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

Calculus

By Takafumi Murai

ISBN-10: 3540190910

ISBN-13: 9783540190912

This learn monograph stories the Cauchy rework on curves with the thing of formulating an actual estimate of analytic means. 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 approach for the Cauchy remodel is given utilizing merely the emerging solar lemma. the ultimate and relevant bankruptcy makes use of the strategy of the second one bankruptcy to match analytic skill with integral-geometric amounts. the necessities for studying this publication are simple wisdom of singular integrals and serve as idea. It addresses experts and graduate scholars in functionality thought and in fluid dynamics.

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In this section, we show Theorem C ([44]). 39) IIC[a]I12,2~ Const (i +~IIalIBMO). 38) was given with was given with __~MO replaced by [44] via [42], [43], [50]. 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.

1) where A C[a](x,y) = i/{(x-y) + i(A(x) - A(y))}, is a primitive of a. 1). 2) = {a E L®; the singular integral We put a is real-valued}. , Tn[a ] Tn[a](x,y) C[a] This is called the Cauchy transform of {(x, A(x)); x E ~}. Coifman-McIntosh-Meyer Theorem B ([7]). where a, we define a kernel by and Tn[a] is the n-th Coifman- is an operator defined by = (A(x) - A(y))n/(x-y) n+l. Prior to this theorem, the following three theorems were shown. 3) IITn[a]II2,2 ~ Const n! 4) IIC[a]ll2,2 is bounded if llall~ (a E ereal) is small enough.

4. A-priori estimates In this section, we show some inequalities which play important roles later. 18) $(I, T, f) = fl IT(Xlf)(x)I 2 f(x)dx. 19) For a ~ with I~I < ~ , we put ~(T;~) = sup { 8-standard kernel ~(l,T,f); K(x,y), f ( Lreal,l, we have easily I component of ~} . , ~ on (f,g) ~(K) I/2 + C 8 0~8(K). ) d~. ) the inner product with (In the case of the 1-dimension Here is an inequality necessary for the proof of Theorem B. 6. Let open set in I I be an (open) interval, and let K(x,y), T(x,y) K(x,y) = T(x,y) for any supported on and a non-negative measure ~ l(Ku,v) ~ = Uk= 1 I k (I k = I2,k) x, y ~ I - ~, x # y.

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


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