By Takafumi Murai

ISBN-10: 3540190910

ISBN-13: 9783540190912

This study monograph reviews the Cauchy remodel on curves with the thing of formulating an exact estimate of analytic potential. The notice 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 in basic terms the emerging solar lemma. the ultimate and central bankruptcy makes use of the strategy of the second one bankruptcy to match analytic skill with integral-geometric amounts. the must haves for studying this publication are simple wisdom of singular integrals and serve as conception. It addresses experts and graduate scholars in functionality conception and in fluid dynamics.

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Example text

I. 27) give the required inequality in our lermna. 5. 9. ) we define S[a](x,y) = ks(x-y ) T[a](x,y), 48 where (IsIl~) ( ~ < - i g (s) = (i - s l s l ) We shall show that gives Theorem A. s I --< ll(2s)) (1/(2s) < IIS[a]ll2,2 $ Const flail, . Isl <_- l/s). 5 shows that a ( Lreal,l. 28) ~S = sup {o(S[a]); a ( Lreal,l } < Const . ) ~(I, S[a]i , f); is defined in the same manner as Shwartz's inequality shows that is finite. 9. ], @ (2) S 2 )4 3 ~ 3 =< {( ~ + 7 } ~S(2) + Const {~S + i}. if necessary, we may assume that = {x ( I; A(x) # B(x)} a, f ( Lreal,l , I interval}, (a) I $ O.

D S®-I/~ s J }dy 2 . 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.

B) I on I, b(x) = - ~13 on ~ , 3 we have K = E[a], T = E[b], 5 = i, ~(I, E[a], f) _~ a(l, E[b], f) + Z k=l O(Ik, E[a], f) + Const {$(E[b];e) I/2 + ~01(E[a]) + ~Ol(E[b])} Ill, where I k = I~, k ~(I, E[b],f) = Since (k >_- i). ~(I, E[~], f). b(x) = -~/3 on Put ~ = b - (6/3). We have Then II{II~ =< 2~/3 ~, we have D(E[b]; ~) = ~2~(H;~) =< Const. Thus ~(I, E[a], f) ~ OE ( ) + ~E(~) + Const 6 , which yields that ~E(6) _-< OE( that is, ) +~ and 001(E[a]) + O~l(E[b]) =< Const ~ . 37) Let n OE(~) < 4 ~E(23 ~) + Const ~ .

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

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