We survey recent developments relating the notions of the Bergman kernel and pluripotential theory and indicate some open problems.

This is a review of research on geometric properties of Neumann eigenfunctions related to the "hot spots" conjecture of Jeff Rauch. The paper also presents, in an informal way, some probabilistic techniques used in the proofs.

This is a survey of results on the exit time and the exit position of Brownian motion from cones and parabola-shaped regions in Euclidean space. The paper begins with a section on harmonic measure.

A classical result of Alice Roth characterizes those functions on the unit circle that can arise from taking radial limits of entire functions. This paper describes recent progress on the characterization of radial limit functions of harmonic functions defined either in the unit ball or the whole of space. Some related open problems are posed.

We consider a subsemigroup $T$ of the linear group $G$ of the $d$-dimensional Euclidean space $V$, which is "sufficiently large". We study the orbit closures of $T$ in $V$ and we apply the results to semigroups of endomorphisms of the $d$-dimensional torus. The method uses the knowledge of the potential kernel of the Markov chain on $V$ defined by a probability measure supported on $T$. The condition of being "large" is satisfied for example by a subsemigroup of $SL(V)$, Zariski-dense in $SL(V)$.

Several notions of densities related to zero sequences, interpolating sequences and sampling sequences of holomorphic functions are presented. Some ties with harmonic measure estimates are shown.

The aim of this note is to summarise some of the approaches to extending Sobolev space theory to metric measure spaces. In particular, we will give a brief survey of Hajłasz-Sobolev, Newton-Sobolev, and the Korevaar-Schoen type Sobolev spaces on metric measure spaces.

We discuss quasisymmetric extension of embeddings that are close to similarities, due to Tukia and Väisälä, and smooth quasiconformal approximation of such extensions. The smoothing is done by convolution with a variable kernel in conjunction with the Tukia-Väisälä extension procedure. We can apply these to the study of branch sets of smooth quasiregular maps, and quasiconformal dimension of self-similar fractals.

We show that for Cheeger $p$-harmonic functions on doubling metric measure spaces supporting a Poincaré inequality, the Wiener criterion is necessary and sufficient for regularity of boundary points.

In this survey, we describe joint work in collaboration with A. Stokolos, O. Svensson and T. Weiss. We consider the following question: How sharp is the Stolz approach region condition for the almost everywhere convergence of bounded harmonic functions? The issue was first settled in the rotation invariant case in the unit disc by Littlewood in 1927 and later examined, under less stringent conditions, by Aikawa in 1991. We show that our results are, in a precise sense, sharp.

Our aim in this paper is to deal with continuity properties for weakly monotone Sobolev functions of variable exponent.

This note consists of our recent researches on Martin kernels of general domains. In particular, minimal Martin boundary points of a John domain, the boundary behavior of quotients of Martin kernels, and comparison estimates for the Green function and the Martin kernel are studied.

Let $s = |s| e^{2\pi \lambda i}$ be a complex constant satisfying $|s| \gt 1$ and $\lambda \notin \mathbb{Q}$. We show that for a transcendental meromorphic solution $f(z)$ of some non-autonomous Schröder equation $f(sz) = R(z, f(z))$, any direction is a Borel direction.

We study space-time excessive functions with respect to a basic submarkovian semigroup $\mathbb{P}$. It is shown that under some regularity assumptions many space-time excessive functions on a half-space have a Choquet-type integral represention by suitably choosen densities of the adjoint semigroup $\mathbb{P}^*$. If $\mathbb{P}$ is a convolution semigroup which is absolutely continuous with respect to the Haar measure, then all space-time excessive functions admit such an integral representation.

Let $\mathcal{D} (R^n)$ be the class of all $C^{\infty}$–functions on $R^n$ with compact support. For a multi-index $\alpha$ we denote $\mathcal{D}^{\alpha} (R^n) = \{\mathcal{D}^{\alpha} \varphi : \varphi \in \mathcal{D} (R^n)\}$. We give a direct sum decomposition of $\mathcal{D} (R^n)$ by $\mathcal{D}^{\alpha} (R^n)$ and its complementary space.

We show that $f \in L^p (X ; m)$ implies $|f| dm \in S_{K}^{1}$ for $p \gt D$ with $D \gt 0$, where $S_{K}^{1}$ is a subfamily of Kato class measures relative to a semigroup kernel $p_t (x, y)$ of a Markov process associated with a (non-symmetric) Dirichlet form on $L^2 (X ; m)$. We only assume that $p_t (x, y)$ satisfies the Nash type estimate of small time depending on $D$. No concrete expression of $p_t (x,y)$ is needed for the result.

We estimate the operator norm of the composition operators mapping Bloch space boundedly into Hardy spaces, $BMOA$ space, Lipschitz spaces and mean Lipschitz spaces respectively.

Let $R$ and $R'$ be $p$-sheeted unlimited covering surfaces of the once punctured Riemann sphere $\hat{\mathbb{C}} \setminus \{0\}$ of Heins type which are quasiconformal equivalent to each other. Then the cardinal numbers of minimal Martin boundaries of $R$ and $R'$ are same.

Let $R$ be a 2-sheeted unlimited covering surface of the once punctured Riemann sphere $\hat{\mathbb{C}} \setminus \{0\}$ of Heins type and $R'$ be an open Riemann surface. If $R$ and $R'$ are quasiconformal equivalent to each other and the set of branch points of $R$ satisfies a condition, then the cardinal numbers of minimal Martin boundaries of $R$ and $R'$ are same.

Let $R$ be an open Riemann surface with Green's functions. It is proved that there exist no unbounded positive harmonic functions on $R$ if and only if the minimal Martin boundary of $R$ consists of finitely many points with positive harmonic measure.

This paper gives a quantitative property of rarefied sets at $\infty$ of a cone. The proof is based on the fact in which the estimations of Green potential and Poisson integral with measures are connected with a kind of densities of the measures modified from the measures.

We consider elliptic systems of divergence form in $\mathbb{R}^n$ under the limited smoothness assumptions on the coefficients. We construct $L^p$ resolvents with evaluation of their operator norms, and derive the Gaussian bounds for heat kernels and estimates for resolvent kernels. These results extend those for single operators.

Our aim in this paper is to deal with the boundedness of maximal functions in Lebesgue spaces with variable exponent. Our result extends the recent work of Diening [4], Cruz-Uribe, Fiorenza and Neugebauer [3] and the authors [8]. As an application of the boundedness of maximal functions, we show Sobolev's inequality for Riesz potentials with variable exponent.

Fix two disjoint nondegenerate continua $A$ and $B$ in the complex plane $\mathbb{C}$ with connected complements and choose a simple arc $\gamma$ in the complex sphere $\widehat{\mathbb{C}}$ disjoint from $A \cup B$, which we call a pasting arc for $A$ and $B$. Then form a covering Riemann surface $\widehat{\mathbb{C}}_{\gamma}$ over $\widehat{\mathbb{C}}$ by pasting two copies of $\widehat{\mathbb{C}} \setminus \gamma$ crosswise along the arc $\gamma$. Viewing $A$ and $B$ as embedded in the different two sheets $\widehat{\mathbb{C}} \setminus \gamma$ of $\widehat{\mathbb{C}}_{\gamma}$, consider the variational 2-capacity $\mathrm{cap} (A, \widehat{\mathbb{C}}_{\gamma} \setminus B)$ of the set $A$ in $\widehat{\mathbb{C}}_{\gamma}$ with respect to the open subset $\widehat{\mathbb{C}}_{\gamma} \setminus B$ containing $A$. We are interested in the comparison of $\mathrm{cap} (A, \widehat{\mathbb{C}}_{\gamma} \setminus B)$ with $\mathrm{cap} (A, \widehat{\mathbb{C}} \setminus B)$. We say that the pasting arc $\gamma$ for $A$ and $B$ is subcritical, critical, or supercritical according as $\mathrm{cap} (A, \widehat{\mathbb{C}}_{\gamma} \setminus B)$ is less than, equal to, or greater than $\mathrm{cap} (A, \widehat{\mathbb{C}} \setminus B)$, respectively. The purpose of this paper is to show the existence of subcritical arc $\gamma$ for any arbitrarily given general pair of admissible $A$ and $B$ and then the existences of critical and also supercritical arcs $\gamma$ under the additional condition imposed upon $A$ and $B$ that each of $A$ and $B$ is symmetric about a common straight line in $\widehat{\mathbb{C}}$, which is the case e.g. if $A$ and $B$ are disjoint closed discs.

We consider the $\alpha$-parabolic Bergman spaces on strip domains. The Bergman kernel is given by a series of derivatives of the fundamental solution. We prove the $L^p$-boundedness of the projection defined by the Bergman kernel and obtain the duality theorem for $1 \lt p \lt \infty$. At the same time, we give a new proof of the Huygens property, which enable us to verify all the results in [3] also for $n = 1$.

We consider quasi-linear second order elliptic differential equations with measures date on the right hand side. In this talk, we investigate Hölder continuity of solutions of such equations.

We study strong ratio limit properties and the exact long time asymptotics of the heat kernel of a general second-order parabolic operator which is defined on a noncompact Riemannian manifold.

The greatest harmonic minorant of a superharmonic function is determined as the limit of a sequence of solutions for discrete Dirichlet problems on finite subnetworks. Without using the Green kernel explicitly, a positive superharmonic function is decomposed uniquely as a sum of a potential and a harmonic function. The infimum of a left directed family of harmonic functions is shown to be either $-\infty$ or harmonic. As applications, we study the reduced functions and their properties. We show the existence of the Green kernel with the aid of our reduced function.

For the unit disc $\mathbb{D}$ in $\mathbb{C}$, the harmonic Hardy spaces $\mathcal{H}^p$, $1 \le p \lt \infty$, are defined as the set of harmonic functions $h$ on $\mathbb{D}$ satisfying $$ ||h||_{p}^{p} = \sup_{0 \lt r \lt 1} \frac{1}{2\pi} \int_{0}^{2\pi} |h(re^{i\theta})|^p d\theta \lt \infty. $$ The classical Littlewood-Paley inequalities for harmonic functions [3] in $\mathbb{D}$ are as follows: Let $h$ be harmonic on $\mathbb{D}$. Then there exist positive constants $C_1$, $C_2$, independent of $h$, such that

(a) for $1 \lt p \le 2$, $$ ||h||_{p}^{p} \le C_1 \left[ |h(0)|^p + \iint_{\mathbb{D}} (1-|z|)^{p-1} |\nabla h(z)|^p dx\, dy \right]. $$

(b) For $p \ge 2$, if $h \in \mathcal{H}^p$, then $$ \iint_{\mathbb{D}} (1-|z|)^{p-1} |\nabla h(z)|^p dx\, dy \le C_2 ||h||_{p}^{p}. $$

In the paper we consider generalizations of these inequalities to Hardy-Orlicz spaces $\mathcal{H}_{\psi}$ of harmonic functions on domains $\Omega \varsubsetneq \mathbb{R}^n$, $n \ge 2$, with Green function $G$ satisfying the following: There exist constants $\alpha$ and $\beta$, $0 \lt \beta \le 1 \le \alpha \lt \infty$, such that for fixed $t_o \in \Omega$, there exist constants $C_1$ and $C_2$, depending only on $t_o$, such that $C_1 \delta (x)^{\alpha} \le G(t_o, x)$ for all $x \in \Omega$, and $G(t_o, x) \le C_2 \delta (x)^{\beta}$ for all $x \in \Omega \setminus B(t_o, \frac{1}{2} \delta(t_o))$.

Let $M$ be the Hardy-Littlewood maximal operator in a quasimetric space $X$. We give the estimates of $Mf$ with weak type and strong type with respect to the $\alpha$-Hausdorff content. To do these, we use the dyadic balls introduced by E. Sawyer and R. L. Wheeden.

The parabolic Bergman space is the Banach space of solutions of some parabolic equations on the upper half space which have finite $L^p$ norms. We introduce and study $L^{(\alpha)}$-harmonic conjugates of parabolic Bergman functions, and give a sufficient condition for a parabolic Bergman space to have unique $L^{(\alpha)}$-harmonic conjugates.

This paper shows that a positive superharmonic function on a cone behaves regularly outside an $\alpha$-minimally thin set in a cone. This fact is known for a half space which is a special cone.