By L P, Baym, G Kadanoff
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Additional info for Quantum statistical mechanics; Green's function methods in equilibrium and nonequilibrium problems
Sample text
V/. V/ ; 1=2;1 s D k C 1; (8) where k 2 N. Besides these isotropic spaces we also need anisotropic versions adapted to parabolic problems. V J/ is naturally identified with space valued distributions on J. V J/ D Lp V J; dvdt . V/ if s … 2N, and a Zygmund space for s 2 2N, of Banach space valued functions on J (see Lunardi [28], for example). V/ ; recalling that J is compact. Although Sobolev-Slobodeckii spaces, respectively Besov-Hölder spaces, are well-defined for each s 2 RC , respectively s > 0, they are not too useful on general Riemannian manifolds since, for example, the fundamental Sobolev type embedding theorems may not hold in general.
Now we illustrate the strength of our results by means of relatively simple examples. For this we assume that is a smooth open subset of Rm with a compact smooth boundary, that is, N is a smooth m-dimensional submanifold of Rm . ii/ if 1 Ä ` Ä m 1 and \ @ 2, then ¤ ;, then @ : : S Then M WD N n f I 2 g, endowed with the Euclidean metric, is an m-dimensional Riemannian submanifold of Rm whose boundary @M equals S @ n f I 2 g. x/ the (Euclidean) distance from x to . Then ı is, sufficiently close to , a well-defined strictly positive smooth function.
T u 30 H. Abels and J. 5 Let  Rd , d D 2; 3 be a bounded domain with C3 -boundary and F defined as in (14). a; b/. @L2 F/ D c 2 H 2 . 0/ . c/ 2 L2 . c/jrcj2 2 L1 . 0/ Furthermore it holds kck2H 2 . c/k2L2 . c/k2L2 . C kck2L2 . @L2 F/. 0/ The proof can be found in [2], cf. 3. 0/ . 0/ . / in the standard way. But we 1 . / via the Riesz isomorphism. 0/1 . 0/ Equation (18) provides a characterization of @L2 F. 0/ between @H 1 F and @L2 F. 7 Let F be as in (14). c/. c/. 0/ hw; c0 ciL2 . 0/ /0 ;L2 .