Oleksiy Karlovych

We find Fredholm criteria and a formula for the index of an arbitrary operator in the Banach algebra of singular integral operators with piecewise continuous coefficients on Nakano spaces (generalized Lebesgue spaces with variable exponent) with Khvedelidze weights over either Lyapunov curves or Radon curves without cusps. These results ``localize'' the Gohberg-Krupnik Fredhohn theory with respect to the variable exponent.

We prove asymptotic formulas for block Toeplitz matrices with symbols admitting right and left Wiener-Hopf factorizations such that all partial indices are equal to some integer number. We consider symbols and Wiener-Hopf factorizations in Wiener algebras with weights satisfying natural submultiplicativity, monotonicity, and regularity conditions. Our results complement known formulas for Holder continuous symbols due to Bottcher and Silbermann.

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 prove necessary conditions for the Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces. These conditions are formulated in terms of indices of submultiplicative functions associated with local properties of the space, of the curve, and of the weight. As an example, we consider weighted Nakano spaces \(L^{p(\cdot)}_w\) (weighted Lebesgue spaces with variable exponent). Moreover, our necessary conditions become also sufficient for weighted Nakano spaces over nice curves whenever \(w\) is a Khvedelidze weight, and the variable exponent \(p(t)\) satisfies the estimate \(|p(\tau)-p(t)|\le A/(-\log|\tau-t|)\).

We prove criteria for the invertibility of the binomial functional operator
\[
A=aI-bW_\alpha
\]
in the Lebesgue spaces \(L^p(0,1)\), \( 1 < p < \infty\), where \(a\) and \(b\) are continuous functions on \((0,1)\), \(I\) is the identity operator, \(W_\alpha\) is the shift operator, \(W_\alpha f=f\circ\alpha\), generated by a non-Carleman shift \(\alpha:[0,1]\to[0,1]\) which has only two fixed points \(0\) and \(1\). We suppose that \(\log\alpha'\) is bounded and continuous on \((0,1)\) and that \(a,b,\alpha'\) slowly oscillate at \(0\) and \(1\). The main difficulty connected with slow oscillation is overcome by using the method of limit operators.

We prove sufficient conditions for the boundedness of the maximal operator on variable Lebesgue spaces with weights \(\varphi_{t,\gamma}(\tau)=|(\tau-t)^\gamma|\), where \(\gamma\) is a complex number, over arbitrary Carleson curves. If the curve has different spirality indices at the point $t$ and \(\gamma\) is not real, then \(\varphi_{t,\gamma}\) is an oscillating weight lying beyond the class of radial oscillating weights considered recently by V. Kokilashvili, N. Samko, and S. Samko.

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 ``disintegration condition'' 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 ``disintegration condition'' is satisfied. One of such spaces is a Lebesgue space with a general Muckenhoupt weight over an arbitrary Carleson curve.

We show that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space \(X(\mathbb{R}^n)\) and on its associate space \(X'(\mathbb{R}^n)\), then a pseudodifferential operator \(\operatorname{Op}(a)\) is bounded on \(X(\mathbb{R}^n)\) whenever the symbol \(a\) belongs to the Hörmander class \(S_{\rho,\delta}^{n(\rho-1)}\) with \(0<\rho\le 1\), \(0\le\delta<1\) or to the the Miyachi class \(S_{\rho,\delta}^{n(\rho-1)}(\varkappa,n)\) with \(0\le\delta\le\rho\le 1\), \(0\le\delta<1\), and \(\varkappa>0\). This result is applied to the case of variable Lebesgue spaces \(L^{p(\cdot)}(\mathbb{R}^n)\).

We prove a higher order asymptotic formula for traces of finite block Toeplitz matrices with symbols belonging to Hölder-Zygmund spaces. The remainder in this formula goes to zero very rapidly for very smooth symbols. This formula refines previous asymptotic trace formulas by Szegő and Widom and complement higher order asymptotic formulas for determinants of finite block Toeplitz matrices due to Böttcher and Silbermann.

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 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 \(C=1\).

We prove the Stechkin inequality for Fourier multipliers on variable Lebesgue spaces under some natural assumptions on variable exponents.

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 \(C^\infty(\mathbb{R}_+\times\mathbb{R})\).

In 1968, Israel Gohberg and Naum Krupnik discovered that local spectra of singular integral operators with piecewise continuous coefficients on Lebesgue spaces \(L^p(\Gamma)\) over Lyapunov curves have the shape of circular arcs. About 25 years later, Albrecht Böttcher and Yuri Karlovich realized that these circular arcs metamorphose to so-called logarithmic leaves with a median separating point when Lyapunov curves metamorphose to arbitrary Carleson curves. We show that this result remains valid in a more general setting of variable Lebesgue spaces \(L^{p(\cdot)}(\Gamma)\) where \(p:\Gamma\to(1,\infty)\) satisfies the Dini-Lipschitz condition. One of the main ingredients of the proof is a new condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with weights related to oscillations of Carleson curves.

We consider singular integral operators with piecewise continuous coefficients on reflexive Orlicz spaces \(L_M(\Gamma)\), which are generalizations of the Lebesgue spaces \(L_p(\Gamma)\), \(1 < p < \infty\). We suppose that \(\Gamma\) belongs to a large class of Carleson curves, including curves with corners and cusps as well as curves that look locally like two logarithmic spirals scrolling up at the same point. For the singular integral operator associated with the Riemann boundary value problem with a piecewise continuous coefficient \(G\), we establish a Fredholm criterion and an index formula in terms of the essential range of \(G\) complemented by spiralic horns depending on the Boyd indices of \(L_M(\Gamma)\) and contour properties. Our main result is a symbol calculus for the closed algebra of singular integral operators with piecewise continuous matrix-valued coefficients on \(L_M^n(\Gamma)\).