By Prof. Richard S. Hamilton (auth.)

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**Extra info for Harmonic Maps of Manifolds with Boundary**

**Example text**

A/On)l/p. To be precise this holds for ~(X) which is dense in Lk(X), For a general f € ~(X) choose a sequence fm € ~(X) converging to f. Then s~p I fm(X) -fm, (x) I ~ cll fm-fm' II Lk(x) and fm Therefore is Cauchy in f € ik(X) a function in Co(X) ik(X) so fm is also Cauchy in eo(X). is equal (as a distribution in -,g(X)*) to and the inclusion 50 tk(X) ~ Co(X) is continuous. 14. By the same methods we can handle corners. denote the space of the variables Let {xn' ... 'Xn'y,z} X XY X Z with the last two distinguished, and consider the four corners with positive or negative.

C(11 s~ DYfllLP + II DjS~DYfIILP) Thus S. ell fllJ\p. •• kj, ... ,o) with the only non-zero entry k j in th the j-- place. where k j is the largest integer less than SOj/a. Then II fll P LrCW j since < cil ) - fll p' I\s W is eq",iva1ent to II fllLP r S. ell rll p' J\s and Since this is true for all W'1+",+Wn' 1\;:: L~. it follows that j. and 29 7. Next we show how to use the Besov spaces ror non-linear estimates. Multiplication Theorem. Suppose 0 < a < 1, pi ~ p, ql ~ q, r l ~ rand If f € pi n Ad.

60 Kj(s,y) The kernel sense that for t >0 Kj(t is skew-heterogeneous of weight 0/a 1 r Moreover the same path II s II = 1. + J < - ~ 2. for II s /I 2.. 3). 1 for II s II Then for all = 1. ,;.. 1. Then and W(~) are comparable for II S II 2.. 1. • ,t in the -r. If ,y) 61 f(S ,y) = J e-i(S ,x) f(x,y)dX. This is an ~somorphism of Rff(X X y+) onto deB x y+). Define the operator by 4. g (XXy+) : ~(x) denote the direct sum of ~(x) j"'l times. Then Bf = (Blf, ••• ,BMf) defines a map Let B: -J (XxY+) ~ : j=l ~ -i(X)(Y+) • with itself m ,&x).