Weighted estimates for integral operators on local type spaces

Abstract

We prove the weighted boundedness for a family of integral operators T α on Lebesgue spaces and local B M O type spaces. To this end we show that T α can be controlled by the Calderón operator and a local maximal operator. This approach allows us to characterize the power weighted boundedness on Lebesgue spaces.

Necessary condition on the weight for maximal and integral operators with rough kernels

“…Remark 1.10. This result contains Theorem 3.2 and Corollary 3.3 in [4], where the authors consider p = q, α = 0, w p (x) = |x| β ∈ A p , A 1 = −I and A 2 = I. Remark 1.11.…”

On commutators of certain fractional type integrals with Lipschitz functions

“…, and for each 1 ≤ i ≤ m, k i satisfies (nα i )-order fractional size condition, A i is a matrix such that (H) A i is invertible and A i -A j is invertible for i = j, 1 ≤ i, j ≤ m. Clearly, T α,1 = I α , the Riesz potential, for m = 1, A 1 is the n-order identity matrix, and k 1 (x -A 1 y) = 1/|x -y| α . For general m and certain k i , T 0,m behaves like a singular integral operator and T α,m has been studied in [1][2][3][4][5][6][7][8][9][10]. In particular, Riveros and Urciuolo [5,6,11] considered each k i as a rough fractional kernel, and each k i satisfies an L α i ,γ i -Hörmander regular condition, or more general k i ∈ H α,γ i , that is, for all x ∈ R n and |x| < R,…”

“…A function Ψ : [0, ∞) → [0, ∞) is said to be a Young function if Ψ is continuous, convex, nondecreasing and satisfies Ψ (0) = 0 and lim t→∞ Ψ (t) = ∞. For f ∈ L 1 loc (R n ) and each Young function Ψ , we can induce an average of the Luxemburg norm of a function f in the ball B defined by Next, we recall the definitions of the fractional size condition and the generalized fractional Hörmander condition. Normally, we use |x| ∼ s to represent s < |x| ≤ 2s.…”

A Brief Look at the Calderón and Hilbert Operators

“…Both operators S α and H α appear in several different contexts and applications, see for instance [4,[11][12][13][14][15][16][17].…”