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Spectral perturbation theory and the two weights problem. (English) Zbl 1312.47057

The paper under review deals with the so-called “two weights problem” for singular integral operators acting on functions on the unit circle \({\mathbb T} \cong (0,2\pi)\). With the projections \(P_{\pm}\) on \(L^2({\mathbb T})\) defined by \( (P_{\pm}f)(e^{i\theta}) := \pm \lim_{r\rightarrow 1\mp 0} \int_0^{2\pi} \frac{f(e^{it})}{1-re^{i(\theta-t)}}\frac{dt}{2\pi}, \) it says:
Characterize all possible pairs of weights \(v_0,v_1 : {\mathbb T} \rightarrow [0,\infty)\) such that the linear operator \(P_+ : L^2({\mathbb T}, v_0(e^{it})dt) \rightarrow L^2({\mathbb T}, v_1(e^{it})dt)\) is bounded (and equally for \(P_-\)).
Apart from some partial results, e.g., on the “one weight problem” with \(v_0=v_1\), no complete answer seems to have yet been obtained. In this paper, the authors pay attention to a result of P. Koosis [C. R. Acad. Sci., Paris, Sér. A 291, 255–257 (1980; Zbl 0454.42009)]. Instead of \(P_{\pm}\), they use the weighted projections \(P_{\pm}^{(w_0)}\) (with weight \(w_0 \geq 0\)) defined by \[ (P_{\pm}^{(w_0)}f)(e^{i\theta}) := \pm \lim_{r\rightarrow 1\mp 0} \int_0^{2\pi} \frac{w_0(e^{it})f(e^{it})}{1-re^{i(\theta-t)}}\frac{dt}{2\pi},\tag{*} \] to reformulate Koosis’ theorem in an equivalent form convenient for their purpose as follows : For each weight \(w_0 \geq 1\) with \(w_0 \in L^1({\mathbb T})\), there exists another weight \(w_1\geq 0\) with \(w_1w_0 \leq 1\) and \(\log w_1 \in L^1({\mathbb T})\) such that the linear operator \(P_+^{(w_0)} : L^2({\mathbb T}, w_0(e^{it})dt) \rightarrow L^2({\mathbb T}, w_1(e^{it})dt) \) is bounded. Then they show an operator-valued analogue to and a more general statement than Koosis’ theorem.
For \(1\leq p <\infty\), \({\mathbf S}_p\) stands for the Schatten class of compact operators which is a subspace of \({\mathcal B}({\mathcal K})\), the Banach space of all bounded linear operators on a Hilbert space \({\mathcal K}\). The main result is: Let in turn \(w_0: {\mathbb T} \rightarrow {\mathbf S}_p\) be a Borel measurable nonnegative (a.e.) weight function satisfying \(\int_0^{2\pi} \|w_0(e^{it})\|_p \frac{dt}{2\pi} =1\). Then, first, for each ‘simple’ \({\mathcal K}\)-valued function \(f\) on \({\mathbb T}\) (which means it can be written as \(f(\mu) = \sum_j (\mu-z_j)^{-1}\chi_j\) (finite sum), \(\mu \in {\mathbb T}\), \(\chi_j \in {\mathcal K}\), \(|z_j| \not= 1\)) and for a.e.\(\theta \in (0,2\pi)\), the limits (*) exist in the strong topology on \({\mathcal K}\). Second, there exists another nontrivial Borel measurable nonnegative weight function \(w_1 : {\mathbb T} \rightarrow {\mathcal B}({\mathcal K})\) with \( \int_0^{2\pi} (w_1(e^{it})\chi,\chi)\,\frac{dt}{2\pi} \leq \|\chi\|^2\), \(\chi \in {\mathcal K}\), and there exist contractions \(X, Y_+, Y_- : L^2(w_0) \rightarrow L^2(w_1)\) such that \(P_{\pm}^{(w_0)}\) is represented as \(P_{\pm}^{(w_0)} = \pm \frac{i}2(X-Y_{\pm})\) and hence, in particular, \(P_{\pm}^{(w_0)} : L^2(w_0) \rightarrow L^2(w_1)\) become contractions. Here, \(L(w)\) is the linear space of \({\mathcal K}\)-valued functions \(g\) with \(\|g\|_{L^2(w)}^2 := \int_0^{2\pi} (w((e^{it})g(e^{it}),g(e^{it}))\, \frac{dt}{2\pi} < \infty \), which is precisely the (Hilbert) space obtained as quotient over the subspace of functions \(g\) with \(\|g\|_{L^2(w)} =0\). The set of such ‘simple’ functions \(f\) as above is dense in \(L(w)\).
The characteristic feature is that the proof is carried out in the formalism of spectral perturbation theory or scattering theory by way of L. de Branges [Am. J. Math. 84, 543–560 (1962; Zbl 0134.11801)], S. T. Kuroda [Bull. Am. Math. Soc. 70, 556–560 (1964; Zbl 0119.31903)].

MSC:

47G10 Integral operators
47A40 Scattering theory of linear operators
30H10 Hardy spaces
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