Net Spaces and Boundedness of Integral Operators

In this paper we introduce new functional spaces which we call net spaces. Using their properties, necessary and sufficient conditions for the integral operators to be of strong or weak type are obtained. Estimates of the norm of the convolution operator in weighted Lebesgue spaces are presented.     

Exercising spatiotemporal control of cell attachment with optically transparent microelectrodes

This paper describes a novel approach of controlling cell-surface interactions through an electrochemical "switching" of biointerfacial properties of optically transparent microelectrodes. The indium tin oxide (ITO) microelectrodes, fabricated on glass substrates, were modified with poly(ethylene glycol) (PEG) silane to make glass and ITO regions resistant to protein and cell adhesion. Cyclic voltammetry, with potassium ferricyanide serving as a redox reporter molecule, was used to monitor electron transfer across the electrolyte-ITO interface. PEG silane modification of ITO correlated with diminished electron transfer, judged by the disappearance of ferricyanide redox activity. Importantly, application of reductive potential (-1.4 V vs Ag/AgCl reference) corresponded with reappearance of typical ferricyanide redox peaks, thus pointing to desorption of an insulating PEG silane layer. Time-of-flight secondary ion mass spectrometry (ToF-SIMS) characterization of the silanized ITO surfaces after electrical stimulation indicated complete removal of the silane layer. Significantly, electrical stimulation allowed to "switch" chosen electrodes from nonfouling to protein-adhesive while leaving other ITO and glass regions protected by a nonfouling PEG silane layer. The spatial and temporal control of biointerfacial properties afforded by our approach was utilized to micropattern proteins and cells and to construct micropatterned co-cultures. In the future, control of the biointerfacial properties afforded by this novel approach may allow the organization of multiple cell types into precise geometric configurations in order to create better in vitro mimics of cellular complexity of the native tissues.     

On summability of Fourier coefficients of functions from Lebesgue space

In this paper we state new theorems of Hardy–Littlewood type for functions with general monotone Fourier coefficients. Sharpness of stated results is discusses.     

Convolution Inequalities in Lorentz Spaces

In this paper we study boundedness of the convolution operator in different Lorentz spaces. We obtain the limit case of the Young-O’Neil inequality in the classical Lorentz spaces. We also investigate the convolution operator in the weighted Lorentz spaces.     

Global and local smoothness of the Hilbert transforms

We study the smoothness properties of the Hilbert transform in terms of the smoothness of the function itself. We prove global and local analogues of Privalov’s theorem. The sharpness of these results is shown.     

Norm inequalities for convolution operators

We study norm convolution inequalities in Lebesgue and Lorentz spaces. First, we improve the well-known O'Neil's inequality for the convolution operators and prove corresponding estimate from below. Second, we obtain Young–O'Neil-type estimate in the Lorentz spaces for the limit value parameters, i.e., $6K1f{6}_{L(p,h_1)\to L(p,h_2)}$. Finally, similar estimates in the weighted Lorentz spaces are presented. To cite this article: E. Nursultanov et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).