# Download Sobolev Spaces by Professor Vladimir G. Maz’ja (auth.) PDF

By Professor Vladimir G. Maz’ja (auth.)

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Setting f = 0 on the support of the singular part of the measure v, we obtain that (1) is equivalent to [ ~ ~f(t)dt Iqd/l(x) 001 X J/q ::;;; C [X~ If(x) IP d:x* dxJ1IP 1 The estimate B::;;; C can be derived in the same way as in the proof of Theorem 2, if Iv(x) IP is replaced by dv*/dx and 00 J Iw(x) Iqdx by /l([r, 00». r Now we establish the upper bound for C. We may assumef~ O. Let {gn} be a sequence of decreasing absolutely continuous functions on [0,00) satisfying 0::;;; gn(x)::;;; gn+l(X)::;;; /l([x, 00» , lim gn(x) = /l([x, n-+oo for almost all x.

Now, using the diagonalization process, we can construct a sequence of functions in COO(Q) which approxim~es U in W~(Q). Thus we proved the density of COO(Q) in W~(Q). The spaces L~(Q) and V~(Q) can be considered in an analogous manner. Theorem 2. Let Q be a domain with compact closure oj the class C. This means that every XEaQ has a neighborhood Ol! such that Q n Ol! has the representation xn

Let u eL~(Q) and let {lh} be a sequence of positive numbers which monotonically tends to zero so that the sequence of balls {(1 + (2k) 8dk} has the same properties as {8dk}' If 8dk = Bix), then by definition we put c 8d k = Bce(x). krk' Clearly, W = L Wk belongs to CCX>(Q). k to satisfy On any bounded open set w, OJ C Q, we have where the sum contains a finite number of terms. Hence, Ilu-wIILb(w)::;:; L Il uk- wkIILb(w)::;:;e(1-e)-1. Therefore, weL~(Q) n CCX>(Q) and II u - W IILb(,Q) ::;:; 2 e .