David Kalaj

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math.CV2026

A bounded globally univalent quasiconformal harmonic map whose analytic part is unbounded

David Kalaj

We construct, for every $0<k<1$, a bounded globally univalent harmonic mapping \[ f=h+\overline g \colon \D\to\C \] such that \[ |g'(z)|\le k|h'(z)|,\qquad z\in\D, \] while the a…

math.CV2026

The $L^{2}$ Norm of the Interior Cauchy Transform: Beyond the First Dirichlet Eigenvalue

David Kalaj

We study sharp $L^2$ bounds for the interior Cauchy transform $C_D$ on a bounded planar domain $D$ and clarify its connection with the Dirichlet spectrum. We analyze an appro…

math.CV2026

The $L^{2}$ -Norm of the Cauchy transform on circular annuli

David Kalaj

We compute the exact $L^{2}$ operator norm of the Cauchy transform \[ (C_Î©f)(z)=\frac1Ï\int_Î©\frac{f(w)}{z-w}\,dA(w) \] on a circular annulus $I^c ◯ = A (r, R) = {r < ∣ z ∣ < R}$ . Exploiting r…

math.CV2025

On generalized M. Riesz conjugate function theorem for harmonic mappings

Anton Gjokaj, David Kalaj, Djordjije Vujadinovic

Let $L^{p} (T)$ be the Lesbegue space of complex-valued functions defined in the unit circle $T = {z : ∣ z ∣ = 1} \subseteq C$ . In this paper, we address the prob…

math.CV2025

Zygmund theorem for harmonic quasiregular mappings

David Kalaj

Let $K \geq 1$ . We prove Zygmund theorem for $K -$ quasiregular harmonic mappings in the unit disk $D$ in the complex plane by providing a constant $C (K)$ in the inequality $$…

A bounded globally univalent quasiconformal harmonic map whose analytic part is unbounded

The L2 Norm of the Interior Cauchy Transform: Beyond the First Dirichlet Eigenvalue

The L2-Norm of the Cauchy transform on circular annuli

On generalized M. Riesz conjugate function theorem for harmonic mappings

Zygmund theorem for harmonic quasiregular mappings

The $L^{2}$ Norm of the Interior Cauchy Transform: Beyond the First Dirichlet Eigenvalue

The $L^{2}$ -Norm of the Cauchy transform on circular annuli