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Asymptotic growth of Cauchy transforms. (English) Zbl 1062.30043

Let \(M(R)\) be the space of those complex measures \(\mu\) on the real line \(R\) such that \[ \int_{R} \frac {d| \mu| (t)} {1 + | t| } < \infty . \] For \(\mu \in M(R)\), the Poisson integral \(P\mu\) and the conjugate Poisson integral \(Q\mu\) are defined on the upper half plane \(C_{+}\) by \[ P\mu(x + iy) = \int_{R} \frac {y} {(x - t)^{2} + y^{2}} \;d\mu(t) \quad \text{and } Q\mu(x + iy) = \int_{R} \frac {x - t} {(x - t)^{2} + y^{2}} \;d\mu(t) . \] For \(\mu \in M(r)\), let \(\mu_{s}\) denote the singular part of \(\mu\) relative to Lebesgue measure. The authors’ basic result is that if \(\mu \in M(R)\) and \(\Sigma\) is a Borel subset of \(R\), then the following conditions are equivalent: (1) \(Q\mu(x + iy) = o(P\mu(x + iy))\) as \(y \to 0+\) for almost every \(x \in \Sigma\) (relative to \(\mu_{s}\)); (2) the restriction of \(\mu_{s}\) on \(\Sigma\) is discrete.
Let \(D\) denote the open unit disc in the complex plane. For a subset \(\Sigma\) of \(T = \partial D\), the inner function \(\theta\) is said to be radial near \(\Sigma\) if it maps every radius that ends at a point of \(\Sigma\) into a curve that is tangential to a radius of \(D\) (at a point of \(T\)). Applying the basic result to inner functions, they prove that an inner function \(\theta\) in \(D\) is radial near a set \(\Sigma \subset T\) if and only if \(\theta\) has non-tangential (angular) derivatives almost everywhere on \(\Sigma\). The authors also give some generalizations of the basic result. Let \(M(R^{n-1})\) be the space of all measures \(\mu\) in the hyperplane \(R^{n-1}\) such that \[ \int_{R^{n-1}} \frac {1} {1 + | x| ^{n-1}} \;d\mu(x) < \infty, \] and, for \(\mu \in M(R^{n-1})\) and \(x \in R^{n}_{+} = \{(x_{1}, x_{2}, \dots, x_{n}): x_{n} > 0 \}\) define \[ R_{j}\mu(x) = \int_{R^{n-1}} \frac {x_{j} - y_{j}} {| x - y| ^{n}} \;d\mu(y), \quad j = 1, 2, \dots , n, \] with \(y_{n} = 0\). It is proved that if \(\mu \in M(R^{n-1})\) and \(\Sigma \subset R^{n-1}\) are such that \[ | <R_{1}\mu(y), R_{2}\mu(y), \dots, R_{n-1}\mu(y)>| = o(R_{n}\mu(y)) \] as \(y \to x \in \Sigma\) non-tangentially almost everywhere (relative to \(\mu\)) on \(\Sigma\), then the restriction of \(\mu\) to \(\Sigma\) is absolutely continuous with respect to Lebesgue measure on \(R^{n-1}\). Also, there are some results given relating to a result of P. Mattila [Adv. Math. 115, No. 1, 1–34 (1995; Zbl 0842.30029)] stating that for a measure \(\mu \in M(R^{2})\) with appropriate local properties (given in terms of limits of integrals), the measure \(\mu\) is the sum of a discrete measure and a measure absolutely continuous with respect to one dimensional Hausdorff measure concentrated on a countable union of \(C^{1}\) curves in \(R^{2}\).

MSC:

30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
31B99 Higher-dimensional potential theory

Citations:

Zbl 0842.30029
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