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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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