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On Stein’s extension operator preserving Sobolev-Morrey spaces. (English) Zbl 1445.42017

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)].

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
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