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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 $$M^{q, \lambda}(\Omega)$$.

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
##### Keywords:
Morrey space; Sobolev space; multiplication operator
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##### References:
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