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On the existence of principal values for the Cauchy integral on weighted Lebesgue spaces for non-doubling measures. (English) Zbl 1002.42010

Let \(\mu\) be a Radon measure on a metric space \((\mathbf{X},d)\) which may not be doubling, but should satisfy the growth condition \(\mu(B(x,r))\leq Cr^{n}\) for all \(r\) and \(x\). Let \(T\) be a Calderón-Zygmund operator with \(n\)-dimensional kernel. The author studies weighted inequalities for \(T_{*}\), the supremum of the truncated operator associated with \(T\), that is to find conditions on \(v\) and \(u\) so that \[ \int_{\mathbf{X}}(T_{*}f(x))^{p}u(x)d\mu (x)\leq \int_{\mathbf{X}} |f(x)|^{p}v(x)d\mu (x). \] The main tool to deal with this problem is the theory of vector-valued inequalities. Their main example is the Cauchy integral operator, and the authors use these results to characterize the existence of principal values for functions in weighted \(L^{p}\).

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
42B25 Maximal functions, Littlewood-Paley theory
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References:

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