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ALGEBRAS OF SINGULAR INTEGRAL OPERATORS

IN REARRANGEMENT-INVARIANT

SPACES WITH MUCKENHOUPT WEIGHTS
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Alexei Karlovich

Departamento de Matem\'atica,

Instituto Superior T\'ecnico,

Av. Rovisco Pais, 1049-001, Lisbon, Portugal

E-mail: akarlov@math.ist.utl.pt


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In this talk we study Fredholmness of singular integral
operators with piecewise continuous coefficients in reflexive
rearrangement-invariant spaces with weights $X(\Gamma,w)$
over arbitrary Carleson curves $\Gamma$.
Suppose a weight $w$ belongs to the Muckenhoupt classes
$A_{\frac{1}{\alpha_X}}(\Gamma)$ and $A_{\frac{1}{\beta_X}}(\Gamma)$,
where $\alpha_X$ and $\beta_X$ are Boyd indices of 
a rearrangement-invariant space $X(\Gamma)$.
We prove that these conditions guarantee the boundedness of the Cauchy
singular integral operator $S$ in the weighted rearrangement-invariant
space $X(\Gamma,w)$. Under some ``disintegration condition'' we construct
a symbol calculus for the Banach algebra generated by singular
integral operators with matrix-valued piecewise continuous coefficients.
We prove criteria for Fredholmness and get a formula for the index
of any operator from this algebra in terms of its symbol.
We give nontrivial examples of spaces, for which this ``disintegration
condition'' is satisfied. One of such spaces is a Lebesgue space with 
a general Muckenhoupt weight over an arbitrary Carleson curve. Another
nontrivial example of space satisfying this ``disintegration condition''
is a reflexive Orlicz space with distinct Boyd indices over a
smooth curve.
Note that for weighted Lebesgue spaces corresponding results
were obtained by I. Gohberg and N. Krupnik for power weights 
(the end of sixties);
and I. Spitkovsky, A. B\"ottcher and Yu. I. Karlovich for arbitrary
Muckenhoupt weights (the middle of nineties).
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