The sharp square function estimate with matrix weight

The sharp square function estimate with matrix weight, Discrete Analysis 2019:2, 8 pp.

A central theme in harmonic analysis is to determine whether naturally occurring operators are bounded with respect to naturally occurring norms. One class of norms that is of particular importance is that of weighted $L_2$-norms, that is, norms of the form

$$\|f\|_{L_2(w)}=\Bigl(\int |f|^2w\Bigr)^{1/2},$$

where $w$ is a non-negative function. A condition on $w$ that is necessary and sufficient for several important linear operators to be bounded is that it should be what is known as an $A_2$ weight. Writing $\langle w\rangle_I$ for the average of $w$ over an interval $I$, we say that $w$ is an $A_2$ weight if the supremum of $\langle w\rangle_I\langle w^{-1}\rangle_I$ is finite, where $I$ ranges over all intervals. In recent years, there has been a flurry of activity aimed at a precise understanding of the dependence of the norms of these linear operators on the $A_2$ characteristic, which has led to several sharp results.

This paper is about generalizing such results to the case of matrix weights. Here one takes a matrix-valued function $W$ that is positive semidefinite everywhere, and if $f$ is a vector-valued function, then

$$\|f\|_{A_2(W)}=\Bigl(\int_{\mathbb C^d}\langle W(x)f(x),f(x)\rangle\Bigr)^{1/2}.$$

The matrix analogue of the scalar weight is not quite obvious: it is the supremum over all intervals $I$ of the quantity $\|\langle W\rangle_I^{1/2}\langle W^{-1}\rangle_I^{1/2}\|^2$.

The main result concerns a nonlinear operator $S_W$, which is a matrix analogue of a type of operator known as a square function. It proves that the norm of $S_W$ is bounded above by a constant times the matrix $A_2$ characteristic of $W$. Since a matching lower bound is known, this is a sharp result, and it is the first sharp result in the matrix case. In fact, as the authors also prove, an even better result is possible if one allows the bound to depend not just on $[W]_{A_2}$ but also on a quantity $[W]_{A_\infty}$.