Extension and Refinement of Spectral Gap Inequalities for Laguerre and Uniform Measures

This paper investigates the Laguerre semigoup (Pₜα)t≥ 0,\, α>-1, generated by the heat semigroup Lα:=xd²dx²+(α+1 - x)d/dx. Our studies conducted withen the framework of a weighted Hilbert space L²([0,+∞[,dμα) associated with the normalized Laguerre probability measure μα(dx):=cαxαe⁻ˣdx. A central aspect of our methodology involves exploiting a fundamental commutation relation between the semigroup and the derivation operator, this approach enables us to establish a new family of sharp integral inequalities related to this operator with the Dirichlet form ∫₀+∞xf²dμα. We also recall the Sturm–Liouville operator to derive new integral inequalities for the uniform measure on [0, 1], associated with the Dirichlet form ∫₀¹f'²dx. These estimates naturally extend and refine the classical spectral gap inequality.