By Hans Triebel
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Additional info for Function spaces and wavelets on domains
As a consequence one has now only a para-basis. 4. 92). 27. One has always the continuous embedding s;rloc s Fpq . / ,! Fzpq . 110) s . 1. 23. It will be one of the main points of what follows to show that s;rloc s Fpq . / D Fzpq . 111) 46 2 Spaces on arbitrary domains (q D 1 if p D 1) for huge classes of domains, the so-called E-thick domains, including bounded C 1 domains and bounded Lipschitz domains. 82). This observation will pave the way to develop a s s theory of wavelet bases for large classes of the spaces AQpq .
40). But afterwards one can follow the arguments in [T06] verbatim. j;G;m/2S s;rloc for f 2 Fpq . /. We do not repeat the details. But we fix the outcome. Recall that j the functions ‰G;m are real. 99) and f is extended outside of by zero. 89). 89). 18. j;G;m/2S s s is finite. Recall that F11 D B11 . ˇ 2jsq ˇ j;G m j m. ˇq Á1=q /ˇ jLp . 23. Let be an arbitrary domain in Rn with 6D Rn . with q D 1 if p D 1/ and pq < s < u 2 N. 87). 71). Then f 2 Lv . / is an element of Fpq . j;G;m/2S Furthermore, s;rloc .
89) (recall that S0 D ;). They fit in the above scheme and need not to be considered. 92). These are boundary elements. 4 (iii). 0 Step 2. First we deal with the one-dimensional model case D . 1; 0/. 130) be an interval centred at 2 r for some suitable negative integer r and of side-length 2 l . 137) with G D F (scaling function) or G D M (wavelet). Of interest is the case 0 where the right endpoint of Qlr , hence xl;r D 2 l r C 2 l 1 , is the left endpoint of an 0 admitted Whitney interval QlC1;r 0 .