Related papers: On a Theorem of Wolff Revisited

We characterize $p-$harmonic functions in the Heisenberg group in terms of an asymptotic mean value property, where $1<p<\infty$, following the scheme described in Manfredi et al. (2009) for the Euclidean case. The new tool that allows us…

Several unitarily invariant norm inequalities and numerical radius inequalities for Hilbert space operators are studied. We investigate some necessary and sufficient conditions for the parallelism of two bounded operators. For a finite rank…

The aim of this work is to derive new explicit solutions to the $\infty$-Laplace equation, the fundamental PDE arising in Calculus of Variations in the space $L^\infty$. These solutions obey certain symmetry conditions and are derived in…

We study existence and regularity of weak solutions for the following $p$-Laplacian system \begin{cases} -\Delta_p u+A\varphi^{\theta+1}|u|^{r-2}u=f, \ &u\in W_0^{1,p}(\Omega),\\-\Delta_p \varphi=|u|^r\varphi^\theta, \ &\varphi\in…

Let G:=-((d/dx)^2+x^2(d/du)^2) denote the Grusin operator on R^2. Consider the Cauchy problem for the associated wave equation on R x R^2, given by ((d/dt)^2+G)v =0, v(0,.)=f, d/dt v(0,.)=g, where t denotes time and f, g are suitable…

For $X$, $Y$, $Z$ and $W$ compact metric spaces, consider two uniformly contractive IFS $\{\tau_x: Z\to Z,\, x\in x\}$ and $\{\tau_y:W\to W,\, y\in Y\}$. For a fixed $\alpha \in \mathcal{P}(X)$ with $supp(\alpha)=X$ we define the entropy of…

Let $\theta \in(0,1)$ and $(\mathcal{M},\tau)$ be a semifinite von Neumann algebra. We consider the function spaces introduced by Sobolev (denoted by $S_{d,\theta}$), showing that there exists a constant $d>0 $ depending on $p$, $0<p\le…

Let $p>1$. The Harnack inequality and H\"older continuity for $p$-harmonic functions in bounded domains in $\mathbb{R}^d$ are usually proved via Moser iteration. In 2013 Luiro, Parviainen and Saksman showed that tug-of-war games can also be…

The Perron method for solving the Dirichlet problem for $p$-harmonic functions is extended to unbounded open sets in the setting of a complete metric space with a doubling measure supporting a $p$-Poincar\'e inequality, $1<p<\infty$. The…

We obtain inequalities of the form $$\int_C |f(z)|^p |dz| \leq A(p) \int_{\mathbb{T}} |f(z)|^p |dz|, \quad (p>1)$$ where $f$ is harmonic in the unit disk $\mathbb{D}$, $\mathbb{T}$ is the unit circle, and $C$ is any convex curve in…

We consider two balayage constructions on the complex plane $\mathbb C$ with real axis $\mathbb R$ for $0\leq b\in \mathbb R$. Let $u\not\equiv -\infty$ be a subharmonic function on $\mathbb C$ of order…

The invertibility of Wiener-Hopf plus Hankel operators $W(a)+H(b)$ acting on the spaces $L^p(\mathbb{R}^+)$, $1 < p<\infty$ is studied. If $a$ and $b$ belong to a subalgebra of $L^\infty(\mathbb{R})$ and satisfy the condition…

Using Maz'ya type integral identities with power weights, we obtain new boundary estimates for biharmonic functions on Lipschitz and convex domains in $R^n$. For $n\ge 8$, combined with a result in \cite{S2}, these estimates lead to the…

We construct explicit complex-valued $p$-harmonic functions and harmonic morphisms on the classical compact symmetric complex and quaternionic Grassmannians. The ingredients for our construction method are joint eigenfunctions of the…

In this paper we give an extension of the classical Caffarelli-Kohn-Nirenberg inequalities: we show that for $1<p,q<\infty$, $0<r<\infty$ with $p+q\geq r$, $\delta\in[0,1]\cap\left[\frac{r-q}{r},\frac{p}{r}\right]$ with $\frac{\delta…

Let $u\not\equiv -\infty$ be a subharmonic function on the complex plane $\mathbb C$. In 2016, we obtained a result on the existence of an entire function $f\neq 0$ satisfying the estimate $\log|f|\leq {\sf B}_u$ on $\mathbb C$, where…

We prove existence and uniqueness of a solution of the Dirichlet problem for separately $(\alpha, \beta)$ - harmonic functions on the unit polydisc $\mathbb D^n$ with boundary data in $C(\mathbb T^n)$ using $(\alpha, \beta)$ - Poisson…

We obtain several inequalities on the generalized means of dependent p-values. In particular, the weighted harmonic mean of p-values is strictly sub-uniform under several dependence assumptions of p-values, including independence, negative…

In this article we consider the class $\mathcal{A}(p)$ which consists of functions that are meromorphic in the unit disc $\ID$ having a simple pole at $z=p\in (0,1)$ with the normalization $f(0)=0=f'(0)-1 $. First we prove some sufficient…

Motivated by Thurston and Daskalopoulos--Uhlenbeck's approach to Teichm\"uller theory, we study the behavior of $q$-harmonic functions and their $p$-harmonic conjugates in the limit as $q \to 1$, where $1/p + 1/q = 1$. The $1$-Laplacian is…