## Matrix $$A_p$$ weights via maximal functions.(English)Zbl 1065.42013

For a matrix weight $$W$$, i.e., a function on $$\mathbb{R}^n$$ taking values in $$d\times d$$ positive definite matrices, let $$L^p(W)$$ be the weighted norm space whose norm is defined by $$\left\| f\right\| _{L^p(W)}^p=\int\nolimits_{ \mathbb{R}^n}\left| W^{1/p}f\right| ^pdx.$$ For a given matrix weight $$W$$ and a ball $$B\subseteq \mathbb{R}^n$$, let $$X_B$$ be the Banach space norm on $$\mathbb{C}^d$$ defined by considering the $$L^p(W)$$ norm of characteristic functions on $$B.$$ A matrix weight $$W$$ satisfies the matrix $$A_p$$ condition if $$V_BV_B^{\prime }$$ are uniformly bounded as operators on $$\mathbb{C}^d$$, that is $$\left\| V_BV_B^{\prime }\right\| \leq C<\infty$$ for all balls $$B\subseteq \mathbb{R}^n$$. Here, $$V_B$$ is the “$$L^p$$ average” of $$W^{1/p}$$ over $$B$$ and $$V_B^{\prime }$$ is the $$L^{\frac p{p-1}}$$ average of $$W^{-1/p}$$.
Let $$T$$ be a singular integral operator associated to the kernel $$K(x)$$ in the sense that $$Tf(x)=f*K(x)$$ for almost every $$x$$ outside the support of $$f$$. The kernel $$K$$ is assumed to satisfy the regularity hypothesis $\left| K(x)\right| \leq C\left| x\right| ^{-n}\text{ and }\left| \triangledown K(x)\right| \leq C\left| x\right| ^{-n-1}$ and the operator $$T$$ is assumed to satisfy the additional assumption that for some $$1<p<\infty$$, the bound $$\left\| Tf\right\| _p\leq A\left\| f\right\| _p$$ holds for all $$f\in L^p$$.
In this paper, it is shown that the matrix $$A_p$$ condition leads to $$L^p$$-boundedness of a Hardy-Littlewood maximal function and certain weighted $$L^p$$ bounds of singular integral operators. Among several results obtained in this paper, the author proves the following:
(1) If $$W$$ is a matrix $$A_p$$ weight, then there exists $$\delta >0$$ such that the vector Hardy-Littlewood maximal function $$M_w$$ defined by $M_wg(x)=\sup\limits_{x\in B}\left| B\right| ^{-1}\int\nolimits_B\left| W^{1/p}(x)W^{-1/p}(y)g(y)\right| dy$ is a bounded operator from $$L^q(\mathbb{R}^n,\mathbb{C}^d)$$ to $$L^q(\mathbb{R},\mathbb{R})$$ whenever $$\left| p-q\right| <\delta$$.
(2) Given a singular integral operator $$T$$ as above, and a weight $$W\in A_p$$, then
(a) there exists $$\delta >0$$ such that $$(W^{1/p}T)_{*}W^{-1/p}$$ is a bounded operator from $$L^q(\mathbb{R}^n,\mathbb{C}^d)$$ to $$L^q(\mathbb{R},\mathbb{R})$$ whenever $$\left| p-q\right| <\delta$$
(b) $$W^{1/p}TW^{-1/p}$$ is bounded on $$L^q(\mathbb{R}^n,\mathbb{C}^d)$$ whenever $$\left| p-q\right| <\delta$$.
(c) $$T$$ is bounded on $$L^p(W)$$ if $$W\in A_p$$. With one additional hypothesis on the structure of $$T$$, the converse statement is also true.

### MSC:

 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25 Maximal functions, Littlewood-Paley theory
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