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Register a new userBump conditions and two-weight inequalities for commutators of fractional integrals
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This paper gives new two-weight bump conditions for the sparse operators related to iterated commutators of fractional integrals. As applications, the two-weight bounds for iterated commutators of fractional integrals under more general bump conditions are obtained. Meanwhile, the necessity of two-weight bump conditions as well as the converse of Bloom type estimates for iterated commutators of fractional integrals are also given.
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This paper gives the pointwise sparse dominations for variation operators of singular integrals and commutators with kernels satisfying the $L^r$-H{o}rmander conditions. As applications, we obtain the strong type quantitative weighted bounds for such variation operators as well as the weak-type quantitative weighted bounds for the variation operators of singular integrals and the quantitative weighted weak-type endpoint estimates for variation operators of commutators, which are completely new even in the unweighted case. In addition, we also obtain the local exponential decay estimates for such variation operators.
For the maximal operator $ M $ on $ mathbb R ^{d}$, and $ 1< p , rho < infty $, there is a finite constant $ D = D _{p, rho }$ so that this holds. For all weights $ w, sigma $ on $ mathbb R ^{d}$, the operator $ M (sigma cdot )$ is bounded from $ L ^{p} (sigma ) to L ^{p} (w)$ if and only the pair of weights $ (w, sigma )$ satisfy the two weight $ A _{p}$ condition, and this testing inequality holds: begin{equation*} int _{Q} M (sigma mathbf 1_{Q} ) ^{p} ; d w lesssim sigma ( Q), end{equation*} for all cubes $ Q$ for which there is a cube $ P supset Q$ satisfying $ sigma (P) < D sigma (Q)$, and $ ell P = rho ell Q$. This was recently proved by Kangwei Li and Eric Sawyer. We give a short proof, which is easily seen to hold for several closely related operators.
We study the commutators $[b,T]$ of pointwise multiplications and bi-parameter Calderon-Zygmund operators and characterize their off-diagonal $L^{p_1}L^{p_2} to L^{q_1}L^{q_2}$ boundedness in the range $(1,infty)$ for several of the mixed norm integrability exponents.
We prove in this note one weight norm inequalities for some positive Bergman-type operators.
We prove a sharp Hardy inequality for fractional integrals for functions that are supported on a general domain. The constant is the same as the one for the half-space and hence our result settles a recent conjecture of Bogdan and Dyda.
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