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Some remarks on spaces of Morrey type. (English) Zbl 1208.42006
Let \(\Omega\) be an unbounded open subset on \(\mathbb{R}^n, n \geq 2\). For \(p \in [1, \infty [\) and \(\lambda \in [0, n[\), the Morrey type space \(M^{p, \lambda}(\Omega)\) is the set of the functions \(f\) such that \[ \| f \|_{M^{p,\lambda}(\Omega)}^p = \sup_{x \in \Omega, r \in ]0,1]} \frac{1}{r^{\lambda}} \int_{\Omega \cap B(x,r)} | f(y) |^p dy < \infty, \] where \(B(x,r)\) is the open ball with center \(x\) and radius \(r\).
P. Cavaliere, M. Longobardi and A. Vitolo [Matematiche 51, No. 1, 87–104 (1996; Zbl 0905.35017)] proved the boundedness of a multiplication operator on Sobolev spaces: If \(\Omega\) satisfies the cone property and \(1<p < q \leq n, \lambda = n-q\), then \[ \| fu \|_{L^p(\Omega)} \leq C \| f \|_{M^{q, \lambda}(\Omega)} \| u \|_{W^{1, p}(\Omega)}. \] The authors prove the compactness of a multiplication operator as follows. Fix \(f \in M^{q, \lambda}_0(\Omega)\). Then the operator \[ u \in W^{1, p}(\Omega) \to fu \in L^p(\Omega) \] is compact, where \(M^{q, \lambda}_0(\Omega)\) is the closure of \(C_0^{\infty}(\Omega)\) in \(M^{q, \lambda}(\Omega)\).

42B25 Maximal functions, Littlewood-Paley theory
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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