Continuous frame decomposition and a vector Hunt-Muckenhoupt-Wheeden theorem. (English) Zbl 0961.44006

From the introduction: This paper deals with the weighted norm inequalities for the Hilbert transform with matrix-valued weights. The main problem can be formulated as follows. Let \(W\) be a \(d\times d\) matrix weight, i.e. an \(L^1\) function whose values are selfadjoint nonnegative \(d\times d\) matrices. We suppose that the weight \(W\) is defined on the unit circle \(T=\{z \in\mathbb{C}: |z|= 1\}\).
Let \(L^2=L^2 (\mathbb{C}^d)\) be the space of square summable functions on \(T\) with values in \(\mathbb{C}^d\), let \(H^2=H^2 (\mathbb{C}^d)\) be the corresponding Hardy space of analytic functions, and let \(P_+\) be the orthogonal projection in \(L^2\) onto \(H^2\). Let \(T\) denote the Hilbert transform, \(T=-iP_+ +i(I-P_+)\).
The question we are interested in is under what conditions on \(W\) the following weighted norm inequality for \(T\) holds (say for all \(f\in L^2\cap L^\infty)\), \[ \int_T\bigl(W(\xi) Tf(\xi), Tf(t)\bigr)dt\leq C \int_T\bigl(W(\xi) f(\xi), f(\xi)\bigr) dm(\xi), \] where \(m\) is the normalized \((m(T)=1)\) Lebesgue measure on \(T\).


44A15 Special integral transforms (Legendre, Hilbert, etc.)
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
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