Karlovich, Alexei Yu., and Ilya M. Spitkovsky. "
Pseudodifferential operators on variable Lebesgue spaces."
Operator Theory, Pseudo-Differential Equations, and Mathematical Physics. Operator Theory: Advances and Applications, 228. Eds. Yuri I. Karlovich, Luigi Rodino, Bernd Silbermann, and Ilya M. Spitkovsky. Basel: Birkhäuser, 2013. 173-183.
AbstractLet \(\mathcal{M}(\mathbb{R}^n)\) 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 \(\mathcal{M}^*(\mathbb{R}^n)\).
Karlovich, Alexei Yu. "
Algebras of singular integral operators with piecewise continuous coefficients on weighted Nakano spaces."
The Extended Field of Operator Theory. Operator Theory: Advances and Applications, 171. Ed. Michael A. Dritschel. Basel: Birkhäuser, 2007. 171-188.
AbstractWe 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.
Karlovich, Alexei Yu., Yuri I. Karlovich, and Amarino B. Lebre. "
Semi-Fredholmness of weighted singular integral operators with shifts and slowly oscillating data."
Operator Theory, Operator Algebras, and Matrix Theory. Operator Theory: Advances and Applications, vol. 267. Eds. Carlos André, Maria Amélia Bastos, Alexei Yu. Karlovich, Bernd Silbermann, and Ion Zaballa. Basel: Birkhäuser, 2018. 221-246.
Karlovich, Alexei Yu. "
Boundedness of pseudodifferential operators on Banach function spaces."
Operator Theory, Operator Algebras and Applications. Operator Theory: Advances and Applications, 242. Eds. Maria Amélia Bastos, Amarino Lebre, Stefan Samko, and Ilya M. Spitkovsky. Basel: Birkhäuser/Springer, 2014. 185-195.
AbstractWe 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 \(L^{p(\cdot)}(\mathbb{R}^n)\).
Karlovich, Alexei Yu. "
Singular integral operators on variable Lebesgue spaces with radial oscillating weights."
Operator Algebras, Operator Theory and Applications.Operator Theory Advances and Applications, 195 . Eds. JJ Grobler, LE Labuschagne, and M. Möller. Basel: Birkhäuser, 2010. 185-212.
AbstractWe 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.