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Characterization of functions via commutators of bilinear fractional integrals on Morrey spaces
Suzhen Mao,Huoxiong Wu 대한수학회 2016 대한수학회보 Vol.53 No.4
For $b\in L^1_{\rm loc}(\mathbb{R}^n)$, let $\mathcal{I}_\alpha$ be the bilinear fractional integral operator, and $[b,\,\mathcal{I}_\alpha]_i$ be the commutator of $\mathcal{I}_\alpha$ with pointwise multiplication $b$ ($i=1,\,2$). This paper shows that if the commutator $[b, \mathcal{I}_\alpha]_i$ for $i=1$ or $2$ is bounded from the product Morrey spaces $L^{p_1, \lambda_1}(\mathbb R^n)\times L^{p_2, \lambda_2}(\mathbb R^n)$ to the Morrey space $L^{q,\lambda}(\mathbb{R}^n)$ for some suitable indexes $\lambda,\,\lambda_1$, $\lambda_2$ and $p_1,\,p_2,\,q$, then $b\in BMO(\mathbb{R}^n)$, as well as that the compactness of $[b,\,\mathcal{I}_\alpha]_i$ for $i=1$ or $2$ from $L^{p_1, \lambda_1}(\mathbb R^n)\times L^{p_2, \lambda_2}(\mathbb R^n)$ to $L^{q,\lambda}(\mathbb{R}^n)$ implies that $b\in CMO(\mathbb{R}^n)$ (the closure in $BMO(\mathbb{R}^n)$ of the space of $C^\infty(\mathbb{R}^n)$ functions with compact support). These results together with some previous ones give a new characterization of $BMO(\mathbb{R}^n)$ functions or $CMO(\mathbb{R}^n)$ functions in essential ways.
CHARACTERIZATION OF FUNCTIONS VIA COMMUTATORS OF BILINEAR FRACTIONAL INTEGRALS ON MORREY SPACES
Mao, Suzhen,Wu, Huoxiong Korean Mathematical Society 2016 대한수학회보 Vol.53 No.4
For $b{\in}L^1_{loc}({\mathbb{R}}^n)$, let ${\mathcal{I}}_{\alpha}$ be the bilinear fractional integral operator, and $[b,{\mathcal{I}}_{\alpha}]_i$ be the commutator of ${\mathcal{I}}_{\alpha}$ with pointwise multiplication b (i = 1, 2). This paper shows that if the commutator $[b,{\mathcal{I}}_{\alpha}]_i$ for i = 1 or 2 is bounded from the product Morrey spaces $L^{p_1,{\lambda}_1}({\mathbb{R}}^n){\times}L^{p_2,{\lambda}_2}({\mathbb{R}}^n)$ to the Morrey space $L^{q,{\lambda}}({\mathbb{R}}^n)$ for some suitable indexes ${\lambda}$, ${\lambda}_1$, ${\lambda}_2$ and $p_1$, $p_2$, q, then $b{\in}BMO({\mathbb{R}}^n)$, as well as that the compactness of $[b,{\mathcal{I}}_{\alpha}]_i$ for i = 1 or 2 from $L^{p_1,{\lambda}_1}({\mathbb{R}}^n){\times}L^{p_2,{\lambda}_2}({\mathbb{R}}^n)$ to $L^{q,{\lambda}}({\mathbb{R}}^n)$ implies that $b{\in}CMO({\mathbb{R}}^n)$ (the closure in $BMO({\mathbb{R}}^n)$of the space of $C^{\infty}({\mathbb{R}}^n)$ functions with compact support). These results together with some previous ones give a new characterization of $BMO({\mathbb{R}}^n)$ functions or $CMO({\mathbb{R}}^n)$ functions in essential ways.
Jialin Li,Fuhua Xie,Xiuzhi Zhu,Haibo Hu,Xiansong Fang,Zhiwei Huang,Zhiping Liu,Suzhen Wu 한국식품영양과학회 2020 Journal of medicinal food Vol.23 No.12
The fleshy fruits of Camellia oleifera Abel are the immature fruits of C. oleifera, which are infected by Exobasidium vexans Massee and then turn to be intumescent and hollowed. They contain rich trace elements and vitamin C and are eaten directly as wild fruits in the Chinese countryside. Recent studies report that C. oleifera has anti-inflammatory and antioxidative effects. The current study, for the first time, evaluates the renal protective capacity of polysaccharides from the fleshy fruits of C. oleifera (CFFP) in streptozotocin-induced diabetic mice fed high-fat diets. The diabetic mice were orally administered CFFP for 3 months to evaluate the renoprotective function of CFFP. Our results indicated that 250 mg/kg CFFP significantly alleviated diabetes-induced renal injury by decreasing serum creatine, blood urea nitrogen levels, the kidney/body weight ratio, expression of fibronectin and collagen, as well as the secretion of tumor necrosis factor-α and interleukin-6. Additionally, 250 mg/kg CFFP could significantly ameliorate renal oxidative stress through increasing glutathione levels and lowering malondialdehyde contents. We confirmed that CFFP could exert antioxidative, anti-inflammatory, and antifibrosis activities. CFFP might be a potential therapeutic agent, and the fleshy fruits of C. oleifera might be a diet therapy for diabetic patients in the future.