1,490 papers
2608.04666Dirichlet symbols and the nonlinear wave equation
We study the operator symbols of Dirichlet type introduced by Hedenmalm and Shimorin (2020), in connection with a given contraction on $L^2$ of the unit disk. They are always holomorphic functions on the bidisk. Such Dirichlet symbols associated with the Grunsky operator of a univalent function on the disk or exterior disk are of particular significance. From the work of Hedenmalm and Shimorin, we know they are characterized as solutions of a certain nonlinear wave equation. We perform a local analysis of such symbols near the diagonal on the bidisk, and in so doing, we provide alternative chart coordinates for the infinite-dimensional manifolds of univalent functions of the disk or the exterior disk. Those coordinates allow us to characterize $\logψ'$ for $ψ$ in the class $Σ$ of normalized univalent functions without explicitly touching the univalence property. Moreover, those manifolds extend the universal Teichmüller space of Lipman Bers beyond the quasicircle boundary setting, allowing for more fractality. The fractality of harmonic measure for the domain associated with the given univalent function can be studied in terms of the asymptotic variance introduced by McMullen (2008). The asymptotic variance captures the $L^2$ average amplitude of the nonlinearity. We introduce the new concept of Schwarzian asymptotic variance, which measures the average amplitude of the Schwarzian derivative in place of the nonlinearity. For this new Schwarzian asymptotic variance, we find that the effectiveaverage amplitude of $(1-|z|^2)^2|\Sop(\vp)|^2$ on the disk in the hyperbolic metric sense is at most $72/5=14.4$, considerably smaller than the maximum amplitude of $36$. Here, $\Sop(\vp)$ is the Schwarzian derivative of $φ\in\mathscr{S}$, and the analogous statement is valid for $ψ\inΣ$ as well.
2608.04666Aug 2026
View2604.04504Weighted $L^2$ theory for the Euclidean Dirac operator in higher dimensions
We study weighted $L^{2}$ solvability for the Euclidean Dirac operator in dimensions $n\ge 3$. We prove that, on the exterior domain $\mathbb{R}^{n}\setminus\overline{B(0,1)}$ with logarithmic weight $\varphi=n\log|x|$, no higher-dimensional analogue of the two-dimensional Hörmander estimate can be controlled solely by $Δ\varphi$; we then establish weighted solvability for the weights $|x|^{m}$ with $m\neq 0$, for the quadratic weight $x_{1}^{2}$, and for sufficiently small anisotropic perturbations of the Gaussian weight, with sharp constant $1/4$ in the Gaussian case. The obstruction arises because, in dimensions $n\ge 3$, the classical weighted identity is coercive only under a structural relation between $Δ\varphi$ and $|\nabla\varphi|^{2}$, a condition that excludes the Gaussian weight and many polynomial weights. The method is based on a weighted identity for the conjugated unknown $U:=ue^{-\varphi/2}$, together with suitable scalar and Clifford-valued multipliers; this identity yields the required coercive estimates and also gives weighted $L^{2}$ solvability for the Poisson equation through the factorization $Δ=-D^{2}$.
2604.04504Apr 2026
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Laplace measure transitions and ghosts for meromorphic functions
We study the measure transition problem for bilateral Laplace transforms of meromorphic functions on vertical strips. Given a meromorphic function F admitting Laplace representations on two adjacent strips separated by a vertical line, we investigate how the corresponding determining measures are related. Our first result shows that in the absence of poles on the separatrix the determining measures coincide. We next derive explicit transition formulas for the case of finitely many poles and obtain sufficient conditions under which these formulas remain valid for infinitely many poles. Applications are given to the analytic continuation of the zeta function, periodic and almost periodic functions, and quotients of Gamma functions related to the confluent hypergeometric function. Finally, using generalized Cauchy integrals, we construct an entire function admitting distinct Laplace representations on the right and left half-planes, thereby producing a ghost transition. This provides a counterexample to uniqueness of solutions of the Cauchy problem for the heat equation.
2604.04162Apr 2026
View2604.04054Composition operators between model and Hardy spaces
Let $n\ge 1$ and $\varphi: \mathbb{D}^n\to\mathbb{D}$ be a holomorphic function, where $\mathbb{D}$ denotes the open unit disk of $\mathbb{C}$. Let $Θ: \mathbb{D} \to \mathbb{D}$ be an inner function and $K^p_Θ$, $p>0$, denote the corresponding model space. We obtain characterizations of the compact composition operators $C_\varphi: K^p_Θ\to H^p(\mathbb{D}^n)$, $1<p<\infty$, where $H^p(\mathbb{D}^n)$ denotes the Hardy space.
2604.04054Apr 2026
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Electrostatic skeletons and condition of strict descent
Given a precompact domain $Ω\subseteq\mathbb{R}^2$, the electrostatic skeleton of $Ω$ is defined as a positive measure inside $Ω$, supported on a set with no simple loops, which generates $\partial Ω$ as an equipotential curve. Eremenko conjectured that every convex polygon admits a unique electrostatic skeleton. This conjecture has since been proven for triangles and regular polygons. In this paper, we will prove the conjecture for quadrilaterals with a line of symmetry using arguments from conformal geometry. We will also discuss a natural condition that implies the existence of electrostatic skeletons.
2604.03861Apr 2026
View2604.03854Hadamard-Type Asymptotics for Products of Best Rational Approximation Errors
Let $ρ_{n,m}(f;E)$ denote the error of best uniform rational approximation to a function $f$ analytic on a compact set $E\subset \mathbb{C}$ by rational functions whose numerator and denominator have degrees at most $n$ and $m$, respectively. Motivated by Hadamard's classical theorem on Hankel determinants and by Gonchar's theorem on rows of the Walsh table, we study, for each fixed $m\ge 0$, the asymptotic behavior as $n\to\infty$ of the products $$ \prod_{k=0}^{m}ρ_{n-m+k,k}(f;E). $$ We establish Hadamard-type asymptotic formulas for these products on the closed unit disc and, more generally, on continua with connected complement and Jordan boundary. In the disc case, our approach combines Hadamard's classical theorem and Gonchar's theorem with weighted Hankel operators and an AAK-type theorem for meromorphic approximation. We also show that there exists a common subsequence along which the extremal exponential behavior of these products and of the corresponding products on the closed Green sublevel sets $E_R$ is attained.
2604.03854Apr 2026
View2604.03223On Picard's Problem via Nevanlinna Theory II
This work continues the author's earlier work (2026, Studia Mathematica) on Picard's problem: is every meromorphic function on a complete noncompact Kähler manifold with nonnegative Ricci curvature necessarily a constant, if it avoids 3 distinct values? In that prior work, a positive answer was obtained under a growth condition for non-parabolic manifolds. In this paper, we give a full solution to the non-parabolic case by removing this growth condition via a global Green function approach. For the parabolic case, to overcome the obstacle arising from the absence of a positive global Green function, we introduce a heat kernel approach to Nevanlinna theory. Based on it, we develop a Carlson-Griffiths theory, which gives the first systematic result in Nevanlinna theory for parabolic Kähler manifolds. As a direct application, we confirm the parabolic case of Picard's problem under a weak growth condition.
2604.03223Apr 2026
View2604.03036A note on the Erdös minimal area problem
We answer a question of Erdös, Herzog, and Piranian on the minimal area of polynomial lemniscates when all the zeros of the polynomial are constrained to lie on a compact set K whose logarithmic capacity is strictly larger than 1.
2604.03036Apr 2026
View2604.02462Remote temperature sensing in 2D and the Bergman kernel
We explore the problem of estimating the steady state temperature in a two-dimensional domain at a point knowing the temperature to high order at another point. We find connections to the Bergman kernel of the domain, Runge's theorem, and approximate null quadrature identities.
2604.02462Apr 2026
View2604.02453Zeros of random $P$-polynomials in $\mathbb{C}^d$ with exponential profiles
We study random multivariate $P$-polynomials in $\mathbb{C}^d$ with monomial supports constrained to $nP\cap\mathbb{Z}_+^d$ for a convex body $P\subset\mathbb{R}_+^d$, and deterministic coefficients admitting a uniform exponential profile $f$ on $P$. Assuming the tail condition $\mathbb{P}(\log(1+|ξ_0|)>t)=o(t^{-d})$ on the i.i.d. complex coefficients, we prove that the normalized potentials $\frac1n\log|\mathbf{P}_n|$ converge in probability in $L^1_{\mathrm{loc}}(\mathbb{C}^d)$ to a deterministic toric plurisubharmonic function $Φ_{P,f}$, and consequently the normalized zero currents $\frac1n[Z_{\mathbf{P}_n}]$ converge weakly to the closed positive $(1,1)$-current $dd^cΦ_{P,f}$. Under the stronger logarithmic moment assumption $\mathbb{E}[(\log(1+|ξ_0|))^d]<\infty$, we prove almost sure weak convergence of the zero currents along the full sequence for $d>2$, and along sparse subsequences for $d \le 2$. On $(\mathbb{C}^*)^d$, the limiting potential is given by $Φ_{P,f}(z)=I_{P,f}(\operatorname{Log} z)$, where $I_{P,f}$ is the Legendre-Fenchel transform of the profile over $P$ and $\operatorname{Log} (z)=(\log|z_1|,\dots,\log|z_d|)$. These results extend the exponential-profile mechanism of Kabluchko and Zaporozhets from one complex variable to the genuinely multivariate $P$-polynomial setting under relaxed probabilistic assumptions, directly connecting random zero hypersurfaces with convex-analytic data determined by $(P,f)$.
2604.02453Apr 2026
View2604.01887Hölder regularity of Siciak-Zaharjuta extremal functions on compact Hermitian manifolds
For a compact subset in a compact Hermitian manifold, we prove that the Hölder continuity of the extremal function at a given point in the set is a local property and that the Hölder continuity of a weighted extremal function follows from the Hölder continuities of the extremal function and the weight function with a uniform density in capacity. The second result can be seen as a continuation of a result of Lu, Phung and Tô \cite{LPT21}. Moreover, for a compact subset in a compact Hermitian manifold, we prove that the Hölder continuity of the extremal function with the uniform density in capacity is equivalent to the local Hölder continuity property, which is also equivalent to the weak local Hölder continuity property. These results are generalizations of the results of Nguyen \cite{Ng24} on compact Kähler manifolds. We also show that the \(μ\)-Hölder continuity property of a convex compact subset in \(\mathbb{C}^n\) implies the local \(μ\)-Hölder continuity property of order \(1\).
2604.01887Apr 2026
View2604.01885Characterization of continuity of Siciak-Zaharjuta extremal functions on compact Hermitian manifolds
For a compact subset in a compact Hermitian manifold, we prove that the continuity of the extremal function at a given point in the set is a local property and that the continuity of a weighted extremal function follows from the continuities of the extremal function and the weight function. These results are generalizations of the results of Nguyen \cite{Ng24} on compact Kähler manifolds. Moreover, for a compact subset in a compact Hermitian manifold, we characterize the continuity of the extremal function via the local \(L\)-regularity, which is equivalent to the weak local \(L\)-regularity.
2604.01885Apr 2026
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Energy estimates for level sets of holomorphic functions and counterexamples to Calderón-Zygmund theory
We demonstrate that the failure of $L^1$ regularity in Calderón-Zygmund theory is a universal phenomenon: every non-constant holomorphic function in $\C^n$ generates a counterexample to the Poisson equation. Using Hironaka's resolution of singularities and the Łojasiewicz gradient inequality, we establish sharp level-set estimates that link harmonic analysis to the geometry of complex structure, providing results of independent interest in algebraic geometry.
2604.01559Apr 2026
View2604.00693Identification of L-functions in the extended Selberg class by preimages of finite sets
In 2023, Li, Du, Yi proved a uniqueness theorem for L functions in the extended Selberg class under the assumptions of positive degree, a shared functional equation, and the sharing of three complex values. This was later strengthened by the present authors, who showed that sharing an arbitrary finite set of complex values, counted with multiplicities, still forces equality of the two L functions, again under the assumption that they satisfy the same functional equation. In this paper, we significantly improve all these results. We completely remove the requirement that the two L functions satisfy the same functional equation, yet we still obtain the same strong uniqueness conclusion under far weaker hypotheses. As a major consequence, we prove that every polynomial with distinct zeros is a strong uniqueness polynomial for L functions.
2604.00693Apr 2026
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Any six points on the Riemann sphere can be split into three pairs by a triple of disjoint discs
We prove that for any six points on the Riemann sphere there exist three disjoint closed (or open) discs, each of which contains exactly two of the six distinguished points. This statement shows that recently proposed method to numerically evaluate Kleinian hyperelliptic functions of genus 2 is applicable to any complex curve of genus 2.
2604.00351Apr 2026
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Harmonic mappings, univalence criteria and a theorem of Lehtinen
The harmonic inner radius $σ_H(Ω)$ of a planar domain $Ω$ is the largest constant with which a univalence criterion via the Schwarzian derivative holds for harmonic mappings. We show that $σ_H(Ω)\leqσ_H(\mathbb{D})\leq 3/2$ for the unit disk $\mathbb{D}$ and for every domain $Ω$ that omits an open set. This is an analogue of a theorem of Lehtinen in the setting of holomorphic functions. We provide two related univalence criteria for harmonic mappings.
2604.00269Mar 2026
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Rigidity of entire functions sharing a finite set with their partial derivatives in C^n
This paper investigates certain classes of entire functions in C^n that, together with their partial derivatives, share a finite set consisting of three elements. By employing normality criteria, we study the behaviour of such functions and derive the necessary conditions governing their existence. Our results extend those of [4], originally established for functions of a single complex variable, to the setting of several complex variables, thereby providing a comprehensive generalization of the earlier result in a direction not previously explored.
2603.29412Mar 2026
View2603.29195The Lee--Yang Edge Exponent via Logarithmic Averaging
Let $F$ be the thermodynamic free energy of a ferromagnetic Ising model,analytic on $\mathbb{C}^{*}\setminus\mathcal{Z}_β$. The Lee--Yang edge at $z_c\in\partial\mathcal{Z}_β$ is characterised by $F(z)=F(z_c)+B(z-z_c)^{σ+1}+o(|z-z_c|^{σ+1})$ with $σ\in(-1,0)$ and $B\neq 0$. We prove three results: Theorem A (Jensen slope): defining the Jensen average $\widetilde{N}(x)=\frac{1}{2π}\int_0^{2π}\log|\widetilde{F}(e^{x+iθ})|\,dθ$ of $\widetilde{F}=F-F(z_c)$, the edge exponent satisfies $\widetilde{N}'(0^+)=σ+1$. The proof is a direct application of Jensen's formula. Theorem B (Monodromy): the monodromy of $F$ around $z_c$ multiplies the singular part by $e^{2πi(σ+1)}$, a primitive $q$-th root of unity when $σ+1=p/q$. Theorem C (Kac monodromy): for any 2D CFT at an RG fixed point with relevant operator $φ$ of weight $h_φ<0$ satisfying the Lee--Yang property, the RG scaling equation forces $σ=h_φ/(1-h_φ)$ and monodromy order $q=\mathrm{denom}(1/(1-h_φ))$. We also prove that the edge expansion follows from the density asymptotics $ρ(θ)\sim A|θ-θ_c|^σ$ via a Mellin-transform calculation, making all three theorems unconditional for the $d=2$ Ising model.
2603.29195Mar 2026
View2603.27210Arithmetic Uniformization of Rigid Elliptic Structures: From Rigid to Standard Vekua without the Beltrami Equation
For the rigid subclass of variable elliptic structures -- characterized equivalently by the inviscid Burgers law $λ_x+λλ_y=0$ or the self-dilatation $μ_{\bar z}=μμ_z$ -- we show that the auxiliary Beltrami equation in the classical Vekua pipeline is unnecessary. The canonical coordinate $ξ=y-λx$, computed by arithmetic from the spectral parameter $λ$, reduces every rigid variable-algebra Vekua equation to a standard Vekua equation in $ξ$ on any open set where the characteristic Jacobian $Φ=\barξ_x+λ\barξ_y$ does not vanish, with global reduction on domains where $ξ$ is injective. No PDE is solved at any stage.
2603.27210Mar 2026
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Sharp Landau-Type Theorems and Schlicht Disc Radii for certain Subclasses of Harmonic Mappings
Let $\mathcal{H}$ be the class of all complex-valued harmonic mappings $f=h+\overline{g}$ defined on the unit disc $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $h(0)=0=h'(0)-1$, here $h$ and $g$ are analytic functions in $\mathbb{D}$. In this paper, we investigates Landau-type theorems for several significant subclasses of sense-preserving harmonic mappings. Specifically, we establish sharp Landau-type theorems for the class $\mathcal{P}_{\mathcal{H}}^{0}(M)$ and the parameterized class $\mathcal{W}_{\mathcal{H}}^{0}(α)$ for $α\ge 0$. For mappings in $\mathcal{W}_{\mathcal{H}}^{0}(α)$, we derive the radii of univalence and the radii of the largest schlicht discs contained in the images of the unit disc, expressing these results in terms of the Lerch Transcendent function $Φ(z,s,a)$ and the Dilogarithm function ${\rm Li}_2(z)$. The sharpness of the obtained radii is demonstrated by constructing appropriate extremal functions for each class. These results generalize and extend various known Landau-type theorems in the theory of harmonic mappings.
2603.28806Mar 2026
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