Lp−Lr estimates for the heat equation with the Grushin operator

In this work, we rely on the expression of the heat kernel associated to the Grushin operator and prove L p − L r estimates, for 1 ≤ p ≤ r ≤ ∞ . Such estimates are key to describing the behavior of the heat semigroup associated to the Grushin operator between Lebesgue spaces, and to studying the existence of solutions for the semilinear Cauchy problem with power-type nonlinearity.