1. CMB 2010 (vol 53 pp. 327)
 Luor, DahChin

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 starshaped 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 LevinCochranLee type inequalities.
Keywords:multidimensional inequalities, geometric mean operators, exponential inequalities, starshaped regions Categories:26D15, 26D10 

2. CMB 2004 (vol 47 pp. 540)
 Jain, Pankaj; Jain, Pawan K.; Gupta, Babita

Compactness of HardyType Operators over StarShaped 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 starshaped region. The cases covered are (i) when the average is
taken over a difference of two dilations of a starshaped region in
$\RR^N$, and (ii) when the average is taken over all dilations of
starshaped regions in $\RR^N$. These cases include, respectively,
the average over annuli and the average over balls centered at origin.
Keywords:Hardy operator, HardySteklov operator, compactness, boundedness, starshaped regions Categories:46E35, 26D10 
