It is well known that, \noindent Theorem 1. For $f\in L^{2}(\mathbb H_{n}=\text{The Heisenberg group of dimesion } 2n+1)$ we have the expansion $$f(z, s)= (2\pi)^{-n} \sum_{k=0}^{ …

In their 1983 JFA-paper Brezis and Lieb have shown, among many other things, a Poincaré-type inequality: in the case of a harmonic function $f$ on a bounded domain $\Omega$, their …

For any real-valued smooth function $u$, we have the Kato inequality $|D|Du||^2\leq(det(Hess(u)))^2$, which holds when $|Du|\neq0$. If moreover $u$ is harmonic (in an open set i …

Is a family of bounded bi-harmonic functions defined in the unit disk an equicontinuous family of functions on compacts? A bi-harmonic function $u$ is a solution of the equation $\ …

A bi-harmonic function $u:U\to C$, where $U$ is an open subset of the complex plane $C$ is a solution of the equation $\Delta^2u=0$. Can a nonconstant bi-harmonic mapping have a co …

Let us consider the unit ball $B^n$ of $\mathbf{R}^n$ and $K = B^{n-1}(0,1/2) \times {0}$ the ball of dimension $n-1$ and radius $1/2$ lying on the equator. We wish to solve the D …

I have a question regarding harmonic maps from all of ${\Bbb R}^2$ into a domain in ${\Bbb R}^2$. Before stating my question in full generality, let me ask a special case of the qu …

Is the modulus of a harmonic map $u$ of the unit disk onto itself w.r.t. to a certain conformal metric $\rho$ (not necessary euclidean or hyperbolic metric), a subharmonic function …

Let ${\rm Harm}_n^d$ be the space of real harmonic polynomials in $n$ variables, homogeneous of degree $d$. If $P\in{\rm Harm}_n^d$, then $$\left(\frac{\partial^2}{\partial x_1^2}+ …