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In the present paper, we concern on discussing some results of the fractional Laplacian with respect to a given function μ. In addition, we also discuss the Nemytskii operators Ψ ↦ | Ψ | {\Psi\mapsto|\Psi|} and Ψ ↦ Ψ ± {\Psi\mapsto\Psi^{\pm}} in μ-fractional Sobolev–Slobodetskii space. [ABSTRACT FROM AUTHOR]

A non-equilibrium plasma is created in a micro-capillary quartz tube (800 µ m of internal diameter), by a DC-pulsed micro-dielectric barrier discharge (micro-DBD) and the propagation of an ionisation wave, in mixtures of He/H2O/VOC at atmospheric pressure where the studied volatile organic compounds (VOCs) are representative of molecules belonging to different chemical families: alcohols (methanol, ethanol, isopropanol, tert-butanol), ketones (acetone), nitriles (acetonitrile), and aromatic hydrocarbons (toluene). The conversion efficiency of these VOCs is studied as a function of the applied voltage on the micro-DBD (or electrical energy deposited in the plasma) and of the initial concentration of the molecules in the range from 1 ppm up to 3000 ppm (depending on the molecule), with the help of high-resolution real-time mass spectrometry Fourier transform ion cyclotron resonance associated to chemical ionisation (CI-FTICR) using H3O+ as precursor ion. A variety of by-products resulting from the conversion of VOCs are identified and quantified, emphasising that the micro-capillary plasma is able to induce a complex chemical reactivity. A qualitative analysis of the involved kinetics, based on the existing literature, reveals that helium species (ions and metastable states) and radicals coming from the dissociation of the water molecules (O and OH) are the most probable candidates to explain the formation of all compounds detected by the CI-FTICR apparatus. Quenching processes of the metastable He(23S) by the VOCs, leading to the dissociation of the molecules, are suggested to explain some of the experimental results. [ABSTRACT FROM AUTHOR]