Sharp Hardy’s Inequality for Orthogonal Expansions in $$H^p$$ Spaces

Abstract

AbstractHardy’s inequality on $$H^p$$ H p spaces, $$p\in (0,1]$$ p ∈ ( 0 , 1 ] , in the context of orthogonal expansions is investigated for general bases on a wide class of domains in $$\mathbb {R}^d$$ R d with Lebesgue measure. The obtained result is applied to various Hermite, Laguerre, and Jacobi expansions. For that purpose some delicate estimates of the higher order derivatives for the underlying functions and of the associated heat or Poison kernels are proved. Moreover, sharpness of studied Hardy’s inequalities is justified by a construction of an explicit counterexample, which is adjusted to all considered settings.