Title: The $p$-harmonic transform beyond its natural domain of definition

Authors: Luigi D'Onofrio and Tadeusz Iwaniec

Issue: Volume 53 (2004), Issue 3, 683-718


The $p$-harmonic transforms are the most natural nonlinear counterparts of the Riesz transforms in $\Real^n$. They originate from the study of the $p$-harmonic type equation \[\mbox{\upshape div}|\nabla u|^{p-2}\nabla u= \mbox{\upshape div}\mathfrak{f},\] where $\mathfrak{f}:\Omega\longrightarrow\Real^n$ is a given vector field in $\mathscr{L}^q(\Omega,\Real^n)$ and $u$ is an unknown function of Sobolev class $\mathscr{W}_0^{1,p}(\Omega,\Real^n)$, $p+q=pq$. The $p$-harmonic transform $\mathscr{H}_p:\mathscr{L}^q(\Omega,\Real^n)\to\mathscr{L}^p(\Omega,\Real^n)$ assigns to $\mathfrak{f}$ the gradient of the solution: $\mathscr{H}_p\mathfrak{f}=\nabla u\in\mathscr{L}^p(\Omega,\Real^n)$. More general PDE's and the corresponding nonlinear operators are also considered. We investigate the extension and continuity properties of the $p$-harmonic transform beyond its natural domain of definition. In particular, we identify the exponents $\lambda>1$ for which the operator $\mathscr{H}_p:\mathscr{L}^{\lambda q}(\Omega,\Real^n)\longrightarrow\mathscr{L}^{\lambda p}(\Omega,\Real^n)$ is well defined and remains continuous. Rather surprisingly, the uniqueness of the solution $\nabla u\in\mathscr{L}^{\lambda p}(\Omega,\Real^n)$ fails when $\lambda$ exceeds certain critical value. In case $p=n=\dim\Omega$, there is no uniqueness in $\mathscr{W}^{1,\lambda n}(\Real^n)$ for any $\lambda>1$.