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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 $$\mathbb{R}^n$$. They originate from the study of the $$p$$-harmonic type equation $\text{div} |\nabla u|^{p-2}\nabla u=\operatorname{div}{\mathfrak f},$ where $${\mathfrak f}: \Omega\to\mathbb{R}^n$$ is a given vector field in $${\mathfrak L}^q(\Omega, \mathbb{R}^n)$$ and $$u$$ is an unknown function of Sobolev class $${\mathcal W}_0^{1,p}(\Omega,\mathbb{R}^n)$$, $$p+q=pq$$. The $$p$$-harmonic transform $${\mathcal H}_p: {\mathcal L}^p(\Omega,\mathbb{R}^n)$$ assigns to $${\mathfrak f}$$ the gradient of the solution: $${\mathcal H}_p{\mathfrak f}=\nabla u\in{\mathcal L}^p(\Omega,\mathbb{R}^n)$$. 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 $${\mathcal H}_p:{\mathcal L}^{\lambda q} (\Omega,\mathbb{R}^n)\to {\mathcal L}^{\lambda p}(\Omega,\mathbb{R}^n)$$ is well defined and remains continuous. Rather surprisingly, the uniqueness of the solution $$\nabla u\in{\mathcal L}^{\lambda p}(\Omega,\mathbb{R}^n)$$ fails when $$\lambda$$ exceeds certain critical value. In case $$p=n=\dim \Omega$$, there is no uniqueness in $${\mathcal W}^{1,\lambda n}(\mathbb{R}^n)$$ for any $$\lambda>1$$.

##### MSC:
 35J99 Elliptic equations and elliptic systems 44A15 Special integral transforms (Legendre, Hilbert, etc.) 47H99 Nonlinear operators and their properties
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