A Boundedness Criterion via Atoms for Linear Operators in by Yang D., Zhou Y.

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By Yang D., Zhou Y.

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1 ( 0) and y 1 ( 0) , where p. 4). 3. 7 Tbe level hypersurfaces M , locally can be described as an "angu- lar product" of s1 with S~+l, 2 s , where the angle of incidence is 2 s - 11/2, and the plane of incidence is spanned by the tangent vectors to the curve p. ( u) of ( 2. 3. 4) and the curve y ( u) of ( 2. 3. 6). Example 2. 3. P. P. = 2 6 .. •• , n. x,x> 2 ,xER 21 , i 1 defines an isoparametric function f = F 1s21-1 on s 21 - 1 of degree 4, with the multiplicities of the principal curvatures being (m 1 ,m2 ).

Proof Let i : M c ~ 0 M be the inclusion map. (f 0 i) df(~i) +trace V'df(di,di) ~f- ldfl (mean curvature of M ) + c where ~ Then = V'f/ ldf I (df is non-zero on M c V'df(CO, since M c is a hypersurface). Now, using Lemma 2 . 1 . 1 , V'df(CE) d f ( '\7 ~0 + '\7 d~ ( 0 d f (0 (V'f)ldfl. 3) f which is a function of f. 2); the level hypersurfaces of f form a parallel family of hypersurfaces of constant mean curvature. Conversely, from equation ( 2. 1. 3) and by reversing the proof of Lemma 2. 1. 1, it is not hard to show that given such a family which are the level hypersurfaces of a function f: M - - R, then f satisfies equations (2.

Parameter choice for a . The physical moden can now be appro- 0 xi mated by a pendulum, and we pick a a 1 (a ) 0 the pendulum just reaches 0 as u (3 11 = G(u,(3,(3 1 ), --co 0 = 0! 1 ( u ) to be the velocity such that 0 in backward time (equations of the form where G, fJG/f'fJ,~G/8(1 1 are continuous, have unique solutions through each point (u 0 , p 0 ,p~), [9 )). In Chapter 6 we will demonstrate precisely the existence of non-trivial solutions of equation ( 1. 6. 3) which exist for all time.

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