Parabolic Equations Involving Laguerre Operators And Weighted Mixed-norm Estimates

In this paper we study the equation?tu =-Lαu+f and the corresponding Cauchy problem,where Lα represents the Laguerre operator Lα = 1/2((-d2/dx2+-x2+1/x2(α2-1/4),for every α≥-1/2.It is novel that we regard?t+Lα as an integral operator,and consider relevant parabolic heat-diffusion semigroup {e-τ(?t+Lα｝τ＞0.We get explicit pointwise formulas for the classical solution and its derivatives by taking advantages that the Laguerre operators Lα are nice(in a suitable sense)perturbations of the Her-mite operator.In addiction,we reveal weighted mixed-norm estimates for solution to degenerate parabolic extension problem arising in connection with the fractional space-time nonlocal equation(?t+Lα)su=f,for 0<s<1.