Oleksiy Karlovych

Let \(\alpha,\beta\) be orientation-preserving diffeomorphism (shifts) of \(\mathbb{R}_+=(0,\infty)\) onto itself with the only fixed points \(0\) and \(\infty\) and \(U_\alpha,U_\beta\) be the isometric shift operators on \(L^p(\mathbb{R}_+)\) given by \(U_\alpha f=(\alpha')^{1/p}(f\circ\alpha)\), \(U_\beta f=(\beta')^{1/p}(f\circ\beta)\), and \(P_2^\pm=(I\pm S_2)/2\) where \[ (S_2 f)(t):=\frac{1}{\pi i}\int\limits_0^\infty \left(\frac{t}{\tau}\right)^{1/2-1/p}\frac{f(\tau)}{\tau-t}\,d\tau, \quad t\in\mathbb{R}_+, \]
is the weighted Cauchy singular integral operator. We prove that if \(\alpha',\beta'\) and \(c,d\) are continuous on \(\mathbb{R}_+\) and slowly oscillating at \(0\) and \(\ infty\), and \[ \limsup_{t\to s}|c(t)|<1,\quad \limsup_{t\to s}|d(t)|<1, \quad s\in\{0,\infty\}, \] then the operator \((I-cU_\alpha)P_2^++(I-dU_\beta)P_2^-\) is Fredholm on \(L^p(\mathbb{R}_+)\) and its index is equal to zero. Moreover, its regularizers are described.

We prove Szegö's strong limit theorem for Toeplitz determinants with a symbol having a nonstandard smoothness. We assume that the symbol belongs to the Wiener algebra and, moreover, the sequences of Fourier coefficients of the symbol with negative and nonnegative indices belong to weighted Orlicz classes generated by complementary \(N\)-functions both satisfying the \(\Delta_2^0\)-condition and by weight sequences satisfying some regularity, and compatibility conditions.

We study Mellin pseudodifferential operators (shortly, Mellin PDO's) with symbols in the algebra \(\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))\) of slowly oscillating functions of limited smoothness introduced in [K09]. We show that if \(\mathfrak{a}\in\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))\) does not degenerate on the ``boundary" of \(\mathbb{R}_+\times\mathbb{R}\) in a certain sense, then the Mellin PDO \(\operatorname{Op}(\mathfrak{a})\) is Fredholm on the space \(L^p\) for \(p\in(1,\infty)\) and each its regularizer is of the form \(\operatorname{Op}(\mathfrak{b})+K\) where \(K\) is a compact operator on \(L^p\) and \(\mathfrak{b}\) is a certain explicitly constructed function in the same algebra \(\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))\) such that \(\mathfrak{b}=1/\mathfrak{a}\) on the ``boundary" of \(\mathbb{R}_+\times\mathbb{R}\). This result complements the known Fredholm criterion from [K09] for Mellin PDO's with symbols in the closure of \(\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))\) and extends the corresponding result by V.S. Rabinovich (see [R98]) on Mellin PDO's with slowly oscillating symbols in \(C^\infty(\mathbb{R}_+\times\mathbb{R})\).

Let \(a\) be a semi-almost periodic matrix function with the almost periodic representatives \(a_l\) and \(a_r\) at \(-\infty\) and \(+\infty\), respectively. Suppose \(p:\mathbb{R}\to(1,\infty)\) is a slowly oscillating exponent such that the Cauchy singular integral operator \(S\) is bounded on the variable Lebesgue space \(L^{p(\cdot)}(\mathbb{R})\). We prove that if the operator \(aP+Q\) with \(P=(I+S)/2\) and \(Q=(I-S)/2\) is Fredholm on the variable Lebesgue space \(L_N^{p(\cdot)}(\mathbb{R})\), then the operators \(a_lP+Q\) and \(a_rP+Q\) are invertible on standard Lebesgue spaces \(L_N^{q_l}(\mathbb{R})\) and \(L_N^{q_r}(\mathbb{R})\) with some exponents \(q_l\) and \(q_r\) lying in the segments between the lower and the upper limits of \(p\) at \(-\infty\) and \(+\infty\), respectively.

In this paper we extend necessary conditions for Fredholmness of singular integral operators with piecewise continuous coefficients in rearrangement-invariant spaces [K98] to the weighted case \(X(\Gamma,w)\). These conditions are formulated in terms of indices \(\alpha(Q_tw)\) and \(\beta(Q_tw)\) of a submultiplicative function \(Q_tw\), which is associated with local properties of the space, of the curve, and of the weight at the point \(t\Gamma\). Using these results we obtain a lower estimate for the essential norm \(S\) of the Cauchy singular integral operator \(S\) in reflexive weighted rearrangement-invariant spaces \(X(\Gamma, w)\) over arbitrary Carleson curves \(\Gamma\):
\[
|S|\ge\cot(\pi\lambda_{\Gamma,w}/2)
\]
where \(\lambda_{\Gamma,w} :=inf_{t\in\Gamma} min\{\alpha(Q_tw), 1 - \beta(Q_tw)\}\). In some cases we give formulas for computation of \(\alpha(Q_tw)\) and \(\beta(Q_tw)\).

In this paper we deal with the interpolation from Lebesgue spaces \(L^p\) and \(L^q\), into an Orlicz space \(L^\varphi\), where \( 1 \le p < q \le \infty \) and \(\varphi^{-1}(t)=t^{1/p}\rho(t^{1/q-1/p})\) for some concave function \(\rho\), with the special attention to the interpolation constant \(C\). For a bounded linear operator \(T\) in \(L^p\) and \(L^q\), we prove modular inequalities, which allow us to get the estimate, for both, the Orlicz norm and the Luxemburg norm,
\[
\|T\|_{L^\varphi\to L^\varphi}
\le C\max\Big\{
\|T\|_{L^p\to L^p},
\|T\|_{L^q\to L^q}
\Big\},
\]
where the interpolation constant \(C\) depends only on \(p\) and \(q\). We give estimates for \(C\), which imply \(C<4\). Moreover, if either \( 1 < p < q \le 2\) or \( 2 \le p < q < \infty\), then \(C< 2\). If \(q=\infty\), then \(C\le 2^{1-1/p}\), and, in particular, for the case \(p=1\) this gives the classical Orlicz interpolation theorem with the constant \(C=1\).

Let \(\Gamma\) be an oriented Jordan smooth curve and \(\alpha\) a diffeomorphism of $\Gamma$ onto itself which has an arbitrary nonempty set of periodic points. We prove criteria for one-sided invertibility of the binomial functional operator \(A=aI-bW\) where $a$ and $b$ are continuous functions, \(I\) is the identity operator, \(W\) is the shift operator, \(Wf=f\circ\alpha\), on a reflexive rearrangement-invariant space \(X(\Gamma)\) with Boyd indices \(\alpha_X,\beta_X\) and Zippin indices \(p_X,q_X\) satisfying inequalities
\[
0<\alpha_X=p_X\le q_X=\beta_X<1.
\]