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From the Sobolev vector space point of view, the following results are well known (see, e.g., (4, Chapter 1, Section 8-10]):

IIUIIx = sup IV(x)l :S Mk IIUIIik) for all U in '2ll(X, C§) (1. 8) xEX

Let U be in '2ll(X, C&);

(**)

Let (Hq)q be an orthonormal basis for C§ with respect to the scalar product ( , ), and let U = "'Lqfq ® Hq and U' = "'Lqf~ ® Hq be t\vo elements in W~k>(X, C§). For each pair (q, q'), I :::::; q :::::; dim(C§), I :::::; q' :::; dim(C§), there exists a sequence (a)q, q'))1 in /2 such that [Hq, Hq.] = "L1 a1(q, q')Hl' from which it follows that

In particular A(U) belongs to S!ll(X, C§). PROOF. It suffices to prove the assertion when X is an open subset