# Download Theory of functions of real variable and theory of Fourier by E. W. Hobson PDF

a,>,(ü, u) is coercive on Wi,B for so~e A ;:=: 0. This is, however, in generalnot the case for (A, B) E E(n) if N > 1. 19) is not accessible by 'variational techniques' if N > 1. 13), namely [Sim72]. 15) for Dirichlet forms with continuous coefficients and arbitrary p E (1, oo ), but in the case of a single equation (N = 1), by a completely different technique. (As mentioned earlier, we restriet ourselves here to the case of second order problems.

C(s;~, s;:ß- 2)) . 20) that is, A a-1 ·= . 15) if 6 = 0 and 1/p < 2o: < 1 + 1/p). Of course, the s;:ß- 2-realization Aa-l of (A, B) depends on SE {H, W} too. But we do not indicate this in order not to overburden the notation. The foilowing theorem justifies the notationAa-l for the s;:ß- 2-realization of (A, B). 9). J. ~- 2 -realization of (A, B) otherwise . For the proof of this theorem it suffices to observe that, thanks to (8. 16) gives (v, A 0 u ) = (A # v, u ) , ( v, u ) E s2-2a p',l3# x w2 p,a , 0< _ 2o: < 1/p.

Thus we do not indicate the wdependence. (c) Suppose that -1 :S ß < a < oo. 5) that (Eß,Ea) is a densely injected Banach couple. Thus, given any e E (0, 1) and any other admissible interpolation functor ((·, ·))o, the interpolation space ((Eß, Ea))o is weil defined. In general, ((Eß, Ea))o will not coincide with any one of the spaces E-y, ß < 1 < a. The following 'sandwich property' shows, however, that ((Eß, Ea))o can be arbitrarily closely approximated by spaces of the original interpolation-extrapolation scale.