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Register a new userWeak Type Endpoint Estimates for the Commutators of Rough Singular Integral Operators
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Let $Omega$ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{n-1}$, $T_{Omega}$ be the convolution singular integral operator with kernel $frac{Omega(x)}{|x|^n}$. For $bin{rm BMO}(mathbb{R}^n)$, let $T_{Omega,,b}$ be the commutator of $T_{Omega}$. In this paper, by establishing suitable sparse dominations, the authors establish some weak type endpoint estimates of $Llog L$ type for $T_{Omega,,b}$ when $Omegain L^q(S^{n-1})$ for some $qin (1,,infty]$.
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Let $Omega$ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{d-1}$, $T_{Omega}$ be the homogeneous singular integral operator with kernel $frac{Omega(x)}{|x|^d}$ and $T_{Omega,,b}$ be the commutator of $T_{Omega}$ with symbol $b$. In this paper, we prove that if $Omegain L(log L)^2(S^{d-1})$, then for $bin {rm BMO}(mathbb{R}^d)$, $T_{Omega,,b}$ satisfies an endpoint estimate of $Llog L$ type.
Let $Omega_1,Omega_2$ be functions of homogeneous of degree $0$ and $vecOmega=(Omega_1,Omega_2)in Llog L(mathbb{S}^{n-1})times Llog L(mathbb{S}^{n-1})$. In this paper, we investigate the limiting weak-type behavior for bilinear maximal function $M_{vecOmega}$ and bilinear singular integral $T_{vecOmega}$ associated with rough kernel $vecOmega$. For all $f,gin L^1(mathbb{R}^n)$, we show that $$lim_{lambdato 0^+}lambda |big{ xinmathbb{R}^n:M_{vecOmega}(f_1,f_2)(x)>lambdabig}|^2 = frac{|Omega_1Omega_2|_{L^{1/2}(mathbb{S}^{n-1})}}{omega_{n-1}^2}prodlimits_{i=1}^2| f_i|_{L^1}$$ and $$lim_{lambdato 0^+}lambda|big{ xinmathbb{R}^n:| T_{vecOmega}(f_1,f_2)(x)|>lambdabig}|^{2} = frac{|Omega_1Omega_2|_{L^{1/2}(mathbb{S}^{n-1})}}{n^2}prodlimits_{i=1}^2| f_i|_{L^1}.$$ As consequences, the lower bounds of weak-type norms of $M_{vecOmega}$ and $T_{vecOmega}$ are obtained. These results are new even in the linear case. The corresponding results for rough bilinear fractional maximal function and fractional integral operator are also discussed.
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.
This article develops a novel approach to the representation of singular integral operators of Calderon-Zygmund type in terms of continuous model operators, in both the classical and the bi-parametric setting. The representation is realized as a finite sum of averages of wavelet projections of either cancellative or noncancellative type, which are themselves Calderon-Zygmund operators. Both properties are out of reach for the established dyadic-probabilistic technique. Unlike their dyadic counterparts, our representation reflects the additional kernel smoothness of the operator being analyzed. Our representation formulas lead naturally to a new family of $T(1)$ theorems on weighted Sobolev spaces whose smoothness index is naturally related to kernel smoothness. In the one parameter case, we obtain the Sobolev space analogue of the $A_2$ theorem; that is, sharp dependence of the Sobolev norm of $T$ on the weight characteristic is obtained in the full range of exponents. In the bi-parametric setting, where local average sparse domination is not generally available, we obtain quantitative $A_p$ estimates which are best known, and sharp in the range $max{p,p}geq 3$ for the fully cancellative case.
By a reduction method, the limiting weak-type behaviors of factional maximal operators and fractional integrals are established without any smoothness assumption on the kernel, which essentially improve and extend previous results. As a byproduct, we characterize the boundedness of several operators by the membership of their kernel in Lebesgue space on sphere.
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