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1. CMB 2011 (vol 56 pp. 148)
On the Gras Conjecture for Imaginary Quadratic Fields In this paper we extend K. Rubin's methods to prove the Gras conjecture
for abelian extensions of a given imaginary quadratic field $k$ and
prime numbers $p$ that divide the number of roots of unity in $k$.
Keywords:elliptic units, Stark units, Gras conjecture, Euler systems Categories:11R27, 11R29, 11G16 |
2. CMB 2010 (vol 53 pp. 327)
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 regions Categories:26D15, 26D10 |
3. CMB 2004 (vol 47 pp. 540)
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 regions Categories:46E35, 26D10 |