ÎHuu weakly in H and (ii) |u – u| r W we have that

Next, we need a lemma which deals with the Fourier analytic properties associated with the eigenfunctions {jn}¥n=1 and eigenvalues {ln}¥n=1 in ( V L – 1) and ( V L – 2) above. In particular, the following lemma holds:

f(n) =áf,jn ñr. (2.11) Then f Î H1p,r if and only if S¥n=1 ln |f(n)|2 < . Furthermore, if f Î

H1p,r, then L(f,f) = S¥n=1 ln |f(n)|2 . For a proof of the above lemma, we refer the reader to [Sh, p. 37].