Two-weight norm inequalities for the Cesaro means of Hermite expansions

An accurate estimate is obtained of the Cesaro kernel for Hermite expansions. This is used to prove two-weight norm inequalities for Cesaro means of Hermite polynomial series and for the supremum of these means. These extend known norm inequalities, even in the single power weight and unweighted cases. An almost everywhere convergence result is obtained as a corollary. It is also shown that the conditions used to prove norm boundedness of the means and most of the conditions used to prove the boundedness of the Cesaro supremum of the means are necessary.