Anisotropic Singular Integrals in Product Spaces

Let $A_i$ for $i=1, 2$ be an expansive dilation, respectively, on ${\mathbb R}^n$ and ${\mathbb R}^m$ and $\vec A\equiv(A_1, A_2)$. Denote by {\mathcal A}_\infty(\rnm; \vec A) the class of Muckenhoupt weights associated with $\vec A$. The authors introduce a class of anisotropic singular integrals on $\mathbb R^n\times\mathbb R^m$, whose kernels are adapted to $\vec A$ in the sense of Bownik and have vanishing moments defined via bump functions in the sense of Stein. Then the authors establish the boundedness of these anisotropic singular integrals on $L^q_w(\mathbb R^n\times\mathbb R^m)$ with $q\in(1, \infty)$ and $w\in\mathcal A_q(\mathbb R^n\times\mathbb R^m; \vec A)$ or on $H^p_w(\mathbb R^n\times\mathbb R^m; \vec A)$ with $p\in(0, 1]$ and $w\in\mathcal A_\infty(\mathbb R^n \times\mathbb R^m; \vec A)$. These results are also new even when $w=1$.Comment: Sci. China Math., to appea

Degree of urbanization and gender differences in substance use among Slovak adolescents

Substance use among adolescents varies with gender and between countries. Urbanization may contribute to this. The aim of our study is to explore the association between the degree of urbanization and gender differences in adolescent smoking, binge drinking, and cannabis use. A cross-sectional questionnaire survey of Slovak adolescents was used (N = 3,493; mean age = 14.33), stratified by degree of urbanization. The effects of gender and urbanization of the area and their interaction on substance use (smoking, binge drinking, and cannabis) were analyzed using a logistic regression model adjusted for age. Gender and area and their interaction had statistically significant (p <0.01) associations with substance use. The lower the urbanization of the area, the less riskily females behaved. An exception was found in the case of binge drinking where the results of the interaction of gender and degree of urbanization were not significant for the second least urbanized area. Prevalence rate of substance use among girls increased along with an increasing degree of urbanization, while the prevalence rate of substance use among boys remained constant