Correlation of boundary behavior of conjugate harmonic functions

Vladimir Ryazanov

Home/Archive/arXiv

arXiv · 1710.00323

Correlation of boundary behavior of conjugate harmonic functions

Vladimir Ryazanov

View original source ↗Read source document ↗Export metadata ↓

Abstract

It is established that if a harmonic function $u$ on the unit disk $\mathbb D$ in $\mathbb C$ has angular limits on a measurable set $E$ of the unit circle $\partial\mathbb D$, then its conjugate harmonic function $v$ in $\mathbb D$ also has angular limits a.e. on $E$ and both boundary functions are finite a.e. and measurable on $E$. The result is extended to arbitrary Jordan domains with rectifiable boundaries in terms of angular limits and of the natural parameter.

Explore related subjects

math.CV

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Vladimir Ryazanov. 2018-03-03. Correlation of boundary behavior of conjugate harmonic functions. https://arxiv.org/abs/1710.00323

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Product Principle for Harmonic Schwarz Lemmas: Boxes, Polydiscs, and Metric Geometry

We establish an exact product principle for the Euclidean operator norm of differentials of harmonic maps. For a bounded domain $G\subset\R^m$ and $p\in G$, let $M_G(p)$ denote the supremum of $\|dF_0\|$ over harmonic maps $F:\D\to G$ with $F(0)=p$. For bounded domains $G_j\subset\R^{m_j}$, we prove \[ M_{G_1\times\cdots\times G_N}(p_1,\ldots,p_N)^2 =\sum_{j=1}^N M_{G_j}(p_j)^2. \] The theorem separates the geometry of the individual factors from the Euclidean geometry of the product: the factor extremal constants combine by a sum-of-squares law, while equality is governed by a single compatibility condition, namely a common maximizing direction for the component differentials. Neither convexity nor attainment of the factor suprema is required. Combining the product principle with the sharp interval and disk factor problems yields exact operator-norm estimates and all equality cases for harmonic maps into boxes and polydiscs. In both families, for every extremal map, the real differential at the origin has one-dimensional image. The same factor constants also define coordinatewise metrics for which the harmonic contraction estimate is sharp when the source disk is equipped with its Poincaré metric of curvature $-1$. For boxes, the resulting metric is complete and equals twice the restriction of the Kobayashi-Royden metric of the product of vertical strips. For polydiscs, the harmonic product metric is pointwise maximal among contracting metrics of the form $\max_j a_j(p)|v_j|$, with $a_j(p)>0$. It is strictly smaller than twice the Kobayashi-Royden metric on every nonzero tangent vector, and its induced path metric is incomplete.

math.CV↗

arXiv2026

bi-Lipschitz versus Analytic equivalence of two variable complex quasihomogeneous function-germs

In this paper we address the problem of classifying complex (non-homogeneous) quasihomogeneous polynomials in two variables under bi-Lipschitz equivalence. We prove that pairs of such polynomials are (right) bi-Lipschitz equivalent as function-germs at $0\in\mathbb{C}^{2}$ iff they are analytically equivalent.

math.CV↗

arXiv2026

Convolution Regularization Preserves the $L^2$-Estimate Property for $(1,n)$-Forms

In this paper, we prove that the $L^2$-estimate property for $(1,n)$-forms is preserved under the standard convolution regularization. As applications, we show that any singular Hermitian metric satisfying the optimal or multiple coarse $L^2$-estimate property for $(n,1)$ or $(1,n)$-forms is Griffiths semi-positive. This resolves a question posed by Deng--Ning--Wang and a question by Inayama.

math.CV↗