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 $$T$$.

MSC:

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

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