Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a “dimension” n. For α ∈ (0, ∞) denote by H α p (X), H d p (X...Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a “dimension” n. For α ∈ (0, ∞) denote by H α p (X), H d p (X), and H *,p (X) the corresponding Hardy spaces on X defined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calderón reproducing formula, it is shown that all these Hardy spaces coincide with L p (X) when p ∈ (1,∞] and with each other when p ∈ (n/(n + 1), 1]. An atomic characterization for H ?,p (X) with p ∈ (n/(n + 1), 1] is also established; moreover, in the range p ∈ (n/(n + 1),1], it is proved that the space H *,p (X), the Hardy space H p (X) defined via the Littlewood-Paley function, and the atomic Hardy space of Coifman andWeiss coincide. Furthermore, it is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from H p (X) to some quasi-Banach space B if and only if T maps all (p, q)-atoms when q ∈ (p, ∞)∩[1, ∞) or continuous (p, ∞)-atoms into uniformly bounded elements of B.展开更多
Let X = （X, d,μ） The purpose of this paper is to be a space of homogeneous type in the sense of Coifman and Weiss. generalize the definition of Hardy space H^P（X） and prove that the generalized Hardy spaces have...Let X = （X, d,μ） The purpose of this paper is to be a space of homogeneous type in the sense of Coifman and Weiss. generalize the definition of Hardy space H^P（X） and prove that the generalized Hardy spaces have the same property as H^P（X）. Our definition includes a kind of Hardy- Orlicz spaces and a kind of Hardy spaces with variable exponent. The results are new even for the R^n case. Let （X, δ, μ） be the normalized space of （X, d, μ） in the sense of Macias and Segovia. We also study the relations of our function spaces for （X, d, μ） and （X, δ,μ）.展开更多
BACKGROUND Dysfunction of the glymphatic system in the brain in different stages of altered glucose metabolism and its influencing factors are not well characterized.AIM To investigate the function of the glymphatic s...BACKGROUND Dysfunction of the glymphatic system in the brain in different stages of altered glucose metabolism and its influencing factors are not well characterized.AIM To investigate the function of the glymphatic system and its clinical correlates in patients with different glucose metabolism states,the present study employed diffusion tensor imaging along the perivascular space(DTI-ALPS)index.METHODS Sample size was calculated using the pwr package in R software.This crosssectional study enrolled 22 patients with normal glucose metabolism(NGM),20 patients with prediabetes,and 22 patients with type 2 diabetes mellitus(T2DM).A 3.0T magnetic resonance imaging was used to evaluate the function of the glymphatic system.The mini-mental state examination(MMSE)was used to assess general cognitive function.The DTI-ALPS index of bilateral basal ganglia and the mean DTI-ALPS index was calculated.Further,the correlation between DTI-ALPS and clinical features was assessed.RESULTS The left-side,right-side,and mean DTI-ALPS index in the T2DM group were significantly lower than that in the NGM group.The right-side DTI-ALPS and mean DTI-ALPS index in the T2DM group were significantly lower than those in the prediabetes group.DTI-ALPS index lateralization was not observed.The MMSE score in the T2DM group was significantly lower than that in the NGM and prediabetes group.After controlling for sex,the left-side DTI-ALPS and mean DTI-ALPS index in the prediabetes group were positively correlated with 2-hour postprandial blood glucose level;the left-side DTI-ALPS index was negatively correlated with total cholesterol and low-density lipoprotein level.The right-side DTI-ALPS and mean DTI-ALPS index were negatively correlated with the glycosylated hemoglobin level and waist-to-hip ratio in the prediabetes group.The left-side,right-side,and mean DTI-ALPS index in the T2DM group were positively correlated with height.The left-side and mean DTIALPS index in the T2DM group were negatively correlated with high-density lipoprotein levels.展开更多
基金supported by the National Science Foundation of USA (Grant No. DMS 0400387)the University of Missouri Research Council (Grant No. URC-07-067)+1 种基金the National Science Foundation for Distinguished Young Scholars of China (Grant No. 10425106)the Program for New Century Excellent Talents in University of the Ministry of Education of China (Grant No. 04-0142)
文摘Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a “dimension” n. For α ∈ (0, ∞) denote by H α p (X), H d p (X), and H *,p (X) the corresponding Hardy spaces on X defined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calderón reproducing formula, it is shown that all these Hardy spaces coincide with L p (X) when p ∈ (1,∞] and with each other when p ∈ (n/(n + 1), 1]. An atomic characterization for H ?,p (X) with p ∈ (n/(n + 1), 1] is also established; moreover, in the range p ∈ (n/(n + 1),1], it is proved that the space H *,p (X), the Hardy space H p (X) defined via the Littlewood-Paley function, and the atomic Hardy space of Coifman andWeiss coincide. Furthermore, it is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from H p (X) to some quasi-Banach space B if and only if T maps all (p, q)-atoms when q ∈ (p, ∞)∩[1, ∞) or continuous (p, ∞)-atoms into uniformly bounded elements of B.
基金Grant-in-Aid for Exploratory Research,No.17654033 the Ministry of Education,Culture,Sports,Science and Technology,JapanGrant-in-Aid for Scientific Research (C),No.20540167,Japan Society for the Promotion of Science
文摘Let X = （X, d,μ） The purpose of this paper is to be a space of homogeneous type in the sense of Coifman and Weiss. generalize the definition of Hardy space H^P（X） and prove that the generalized Hardy spaces have the same property as H^P（X）. Our definition includes a kind of Hardy- Orlicz spaces and a kind of Hardy spaces with variable exponent. The results are new even for the R^n case. Let （X, δ, μ） be the normalized space of （X, d, μ） in the sense of Macias and Segovia. We also study the relations of our function spaces for （X, d, μ） and （X, δ,μ）.
文摘BACKGROUND Dysfunction of the glymphatic system in the brain in different stages of altered glucose metabolism and its influencing factors are not well characterized.AIM To investigate the function of the glymphatic system and its clinical correlates in patients with different glucose metabolism states,the present study employed diffusion tensor imaging along the perivascular space(DTI-ALPS)index.METHODS Sample size was calculated using the pwr package in R software.This crosssectional study enrolled 22 patients with normal glucose metabolism(NGM),20 patients with prediabetes,and 22 patients with type 2 diabetes mellitus(T2DM).A 3.0T magnetic resonance imaging was used to evaluate the function of the glymphatic system.The mini-mental state examination(MMSE)was used to assess general cognitive function.The DTI-ALPS index of bilateral basal ganglia and the mean DTI-ALPS index was calculated.Further,the correlation between DTI-ALPS and clinical features was assessed.RESULTS The left-side,right-side,and mean DTI-ALPS index in the T2DM group were significantly lower than that in the NGM group.The right-side DTI-ALPS and mean DTI-ALPS index in the T2DM group were significantly lower than those in the prediabetes group.DTI-ALPS index lateralization was not observed.The MMSE score in the T2DM group was significantly lower than that in the NGM and prediabetes group.After controlling for sex,the left-side DTI-ALPS and mean DTI-ALPS index in the prediabetes group were positively correlated with 2-hour postprandial blood glucose level;the left-side DTI-ALPS index was negatively correlated with total cholesterol and low-density lipoprotein level.The right-side DTI-ALPS and mean DTI-ALPS index were negatively correlated with the glycosylated hemoglobin level and waist-to-hip ratio in the prediabetes group.The left-side,right-side,and mean DTI-ALPS index in the T2DM group were positively correlated with height.The left-side and mean DTIALPS index in the T2DM group were negatively correlated with high-density lipoprotein levels.
