REAL-VARIABLE CHARACTERIZATIONS OF HARDY SPACES ASSOCIATED WITH BESSEL OPERATORS

Let λ > 0, p ∈ ((2λ + 1)/(2λ + 2), 1], and [Formula: see text] be the Bessel operator. In this paper, the authors establish the characterizations of atomic Hardy spaces Hp((0,∞),dmλ) associated with △λ in terms of the radial maximal function, the nontangential maximal function, the grand maximal function, the Littlewood–Paley g-function and the Lusin-area function, where dmλ(x) ≡ x2λ dx. As an application, the authors further obtain the Riesz transform characterization of these Hardy spaces.

Estimates for the Multiplicative Square Function of Solutions to Nondivergence Elliptic Equations

We prove distributional inequalities that imply the comparability of theLpnorms of the multiplicative square function ofuand the nontangential maximal function oflogu, whereuis a positive solution of a nondivergence elliptic equation. We also give criteria for singularity and mutual absolute continuity with respect to harmonic measure of any Borel measure defined on a Lipschitz domain based on these distributional inequalities. This extends recent work of M. González and A. Nicolau where the term multiplicative square functions is introduced and where the case whenuis a harmonic function is considered.

On the Neumann Problem for the Schrödinger Equations with Singular Potentials in Lipschitz Domains

We consider the Neumann problem for the Schrödinger equations –Δu + Vu = 0, with singular nonnegative potentials V belonging to the reverse Hölder class ℬn, in a connected Lipschitz domain Ω Rn. Given boundary data g in Hp or Lp for 1 – ɛ < p ≤ 2, where 0 < ɛ < , it is shown that there is a unique solution, u, that solves the Neumann problem for the given data and such that the nontangential maximal function of ▽u is in Lp(∂Ω). Moreover, the uniform estimates are found.