A characterization of two weight norm inequality for Littlewood-Paley gλ∗g_{\lambda}^{*}-function

Let $n\ge 2$ and $g_{\lambda}^{*}$ be the well-known high dimensional Littlewood-Paley function which was defined and studied by E. M. Stein, \begin{align*} g_{\lambda}^{*}(f)(x) =\bigg(\iint_{\mathbb R^{n+1}_{+}} \Big(\frac{t}{t+|x-y|}\Big)^{n\lambda} |\nabla P_tf(y,t)|^2 \frac{dy dt}{t^{n-1}}\bigg)^{1/2}, \ \quad \lambda > 1, \end{align*} where $P_tf(y,t)=p_t*f(y)$, $p_t(y)=t^{-n}p(y/t)$ and $p(x) = (1+|x|^2)^{-(n+1)/2}$, $\nabla =(\frac{\partial}{\partial y_1},\ldots,\frac{\partial}{\partial y_n},\frac{\partial}{\partial t})$. In this paper, we give a characterization of two-weight norm inequality for $g_{\lambda}^{*}$-function. We show that, $\big\| g_{\lambda}^{*}(f \sigma) \big\|_{L^2(w)} \lesssim \big\| f \big\|_{L^2(\sigma)}$ if and only if the two-weight Muchenhoupt $A_2$ condition holds, and a testing condition holds : \begin{align*} \sup_{Q : cubes \ in \mathbb R^n} \frac{1}{\sigma(Q)} \int_{\mathbb R^n} \iint_{\widehat{Q}} \Big(\frac{t}{t+|x-y|}\Big)^{n\lambda}|\nabla P_t(\mathbf{1}_Q \sigma)(y,t)|^2 \frac{w dx dt}{t^{n-1}} dy < \infty, \end{align*} where $\widehat{Q}$ is the Carleson box over $Q$ and $(w, \sigma)$ is a pair of weights. We actually prove this characterization for $g_{\lambda}^{*}$-function associated with more general fractional Poisson kernel $p^\alpha(x) = (1+|x|^2)^{-{(n+\alpha)}/{2}}$. Moreover, the corresponding results for intrinsic $g_{\lambda}^*$-function are also presented.Comment: 21 pages, to appear in Journal of Geometric Analysi

Hierarchical Structure with Highly Ordered Macroporous-Mesoporous Metal-Organic Frameworks as Dual Function for CO2 Fixation.

As a major greenhouse gas, the continuous increase of carbon dioxide (CO2) in the atmosphere has caused serious environmental problems, although CO2 is also an abundant, inexpensive, and nontoxic carbon source. Here, we use metal-organic framework (MOF) with highly ordered hierarchical structure as adsorbent and catalyst for chemical fixation of CO2 at atmospheric pressure, and the CO2 can be converted to the formate in excellent yields. Meanwhile, we have successfully integrated highly ordered macroporous and mesoporous structures into MOFs, and the macro-, meso-, and microporous structures have all been presented in one framework. Based on the unique hierarchical pores, high surface area (592 m2/g), and high CO2 adsorption capacity (49.51 cm3/g), the ordered macroporous-mesoporous MOFs possess high activity for chemical fixation of CO2 (yield of 77%). These results provide a promising route of chemical CO2 fixation through MOF materials

Comparison of the efficacy of half ticagrelor loading doses and clopidogrel in elderly acute coronary syndrome patients in China

Purpose: To evaluate the effects of half-load doses (HLD) of ticagrelor and clopidogrel on elderly acute coronary syndrome patients (ACS) over a period of 90 days. Methods: Seventy-four patients diagnosed as ACS were included in this trial. The patients were randomly distributed into group 1 (treated with HLD ticagrelor, 90 mg LD) and group 2 (treated with clopidogrel, 300 mg LD). The interaction of treatment effect was evaluated using Multivariate Cox proportional hazards regression models. Results: Within three months, a total of 12 patients (16.21 %) died of myocardial infarction or stroke. The endpoint of HLD ticagrelor-treated elderly ACS patients was 20 %, and the incidence of clopidogreltreated endpoints was 14.81 %. Conclusion: In the first 45 patients treated with HLD ticagrelor, their cumulative incidence of cardiac events was relatively high. However, there were no considerable changes in the therapeutic benefits of these two drugs in elderly ACS patients. Keywords: Elder patients, Acute coronary syndrome, Ticagrelor, Clopidogre