Related papers: Boundary multifractal behaviour for harmonic funct…

It was recently shown that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a domain with an $n-1$ dimensional uniformly rectifiable boundary, in the presence of now well understood additional…

For a certain class of genuinely nonlinear two-by-two planar hyperbolic systems we show that any classical solution on a smoothly bounded domain has nontangential boundary limits except on a set whose Hausdorff dimension is bounded by some…

We study harmonic functions which admit a certain majorant in the unit ball in $\R^m $. We prove that when the majorant fulfills a doubling condition, the extremal growth or decay may occur only along small sets of radii, and we give…

We consider harmonic functions in the unit ball of $\mathbb{R}^{n+1}$ that are unbounded near the boundary but can be estimated from above by some (rapidly increasing) radial weight $w$. Our main result gives some conditions on $w$ that…

We develop the theory of multiresolutions in the context of Hausdorff measure of fractional dimension between 0 and 1. While our fractal wavelet theory has points of similarity that it shares with the standard case of Lebesgue measure on…

Using the characterizations in terms of various differential operators including partial, normal, and tangential derivatives, we extend the family of Bergman spaces of $\mathcal H$-harmonic functions on the real hyperbolic ball from…

As well known, harmonic functions satisfy the mean value property, namely the average of the function over a ball is equal to its value at the center. This fact naturally raises the question on whether this is a characterizing feature of…

Consider all the level sets of a real function. We can group these level sets according to their Hausdorff dimensions. We show that the Hausdorff dimension of the collection of all level sets of a given Hausdorff dimension can be…

We show that there exists an integrable function on the $n$-sphere $(n\ge 2)$, whose Ces\`aro (C,$\frac{n-1}{2}$) means with respect to the spherical harmonic expansion diverge unboundedly almost everywhere. By studying equivalence…

We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open set $D$. This yields a unique representation of such functions as integrals against measures on $D^c\cup…

Let $u$ be a pluriharmonic function on the unit ball in $\mathbb{C}^n$. I consider the relationship between the set of points $L_u$ on the boundary of the ball at which $u$ converges nontangentially and the set of points $\mathcal{L}_u$ at…

In this work, we revisit the following estimate due to Dahlberg \cite{Dahl}. Let $\textit{\textbf x}_0$ a fixed point in a bounded Lipschitz domain $\Omega$. Then there exists a constant $C > 0$ such that if $u$ is a harmonic function in…

We consider digits-deleted sets or Cantor-type sets with $\beta$-expansions. We calculate the Hausdorff dimension $d$ of these sets and show that $d$ is continuous with respect to $\beta$. The $d$-dimentional Hausdorff measure of these sets…

We adapt the results of Part 1 to include the unit ball in the Heisenberg group, the model domain with characteristic boundary points. In particular, we construct function spaces on which the Kohn Laplacian with the \bar{\partial}_b-Neumann…

We study the Hausdorff dimension of the sets on which the pointwise convergence of the solutions to the fractional Schr\"odinger equation $e^{it(-\Delta)^\frac m2}f$ fails when $m\in(0,1)$ in one spatial dimension. The pointwise convergence…

A well-known problem in holomorphic dynamics is to obtain Denjoy--Wolff-type results for compositions of self-maps of the unit disc. Here, we tackle the particular case of inner functions: if $f_n:\mathbb{D}\to\mathbb{D}$ are inner…

In this paper we study a p harmonic measure, associated with a positive p harmonic function \hat{u} defined in an open set O, subset of R^n, and vanishing on a portion \Gamma of boundary of O. If p>n we show that this p harmonic measure is…

This paper is devoted to the study of semi-stable radial solutions $u\in H^1(B_1)$ of $-\Delta u=\vert x\vert^\alpha f(u) \mbox{ in } B_1\setminus\lbrace0\rbrace$, where $f\in C^1(\mathbb{R})$ is a general nonlinearity, $\alpha>-2$ and…

Inspired by recent work of Ferone and Volzone arXiv:2007.13195, we derive sufficient conditions for the validity and non-validity of a boundary version of Talenti's comparison principle in the context of Dirichlet-Poisson problems for the…

In this paper, we study functions of bounded variation on a complete and connected metric space with finite one-dimensional Hausdorff measure. The definition of BV functions on a compact interval based on pointwise variation is extended to…