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The $p$-harmonic transform beyond its natural domain of definition. (English) Zbl 1081.35048
Summary: The $p$-harmonic transforms are the most natural nonlinear counterparts of the Riesz transforms in $\bbfR^n$. They originate from the study of the $p$-harmonic type equation $$\text{div} |\nabla u|^{p-2}\nabla u=\operatorname{div}{\germ f},$$ where ${\germ f}: \Omega\to\bbfR^n$ is a given vector field in ${\germ L}^q(\Omega, \bbfR^n)$ and $u$ is an unknown function of Sobolev class ${\cal W}_0^{1,p}(\Omega,\bbfR^n)$, $p+q=pq$. The $p$-harmonic transform ${\cal H}_p: {\cal L}^p(\Omega,\bbfR^n)$ assigns to ${\germ f}$ the gradient of the solution: ${\cal H}_p{\germ f}=\nabla u\in{\cal L}^p(\Omega,\bbfR^n)$. 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 ${\cal H}_p:{\cal L}^{\lambda q} (\Omega,\bbfR^n)\to {\cal L}^{\lambda p}(\Omega,\bbfR^n)$ is well defined and remains continuous. Rather surprisingly, the uniqueness of the solution $\nabla u\in{\cal L}^{\lambda p}(\Omega,\bbfR^n)$ fails when $\lambda$ exceeds certain critical value. In case $p=n=\dim \Omega$, there is no uniqueness in ${\cal W}^{1,\lambda n}(\bbfR^n)$ for any $\lambda>1$.

35J99Elliptic equations and systems
44A15Special transforms (Legendre, Hilbert, etc.)
47H99Nonlinear operators
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