Coelho, T., De Oliveira, R., Cardoso, T., Rosas, J., and Rybarczyk Y. "
Body ownership of virtual avatars: an affordance approach of telepresence."
Innovative and Creative Developments in Multimodal Interaction Systems. Eds. Y. Rybarczyk, T. Cardoso, J. Rosas, and L. Camarinha-Matos. Heidelberg: Springer, 2014. 3-19.
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., and Ilya M. Spitkovsky. "
The Cauchy singular integral operator on weighted variable Lebesgue spaces."
Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation. Operator Theory: Advances and Applications, 236. Eds. Manuel Cepedello Boiso, Håkan Hedenmalm, Marinus A. Kaashoek, Alfonso Montes Rodríguez, and Sergei Treil. Basel: Birkhäuser, 2014. 275-291.
AbstractLet \(p:\mathbb{R}\to(1,\infty)\) be a globally log-Hölder continuous variable exponent and \(w:\mathbb{R}\to[0,\infty]\) be a weight. We prove that the Cauchy singular integral operator \(S\) is bounded on the weighted variable Lebesgue space \(L^{p(\cdot)}(\mathbb{R},w)=\{f:fw\in L^{p(\cdot)}(\mathbb{R})\}\) if and only if the weight \(w\) satisfies $$ \sup_{-\infty < a < b < \infty} \frac{1}{b-a} \|w\chi_{(a,b)}\|_{p(\cdot)} \|w^{-1}\chi_{(a,b)}\|_{p'(\cdot)}<\infty \quad (1/p(x)+1/p'(x)=1). $$