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 Poincare inequalities as particular cases.