On the connection between Kronecker and Hadamard convolution products of matrices and some applications.

We are concerned with Kronecker and Hadamard convolution products and present some important connections between these two products. Further we establish some attractive inequalities for Hadamard convolution product. It is also proved that the results can be extended to the finite number of matrices, and some basic properties of matrix convolution products are also derived

Про згортки на просторах конфігурацій. І. Простори скінченних конфігурацій

Рассмотрены два типа сверток (∗ и ⋆) функций на пространствах конечных конфигураций (конечных подмножеств фазового пространства), исследованы некоторые их свойства. Показана связь ∗-свертки со сверткой мер на пространствах конечных конфигураций. Изучены свойства операторов умножения и дифференцирования относительно ∗-свертки. Найдены условия, при которых ∗-свертка функций положительно определена относительно ⋆-свертки.We consider two types of convolutions (* and ★) of functions on spaces of finite configurations (finite subsets of a phase space) and study some of their properties. A relationship between the *-convolution and the convolution of measures on spaces of finite configurations is described. Properties of the operators of multiplication and derivation with respect to the *-convolution are investigated. We also present conditions under which the *-convolution is positive-definite with respect to the ★-convolution

De Giorgi's theorem on the isoperimetric property of the hypersphere

openThe aim of the thesis is to prove the isoperimetric property of the hypersphere as formulated by Ennio De Giorgi in his 1958 article. We will introduce the necessary elements of measure and integration  theory as well as some basic properties of convolution and the general version of the  Gauss-Green theorem. Subsequently, we will define functions of bounded variation in several variables and sets with finite perimeter. Finally, we will discuss some important properties of sets of finite perimeter such as approximation by polygonal domains and the monotonicity of perimeter under Steiner symmetrization. With these tools, we will show the isoperimetric property of the  hypersphere following  De Giorgi's original proof

Negative powers of Laguerre operators

We study negative powers of Laguerre differential operators in $\R$, $d\ge1$. For these operators we prove two-weight $L^p-L^q$ estimates, with ranges of $q$ depending on $p$. The case of the harmonic oscillator (Hermite operator) has recently been treated by Bongioanni and Torrea by using a straightforward approach of kernel estimates. Here these results are applied in certain Laguerre settings. The procedure is fairly direct for Laguerre function expansions of Hermite type, due to some monotonicity properties of the kernels involved. The case of Laguerre function expansions of convolution type is less straightforward. For half-integer type indices $\alpha$ we transfer the desired results from the Hermite setting and then apply an interpolation argument based on a device we call the {\sl convexity principle} to cover the continuous range of $\alpha\in[-1/2,\infty)^d$. Finally, we investigate negative powers of the Dunkl harmonic oscillator in the context of a finite reflection group acting on $\R$ and isomorphic to $\mathbb Z^d_2$. The two weight $L^p-L^q$ estimates we obtain in this setting are essentially consequences of those for Laguerre function expansions of convolution type.Comment: 30 page