A sublinear Sobolev inequality for \(p\)-superharmonic functions (Q2832832)

In his Theorem 1.4 the author establishes a ``sublinear'' Sobolev inequality of the form NEWLINE\[NEWLINE\left(\int_{{\mathbb{R}}^n}u^{\frac{nq}{n-q}}\,dx\right)^{\frac{n-q}{nq}}\leq C\, \left(\int_{{\mathbb{R}}^n}| Du|^q \,dx\right)^{\frac{1}{q}}NEWLINE\]NEWLINE for all global \(p\)-superharmonic (\(1<p<2\)) functions \(u\) in \({\mathbb{R}}^n\), \(n\geq 2\), with \(\inf_{{\mathbb{R}}^n}u=0\) and \(p-1<q<1\), with \(C=C(n,p,q)>0\). As a matter of fact, the author proves the result for the more general class of \({\mathcal{A}}\)-superharmonic functions \(u\) in \({\mathbb{R}}^n\) with \(\inf_{{\mathbb{R}}^n}u=0\), with \text {\(C=C(n,p,q,\alpha ,\beta )>0\),} where \(\alpha\) and \(\beta\) are the structural constants of the function \({\mathcal{A}}: {\mathbb{R}}^n\times {\mathbb{R}}^n \rightarrow {\mathbb{R}}^n\).NEWLINENEWLINEThough the proof is rather detailed, it is by no means easy. It relies, among others, on certain pointwise estimates by Wolff's potentials, obtained by \textit{T. Kilpeläinen} and \textit{J. Malý} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 19, No. 4, 591--613 (1992; Zbl 0797.35052); Acta Math. 172, No. 1, 137--161 (1994; Zbl 0820.35063)], on a convolution identity [\textit{E. M. Stein}, Singular integrals and differentiability properties of functions. Princeton, N.J.: Princeton University Press (1970; Zbl 0207.13501)], and on the Gauss-Green formula for \(L^1_{\mathrm{loc}}\) vector fields with divergence measure [\textit{M. Degiovanni} et al., Arch. Ration. Mech. Anal. 147, No. 3, 197--223 (1999; Zbl 0933.74007), Theorem 5.4]. NEWLINENEWLINENEWLINENEWLINE In his Theorem 3.1, the author gives an even more general version for his above result. Moreover, he also asks for a sublinear Sobolev inequality on bounded domains \(\Omega \subset {\mathbb{R}}^n\) for certain \({\mathcal{A}}\)-superharmonic functions in \(\Omega\). For example, is it true that for \(u\in W^{1,p}_{0}(\Omega )\), \(1<p<2\), with \(-{\mathrm{div}}\,{\mathcal{A}}(x,\nabla u)\geq 0\), and \(p-1<q<1\), the Sobolev inequality NEWLINE\[NEWLINE \left(\int_{\Omega}u^{\frac{nq}{n-q}}\,dx\right)^{\frac{n-q}{nq}}\leq C\, \left(\int_{\Omega}| \nabla u|^q \,dx\right)^{\frac{1}{q}}NEWLINE\]NEWLINE holds with a constant \(C>0\) independent of \(u\)? As the author states, this seems to be open even when \(\Omega\) is a ball in \({\mathbb{R}}^n\).