We show that the set of Julia limiting directions of a transcendental-type K-quasiregular mapping $f:\mathbb {R}^n\to \mathbb {R}^n$ must contain a component of a certain size, depending on the dimension n, the maximal dilatation K, and the order of growth of f. In particular, we show that if the order of growth is small enough, then every direction is a Julia limiting direction. We also show that if every component of the set of Julia limiting directions is a point, then f has infinite order. The main tool in proving these results is a new version of a Phragmén–Lindelöf principle for sub-F-extremals in sectors, where we allow for boundary growth of the form $O( \log |x| )$ instead of the previously considered $O(1)$ bound.

A compactness of the Revuz map is established in the sense that the locally uniform convergence of a sequence of positive continuous additive functionals (PCAFs) is derived in terms of their smooth measures. To this end, we first introduce a metric on the space of measures of finite energy integrals and show some structures of the metric. Then, we show the compactness and give some examples of PCAFs that the convergence holds in terms of the associated smooth measures.

Macroscopically, a Darcian unsaturated moisture flow in the top soil is usually represented by an one-dimensional volume scale of evaporation from a static water table. On the microscale, simple pore-level models posit bundles of small-radius capillary tubes of a constant circular cross-section, fully occupied by mobile water moving in the Hagen–Poiseuille (HP) regime, while large-diameter pores are occupied by stagnant air. In our paper, cross-sections of cylindrical pores are polygonal. Steady, laminar, fully developed two-dimensional flows of Newtonian water in prismatic conduits, driven by a constant pressure gradient along a pore gradient, are more complex than the HP formula; this is based on the fact that the pores are only partially occupied by water and immobile air. The Poisson equation in a circular tetragon, with no-slip or mixed (no-shear-stress) boundary conditions on the two adjacent pore walls and two menisci, is solved by the methods of complex analysis. The velocity distribution is obtained via the Keldysh–Sedov type of singular integrals, and the flow rate is evaluated for several sets of meniscus radii by integrating the velocity over the corresponding tetragons.

We solve a problem posed by Calabi more than 60 years ago, known as the Saint-Venant compatibility problem: Given a compact Riemannian manifold, generally with boundary, find a compatibility operator for Lie derivatives of the metric tensor. This problem is related to other compatibility problems in mathematical physics, and to their inherent gauge freedom. To this end, we develop a framework generalizing the theory of elliptic complexes for sequences of linear differential operators $(A_{\bullet })$ between sections of vector bundles. We call such a sequence an elliptic pre-complex if the operators satisfy overdetermined ellipticity conditions and the order of $A_{k+1}A_k$ does not exceed the order of $A_k$. We show that every elliptic pre-complex $(A_{\bullet })$ can be ‘corrected’ into a complex $({\mathcal {A}}_{\bullet })$ of pseudodifferential operators, where ${\mathcal {A}}_k - A_k$ is a zero-order correction within this class. The induced complex $({\mathcal {A}}_{\bullet })$ yields Hodge-like decompositions, which in turn lead to explicit integrability conditions for overdetermined boundary-value problems, with uniqueness and gauge freedom clauses. We apply the theory on elliptic pre-complexes of exterior covariant derivatives of vector-valued forms and double forms satisfying generalized algebraic Bianchi identities, thus resolving a set of compatibility and gauge problems, among which one is the Saint-Venant problem.

In the article, we investigate Trudinger–Moser type inequalities in presence of logarithmic kernels in dimension N. A sharp threshold, depending on N, is detected for the existence of extremal functions or blow-up, where the domain is the ball or the entire space $\mathbb{R}^N$. We also show that the extremal functions satisfy suitable Euler–Lagrange equations. When the domain is the entire space, such equations can be derived by a N-Laplacian Schrödinger equation strongly coupled with a higher order fractional Poisson’s equation. The results extends [16] to any dimension $N \geq 2$.

We show that March’s criterion for the existence of a bounded nonconstant harmonic function on a weak model (that is, $\mathbb {R}^n$ with a rotationally symmetric metric) is also a necessary and sufficient condition for the solvability of the Dirichlet problem at infinity on a family of metrics that generalise metrics with rotational symmetry on $\mathbb {R}^n$. When the Dirichlet problem at infinity is not solvable, we prove some quantitative estimates on how fast a nonconstant harmonic function must grow.

The manuscript is devoted to the boundary behavior of mappings with bounded and finite distortion. We consider mappings of domains of the Euclidean space that satisfy weighted Poletsky inequality. Assume that, the definition domain is finitely connected on its boundary and, in addition, on the set of all points which are pre-images of the cluster set of this boundary. Then the specified mappings have a continuous boundary extension provided that the majorant in the Poletsky inequality satisfies some integral divergence condition, or has a finite mean oscillation at every boundary point.

Recently, it is proven that positive harmonic functions defined in the unit disc or the upper half-plane in $\mathbb{C}$ are contractions in hyperbolic metrics [14]. Furthermore, the same result does not hold in higher dimensions as shown by given counterexamples [16]. In this paper, we shall show that positive (or bounded) harmonic functions defined in the unit ball in $\mathbb{R}^{n}$ are Lipschitz in hyperbolic metrics. The involved method in main results allows to establish essential improvements of Schwarz type inequalities for monogenic functions in Clifford analysis [24, 25] and octonionic analysis [21] in a unified approach.

In this article, by the use of nth derivative characterization, we obtain several some sufficient conditions for all solutions of the complex linear differential equation

$$ \begin{align*}f^{(n)}+A_{n-1}(z)f^{(n-1)}+\ldots+A_1(z)f'+A_0(z)f=A_n(z) \end{align*} $$

$A_i(z) (i=0,1,\ldots ,n)$

We introduce a relaxed version of the metric definition of quasiconformality that is natural also for mappings of low regularity, including $W_{\mathrm{loc}}^{1,1}({\mathbb R}^n;{\mathbb R}^n)$-mappings. Then we show on the plane that this relaxed definition can be used to prove Sobolev regularity, and that these ‘finely quasiconformal’ mappings are in fact quasiconformal.

In this paper, we will show the Carleson measure characterizations for the boundedness and compactness of the Cesàro-type operator

\begin{equation*}\mathcal{C}_{\mu}(f)(z)=\sum^{\infty}_{n=0}\left( \int_{[0,1)}t^nd\mu(t)\right) \left(\sum^{n}_{k=0}a_k \right)z^n, \quad z\in \mathbb{D},\end{equation*}

acting on a number of important analytic function spaces on $\mathbb{D}$, where µ is a positive finite Borel measure. The function spaces are some newly appeared analytic function spaces (e.g., Bergman–Morrey spaces $A^{p,\lambda}$ and Dirichlet–Morrey spaces $\mathcal{D}_p^{\lambda}$) . This work continues the lines of the previous characterizations by Blasco and Galanopoulos et al. for classical Hardy spaces and weighted Bergman spaces and so forth.

We introduce and study Dirichlet-type spaces $\mathcal D(\mu _1, \mu _2)$ of the unit bidisc $\mathbb D^2,$ where $\mu _1, \mu _2$ are finite positive Borel measures on the unit circle. We show that the coordinate functions $z_1$ and $z_2$ are multipliers for $\mathcal D(\mu _1, \mu _2)$ and the complex polynomials are dense in $\mathcal D(\mu _1, \mu _2).$ Further, we obtain the division property and solve Gleason’s problem for $\mathcal D(\mu _1, \mu _2)$ over a bidisc centered at the origin. In particular, we show that the commuting pair $\mathscr M_z$ of the multiplication operators $\mathscr M_{z_1}, \mathscr M_{z_2}$ on $\mathcal D(\mu _1, \mu _2)$ defines a cyclic toral $2$-isometry and $\mathscr M^*_z$ belongs to the Cowen–Douglas class $\mathbf {B}_1(\mathbb D^2_r)$ for some $r>0.$ Moreover, we formulate a notion of wandering subspace for commuting tuples and use it to obtain a bidisc analog of Richter’s representation theorem for cyclic analytic $2$-isometries. In particular, we show that a cyclic analytic toral $2$-isometric pair T with cyclic vector $f_0$ is unitarily equivalent to $\mathscr M_z$ on $\mathcal D(\mu _1, \mu _2)$ for some $\mu _1,\mu _2$ if and only if $\ker T^*,$ spanned by $f_0,$ is a wandering subspace for $T.$

In this paper, we review some recent results on nonlocal interaction problems. The focus is on interaction kernels that are anisotropic variants of the classical Coulomb kernel. In other words, while preserving the same singularity at zero of the Coulomb kernel, they present preferred directions of interaction. For kernels of this kind and general confinement we will prove existence and uniqueness of minimizers of the corresponding energy. In the case of a quadratic confinement we will review a recent result by Carrillo and Shu about the explicit characterization of minimizers, and present a new proof, which has the advantage of being extendable to higher dimensions. In light of this result, we will re-examine some previous works motivated by applications to dislocation theory in materials science. Finally, we will discuss some related results and open questions.

We verify a long-standing conjecture on the membership of univalent harmonic mappings in the Hardy space, whenever the functions have a “nice” analytic part. We also produce a coefficient estimate for these functions, which is in a sense best possible. The problem is then explored in a new direction, without the additional hypothesis. Interestingly, our ideas extend to certain classes of locally univalent harmonic mappings. Finally, we prove a Baernstein-type extremal result for the function $\log (h'+cg')$, when $f=h+\overline {g}$ is a close-to-convex harmonic function, and c is a constant. This leads to a sharp coefficient inequality for these functions.

In this paper, we study the Dirichlet problem for systems of mean value equations on a regular tree. We deal both with the directed case (the equations verified by the components of the system at a node in the tree only involve values of the unknowns at the successors of the node in the tree) and the undirected case (now the equations also involve the predecessor in the tree). We find necessary and sufficient conditions on the coefficients in order to have existence and uniqueness of solutions for continuous boundary data. In a particular case, we also include an interpretation of such solutions as a limit of value functions of suitable two-players zero-sum games.

For every $n\geq 2$, Bourgain’s constant $b_n$ is the largest number such that the (upper) Hausdorff dimension of harmonic measure is at most $n-b_n$ for every domain in $\mathbb {R}^n$ on which harmonic measure is defined. Jones and Wolff (1988, Acta Mathematica 161, 131–144) proved that $b_2=1$. When $n\geq 3$, Bourgain (1987, Inventiones Mathematicae 87, 477–483) proved that $b_n>0$ and Wolff (1995, Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), Princeton University Press, Princeton, NJ, 321–384) produced examples showing $b_n<1$. Refining Bourgain’s original outline, we prove that

$$\begin{align*}b_n\geq c\,n^{-2n(n-1)}/\ln(n),\end{align*}$$

$n\geq 3$

$c>0$

$b_3\geq 1\times 10^{-15}$

$b_4\geq 2\times 10^{-26}$

For an n-element subset U of $\mathbb {Z}^2$, select x from U according to harmonic measure from infinity, remove x from U and start a random walk from x. If the walk leaves from y when it first enters the rest of U, add y to it. Iterating this procedure constitutes the process we call harmonic activation and transport (HAT).

HAT exhibits a phenomenon we refer to as collapse: Informally, the diameter shrinks to its logarithm over a number of steps which is comparable to this logarithm. Collapse implies the existence of the stationary distribution of HAT, where configurations are viewed up to translation, and the exponential tightness of diameter at stationarity. Additionally, collapse produces a renewal structure with which we establish that the center of mass process, properly rescaled, converges in distribution to two-dimensional Brownian motion.

To characterize the phenomenon of collapse, we address fundamental questions about the extremal behavior of harmonic measure and escape probabilities. Among n-element subsets of $\mathbb {Z}^2$, what is the least positive value of harmonic measure? What is the probability of escape from the set to a distance of, say, d? Concerning the former, examples abound for which the harmonic measure is exponentially small in n. We prove that it can be no smaller than exponential in $n \log n$. Regarding the latter, the escape probability is at most the reciprocal of $\log d$, up to a constant factor. We prove it is always at least this much, up to an n-dependent factor.

Let $\mathcal {M}$ be an Ahlfors $n$-regular Riemannian manifold such that either the Ricci curvature is non-negative or the Ricci curvature is bounded from below together with a bound on the gradient of the heat kernel. In the paper [IMRN, 2022, no. 2, 1245-1269] of Brazke–Schikorra–Sire, the authors characterised the BMO function $u : \mathcal {M} \to \mathbb {R}$ by a Carleson measure condition of its $\sigma$-harmonic extension $U:\mathcal {M}\times \mathbb {R}_+ \to \mathbb {R}$. This paper is concerned with the similar problem under a more general Dirichlet metric measure space setting, and the limiting behaviours of BMO & Carleson measure, where the heat kernel admits only the so-called diagonal upper estimate. More significantly, without the Ricci curvature condition, we relax the Ahlfors regularity to a doubling property, and remove the pointwise bound on the gradient of the heat kernel. Some similar results for the Lipschitz function are also given, and two open problems related to our main result are considered.

In this article, we establish three new versions of Landau-type theorems for bounded bi-analytic functions of the form $F(z)=\bar {z}G(z)+H(z)$, where G and H are analytic in the unit disk with $G(0)=H(0)=0$ and $H'(0)=1$. In particular, two of them are sharp, while the other one either generalizes or improves the corresponding result of Abdulhadi and Hajj. As consequences, several new sharp versions of Landau-type theorems for certain subclasses of bounded biharmonic mappings are proved.

On post-critically finite self-similar sets, whose walk dimensions of diffusions are in general larger than 2, we find a sharp region where two classes of Besov spaces, the heat Besov spaces $B^{p,q}_\sigma (K)$ and the Lipschitz–Besov spaces $\Lambda ^{p,q}_\sigma (K)$, are identical. In particular, we provide concrete examples that $B^{p,q}_\sigma (K)=\Lambda ^{p,q}_\sigma (K)$ with $\sigma>1$. Our method is purely analytical, and does not involve heat kernel estimate.