Riesz transforms on Q-type spaces with application to quasi-geostrophic equation

In this paper, we prove the boundedness of Riesz transforms $\partial_{j}(-\Delta)^{-1/2}$ ($j=1,2,...,n$) on the Q-type spaces $Q_{\alpha}^{\beta}(\mathbb{R}^{n})$. As an application, we get the well-posedness and regularity of the quasi-geostrophic equation with initial data in $Q_{\alpha}^{\beta,-1}(\mathbb{R}^{2}).$Comment: 18 pages, submitte

Several analytic inequalities in some Q−Q-spaces

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 $Q_{\alpha}(\mathbb{R}^{n}).$Comment: 13 pages submitte

Schr\"odinger type operators on generalized Morrey spaces

In this paper we introduce a class of generalized Morrey spaces associated with Schr\"odinger operator $L=-\Delta+V$. Via a pointwise estimate, we obtain the boundedness of the operators $V^{\beta_{2}}(-\Delta+V)^{-\beta_{1}}$ and their dual operators on these Morrey spaces

Analytic version of critical QQ spaces and their properties

In this paper, we establish an analytic version of critical spaces $Q_{\alpha}^{\beta}(\mathbb{R}^{n})$ on unit disc $\mathbb{D}$, denoted by $Q^{\beta}_{p}(\mathbb{D})$. Further we prove a relation between $Q^{\beta}_{p}(\mathbb{D})$ and Morrey spaces. By the boundedness of two integral operators, we give the multiplier spaces of $Q^{\beta}_{p}(\mathbb{D})$

Wavelets, Multiplier spaces and application to Schr\"{o}dinger type operators with non-smooth potentials

In this paper, we employ Meyer wavelets to characterize multiplier spaces between Sobolev spaces without using capacity. Further, we introduce logarithmic Morrey spaces $M^{t,\tau}_{r,p}(\mathbb{R}^{n})$ to establish the inclusion relation between Morrey spaces and multiplier spaces. By wavelet characterization and fractal skills, we construct a counterexample to show that the scope of the index $\tau$ of $M^{t,\tau}_{r,p}(\mathbb{R}^{n})$ is sharp. As an application, we consider a Schr\"odinger type operator with potentials in $M^{t,\tau}_{r,p}(\mathbb{R}^{n})$

Wavelets and Triebel type oscillation spaces

We apply wavelets to identify the Triebel type oscillation spaces with the known Triebel-Lizorkin-Morrey spaces $\dot{F}^{\gamma_1,\gamma_2}_{p,q}(\mathbb{R}^{n})$. Then we establish a characterization of $\dot{F}^{\gamma_1,\gamma_2}_{p,q}(\mathbb{R}^{n})$ via the fractional heat semigroup. Moreover, we prove the continuity of Calder\'on-Zygmund operators on these spaces. The results of this paper also provide necessary tools for the study of well-posedness of Navier-Stokes equations

Global Mild Solutions of Fractional Naiver-Stokes Equations with Small Initial Data in Critical Besov-Q Spaces

In this paper, we establish the global existence and uniqueness of a mild solution of the so-called fractional Navier-Stokes equations with a small initial data in the critical Besov-Q space covering many already known function spaces

Fefferman-Stein decomposition for QQ-spaces and micro-local quantities

In this paper, we consider the Fefferman-Stein decomposition of $Q_{\alpha}(\mathbb{R}^{n})$ and give an affirmative answer to an open problem posed by M. Essen, S. Janson, L. Peng and J. Xiao in 2000. One of our main methods is to study the structure of the predual space of $Q_{\alpha}(\mathbb{R}^{n})$ by the micro-local quantities. This result indicates that the norm of the predual space of $Q_{\alpha}(\mathbb{R}^{n})$ depends on the micro-local structure in a self-correlation way

Generalized Naiver-Stokes equations with initial data in local QQ-type spaces

In this paper, we establish a link between Leray mollified solutions of the three-dimensional generalized Naiver-Stokes equations and mild solutions for initial data in the adherence of the test functions for the norm of $Q^{\beta,-1}_{\alpha, loc}(\mathbb{R}^{3}).$ This result applies to the usual incompressible Navier-Stokes equations and deduces a known link.Comment: 18 page

On weighted compactness of commutators of Schr\"{o}dinger operators

Let $\mathcal{L}=-\Delta+\mathit{V}(x)$ be a Schr\"{o}dinger operator, where $\Delta$ is the Laplacian operator on $\mathbb{R}^{d}$ $(d\geq 3)$, while the nonnegative potential $\mathit{V}(x)$ belongs to the reverse H\"{o}lder class $B_{q}, q>d/2$. In this paper, we study weighted compactness of commutators of some Schr\"{o}dinger operators, which include Riesz transforms, standard Calder\'{o}n-Zygmund operatos and Littlewood-Paley functions. These results generalize substantially some well-know results.Comment: 25page