothness properties of functions
For 0 < λ < n, the fractional integral operator Iλ of a locally integrable function f defined on Rn is given by f (y)
In [6], Muckenhoupt and Wheeden showed that, for 1 < p < n/(n – λ) and 1/q + 1/p = λ/n, the fractional integral operator Iλ is bounded from Lp(wp) to Lq(wq) if and only if w belongs to Ap,q
Summary
|Q|λ/n f (y) dy, Q where the supremum is taken over all cubes Q in Rn with sides parallel to the axes. In [6], Muckenhoupt and Wheeden showed that, for 1 < p < n/(n – λ) and 1/q + 1/p = λ/n, the fractional integral operator Iλ is bounded from Lp(wp) to Lq(wq) if and only if w belongs to Ap,q They proved that the fractional maximal function Mλ is bounded. Moen, Pérez and Torres [7] proved the sharp weighted bound for fractional integral operators. Given 1 < p, q < ∞, we say that a pair of weights (μ, σ ) is in the class Ap,q if [μ, σ ]Ap,q := sup μ(x) dx σ (x) dx With this notation, we can generalize Theorem 1.3 as follows. 2.1 General dyadic grids Let D be a set consisting of cubes in Rn. Recall that D is said to be a general dyadic grid if it satisfies the following three conditions: 1.
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