@article { ISI:000176991300004,
	title = {Algebras of singular integral operators with PC coefficients in rearrangement-invariant spaces with Muckenhoupt weights},
	journal = {Journal of Operator Theory},
	volume = {47},
	number = {{2}},
	year = {2002},
	month = {{SPR}},
	pages = {303-323},
	publisher = {{THETA FOUNDATION}},
	type = {{Article}},
	address = {{C/O INST MATHEMATICS, PO BOX 1-764, BUCHAREST 70700, ROMANIA}},
	abstract = {<script src={\textquoteright}https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML{\textquoteright}></script><p>In this paper we extend results on Fredholmness of singular integral operators with piecewise continuous coefficients in reflexive rearrangement-invariant spaces \(X(\Gamma)\) with nontrivial Boyd indices \(\alpha_X,\beta_X\) [K98] to the weighted case. Suppose a weight \(w\) belongs to the Muckenhoupt classes \(A_{\frac{1}{\alpha_X}}(\Gamma)\) and \(A_{\frac{1}{\beta_X}}(\Gamma)\). 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 a {\textquoteleft}{\textquoteleft}disintegration condition{\textquoteright}{\textquoteright} we construct a symbol calculus for the Banach algebra generated by singular integral operators with matrix-valued piecewise continuous coefficients and get a formula for the index of an arbitrary operator from this algebra. We give nontrivial examples of spaces, for which this {\textquoteleft}{\textquoteleft}disintegration condition{\textquoteright}{\textquoteright} is satisfied. One of such spaces is a Lebesgue space with a general Muckenhoupt weight over an arbitrary Carleson curve.</p>
},
	keywords = {Fredholmness, Muckenhoupt weight}, rearrangement-invariant space, {singular integral operator},
	issn = {{0379-4024}},
	url = {http://www.theta.ro/jot/archive/2002-047-002/2002-047-002-004.html},
	author = {Karlovich, Alexei Yu.}
}
