By W. B. Jones

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That of the be c h o s e n w i t h care: such that curves boundedness satisfactory curvature to group of m a p p i n g s , a n d curve can be carried show that can difficult is a point if its c u r v a t u r e we shall which same way as in the is of synthesis of the result con- average in careful subset the for smooth curves and parallel or a c e r t a i n there are operators uniform of m a p p i n g s same m e t h o d can A~Rn), of to find a good if there if same tangent choice in E then, it is in general is to prove fails the m a p p i n g s to form a family ~n, sequence, in the but its sign m t h e n can be found.

S for u = uI and u = u 2. ~lUl~(r)dt K s (I~)(_u ,~(u2)) e T sf is c l e a r l y This : a bounded Furthermore, when tor. ]2~vI [(F)dt = Eee where E is e x p e c t a t i o n fine the o p e r a t o r , Tsf Let under 7~iu 21 dt • e the c o n d i t i o n s ~(Ul),e(u2). We de- i Ul Ks(~ I, 2 )f( i ) d P ( ~ ) . operator from L2(Ul,dP) to L2(u2,dP). s + ~, T c o n v e r g e s to a c o m p l e t e l y c o n t i n u o u s o p e r a s i m m e d i a t e l y from K b e i n g u n i f o r m l y b o u n d e d and the s follows lemma.

I, and its derivatives of all orders converge Thus, by the lemma, T~hf ~ fiE' ' as : By the density, in A(E), the same result holds for every f E A(E). Furthermore tinuous) it suffices to prove character X. f(x,y) on E, with In fact, = ~ E LIoR2), T~hf : (9) when f in that case, if is a (bounded con- ~IR2 e-i(x~ + Yn)~(~'n)d~ d~ then ~]R T 2 ~h (e -i(x~ + Y n ) ) f ( ~ , n ) d ~ d~ ,58 on E'; and hence liT ~hf! [A(E' ) 5 C Varying f, Thus we o b t a i n we m u s t (9). prove that HT~h×NA(E' ) is u n i f o r m l y ~2, and We and, bounded, h varies shall moreover, case, reviewing various the proof ditions lowing to assume that proved obtain and that ~ e iay stated certain uniform on how the set of c h a r a c t e r s , special boundedness conditions for this general situation uniform bound set of c h a r a c t e r s where in the a > 0}, beginning : (ii) i, and under that of this g(0) xt(o,t) yt(~,-t) < : > 0, 0 < yt(o,t), (13) hold particu- simply depends discussion the m a p p i n g section j(0) for to find the v a l u e an u p p e r of w h i c h [ at bound o by on the is fulfills as w e l l as the the con- fol- ~(0) : 0, (~,t) E [0,I], t ( ]0,2] .