A characterization of weighted local Hardy spaces

In this paper, we give a characterization of weighted local Hardy spaces h^1_\wz(\rz) associated with local weights by using the truncated Reisz transforms, which generalizes the corresponding result of Bui in \cite{b}

Weighted norm inequalities, spectral multipliers and Littlewood-Paley operators in the Schr\"odinger settings

In this paper, we establish a good-\lz inequality with two parameters in the Schr\"odinger settings. As it's applications, we obtain weighted estimates for spectral multipliers and Littlewood-Paley operators and their commutators in the Schr\"odinger settings

Weighted norm inequalities for pseudo-differential operators with smooth symbols and their commutators

We obtain weighted $L^p$ inequalities for pseudo-differential operators with smooth symbols and their commutators by using a class of new weight functions which include Muckenhoupt weight functions. Our results improve essentially some well-known results

Extrapolation from A_\fz^{\rho,\fz}, vector-valued inequalities and applications in the Schr\"odinger settings

In this paper, we generalize the A_\fz extrapolation theorem in \cite{cmp} and the $A_p$ extrapolation theorem of Rubio de Francia to Schr\"odinger settings. In addition, we also establish the weighted vector-valued inequalities for Schr\"odinger type maximal operators by using weights belonging to A_p^{\rho,\tz} which includes $A_p$. As their applications, we establish the weighted vector-valued inequalities for some Sch\"odinger type operators and pseudo-differential operators

Weighted norm inequalities for Schr\"odinger type operators

Let $L=-\Delta+V$ be a Schr\"{o}dinger operator, where $\Delta$ is the Laplacian operator on \rz, while nonnegative potential $V$ belongs to the reverse H\"{o}lder class. In this paper, we establish the weighted norm inequalities for some Schr\"odinger type operators, which include Riesz transforms and fractional integrals and their commutators. These results generalize substantially some well-known results

Weighted local Hardy spaces and their applications

In this paper, we study weighted local Hardy spaces h^p_\wz(\rz) associated with local weights which include the classical Muckenhoupt weights. This setting includes the classical local Hardy space theory of Goldberg \cite{g}, and the weighted Hardy spaces of Bui \cite{bu}.Comment: 40 page

Effect of histamine on the electric activities of cerebellar Purkinje cell

The effect of histamine (HA) on the electric activities of Purkinje cell (PC) is studied on the cerebellum slice. We find that: (1) HA's main effect on PC is excitative (72.9%); there are also a small amount of PC showing inhibitive (10.2%) or no (16.9%) response to HA. (2) Different from the conventional opinion, HA's excitative effect on PC is mutually conducted by H1 and H2 receptors; the antagonist for H1 receptor could weaken HA's excitative effect on PC, while the antagonist for H2 receptor could weaken or even block the excitative effect of HA on PC. (3) PC's reaction to HA is related to its intrinsic discharge frequency; there exists a frequency at which PC is highly sensitive to HA, and well above this frequency PC becomes stable against HA. These results indicate that the histaminergic afferent fibre can adjust PC's electric activities by releasing HA, and thereby influence the global function of the cerebellar cortex; and that just like the $\gamma$ region of cerebrum, cerebellum may also have some sort of characteristic frequency.Comment: 5 pages RevTex, 3 figures, each contains 2-3 eps file

Interior C1,αC^{1,\alpha} regularity on the linearized Monge-Ampeˋ\grave{e}re equation with VMO\mathrm{VMO} type coefficients

In this paper, we establish interior $C^{1,\alpha}$ estimates for solutions of the linearized Monge-Amp$\grave{e}$re equation $$\mathcal{L}_{\phi}u:=\mathrm{tr}[\Phi D^2 u]=f,$$ where the density of the Monge-Amp$\grave{e}$re measure $g:=\mathrm{det}D^2\phi$ satisfies a $\mathrm{VMO}$-type condition and $\Phi:=(\mathrm{det}D^2\phi)(D^2\phi)^{-1}$ is the cofactor matrix of $D^2\phi$.Comment: 9 page

Weighted local Hardy spaces associated to Schr\"{o}dinger operators

In this paper, we characterize the weighted local Hardy spaces $h^p_\rho(\omega)$ related to the critical radius function $\rho$ and weights $\omega\in A_{\infty}^{\rho,\,\infty}(\mathbb{R}^{n})$ which locally behave as Muckenhoupt's weights and actually include them, by the local vertical maximal function, the local nontangential maximal function and the atomic decomposition. By the atomic characterization, we also prove the existence of finite atomic decompositions associated with $h^{p}_{\rho}(\omega)$. Furthermore, we establish boundedness in $h^p_\rho(\omega)$ of quasi- Banach-valued sublinear operators. As their applications, we establish the equivalence of the weighted local Hardy space $h^1_\rho(\omega)$ and the weighted Hardy space $H^1_{\cal L}(\omega)$ associated to Schr\"{o}dinger operators $\cal L$ with $\omega \in A_1^{\rho,\infty}(\mathbb{R}^{n})$Comment: 54 pages. arXiv admin note: substantial text overlap with arXiv:1107.3266, arXiv:1108.2797 by other author

Existence of ground state solutions of Nehari-Pankov type to Schr\"odinger systems

This paper is dedicated to studying the following elliptic system of Hamiltonian type: \left\{ \begin{array}{ll} -\varepsilon^2\triangle u+u+V(x)v=Q(x)F_{v}(u, v), \ \ \ \ x\in \mathbb{R}^N,\\ -\varepsilon^2\triangle v+v+V(x)u=Q(x)F_{u}(u, v), \ \ \ \ x\in \mathbb{R}^N,\\ |u(x)|+|v(x)| \rightarrow 0, \ \ \mbox{as} \ |x|\rightarrow \infty, \end{array}\right. where $N\ge 3$, $V, Q\in \mathcal{C}(\mathbb{R}^N, \mathbb{R})$, $V(x)$ is allowed to be sign-changing and $\inf Q > 0$, and $F\in \mathcal{C}^1(\mathbb{R}^2, \mathbb{R})$ is superquadratic at both $0$ and infinity but subcritical. Instead of the reduction approach used in [Calc Var PDE, 2014, 51: 725-760], we develop a more direct approach -- non-Nehari manifold approach to obtain stronger conclusions but under weaker assumptions than these in [Calc Var PDE, 2014, 51: 725-760]. We can find an $\varepsilon_0>0$ which is determined by terms of $N, V, Q$ and $F$, then we prove the existence of a ground state solution of Nehari-Pankov type to the coupled system for all $\varepsilon\in (0, \varepsilon_0]$