Download e-book for iPad: A Real Variable Method for the Cauchy Transform, and by Takafumi Murai

By Takafumi Murai

ISBN-10: 3540190910

ISBN-13: 9783540190912

This learn monograph reports the Cauchy remodel on curves with the article of formulating an exact estimate of analytic ability. The be aware is split into 3 chapters. the 1st bankruptcy is a overview of the Calderón commutator. within the moment bankruptcy, a true variable procedure for the Cauchy remodel is given utilizing simply the emerging solar lemma. the ultimate and central bankruptcy makes use of the tactic of the second one bankruptcy to check analytic skill with integral-geometric amounts. the necessities for examining this booklet are simple wisdom of singular integrals and serve as conception. It addresses experts and graduate scholars in functionality idea and in fluid dynamics.

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Extra resources for A Real Variable Method for the Cauchy Transform, and Analytic Capacity

Example text

Use RSL repeatedly. a(x), I, ~, ~ and we define analogously aretan y. 36). The lower bound of A(x) B(x) a replaced by 8 IIaIIBMO. 8). 2. e. on (k g i), I, I~I ~ b(x) = 7 (x ( I k, k ~ i), (-~) * (b)l III. (-~) + 7 This RSL is called of Type 2 ( y - h a y , ~ - d g 6 e e ~ ) . independent of the above estimates. e. on !. We define b, ~ --<-b(x) --< 7 In this case @ in the same manner. e. on I, ~ we define such that is B(x) = sup @(x), • ~ A, ~' ~ T Then b(x) = 7 (x ( I k, k >_- i), (-~) + (b) (a)i k > y (k ~ i), I~I =< This RSL is called of Type 3 ( y - r .

Consequently we have Hence L 1 =< Ca, 8 lllfill~ gives the required inequality. Given and Sj {Sjc (K(Y0,S) - K(y,s))ds} dy < we have K(y,s)ds i + 1} . > ly0-yl ~ Proof. + Const, S_l 11(O12 {7o (I'_~ -7 t s ) 2 i + = C a d'_] If(E) l2 d~ = C a IIfll2 . t2E2 t -c~ --I~I1-L dt +dO(f0 ds) l+teiGi2 dt} dK 24 Let f ~ [E ,E_~]. D. 12 is deduced as follows. ~05(K) = i. 16 with s > 0, we can choose gs E E a, a = 6/2. 15. [[Kfi[22 -< C6 [[Kfll~nt,a = C6 /0 B(s'Kf)2 E_a Let K1 = 0 f ~ L 2. ii (INKgs [[12 + s2 [IIKhsIii2-a) ~ds s 2 + s 2 [IlhslIi2a)--2as < C fo ([[[gsIiia = 6 s C 6 IIfII2nt,c~ < B(s'f)2 __ds 2 s c6 IlfIl~.

For -<_Const {o(K) + C 5} • n >-_ I, f ~ Lreal,l and an interval we have 1 ~(I'K(2 n-i )'f) =< { 41 ~f~ = {~ 7i / -= =IK(2n-l)(xif)(x)l 2 dx}i/2 _2 n 12 f(x)~( )(Xlf)(x)dx} / --< { ~ and hence i Tf ~ ~(l,K,f) _<- { ~ i n -n o(I,K (2) f)}2 . o(l,K(2n),f)} I/2 I, 26 For a while, we assume that XI f = Zk=IN ~k Xlk Xlf (l~kl ~ I) is a step function, saying Ik interval, Ik D I g = ~ (k # g)). 17 show that { ~- 1 N ~T < = 171f(x) ~(l,K(2n),f)} 2 ~ ~ K(2~+I) % k=l I:I f(x) Xlk(X)dxl N :on+l, :^n+l. kEllflk Kk~ :X I k (x) dx I + ~5(K ~Z )) ~ = i ~f~ ~(K(2n+l)) (2n+l) + Const N infinity, we have ~k' Ik ~6(K ) ~ N d k%l:l_ik{flk ~ = (i/II I) ~(l,K)f) ~ C 6.

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A Real Variable Method for the Cauchy Transform, and Analytic Capacity by Takafumi Murai

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