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Zhichun Zhai

  1. Several analytic inequalities in some $Q-$spaces.

    Fri, 11/13/2009 - 16:01 — SomNambul
    Authors: Zhichun Zhai, Pengtao Li
    Subjects: Analysis of PDEs
    Abstract

    In this paper, we establish separate necessary and sufficient

    John-Nirenberg (JN) type inequalities for functions in

    $Q_{\alpha}^{\beta}(\mathbb{R}^{n})$ which imply Gagliardo-Nirenberg (GN)
    type inequalities in

    $Q_{\alpha}(\mathbb{R}^{n}).$ Consequently, we obtain Trudinger-Moser type
    inequalities and Brezis-Gallouet-Wainger type inequalities in
    $Q_{\alpha}(\mathbb{R}^{n}).$

  2. Note on affine Gagliardo-Nirenberg inequalities.

    Mon, 08/17/2009 - 18:30 — SomNambul
    Authors: Zhichun Zhai
    Subjects: Functional Analysis
    Abstract

    This note proves sharp affine Gagliardo-Nirenberg inequalities which are
    stronger than all known sharp Euclidean Gagliardo-Nirenberg inequalities and
    imply the affine $L^{p}-$Sobolev inequalities. The logarithmic version of
    affine $L^{p}-$Sobolev inequalities is verified. Moreover, An alternative proof
    of the affine Moser-Trudinger and Morrey-Sobolev inequalities is given. The
    main tools are the equimeasurability of rearrangements and the strengthened
    version of the classical P\'{o}lys-Szeg\"{o} principle.

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