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Search: All articles in the CMB digital archive with keyword star-shaped regions

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1. CMB 2010 (vol 53 pp. 327)

Luor, Dah-Chin
 Multidimensional Exponential Inequalities with Weights We establish sufficient conditions on the weight functions $u$ and $v$ for the validity of the multidimensional weighted inequality $$\Bigl(\int_E \Phi(T_k f(x))^q u(x)\,dx\Bigr)^{1/q} \le C \Bigl (\int_E \Phi(f(x))^p v(x)\,dx\Bigr )^{1/p},$$ where 0<$p$, $q$<$\infty$, $\Phi$ is a logarithmically convex function, and $T_k$ is an integral operator over star-shaped regions. The condition is also necessary for the exponential integral inequality. Moreover, the estimation of $C$ is given and we apply the obtained results to generalize some multidimensional Levin--Cochran-Lee type inequalities. Keywords:multidimensional inequalities, geometric mean operators, exponential inequalities, star-shaped regionsCategories:26D15, 26D10

2. CMB 2004 (vol 47 pp. 540)

Jain, Pankaj; Jain, Pawan K.; Gupta, Babita
 Compactness of Hardy-Type Operators over Star-Shaped Regions in $\mathbb{R}^N$ We study a compactness property of the operators between weighted Lebesgue spaces that average a function over certain domains involving a star-shaped region. The cases covered are (i) when the average is taken over a difference of two dilations of a star-shaped region in $\RR^N$, and (ii) when the average is taken over all dilations of star-shaped regions in $\RR^N$. These cases include, respectively, the average over annuli and the average over balls centered at origin. Keywords:Hardy operator, Hardy-Steklov operator, compactness, boundedness, star-shaped regionsCategories:46E35, 26D10
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