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%*   Abstracts for Conference dedicated to the memory of V.K.Dzjadyk  
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%*********************** AUTHORS ****************************
\def\author{Alexei Yu. Karlovich} 
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\tit     {On the Essential Norm of} 
\titwo  {the Cauchy Singular Integral Operator}
\tithree{in Rearrangement-Invariant Spaces}
                
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\abs {
This talk is devoted to a lower estimate for the essential
norm of the Cauchy singular integral operator $S$
in reflexive weighted rearrangement-invariant spaces $X(\Gamma,w)$
over Carleson (or Ahlfors-David regular) curves $\Gamma$. These
spaces are a wide generalization of classic Lebesgue, Orlicz, and
Lorentz spaces. Using results on Fredholmness of singular
integral operators with piecewise continuous coefficients
in the space $X(\Gamma,w)$, we prove that
$$
|S|:=\inf\|S+compact\|_{{\cal L}(X(\Gamma,w))}\ge
\cot\big({\pi}\lambda/2\big)
\eqno{(1)}
$$
where
$$
\lambda=\inf_{t\in\Gamma}\min\{\alpha(Q_tw),1-\beta(Q_tw)\},
$$
and $0<\alpha(Q_tw)\le\beta(Q_tw)<1$ are the indices of a 
submultiplicative function $(Q_tw)(x):(0,\infty)\to(0,\infty)$, 
which is associated with
local properties of the space, of the curve, and of the weight at 
the point $t\in\Gamma$.
In some cases we give formulas for computation of $\alpha(Q_tw)$
and $\beta(Q_tw)$. In particular, if we consider the non-weighted case 
($w=1$), then $\alpha(Q_t1)$ and $\beta(Q_t1)$ coincide with
Zippin (fundamental) indices [3] of the rearrangement-invariant space 
$X(\Gamma)$. For Lebesgue spaces $L^p(\Gamma), 1\le p\le\infty$, 
Zippin indices coincide and equal $1/p$. So, the 
estimate (1) correlates with the well-known result by Pichorides [2]
and Gohberg-Krupnik (see [1]) for the Lebesgue space $L^p({\bf T}),
1<p<\infty$, over the unit circle ${\bf T}$: 
$$
|S|=\|S\|_{{\cal L}(L^p({\bf T}))}=
\cot\big(\pi/2\cdot\min\{1/p,1-1/p\}\big).
$$
}
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%********************** REFERENCES **************************
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\ref
I.~Gohberg and N.~Krupnik,
{\sl One-Dimensional Linear Singular Integral Equations},
Vols. 1, 2, Birkh\"auser Verlag, Basel,
Boston, Berlin, 1992. Russian original: Shtiintsa, Kishinev, 1973.

\ref 
S.~K.~Pichorides,
{\sl On the best values of the constants in the theorems M. Riesz,
Zygmund and Kolmogorov}, 
Studia Math., {\bf 19}, 2 (1972), 165--179.

\ref
M.~Zippin,
{\sl Interpolation of operators of weak type between rearrangement
invariant spaces},
J. Functional Analysis {\bf 7} (1971), 267--284.
%*********************** ADDRESS ****************************
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% Address : For each of the authors, please 
\def\name{Alexei Yu. Karlovich} 
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\def\affiliate{South Ukrainian State Pedagogical University}
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\def\address{Staroportofrankovskaya str. 26\par
             270020, Odessa\par
             Ukraine}
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\def\email{karlik@paco.net} 
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\name\par\affiliate\par\address\par\email
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\bye 
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