Oleksiy Karlovych

Suppose \(\alpha\) is an orientation preserving diffeomorphism (shift) of \(\mathbb{R}_+=(0,\infty)\) onto itself with the only fixed points \(0\) and \(\infty\). We establish sufficient conditions for the Fredholmness of the singular integral operator with shift \[ (aI-bW_\alpha)P_++(cI-dW_\alpha)P_- \] acting on \(L^p(\mathbb{R}_+)\) with \( 1 < p < \infty \), where \(P_\pm=(I\pm S)/2\), \(S\) is the Cauchy singular integral operator, and \(W_\alpha f=f\circ\alpha\) is the shift operator, under the assumptions that the coefficients \(a,b,c,d\) and the derivative \(\alpha'\) of the shift are bounded and continuous on \(\mathbb{R}_+\) and may admit discontinuities of slowly oscillating type at \(0\) and \(\infty\).

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

In this paper, the author proves that the set of all integrable functions whose sequences of negative (resp. nonnegative) Fourier coefficients belong to \(\ell^1\cap\ell^\Phi_{\varphi,w}\) (resp. to \(\ell^1\cap\ell^\Psi_{\psi,\varrho}\)), where \(\ell^\Phi_{\varphi,w}\) and \(\ell^\Psi_{\psi,\varrho}\) are two-weighted Orlicz sequence spaces, forms an algebra under pointwise multiplication whenever the weight sequences
\[
\varphi=\{\varphi_n\},\quad
\psi=\{\psi_n\},\quad
w=\{w_n\},\quad
\varrho=\{\varrho_n\}
\]
increase and satisfy the \(\Delta_2\)-condition.

The paper is devoted to some only recently uncovered phenomena emerging in the study of singular integral operators (SIO's) with piecewise continuous (PC) coefficients in reflexive rearrangement-invariant spaces over Carleson curves. We deal with several kinds of indices of submultiplicative functions which describe properties of spaces (Boyd and Zippin indices) and curves (spirality indices). We consider some ``disintegration condition{''} which combines properties of spaces and curves, the Boyd and spirality indices. We show that the essential spectrum of SIO associated with the Riemann boundary value problem with PC coefficient arises from the essential range of the coefficient by filling in certain massive connected sets (so-called logarithmic leaves) between the endpoints of jumps. These results combined with the Allan-Douglas local principle and with the two projections theorem enable us to study the Banach algebra \(\mathfrak{A}\) generated by SIO's with matrix-valued piecewise continuous coefficients. We construct a symbol calculus for this Banach algebra which provides a Fredholm criterion and gives a basis for an index formula for arbitrary SIO's from \(\mathfrak{A}\) in terms of their symbols.

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 \(H^\infty+C\).

We prove a Fredholm criterion for operators in the Banach algebra of singular integral operators with matrix piecewise continuous coefficients acting on a variable Lebesgue space with a radial oscillating weight over a logarithmic Carleson curve. The local spectra of these operators are massive and have a shape of spiralic horns depending on the value of the variable exponent, the spirality indices of the curve, and the Matuszewska-Orlicz indices of the weight at each point. These results extend (partially) the results of A. Böttcher, Yu. Karlovich, and V. Rabinovich for standard Lebesgue spaces to the case of variable Lebesgue spaces.

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.

In 1968, Gohberg and Krupnik found a Fredholm criterion for singular integral operators of the form \(aP+bQ\), where \(a,b\) are piecewise continuous functions and \(P,Q\) are complementary projections associated to the Cauchy singular integral operator, acting on Lebesgue spaces over Lyapunov curves. We extend this result to the case of Nakano spaces (also known as variable Lebesgue spaces) with certain weights having finite sets of discontinuities on arbitrary Carleson curves.

Suppose \(\Gamma\) is a Carleson Jordan curve with logarithmic whirl points, \(\varrho\) is a Khvedelidze weight, \(p:\Gamma\to(1,\infty)\) is a continuous function satisfying \(|p(\tau)-p(t)|\le -\mathrm{const}/\log|\tau-t|\) for \(|\tau-t|\le 1/2\), and \(L^{p(\cdot)}(\Gamma,\varrho)\) is a weighted generalized Lebesgue space with variable exponent. We prove that all semi-Fredholm operators in the algebra of singular integral operators with \(N\times N\) matrix piecewise continuous coefficients are Fredholm on \(L_N^{p(\cdot)}(\Gamma,\varrho)\).

Recently V. Kokilashvili, N. Samko, and S. Samko have proved a sufficient condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with radial oscillating weights over Carleson curves. This condition is formulated in terms of Matuszewska-Orlicz indices of weights. We prove a partial converse of their result.

Let \(\mathcal{M}(\mathbb{R}^n)\) be the class of bounded away from one and infinity functions \(p:\mathbb{R}^n\to[1,\infty]\) such that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space \(L^{p(\cdot)}(\mathbb{R}^n)\). We show that if \(a\) belongs to the Hörmander class \(S_{\rho,\delta}^{n(\rho-1)}\) with \(0<\rho\le 1\), \(0\le\delta<1\), then the pseudodifferential operator \(\operatorname{Op}(a)\) is bounded on the variable Lebesgue space \(L^{p(\cdot)}(\mathbb{R}^n)\) provided that \(p\in\mathcal{M}(\mathbb{R}^n)\). Let \(\mathcal{M}^*(\mathbb{R}^n)\) be the class of variable exponents \(p\in\mathcal{M}(\mathbb{R}^n)\) represented as \(1/p(x)=\theta/p_0+(1-\theta)/p_1(x)\) where \(p_0\in(1,\infty)\), \(\theta\in(0,1)\), and \(p_1\in\mathcal{M}(\mathbb{R}^n)\). We prove that if \(a\in S_{1,0}^0\) slowly oscillates at infinity in the first variable, then the condition \[ \lim_{R\to\infty}\inf_{|x|+|\xi|\ge R}|a(x,\xi)|>0 \] is sufficient for the Fredholmness of \(\operatorname{Op}(a)\) on \(L^{p(\cdot)}(\mathbb{R}^n)\) whenever \(p\in\mathcal{M}^*(\mathbb{R}^n)\). Both theorems generalize pioneering results by Rabinovich and Samko [RS08] obtained for globally log-Hölder continuous exponents \(p\), constituting a proper subset of \(\mathcal{M}^*(\mathbb{R}^n)\).