Integral inequalities related to the Tchebychev semigroup

Abstract

We study the semigroup (P t ) t≥0 generated by the operator $$\mathcal{L}:=(1-x^{2}){\frac{d^{2}}{dx^{2}}}-x\,{\frac{d}{dx}}$$ acting on the space \(\mathbb{L}^{2}([-1,+1],\sigma)\) with respect to the probability measure \(\sigma (dx):={\frac{1}{\pi \sqrt{1-x^{2}}}}\ dx.\) We prove the Sobolev and Onofri inequalities by means of an elementary method involving essentially a commutation property between the semigroup and the derivation.