# A Weak-Type Estimate for Commutators

@article{Grafakos2012AWE,
title={A Weak-Type Estimate for Commutators},
author={Loukas Grafakos and Petr Honz{\'i}k},
journal={International Mathematics Research Notices},
year={2012},
volume={2012},
pages={4785-4796}
}
• Published 2012
• Mathematics
• International Mathematics Research Notices
Let K be a smooth Calderón-Zygmund convolution kernel on R and suppose we are given a function a ∈ L∞. The two-dimensional commutator Tf(x) = ∫ K(x− y)f(y) ∫ [x,y] a(z) dz dy was shown to be bounded on L(R), p > 1 by Christ and Journé [1]. In this article, we show that this operator is also of weak type (1, 1).
Weighted weak type bm$(1,1)$estimate for the Christ-Journé type commutator
• Mathematics
• 2018
Let $K$ be the Calderon-Zygmund convolution kernel on $\mathbb{R}^d~(d\geq2)$. Christ and Journe defined the commutator associated with $K$ and $a\in~L^\infty(\mathbb{R}^d)$ by which is an extension
Some jump and variational inequalities for the Calder\'on commutators and related operators
• Mathematics
• 2017
In this paper, we establish jump and variational inequalities for the Calderon commutators, which are typical examples of non-convolution Calderon-Zygmund operators. For this purpose, we also show
BOUND CRITERION FOR SINGULAR INTEGRALS WITH ROUGH KERNEL AND ITS APPLICATIONS
• Mathematics
• 2018
In this paper, a weak type (1,1) bound criterion is established for singular integral operators with rough kernel. As some applications of this criterion, we show that some important operators with
Weak type (1,1) bound criterion for singular integrals with rough kernel and its applications
• Mathematics
Transactions of the American Mathematical Society
• 2018
In this paper, a weak type (1,1) bound criterion is established for singular integral operator with rough kernel. As some applications of this criterion, we prove some important operators with rough
Commutators with fractional differentiation for second-order elliptic operators on ℝn
• Mathematics
Communications in Contemporary Mathematics
• 2019
Let [Formula: see text] be a second-order divergence form elliptic operator and [Formula: see text] an accretive, [Formula: see text] matrix with bounded measurable complex coefficients in [Formula:
A weak type bound for a singular integral
A weak type $(1,1)$ estimate is established for the first order $d$-commutator introduced by Christ and Journ\'e, in dimension $d\ge 2$.
Multilinear Singular Integral Forms of Christ-Journé Type
• Mathematics
Memoirs of the American Mathematical Society
• 2019
We prove $L^{p_1}(\mathbb R^d)\times \dots \times L^{p_{n+2}}(\mathbb R^{d})$ polynomial growth estimates for the Christ-Journ\'e multilinear singular integral forms and suitable generalizations.
On estimate of operator for $0 Operators such as Carleson operator are known to be bounded on L for all 1 < p < ∞, but not from L to weak-L and from H to L for each 0 < p ≤ 1, the object of this article is to give a estimate for Gradient Estimates for Commutators of Square Roots of Elliptic Operators with Complex Bounded Measurable Coefficients • Mathematics • 2017 Let $$L=-\mathrm{div}(A\nabla )$$L=-div(A∇) be a second order divergence form elliptic operator and A an accretive $$n\times n$$n×n matrix with bounded measurable complex coefficients in $${\mathbb Weighted Bound for Commutators • Mathematics • 2015 Let$$K$$K be the Calderón–Zygmund convolution kernel on$$\mathbb {R}^d (d\ge 2)$$Rd(d≥2). Define the commutator associated with$$K$$K and$$a\in L^\infty (\mathbb {R}^d)$\$a∈L∞(Rd) by

## References

SHOWING 1-9 OF 9 REFERENCES
The weak-type (1,1) of L \log L homogeneous convolution operators
We show that a homogeneous convolution kernel on an arbitrary homogeneous group which is L \log L on the unit annulus is bounded on L^p for 1 < p < \infty and is of weak-type (1,1), generalizing the
Weak (1,1) boundedness of singular integrals with nonsmooth kernel
If Q E Lq(Sl) for some q > 1, f.1 Q = 0, and Q is homogeneous of degree 0, then the operator defined in two dimensions by Tef(x) = jV> f(x y)f2(y)IyI2 dy is of weak-type (1, 1) with bound independent
Singular integral operators with rough convolution kernels
in all dimensions, again under the assumption Q e L log L. It is conceivable that a variant of the arguments in [3] for the maximal operator could also work for the singular integral operator; in
Polynomial growth estimates for multilinear singular integral operators
• Mathematics
• 1987
On demontre un theoreme pour des mesures de Carleson et on l'applique a l'operateur de Kato en dimension 1. On demontre un critere de limitation general. On donne une application aux commutateurs de
Journé, Polynomial growth estimates for multilinear singular integral operators
• Acta Math. 159,
• 1987
University of Missouri, Columbia MO 65211, USA E-mail address: grafakosl@missouri
• University of Missouri, Columbia MO 65211, USA E-mail address: grafakosl@missouri
CZ -115 67 Praha 1, Czech Republic E-mail address: honzik@gmail
• CZ -115 67 Praha 1, Czech Republic E-mail address: honzik@gmail