\documentclass{elsart} \usepackage{amsmath} \usepackage{amsfonts} % Karlovich's symbols and commands %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\A}{\mathbb{A}} \newcommand{\C}{\mathbb{C}} \newcommand{\D}{\mathbb{D}} \newcommand{\N}{\mathbb{N}} \newcommand{\T}{\mathbb{T}} \newcommand{\cB}{\mathcal{B}} \newcommand{\cC}{\mathcal{C}} \newcommand{\cD}{\mathcal{D}} \newcommand{\cO}{\mathcal{O}} \newcommand{\wu}{\widetilde{u}} \newcommand{\im}{\operatorname{Im}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}[proposition]{Theorem} \newtheorem{lemma}[proposition]{Lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \runauthor{Karlovich} \begin{frontmatter} \title{Singular integral operators with flip\\ and unbounded coefficients\\ on rearrangement-invariant spaces} \author[Lisboa]{Alexei Karlovich\thanksref{X}} \thanks[X]{The author is partially supported by F.C.T. (Portugal) grants POCTI/ 34222/MAT/2000 and PRAXIS/XXI/BPD/22006/99.} \address[Lisboa]{ Departamento de Matem\'{a}tica, Instituto Superior T\'{e}cnico,\\ Av. Rovisco Pais 1, 1049--001 Lisboa, Portugal} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{abstract} We prove Fredholm criteria for singular integral operators of the form \[ P_++M_bP_-+M_uUP_-, \] where $P_\pm$ are the Riesz projections, $U$ is the flip operator, and $M_b,M_u$ are operators of multiplication by functions $b,u$, respectively, on a reflexive rearrangement-invariant space with nontrivial Boyd indices over the unit circle. We assume a priori that $M_b$ is bounded, but $M_u$ may be unbounded. The function $u$ belongs to a class of, in general, unbounded functions that relates to the Douglas algebra $H^\infty+C$. \end{abstract} \begin{keyword} Rearrangement-invariant space, Boyd indices, Douglas algebra $H^\infty+C$, outer function, Fredholmness, singular integral operator with flip. \end{keyword} \end{frontmatter} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} Let $\T$ be the unit circle equipped with the normalized Lebesgue measure $m$ and let $X:=X(\T,m)$ be a \textit{rearrangement-invariant space} (for the definition and basic properties, see \cite{BS88}). Rearrangement-invariant spaces are wide generalizations of Lebesgue spaces $L^p:=L^p(\T,m),1\le p\le \infty$, as well as Orlicz and Lorentz spaces. The \textit{domain} of a linear operator $T$ from $X$ to $X$ is denoted by $\cD(T)$. The set of all closed (bounded) linear operators from $X$ to $X$ is denoted by $\cC(X)$ (respectively, $\cB(X)$). We shall denote by $I$ the identity operator on $X$ and by $M_a$ the operator of multiplication by a measurable (not necessarily bounded) function $a$ on $\T$. The {\it flip operator} is defined by \[ (U\varphi)(t):=\overline{t}\varphi(\overline{t}) \quad (t\in\T). \] The \textit{Cauchy singular integral} of a function $\varphi\in L^1$ is given by \[ (S\varphi)(t):=\frac{1}{\pi i}p.v.\int_\T\frac{\varphi(\tau)}{\tau-t}d\tau \quad (t\in\T). \] We will always assume that the rearrangement-invariant space $X$ is reflexive and the \textit{Boyd indices} $\alpha_X,\beta_X$ of $X$ (see \cite{Boyd69}) are nontrivial, i.e., $\alpha_X,\beta_X\in (0,1)$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:basic} {\rm (a)} $S,U\in\cB(X)$ and $U^2=I, S^2=I, SU=-US$; {\rm (b)} $M_a\in\cB(X)$ if and only if $a\in L^\infty$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textit{Proof.} (a) Since $S,U\in\cB(L^p)$ for $p\in(1,\infty)$, by Boyd's interpolation theorem \cite{Boyd69}, $S,U\in\cB(X)$ whenever $X$ has nontrivial Boyd indices. The equality $U^2=I$ is obvious. The equality $S^2=I$ is proved in \cite[Lemma~6.5]{K98}. The equality $SU=-US$ is proved similarly. Part (b) is proved in \cite[Theorem~1]{MP89}. \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Put $P_+:=(I+S)/2$ and $P_-:=(I-S)/2$. From Lemma~\ref{le:basic}(a) it follows that %%% \begin{equation}\label{eq:basic} P_+^2=P_+,\quad P_-^2=P_-,\quad U^2=I,\quad P_+U=UP_-,\quad P_-U=UP_+. \end{equation} %%% For a linear operator $T$ from $X$ to $X$, we shall use the notation \[ R(T):=P_++TP_-. \] Following \cite{KLLTmn}, we study Fredholmness of singular integral operators with flip of the form $R(A)$, where \[ A:=M_b+M_uU, \] $b\in L^\infty$, but the function $u$ may be unbounded. We refer to \cite[Ch.~3--4]{Kato} for properties of operators in $\cC(X)$ and to \cite[Ch.~1]{BS90}, \cite[Ch.~4]{gk} for the definition and properties of (bounded) Fredholm operators on Banach spaces. The paper is organized as follows. In Section~\ref{sec:auxiliary}, we prove that $R(A)$ is closed whenever $b\in L^\infty$ and $u$ is measurable and a.e. finite. Further we study the invertibility of two auxiliary operators. In Section~\ref{sec:Fredholmness}, we define the class $\A$ of, in general, unbounded functions $u$ such that $\log\wu\in L^1$ and $u\cO_{1/\wu}$ belongs to the Douglas algebra $H^\infty+C$. Here $\wu:=\max\{1,|u|\}$ and $\cO_{1/\wu}$ is the outer function generated by $1/\wu$. Then we prove the relative compactness of the Hankel operator $P_-M_uP_+$ with respect to the operator $M_uP_+$ assuming $u\in\A$. By using this key result, we obtain Fredholm criteria for the operator $R(A)$, where $b\in L^\infty$ and $u\in\A$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Auxiliary results}\label{sec:auxiliary} \subsection{Closedness of $R(A)$} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:mult-closedness} If $a$ is a measurable a.e. finite function on $\T$, then $M_a\in\cC(X)$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textit{Proof.} The idea of the proof is borrowed from \cite[Ch.~3, Theorem~9.2]{EE87}. Clearly, \[ \cD(M_a)=\{\varphi\in X:\ a\varphi\in X\}. \] For each $n\in\N$, let $\T_n:=\{t\in\T: |a(t)|\le n\}$. Then %%% \begin{equation}\label{eq:mult-closedness-1} \T_n\subset\T_{n+1}, \quad \bigcup\limits_{n=1}^\infty \T_n=\T\setminus\T_\infty, \quad\mbox{where}\quad \T_\infty:=\{t\in\T: a(t)=\infty\}. \end{equation} %%% For every $\varphi\in X$ and $n\in\N$, put $\varphi_n:=\chi_{\T_n}\varphi$. Obviously, \[ |\varphi_n|\le |\varphi|, \quad |a\varphi_n|=|a\chi_{\T_n}\varphi|\le n|\varphi|, \quad n\in\N. \] Hence, by the lattice property (see \cite[Ch.~1, Definition~1.1]{BS88}), $\varphi_n$ and $a\varphi_n$ belong to $X$ whenever $\varphi\in X$ and $n\in\N$. Therefore, $\varphi_n\in \cD(M_a)$ for $n\in\N$. Since $a$ is a.e. finite, we have $m(\T_\infty)=0$. Then $\chi_{\T\setminus\T_n}\to 0$ a.e. on $\T$ as $n\to\infty$. By \cite[Ch.~1, Corollary~4.4]{BS88}, the reflexivity of $X$ implies the absolutely continuity of the norm of $\varphi\in X$. Hence, $\|\varphi-\varphi_n\|_X=\|\chi_{\T\setminus\T_n}\varphi\|_X\to 0$ as $n\to\infty$. This means that $\cD(M_a)$ is dense in $X$. Let us prove that $(M_a)^*=M_{\overline{a}}$. From \cite[Ch.~1, Corollaries~4.3--4.4]{BS88} it follows that $X'=X^*$, where $X'$ is the associate space of $X$ (see \cite[Ch.~1, Section~2]{BS88}). This means that the general form of a functional $G\in X^*$ is given by \[ G(f)=(f,g):=\int_\T f\overline{g}\,dm, \] where $f\in X,g\in X'$ and $\|G\|_{X^*}=\|g\|_{X'}$. Then for all $\varphi\in\cD(M_a)\subset X$ and all $g\in\cD(M_{\overline{a}})\subset X'$, %%% \begin{equation}\label{eq:mult-closedness-2} (M_a\varphi,g) = \int_\T a\varphi\cdot\overline{g}\,dm = \int_\T \varphi\cdot\overline{M_{\overline{a}}g}\,dm = (\varphi, M_{\overline{a}}g). \end{equation} %%% Hence, the operators $M_a$ and $M_{\overline{a}}$ are adjoint to each other. Let us show that any other adjoint to $M_a$ is a restriction of $M_{\overline{a}}$. Assume $T$ is an adjoint to $M_a$, that is, %%% \begin{equation}\label{eq:mult-closedness-3} (M_a\varphi,g)=(\varphi,Tg), \quad \varphi\in \cD(M_a), \quad g\in \cD(T). \end{equation} %%% Above we show that $\chi_{\T_n}\varphi\in\cD(M_a)$ for every $\varphi\in X, n\in\N$. Analogously one can prove that $\chi_{\T_n}g\in\cD(M_{\overline{a}})$ for every $g\in\cD(T)\subset X', n\in\N$. Then from (\ref{eq:mult-closedness-2}) and (\ref{eq:mult-closedness-3}) we get for every $\varphi\in X, g\in\cD(T)$, and $n\in\N$, \[ \int_\T\varphi\chi_{\T_n}\cdot\overline{Tg-M_{\overline{a}}g}\,dm = \Big(\varphi,\chi_{\T_n}(Tg-M_{\overline{a}}g)\Big)=0. \] Therefore, $Tg=M_{\overline{a}}g$ a.e. on $\T_n$ for every $g\in\cD(T)$ and every $n\in\N$. Since $m(\T_\infty)=0$, taking into account (\ref{eq:mult-closedness-1}), we deduce that $Tg=M_{\overline{a}}g$ a.e. on $\T$ for every $g\in\cD(T)$. Thus, $T$ is a restriction of $M_{\overline{a}}$, and $M_{\overline{a}}$ is the adjoint to $M_a$. By \cite[Ch.~1, Corollary~4.4]{BS88}, $(X')^*=X''=X$. In view of just proved, $M_{\overline{a}}$ is densely defined in $X'$ and $(M_{\overline{a}})^*=M_{\overline{\overline{a}}}=M_a$. Since $M_{\overline{a}}$ is densely defined in $X'$, its adjoint $M_a$ is closed in $X$ (see \cite[Ch.~3, Section~5.4]{Kato}). \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:RA-closedness} If $b\in L^\infty$ and $u$ is a measurable a.e. finite function on $\T$, then $R(A),M_uP_+\in \cC(X)$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textit{Proof.} By Lemma~\ref{le:basic}, $M_b,U,P_-,P_+\in\cB(X)$. Lemma~\ref{le:mult-closedness} gives $M_u\in\cC(X)$. By using \cite[Ch.~3, Section~5.2, Problems~5.6--5.7]{Kato}, we subsequently conclude that the operators $M_uU$, $M_uP_+,\ M_b+M_uU,\ (M_b+M_uU)P_-,\ P_++(M_b+M_uU)P_-$ are closed. \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% If $T\in\cC(X)$, then its domain $\cD(T)$ becomes a Banach space with respect to the \textit{graph norm} \[ \|f\|_{\cD(T)}:=\|f\|_X+\|Tf\|_X \] (see, e.g., \cite[Ch.~4, Section~1.1, Remark~1.4]{Kato}). Hence, from Lemma~\ref{le:RA-closedness} it follows that $\cD(R(A))$ equipped with the graph norm is a Banach space. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:domain} If $b\in L^\infty$ and $u$ is a measurable a.e. finite function on $\T$, then %%% \begin{equation}\label{eq:domain} \cD(R(A))=\cD(M_uUP_-)=\cD(P_\pm M_uUP_-)=U\cD(M_uP_+). \end{equation} \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textit{Proof.} By Lemma~\ref{le:basic}, the operators $P_\pm$ and $R(M_b)$ are bounded on $X$. Therefore, using the representation %%% \begin{equation}\label{eq:representation} R(A)=R(M_b)+M_uUP_-, \end{equation} %%% we can easily obtain the equalities $\cD(R(A))=\cD(M_uUP_-)=\cD(P_\pm M_uUP_-)$. The equality $\cD(R(A))=U\cD(M_uP_+)$ is proved as in \cite[Lemma~2.3]{KLLTmn}. \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Invertibility of some auxiliary operators} \begin{lemma}\label{le:invert1} If $b\in L^\infty$ and $u$ is a measurable a.e. finite function on $\T$, then %%% \begin{equation}\label{eq:invert1} I-P_+M_uUP_-:\cD(R(A))\to\cD(R(A)) \end{equation} %%% is a bounded and continuously invertible operator. Its inverse is $I+P_+M_uUP_-$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The proof is straightforward. Consider the usual Hardy spaces $H^p, 1\le p\le\infty$, over $\T$ (see, e.g., \cite{Hoffman}). For a nonnegative function $\Phi$ with $\log\Phi\in L^1$ one can define the corresponding {\it outer function} $\cO_\Phi$ by the formula \[ \cO_\Phi(z) := \exp\left(\int_\T\frac{\tau+z}{\tau-z}\log\Phi(\tau)\,dm(\tau)\right), \quad z\in\D:=\{\zeta\in\C:|\zeta|<1\}. \] It is analytic in $\D$ and has the crucial property that $|\cO_\Phi|=\Phi$ a.e. on $\T$. Moreover, $\cO_\Phi\in H^1$ if and only if $\Phi\in L^1$ (see, e.g., \cite[p.~62]{Hoffman}). For a measurable function $u$ on $\T$, put $\wu:=\max\{1,|u|\}$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{proposition}\label{pr:O-properties} {\rm (a)} If $\log\wu\in L^1$, then $\cO_{1/\wu}\in H^\infty$. {\rm (b)} If $\log\wu\in L^1$, $F\in H^1$, and $\cO_{\wu}F\in L^1$, then $\cO_{\wu}F\in H^1$. {\rm (c)} If $\log\wu\in L^1$ and $\varphi\in\cD(M_uP_+)$, then $\varphi\in\cD(M_{\cO_{\wu}}P_+)$ and %%% \begin{equation}\label{eq:O-properties-1} P_+M_{\cO_{\wu}}P_+\varphi=M_{\cO_{\wu}}P_+\varphi, \quad P_-M_{\cO_{\wu}}P_+\varphi=0. \end{equation} \end{proposition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textit{Proof.} (a) Obviously, $\log\wu\in L^1$ if and only if $\log(1/\wu)\in L^1$. Hence, $\cO_{1/\wu}$ is well defined whenever $\log\wu\in L^1$. From the definition of $\wu$ it follows that $1/\wu\in L^\infty\subset L^1$. Hence, $\cO_{1/\wu}\in L^\infty\cap H^1=H^\infty$. Part (a) is proved. (b) Obviously, %%% \begin{equation}\label{eq:O-properties-2} \cO_{\wu}\cO_{1/\wu}=1. \end{equation} %%% Therefore, $\cO_{\wu}F=F/\cO_{1/\wu}$. The denominator is in $H^1$, in view of Part (a). By the assumption, the numerator is in $H^1$ and the fraction is in $L^1$. Hence, the fraction is in $H^1$, due to \cite[Ch.~5, Exercise~5, p.~75]{Hoffman}. Part (b) is proved. (c) If $\varphi\in\cD(M_uP_+)$, then $M_uP_+\varphi\in X$ and $P_+\varphi\in P_+X\subset H^1$. From properties of outer functions and the definition of $\wu$ it follows that \[ |\cO_{\wu}|=\wu\le 1+|u| \quad\mbox{a.e. on}\quad \T. \] Then $|M_{\cO_{\wu}}P_+\varphi|\le|P_+\varphi|+|M_uP_+\varphi|$ a.e. on $\T$. By the lattice property, %%% \begin{equation}\label{eq:O-properties-3} \|M_{\cO_{\wu}}P_+\varphi\|_X \le \|P_+\varphi\|_X+\|M_uP_+\varphi\|_X. \end{equation} %%% Therefore, $M_{\cO_{\wu}}P_+\varphi\in X\subset L^1$. Applying Part (b) to $F:=P_+\varphi$, we obtain $M_{\cO_{\wu}}P_+\varphi\in X\cap H^1=P_+X$. From the latter inclusion we get (\ref{eq:O-properties-1}). \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:invert2} If $b\in L^\infty$ and $\log\wu\in L^1$, then the operator %%% \begin{equation}\label{eq:invert2-1} R(UM_{\cO_{1/\wu}}U):X\to\cD(R(A)) \end{equation} %%% is bounded and continuously invertible. Its inverse is %%% \begin{equation}\label{eq:invert2-2} R(UM_{\cO_{\wu}}U):\cD(R(A))\to X. \end{equation} \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textit{Proof.} The idea of the proof is borrowed from the proof of \cite[Lemma~2.6]{KLLTmn}. If $\log\wu\in L^1$, then $u$ is a measurable and a.e. finite function. Hence, by Lemma~\ref{le:RA-closedness}, $R(A)$ is closed. Therefore, $\cD(R(A))$ is a Banach space with respect to the graph norm. By Proposition~\ref{pr:O-properties}(a), $\cO_{1/\wu}\in H^\infty$. Then $M_{\cO_{1/\wu}}, R(UM_{\cO_{1/\wu}}U)\in\cB(X)$. %%% By using (\ref{eq:basic}), (\ref{eq:representation}), and $\cO_{1/\wu}\in H^\infty$, we get for $\varphi\in X$, %%% \begin{equation}\label{eq:invert2-3} R(A)R(UM_{\cO_{1/\wu}}U)\varphi = R(M_b)R(UM_{\cO_{1/\wu}}U)\varphi + M_uM_{\cO_{1/\wu}}UP_-\varphi. \end{equation} %%% Since $b\in L^\infty$, we have $R(M_b)\in\cB(X)$, in view of Lemma~\ref{le:basic}. On the other hand, from properties of outer functions and the definition of $\wu$ we get $|u\cO_{1/\wu}|=|u|/\wu\le 1$ a.e. on $\T$. Therefore, by the lattice property (see \cite[Ch.~1, Definition~1.1]{BS88}), %%% \begin{equation}\label{eq:invert2-4} \|M_uM_{\cO_{1/\wu}}\|_{\cB(X)}\le 1. \end{equation} %%% Thus, from (\ref{eq:invert2-3}) and (\ref{eq:invert2-4}) we deduce that the operator (\ref{eq:invert2-1}) is bounded and %%% \begin{equation}\label{eq:invert2-5} \im R(UM_{\cO_{1/\wu}}U)\subset\cD(R(A)). \end{equation} %%% Let us show that %%% \begin{equation}\label{eq:invert2-6} \cD(R(A))\subset\im R(UM_{\cO_{1/\wu}}U). \end{equation} %%% Assume that $f\in\cD(R(A))\subset X$. By Lemma~\ref{le:domain}, $Uf\in\cD(M_uP_+)$. Therefore, by Proposition~\ref{pr:O-properties}(c) and (\ref{eq:basic}), we get %%% \begin{equation}\label{eq:invert2-7} P_+UM_{\cO_{\wu}}UP_-f =0, \quad P_-UM_{\cO_{\wu}}UP_-f = UM_{\cO_{\wu}}UP_-f. \end{equation} %%% Put $\varphi:=P_+f+UM_{\cO_{\wu}}UP_-f$. Then from (\ref{eq:basic}), (\ref{eq:O-properties-2}), and (\ref{eq:invert2-7}) we obtain \[ R(UM_{\cO_{1/\wu}}U)\varphi = P_+f+UM_{\cO_{1/\wu}}U^2M_{\cO_{\wu}}UP_-f=f. \] Therefore, $f\in\im R(UM_{\cO_{1/\wu}}U)$. Hence, (\ref{eq:invert2-6}) holds and %%% \begin{equation}\label{eq:invert2-8} R(UM_{\cO_{1/\wu}}U) R(UM_{\cO_{\wu}}U)f=f, \quad f\in\cD(R(A)). \end{equation} %%% Analogously to (\ref{eq:invert2-8}), by using Proposition~\ref{pr:O-properties}(a), one can show that %%% \begin{equation}\label{eq:invert2-9} R(UM_{\cO_{\wu}}U) R(UM_{\cO_{1/\wu}}U)\varphi=\varphi, \quad \varphi\in X. \end{equation} %%% Equalities (\ref{eq:invert2-8}) and (\ref{eq:invert2-9}) imply that the operator (\ref{eq:invert2-1}) is invertible and its inverse is given by (\ref{eq:invert2-2}). Moreover, we have shown that the operator (\ref{eq:invert2-1}) is bounded (and, therefore, it is closed) and its image coincides with the range space (see (\ref{eq:invert2-5}) and (\ref{eq:invert2-6})). From above and the closed graph theorem (see, e.g., \cite[Ch.~3, Section~5.4, Problem~5.21]{Kato}) it follows that the operator (\ref{eq:invert2-2}) is bounded. \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Fredholm theory}\label{sec:Fredholmness} \subsection{Relative compactness of Hankel operators} By analogy with \cite{KLLTmn} define the class \[ \A :=\Big\{ u\mbox{ is measurable}:\quad \log\wu\in L^1, \quad u\cO_{1/\wu}\in H^\infty+C \Big\}. \] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:rel-compactness} If $u\in\A$, then the operator $P_-M_uP_+$ is relatively compact with respect to the operator $M_uP_+$. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textit{Proof.} This lemma is proved by analogy with \cite[Lemma~2.1]{KLLTmn}. Since $P_-$ is bounded on $X$, we have $\cD(M_uP_+)=\cD(P_-M_uP_+)$. Let $\{\varphi_n\}\subset \cD(M_uP_+)$ be a bounded sequence in $X$ such that the sequence $\{M_uP_+\varphi_n\}$ is also bounded in $X$. Since $P_+$ is bounded on $X$, the sequence $\{P_+\varphi_n\}$ is bounded too. Put $\psi_n:=M_{\cO_{\wu}}P_+\varphi_n$. From (\ref{eq:O-properties-3}) it follows that \[ \|\psi_n\|_X \le \|P_+\varphi_n\|_X+\|M_uP_+\varphi_n\|_X\le C \] for some constant $C>0$ and all $n\in\N$. Thus, $\{\psi_n\}$ is bounded in $X$. Taking into account (\ref{eq:O-properties-1}) and (\ref{eq:O-properties-2}), we obtain %%% \begin{equation}\label{eq:rel-compactness-1} P_-M_uP_+\varphi_n = P_-M_{u\cO_{1/\wu}}P_+M_{\cO_{\wu}}P_+\varphi_n = P_-M_{u\cO_{1/\wu}}P_+\psi_n. \end{equation} %%% Since $u\in\A$, we have $u\cO_{1/\wu}\in H^\infty+C$. By \cite[Theorem~2.54]{BS90}, the operator $P_-M_{u\cO_{1/\wu}}P_+$ is compact on $L^p$ for every $p\in(1,\infty)$. Hence, by \cite[Corollary~1]{Sharp73}, it is compact on $X$. This means that, for the bounded sequence $\{\psi_n\}\subset X$, the corresponding sequence $\{P_-M_{u\cO_{1/\wu}}P_+\psi_n\}$ contains a convergent subsequence. From (\ref{eq:rel-compactness-1}) we deduce that the sequence $\{P_-M_uP_+\varphi_n\}$ contains a convergent subsequence. Thus, the operator $P_-M_uP_+$ is relatively compact with respect to the operator $M_uP_+$. \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Fredholm criteria} \begin{lemma}\label{le:fred1} If $b\in L^\infty$ and $\log\wu\in L^1$, then {\rm (a)} the operators %%% \begin{eqnarray} && R(M_b) : \cD(R(A))\to X, \label{eq:fred1-1} \\[3mm] && R(M_bUM_{\cO_{1/\wu}}U) : X\to X \label{eq:fred1-2} \end{eqnarray} %%% are bounded; {\rm (b)} the operator {\rm (\ref{eq:fred1-1})} is Fredholm if and only if the operator {\rm (\ref{eq:fred1-2})} is Fredholm. If one of these operators is Fredholm, then their indices coincide. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textit{Proof.} This statement is proved similarly to \cite[Lemma~2.7]{KLLTmn}. By analogy with (\ref{eq:invert2-8}) one can prove that %%% \begin{equation}\label{eq:fred1-3} R(M_bUM_{\cO_{1/\wu}}U)R(UM_{\cO_{\wu}}U)\varphi =R(M_b)\varphi, \quad \varphi\in \cD(R(A)). \end{equation} %%% (a) By Proposition~\ref{pr:O-properties}(a), $M_{\cO_{1/\wu}}\in\cB(X)$. Due to Lemma~\ref{le:basic}, $M_b\in\cB(X)$. Thus, the operator (\ref{eq:fred1-2}) is bounded. On the other hand, by Lemma~\ref{le:invert2}, the operator %%% \begin{equation}\label{eq:fred1-4} R(UM_{\cO_{\wu}}U):\cD(R(A))\to X \end{equation} %%% is also bounded. From above and (\ref{eq:fred1-3}) we infer that the operator (\ref{eq:fred1-1}) is bounded too. Part (a) is proved. (b) By Lemma~\ref{le:invert2}, the operator (\ref{eq:fred1-4}) is invertible. Therefore, by \cite[Ch.~4, Theorem~6.1]{gk}, from (\ref{eq:fred1-3}) we conclude that the operators (\ref{eq:fred1-1}) and (\ref{eq:fred1-2}) are Fredholm only simultaneously and their indices coincide. \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{lemma}\label{le:fred2} If $b\in L^\infty$ and $u\in\A$, then the operator %%% \begin{equation}\label{eq:fred2-1} R(A):\cD(R(A))\to X \end{equation} %%% is Fredholm if and only if the operator {\rm (\ref{eq:fred1-1})} is Fredholm. If one of these operators is Fredholm, then their indices coincide. \end{lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textit{Proof.} This statement is proved by analogy with \cite[Lemmas~2.4 and~2.5]{KLLTmn}. By Lemma~\ref{le:rel-compactness}, the operator $P_-M_uP_+$ is relatively compact with respect to the operator $M_uP_+$. From this fact and the definition of relative compactness (see \cite[Ch.~4, Section~1.3]{Kato}) one can easily obtain that the operator $P_-M_uUP_-=P_-M_uP_+U$ is relatively compact with respect to the operator $M_uP_+U=M_uUP_-$. Due to Lemma~\ref{le:basic}, $R(M_b)$ is bounded on $X$. Therefore, it is not difficult to prove that the operator $P_-M_uUP_-$ is relatively compact with respect to the operator $R(A)=R(M_b)+M_uUP_-$. Hence, the operator $P_-M_uUP_-$ is also relatively bounded with respect to the operator $R(A)$ (see \cite[Ch.~4, Section~1.3]{Kato}). Thus, the operator $P_-M_uUP_-:\cD(R(A))\to X$ becomes bounded and compact (see \cite[Ch.~4, Section~1.1, Remark~1.4 and Section~1.3, Remark~1.12]{Kato}). By \cite[Ch.~4, Theorem~6.3]{gk}, the operator (\ref{eq:fred2-1}) is Fredholm if and only if the operator %%% \begin{equation}\label{eq:fred2-2} R(A)-P_-M_uUP_-:\cD(R(A))\to X \end{equation} %%% is Fredholm and the indices of (\ref{eq:fred2-1}) and (\ref{eq:fred2-2}) coincide. From (\ref{eq:basic}), (\ref{eq:domain}), and (\ref{eq:representation}) we get for any $\varphi\in\cD(R(A))$, %%% \begin{equation}\label{eq:fred2-3} (R(A)-P_-M_uUP_-)(I-P_+M_uUP_-)\varphi=R(M_b)\varphi. \end{equation} %%% In view of Lemma~\ref{le:invert1}, the operator (\ref{eq:invert1}) is bounded and continuously invertible. From above and (\ref{eq:fred2-3}) it follows that the operator (\ref{eq:fred2-2}) (and thus, the operator (\ref{eq:fred2-1})) is Fredholm if and only if the operator (\ref{eq:fred1-1}) is Fredholm. In that case the indices of the operators (\ref{eq:fred2-1}), (\ref{eq:fred2-2}), and (\ref{eq:fred1-1}) coincide (see \cite[Ch.~4, Theorem~6.1]{gk}). \rule{2mm}{2mm} Now we are in a position to prove the main result of this paper. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{theorem}\label{th:main} If $b\in L^\infty$ and $u\in\A$, then the operator {\rm (\ref{eq:fred2-1})} is Fredholm if and only if $u\in L^\infty$ and the operator %%% \begin{equation}\label{eq:main} R(M_b):X\to X \end{equation} %%% is Fredholm. If one of these operators is Fredholm, then their indices coincide. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textit{Proof.} The proof is developed by analogy with \cite[Theorem~2.8]{KLLTmn}. \textit{Necessity}. By Lemma~\ref{le:fred1}(a), the operator (\ref{eq:fred1-1}) is bounded. If the operator (\ref{eq:fred2-1}) is Fredholm, then the operator (\ref{eq:fred1-1}) is Fredholm, due to Lemma~\ref{le:fred2}. Therefore, by Lemma~\ref{le:fred1}, the operator (\ref{eq:fred1-2}) is bounded and Fredholm. Hence, by \cite[Theorem~6.8]{K98}, \[ \operatornamewithlimits{ess\,inf}_{t\in\T} \left|b(t)\cO_{1/\wu}(1/t)\right|>0. \] Then \[ 0<\operatornamewithlimits{ess\,inf}_{t\in\T} \left|\cO_{1/\wu}(1/t)\right| = \operatornamewithlimits{ess\,inf}_{t\in\T}\Big(1/\widetilde{u}(t)\Big) = \Big(\operatornamewithlimits{ess\,sup}_{t\in\T}\widetilde{u}(t)\Big)^{-1}. \] The latter condition is obviously equivalent to $\widetilde{u}\in L^\infty$. Therefore, $u\in L^\infty$. In this case $R(A)$ is bounded on $X$. Thus, $\cD(R(A))=X$. So, the operator (\ref{eq:main}) is Fredholm. The necessity part is proved. \textit{Sufficiency.} Assume that $u\in L^\infty$ and the operator (\ref{eq:main}) is Fredholm. Then the first condition guarantees that $R(A)\in\cB(X)$, and so, $\cD(R(A))=X$. This means that the operator (\ref{eq:main}) coincide with the operator (\ref{eq:fred1-1}). By Lemma~\ref{le:fred2}, the operator (\ref{eq:fred2-1}) is Fredholm and its index coincide with the index of the operator (\ref{eq:main}). \rule{2mm}{2mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Theorem~\ref{th:main} generalizes \cite[Theorem~2.8]{KLLTmn}. It is new even for Lebesgue spaces $L^p, 1