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Function spaces on quantum tori and their applications to pseudo-differential operators.

Abstract : This thesis is devoted to the study of the analysis on the spaces hpc(Rd,M), the local version of operator-valued Hardy spaces studied by Tao Mei. The operator-valued local Hardy spaces are defined by the truncated Littlewood-Paley g-functions and the truncated Lusin square functions associated to the Poisson kernel. We develop the Calderón-Zygmund theory on hpc(Rd,M), and study the hpc-bmocq duality and the interpolation. Based on these results, we obtain general characterization of hpc(Rd,M) which states that the Poisson kernel can be replaced by any reasonable test function. This characterization plays an important role in the smooth atomic decomposition of h1c(Rd,M). We also investigate the operator-valued inhomogeneous Triebel-Lizorkin spaces Fpα,c(Rd,M). Like in the classical case, these spaces are connected with the operator-valued local Hardy spaces via Bessel potentials. Then by the aid of the Calderón-Zygmund theory, we obtain the Littlewood-Paley type and the Lusin type characterizations of Fpα,c(Rd,M) by more general kernels. These characterizations allow us to study various properties of Fpα,c(Rd,M), in particular, the smooth atomic decomposition. This is an extension and an improvement of the previous atomic decomposition of h1c(Rd,M). As an important application of this smooth atomic decomposition, we show the boundedness of pseudo-differential operators with regular operator-valued symbols on Triebel-Lizorkin spaces Fpα,c(Rd,M), for α ∈ R and 1 ≤ p ≤ ∞. Finally, by virtue of transference, we obtain the Fpα,c-boundedness of pseudo-differential operators on quantum tori
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Runlian Xia. Function spaces on quantum tori and their applications to pseudo-differential operators.. Functional Analysis [math.FA]. Université Bourgogne Franche-Comté, 2017. English. ⟨NNT : 2017UBFCD084⟩. ⟨tel-01988561⟩

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