# Weighted vector-valued estimates for a non-standard Calder\'on-Zygmund operator

@article{Hu2016WeightedVE, title={Weighted vector-valued estimates for a non-standard Calder\'on-Zygmund operator}, author={G. Hu}, journal={arXiv: Classical Analysis and ODEs}, year={2016} }

In this paper, the author considers the weighted vector-valued estimate for the operator defined by $$T_Af(x)={\rm p.\,v.}\int_{\mathbb{R}^n}\frac{\Omega(x-y)}{|x-y|^{n+1}}\big(A(x)-A(y)-\nabla A(y)\big)f(y){\rm d}y,$$ and the corresponding maximal operator $T_A^*$, where $\Omega$ is homogeneous of degree zero, has vanishing moment of order one, $A$ is a function in $\mathbb{R}^n$ such that $\nabla A\in {\rm BMO}(\mathbb{R}^n)$. By a pointwise estimate for $\|\{T_Af_k(x)\}\|_{l^q}$ and the… Expand

#### 9 Citations

WEIGHTED WEAK TYPE ENDPOINT ESTIMATES FOR THE COMPOSITIONS OF CALDERÓN–ZYGMUND OPERATORS

- Mathematics
- Journal of the Australian Mathematical Society
- 2019

Abstract Let $T_{1}$ , $T_{2}$ be two Calderón–Zygmund operators and $T_{1,b}$ be the commutator of $T_{1}$ with symbol $b\in \text{BMO}(\mathbb{R}^{n})$ . In this paper, by establishing new bilinear… Expand

Quantitative Weighted Bounds for a Class of Singular Integral Operators

- Physics
- 2019

In this article, the authors consider the weighted bounds for the singular integral operator defined by $${T_A}f(x) = {\rm{p}}.{\rm{v}}.\int_{\mathbb{R}^{n}} {{{{\rm{\Omega }}(x - y)} \over… Expand

Weighted estimates for the Calderón commutator

- Mathematics, Physics
- Proceedings of the Edinburgh Mathematical Society
- 2019

Abstract In this paper the authors consider the weighted estimates for the Calderón commutator defined by \mathcal{C}_{m+1, A}(a_1,\ldots,a_{m};f)(x)={\rm p. v.}… Expand

On the Composition of Rough Singular Integral Operators

- Mathematics
- 2018

In this paper, we investigate the behavior of the bounds of the composition for rough singular integral operators on the weighted space. More precisely, we obtain the quantitative weighted bounds of… Expand

The composition of singular integral operators with nonsmooth kernels

- Mathematics
- 2018

Let $T_1$, $T_2$ be two singular integral operators with nonsmooth kernels introduced by Duong and McIntosh. In this paper, by establishing certain bi-sublinear sparse domination, the authors obtain… Expand

θ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space

- Mathematics
- 2018

Abstract The aim of this paper is to deal with the boundedness of the θ-type Calderón-Zygmund operators and their commutators on Herz spaces with two variable exponents p(⋅), q(⋅). It is proved that… Expand

Quantitative weighted bounds for the vector-valued singular integral operators with nonsmooth kernels

- Mathematics
- 2018

Let T be the singular integral operator with nonsmooth kernel which was introduced by Duong and McIntosh, and Tq (q ∈ (1, ∞)) be the vector-valued operator defined by Tqf(x) = (∑∞ k=1 |Tfk(x)| )1/q .… Expand

Weighted vector-valued bounds for the singular integral operators with nonsmooth kernels

- Mathematics
- 2016

Let $T$ be a singular integral operator with non-smooth kernel which were introduced by Duong and McIntosh. In this paper, we prove that this operator and its corresponding grand maximal operator… Expand

Some Remarks on the Pointwise Sparse Domination

- Mathematics
- 2019

We obtain an improved version of the pointwise sparse domination principle established by Lerner (N Y J Math 22:341–349, 2016 ). This allows us to determine nearly minimal assumptions on a singular… Expand

#### References

SHOWING 1-10 OF 33 REFERENCES

Weighted vector-valued bounds for a class of multilinear singular integral operators

- Mathematics
- 2016

In this paper, we investigate the weighted vector-valued bounds for a class of multilinear singular integral operators, and its commutators, from $L^{p_1}(l^{q_1};\,\mathbb{R}^n,w_1)\times\dots\times… Expand

The $L(\log L)^{\epsilon}$ endpoint estimate for maximal singular integral operators

- Mathematics
- 2015

We prove in this paper the following estimate for the maximal operator $T^*$ associated to the singular integral operator $T$: $ \|T^*f\|_{L^{1,\infty}(w)} \lesssim \frac{1}{\epsilon}
… Expand

On pointwise and weighted estimates for commutators of Calder\'on-Zygmund operators

- Mathematics
- 2016

In recent years, it has been well understood that a Calder\'on-Zygmund operator $T$ is pointwise controlled by a finite number of dyadic operators of a very simple structure (called the sparse… Expand

Weighted norm inequalities for the Hardy maximal function

- Mathematics
- 1972

The principal problem considered is the determination of all nonnegative functions, U(x), for which there is a constant, C, such that | [f*(x)rUix)dx g CJ \f(x)\'U(x) dx, where l<p<oo, J is a fixed… Expand

Sharp weighted estimates for classical operators

- Mathematics
- 2012

Abstract We give a general method based on dyadic Calderon–Zygmund theory to prove sharp one- and two-weight norm inequalities for some of the classical operators of harmonic analysis: the Hilbert… Expand

The sharp weighted bound for general Calderón-Zygmund operators

- Mathematics
- 2010

For a general Calderon‐Zygmund operator T on R N , it is shown that kTfkL2(w) C(T) sup Q A Q w Q w 1 a k fkL2(w) for all Muckenhoupt weights w 2 A2. This optimal estimate was known as the A2… Expand

On pointwise estimates involving sparse operators

- Mathematics
- 2015

We obtain an alternative approach to recent results by M. Lacey \cite{La} and T. Hyt\"onen {\it et al.} \cite{HRT} about a pointwise domination of $\omega$-Calder\'on-Zygmund operators by sparse… Expand

On $$A_p$$Ap–$$A_\infty $$A∞ type estimates for square functions

- Mathematics
- 2015

We prove strong-type $$A_p$$Ap–$$A_\infty $$A∞ estimate for square functions, improving on the $$A_p$$Ap bound due to Lerner. Entropy bounds, in the recent innovation of Treil–Volberg, are then… Expand

SHARP BOUNDS FOR GENERAL COMMUTATORS ON WEIGHTED LEBESGUE SPACES

- Mathematics
- 2010

We show that if an operator T is bounded on weighted Lebesgue space L 2 (w) and obeys a linear bound with respect to the A2 constant of the weight, then its commutator (b;T ) with a function b in BMO… Expand

Improved weighted bounds for rough singular integral operators

- Mathematics
- 2017

In this paper, we improve the known $A_p$ type, $A_1$ type estimates as well as the weighted weak $(1,1)$ estimates for rough operators, including the rough homogeneous singular integrals and the… Expand