A Family of Integral Inequalities on the Interval $[-1,1]$

Abstract

We study the heat semigroup $(P^{n}_{t})_{t\ge 0}=\{\exp (tL_{n})\}_{t\ge 0}$ generated by the Gegenbauer operator $L_{n}:=(1-x^{2})\frac{d^{2}}{dx^{2}}-nx\frac{d}{dx}$, on the interval $[-1,1]$ equipped with the probability measure    $\mu _{n}(dx):=c_{n}(1-x^{2})^{\frac{n}{2}-1}$, where $c_{n}$ the normalization constant and n is a strictly positive real number. By means of a simple method involving essentially a commutation property between the semigroup and derivation, we describe a large family of optimal integral inequalities with logarithmic Sobolev and Poincaré inequalities as particular cases.