Oleksiy Karlovych

We study the boundedness of Fourier convolution operators \(W^0(b)\) and the compactness of commutators of \(W^0(b)\) with multiplication operators \(aI\) on some Banach function spaces \(X(\mathbb{R})\) for certain classes of piecewise quasicontinuous functions \(a\in PQC\) and piecewise slowly oscillating Fourier multipliers \(b\in PSO_{X,1}^\diamond\). We suppose that \(X(\mathbb{R})\) is a separable rearrangement-invariant space with nontrivial Boyd indices or a reflexive variable Lebesgue space, in which the Hardy-Littlewood maximal operator is bounded. Our results complement those of Isaac De La Cruz-Rodríguez, Yuri Karlovich, and Iván Loreto Hernández obtained for Lebesgue spaces with Muckenhoupt weights.

Let \(\alpha\) and \(\beta\) be orientation-preserving diffeomorphism (shifts) of \(\mathbb{R}_+=(0,\infty)\) onto itself with the only fixed points \(0\) and \(\infty\), where the derivatives \(\alpha'\) and \(\beta'\) may have discontinuities of slowly oscillating type at \(0\) and \(\infty\). For \(p\in(1,\infty)\), we consider the weighted shift operators \(U_\alpha\) and \(U_\beta\) given on the Lebesgue space \(L^p(\mathbb{R}_+)\) by \(U_\alpha f=(\alpha')^{1/p}(f\circ\alpha)\) and \(U_\beta f=(\beta')^{1/p}(f\circ\beta)\). For \(i,j\in\mathbb{Z}\) we study the simplest weighted singular integral operators with two shifts \(A_{ij}=U_\alpha^i P_\gamma^++U_\beta^j P_\gamma^-\) on \(L^p(\mathbb{R}_+)\), where \(P_\gamma^\pm=(I\pm S_\gamma)/2\) are operators associated to the weighted Cauchy singular integral operator \[ (S_\gamma f)(t)=\frac{1}{\pi i}\int_{\mathbb{R}_+} \left(\frac{t}{\tau}\right)^\gamma\frac{f(\tau)}{\tau-t}d\tau \] with \(\gamma\in\mathbb{C}\) satisfying \(0<1/p+\Re\gamma<1\). We prove that the operator \(A_{ij}\) is a Fredholm operator on \(L^p(\mathbb{R}_+)\) and has zero index if \[ 0<\frac{1}{p}+\Re\gamma+\frac{1}{2\pi}\inf_{t\in\mathbb{R}_+}(\omega_{ij}(t)\Im\gamma), \quad \frac{1}{p}+\Re\gamma+\frac{1}{2\pi}\sup_{t\in\mathbb{R}_+}(\omega_{ij}(t)\Im\gamma)<1, \] where \(\omega_{ij}(t)=\log[\alpha_i(\beta_{-j}(t))/t]\) and \(\alpha_i\), \(\beta_{-j}\) are iterations of \(\alpha\), \(\beta\). This statement extends an earlier result obtained by the author, Yuri Karlovich, and Amarino Lebre for \(\gamma=0\).

We prove that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space \(X(\mathbb{R}^n)\) and on its associate space \(X'(\mathbb{R}^n)\) and a maximally modulated Calderón-Zygmund singular integral operator \(T^{\Phi}\) is of weak type \((r,r)\) for all \(r\in(1,\infty)\), then \(T^{\Phi}\) extends to a bounded operator on \(X(\mathbb{R}^n)\). This theorem implies the boundedness of the maximally modulated Hilbert transform on variable Lebesgue spaces \(L^{p(\cdot)}(\mathbb{R})\) under natural assumptions on the variable exponent \(p:\mathbb{R}\to(1,\infty)\). Applications of the above result to the boundedness and compactness of pseudodifferential operators with \(L^\infty(\mathbb{R},V(\mathbb{R}))\)-symbols on variable Lebesgue spaces \(L^{p(\cdot)}(\mathbb{R})\) are considered. Here the Banach algebra \(L^\infty(\mathbb{R},V(\mathbb{R}))\) consists of all bounded measurable \(V(\mathbb{R})\)-valued functions on \(\mathbb{R}\) where \(V(\mathbb{R})\) is the Banach algebra of all functions of bounded total variation.

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

We compute the generalized Cauchy index of some semi-almost periodic functions, which are important
in the study of the Fredholm index of singular integral operators with shifts and slowly oscillating data.

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.