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

This paper investigates the Laguerre semigoup (Pₓ^) ₓ ₀, \, >-1, generated by the heat semigroup L^: =xd^2{dx^{2}}+ (+1 - x) ddx. Our studies conducted withen the framework of a weighted Hilbert space L^2 (0, +[, d_) associated with the normalized Laguerre probability measure _ (dx): =c_x^e^-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^+xf^2d_. We also recall the Sturm–Liouville operator to derive new integral inequalities for the uniform measure on [0, 1, associated with the Dirichlet form ₀^₁f'^2dx. These estimates naturally extend and refine the classical spectral gap inequality.