By Takafumi Murai

ISBN-10: 3540190910

ISBN-13: 9783540190912

This learn monograph stories the Cauchy rework on curves with the item of formulating an actual estimate of analytic capability. The word is split into 3 chapters. the 1st bankruptcy is a evaluate of the Calderón commutator. within the moment bankruptcy, a true variable technique for the Cauchy rework is given utilizing basically the emerging solar lemma. the ultimate and imperative bankruptcy makes use of the tactic of the second one bankruptcy to check analytic potential with integral-geometric amounts. the must haves for studying this publication are simple wisdom of singular integrals and serve as thought. It addresses experts and graduate scholars in functionality concept and in fluid dynamics.

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

**Sample text**

17. In Chapter I, we showed 8 proofs of the boundedness of the CalderSn commutator T[']. Since the Calder~n commutator is closely related to analycity of functions, it seems necessary to give more proofs and to have a unified understanding. CHAPTER II. 1. 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]).

This completes the proof of Theorem A. 6. Proof of Theorem B by perturbation ([17]) In this section, we deduce Theorem B from the boundedness of H. Here are two lemmas necessary for the proof. 10 (Calder~n [4]). Proof. Fixing IITn[a]ll2,2 $ (Const) n llalI~ 0 < s < 1/2, we put Sn[a](x,y ) = Since el(Sn[a]) ~ Const n (n ~ i). 32) ~S = sup {~(Sn[a]); a ( ereal,l } ~ C O n for some absolute constant So[-] = (-~)H. (0 ~ k ~ n-l). CO Then We d e f i n e (which will be determined later). 33) k+l ffSk ~ C O Suppose that in the same manner as 4 Let a, b, f, I, ~ <~Sn -<-~S (2) " n K = Sn[a], ~(-[ H) ~ ~.

E. e. on Let (Ifl) • X (0,~). = ~ l~l,k (0,®) - ~I " ~I = {x > 0; A(x) # B(x)}. (k ~ i), Then 34 Considering Then f(-x), we obtain, in the same manner, an open set ~2 in (0,~). ~i U {-x; x E ~2 } is the required open set. 2. satisfy Let f be an integrable function in an interval k > (Ifl)l. e. M is defined by the supremum is taken over all intervals M I k > 0 such that I The (non-centered) maximal operator denotes the norm of in and let (k>= i), (Ifl) I =< ~,k on ~ I I Mf(x) = sup(Ifl) I, containing as an operator from Lp x.

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

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