\documentclass[12pt]{article}
\begin{document}
\title{Compactness of commutators arising\\ 
in the  Fredholm theory\\ 
of singular integral operators with shifts}
\author{Alexei Karlovich}
\date{}
\maketitle

Commutators $aS-SaI$ and $W_\alpha S-SW_\alpha$ of the Cauchy singular 
integral operator $S$ with the operator of multiplication $aI$ by a 
function $a\in L^\infty$ and with the shift operator $W_\alpha$ 
defined by $W_\alpha f=f\circ\alpha$ play a very important role in 
the Fredholm theory of singular integral operators with shifts.
The compactness criterion for the first commutator on $L^2({\bf T})$
over the unit circle ${\bf T}$ is well known for a long time, while 
the compactness criterion for the second commutator on $L^2({\bf T})$
was obtained recently (1995) by P. Muhly and J. Xia when $\alpha$ 
is an orientation preserving homeomorphism of ${\bf T}$ onto itself.

Passing from $L^2({\bf T})$ to $L^p({\bf T})$ with the help of the 
Krasnoselskii interpolation theorem for compact operators, and ``cutting''
the unit circle ${\bf T}$, we prove compactness criteria for these 
commutators on Lebesgue spaces $L^p({\bf J})$ where ${\bf J}:=(0,1)$.

\vspace{3mm}
\noindent
{\bf Theorem 1.}
{\it
Suppose $1<p<\infty$ and $a\in L^\infty({\bf J})$. 
The operator $aS-SaI$ is compact on the Lebesgue space
$L^p({\bf J})$ if and only if $a$ has vanishing mean oscillation 
on ${\bf J}$.
}

\vspace{3mm}
\noindent
{\bf Theorem 2.}
{\it
Suppose $1<p<\infty$ and $\alpha$ is an orientation preserving 
homeomorphism of $[0,1]$ onto itself such that $\alpha(0)=0$ and 
$\alpha(1)=1$. Suppose $\log\alpha'\in L^\infty({\bf J})$. 
The operator $W_\alpha S-SW_\alpha$ is compact on the Lebesgue 
space $L^p({\bf J})$ if and only if $\alpha'$ has vanishing mean 
oscillation on ${\bf J}$.
}

\vspace{3mm}
Passage from ${\bf T}$ to ${\bf J}$ (``cutting'') involves an 
operator $R$ with two fixed singularities at the endpoints of 
half-circles. The main difficulty here consists of the 
proof of compactness for the commutators of $R$ with the 
operators of multiplication and with the shift operator.

These results are obtained in collaboration with Yuri Karlovich.
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