Lamberti, Pier Domenico; Violo, Ivan Yuri On Stein’s extension operator preserving Sobolev-Morrey spaces. (English) Zbl 1445.42017 Math. Nachr. 292, No. 8, 1701-1715 (2019). The authors prove that Stein’s extension operator \(T\) for Sobolev spaces on Lipschitz domains [E. M. Stein, Singular integrals and differentiability properties of functions. Princeton: Princeton University Press (1970; Zbl 0207.13501)], also defines a continuous operator on the Sobolev-Morrey spaces (defined by the condition that the derivatives \(D^{\alpha}f\in M_p^{\delta,\phi}\), the Morrey space). Their main results states the following:Theorem 3.3. Let \(1\le p<\infty\), \(l\in\mathbb N\) and \(\phi\) a function from \(\mathbb R^+\) to \(\mathbb R^+\). Let \(\Omega\) be a special Lipschitz domain of \(\mathbb R^n\) with Lipschitz bound \(M\). Let \(T:W^{l,p}(\Omega)\rightarrow W^{l,p}(\mathbb R^n)\) be the Stein’s extension operaotr. Then there exists \(C>0\) depending only on \(n\), \(l\) and \(M\) such that \[ \|D^\alpha Tf\|_{M_p^{\delta,\phi}(\mathbb R^n)}\le C\sum_{|\beta|=|\alpha|}\|D^{\beta}f\|_{M_p^{\delta,\phi}(\Omega)} \] holds for all \(f\in W^{l,p}(\Omega)\), \(\delta>0\), and \(\alpha\in\mathbb N^n_0\) with \(|\alpha|\le l\).Similar results were previously obtained in [M. S. Fanciullo and P. D. Lamberti, Math. Nachr. 290, No. 1, 37–49 (2017; Zbl 1366.42026)] for another local extension operator introduced in [V. I. Burenkov, Dokl. Akad. Nauk SSSR 224, 269–272 (1975; Zbl 0351.46022)]. Reviewer: Javier Soria (Madrid) Cited in 3 Documents MSC: 42B35 Function spaces arising in harmonic analysis 46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems Keywords:extension operator; Lipschitz domains; Sobolev spaces; Morrey spaces Citations:Zbl 0207.13501; Zbl 1366.42026; Zbl 0351.46022 PDFBibTeX XMLCite \textit{P. D. Lamberti} and \textit{I. Y. Violo}, Math. Nachr. 292, No. 8, 1701--1715 (2019; Zbl 1445.42017) Full Text: DOI arXiv