LpL^p norm of truncated Riesz transform and an improved dimension-free LpL^p estimate for maxima Riesz transform

In this paper, we prove that the $L^p(\mathbb{R}^d)$ norm of the maximal truncated Riesz transform in terms of the $L^p(\mathbb{R}^d)$ norm of Riesz transform is dimension-free for any $2\leq p<\infty$, using integration by parts formula for radial Fourier multipliers. Moreover, we show that \|R_j^*f\|_{L^p}\leq \left({2+\frac{1}{\sqrt{2}}}\right)^{\frac{2}{p}}\|R_jf\|_{L^p},\ \ \mbox{for}\ \ p\geq2,\ \ d\geq2. As by products of our calculations, we infer the $L^p$ norm contractivity of the truncated Riesz transforms $R^t_j$ in terms of $R_j$, and their accurate $L^p$ norms. More precisely, we prove: $$\|R^t_jf\|_{L^p}\leq\|R_jf\|_{L^p}$$ and $$\|R^t_j\|_{L^p}=\|R_j\|_{L^p},$$ for all $10.$Comment: 16 page

Retracted: Inhibition of Corneal Neovascularization by Hydrazinocurcumin

This article previously published in Volume 15 Issue 2 of this journal in February 2016 has been retracted in line with the guidelines from the Committee on Publication Ethics (COPE, http://publicationethics.org/resources/guidelines)Retracted: Zhan W, Zhu J, Zhang Y. Inhibition of corneal neovascularization by hydrazinecurcumin. Trop J Pharm Res 2016; 15(2):349-354 doi: http://dx.doi.org/10.4314/tjpr.v15i2.18

Boundary Schwarz lemma for solutions to non-homogeneous biharmonic equations

In this paper, we establish a boundary Schwarz lemma for solutions to non-homogeneous biharmonic equations.Comment: 9 page

Inhibition of Corneal Neovascularization by Hydrazinocurcumin

Purpose: To investigate the effect of hydrazinocurcumin on a human vascular endothelial growth factor (VEGF)-induced corneal neovascularization in rabbit model.Methods: Murine corneal neovascularization (CorNV) was induced via two intrastromal implantations of VEGF polymer 2 mm from the limbus. Hydrazinocurcumin was administered topically on the cornea 4 times daily for 7 days. The therapeutic effects of hydrazinocurcumin were evaluated daily using slitlamp. At the end of the treatment, the corneas were harvested for H&amp;E staining, masson trichrome staining, immuno-histochemical study, and semi quantification reverse transcription polymerase chain reaction (RT-PCR) was utilized for measurement of inflammation-related molecules.Results: Topical application of hydrazinocurcumin had significant therapeutic effects on CorNV Hydrazinocurcumin extract treatment was more effective in suppressing CorNV in terms of vessel length and levels of cluster of differentiation 31 (CD31) proteins or angiogenesis-related genes such as VEGF, matrix metalloproteinase-2 (MMP2) and matrix metalloproteinase-9 (MMP9). The average length of vessels in hydrazinocurcumin-treated group was only 17 % of that in the control group. Hydrazinocurcumin also inhibited inflammation more markedly by more effectively inhibiting mononuclear and polymorphonuclear cell infiltration into the corneal stroma and reducing levels of stromal cell-derived factor-1 (SDF1), tumor necrosis factor-alpha (TNFα) and macrophage inflammatory protein-3 (MIP3a). In addition, the corneas of hydrazinocurcumin group had a more regular and compact architecture of collagen with thinner corneal thickness than those of the untreated group.Conclusion: Hydrazinocurcumin inhibited human vascular endothelial growth factor (VEGF)-induced rabbit corneal neovascularization and thus can potentially be used for its treatment.Keywords: Hydrazinocurcumin, Corneal neovascularization, Inflammation, Vascular endothelial growth factor, Corneal thicknes