In this paper, we present Lappan's five-point type criteria for $\varphi$-normality of sense-preserving harmonic mappings and families of harmonic mappings in the unit disc $\mathbb{D}$. We establish a normality criterion involving the higher-order spherical derivative of order $k$, and $k+2$ distinct test values under suitable conditions. We further reduce the number of test values to $\left[\frac{k}{2}\right]+2$ under an additional differential condition involving the analytic and co-analytic parts. We also prove a generalization in which the spherical derivative is replaced by a polynomial with non-negative real coefficients.
We study the problem of finding the norm of the point evaluation operator in the Paley--Wiener space $PW^{1}(\r^d)$, consisting of $d$-variable functions of spherical exponential type that are integrable on $\r^d$. The extremal functions can be taken radial, which naturally leads us to consider a related extremal probl...
Felipe Gonçalves, D. Radchenko, Antonio Pedro Ramos· 0 citations
We study harmonic mappings between the source domain in the $\alpha$-Grushin plane and the target domain in the $\beta$-Grushin plane for $\alpha$ not necessarily equal to $\beta$ and $\alpha\in [0,1)$. Our harmonic mappings arise as a Euler--Lagrange system of equations related to the Dirichlet energy and are studied...
Tomasz Adamowicz, Krzysztof Chełmiński, Marcin Walicki· 0 citations
Let $k$ be a complete, non-archimedean. We study the dynamics of families of one-variable rational functions parametrized by Berkovich spaces over $k$ and over $\mathbb{C}$ equipped with the hybrid norm. In both settings, we show an analogue of Man\'e--Sad--Sullivan's theorem about the continuity of the Julia set (in t...
Let $F$ be the unramified quadratic extension of $\mathbb{Q}_p$, and let $P_n(T)$ denote the polynomials arising in the $p$-adic Fourier theory of Schneider and Teitelbaum. We determine explicitly the norms of the functions $P_n(x\Omega)$ on the Banach spaces of locally $F$-analytic functions of fixed order, where $\Om...
Using different approaches, Simon Donaldson and Dashen Yan recently constructed $\mathbb Z_2$-harmonic functions on $\mathbb R^3$ and on $\mathbb R^n$ for $n\geqslant 3$, respectively. Their examples have quadratic growth at infinity and $\mathcal{O}(r^{3/2})$ local growth near a smooth codimension-two branching set $\...
Yang Li, Parsa Mashayekhi, Yicheng Zhang· 1 citation
We show that there exist two separable real Banach spaces $X$ and $Y$ such that, for every $\varepsilon>0$, there is a standard exact $\varepsilon$-isometry $f_\varepsilon: X\to Y$ whose distortion of every distance is between $0$ and $\varepsilon$, whereas no isometric embedding of $X$ into $Y$ exists, even without a...
Long-Fa Sun, Yi-Peng Zhang· 0 citations
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