On considère un polynôme $P$, à coefficients réels non négatifs, à deux indéterminées. On montre que la connaissance des pôles des intégrales$${\int }_0^1{\int }_0^1{x}_1^{{\beta }_1-1}{x}_2^{{\beta }_2-1}P(x_1,x_2{)}^{s}d{x}_{1}d{x}_2$$donne des renseignements sur les racines du polynômes de Bernstein de $P$. La détermination des pôles des intégrales peut se faire en utilisant certaines méthodes de Mellin. Des calculs explicites sont donnés.

We establish weighted Hardy-Littlewood inequalities for radial derivative and fractional radial derivatives on bounded symmetric domains.

An RD-space $𝒳$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds. The authors prove that for a space of homogeneous type $𝒳$ having “dimension” $n$, there exists a $p_0\in (n/(n+1),1)$ such that for certain classes of distributions, the $L^p\left(𝒳\right)$ quasi-norms of their radial maximal functions and grand maximal functions are equivalent when $p\in (p_0,\infty ]$. This result yields a radial maximal function characterization for Hardy spaces on $𝒳$.

The aim of this paper is to establish a random coincidence degree theory. This degree theory possesses all the usual properties of the deterministic degree theory such as existence of solutions, excision and Borsuk’s odd mapping theorem. Our degree theory provides a method for proving the existence of random solutions of the equation $Lx\in N(\omega ,x)$ where $L:\mathit{\text{dom}}L\subset X\to Z$ is a linear Fredholm mapping of index zero and $N:\Omega \times \overline{G}\to 2^Z$ is a noncompact Carathéodory mapping. Applications to random differential inclusions are also considered.

Let $(\Omega ,\Sigma )$ be a measurable space and $C$ a nonempty bounded closed convex separable subset of $p$-uniformly convex Banach space $E$ for some $p>1$. We prove random fixed point theorems for a class of mappings $T\Omega \times C\to C$ satisfying: for each $x,y\in C$, $\omega \in \Omega$ and integer $n\ge 1$, $$\parallel T^n(\omega ,x)-T^n(\omega ,y)\parallel \le a\left(\omega \right)·\parallel x-y\parallel +b\left(\omega \right)\{\parallel x-T^n(\omega ,x)\parallel +\parallel y-T^n(\omega ,y)\parallel \}+c\left(\omega \right)\{\parallel x-T^n(\omega ,y)\parallel +\parallel y-T^n(\omega ,x)\parallel \},$$ where $a,b,c\Omega \to [0,\infty )$ are functions satisfying certain conditions and $T^n(\omega ,x)$ is the value at $x$ of the $n$-th iterate of the mapping $T(\omega ,·)$. Further we establish for these mappings some random fixed point theorems in a Hilbert space, in $L^p$ spaces, in Hardy spaces $H^p$ and in Sobolev spaces $H^{k,p}$...