## 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