Weak Hardy Spaces WHₗᵖ(Rⁿ) Associated to Operators Satisfying k-Davies-Gaffney Estimates

الأصل · EN

Let L be a one-to-one operator of type ω having a bounded H∞ functional calculus and satisfying the k-Davies-Gaffney estimates with k∈N. In this paper, the authors introduce the weak Hardy space WHₗᵖ(Rⁿ) associated to L for p∈ (0,1] via the non-tangential square function Sₗ and establish a weak molecular characterization of WHₗᵖ(Rⁿ). Typical examples of such operators include the 2k-order divergence form homogeneous elliptic operator L₁:=(-1)ᵏ∑|α|₌ₖ₌|ᵦ|∂β(aα,ᵦ∂α), where {aα,ᵦ}|α|₌ₖ₌|ᵦ| are complex bounded measurable functions, and the 2k-order Schrödinger type operator L₂:= (-Δ)ᵏ+Vᵏ, where Δ is the Laplacian operator and 0≤ V∈ Lᵏ(Rⁿ). As applications, for i∈{1,2} and p∈(n/n+k,1], the authors prove that the associated Riesz transform ∇ᵏ (Lᵢ⁻¹/²) is bounded from WHᵖₗᵢ(Rⁿ) to the classical weak Hardy space WHᵖ(Rⁿ) and, for all 0<p<r≤1 and α=n(1/p-1/r), the fractional power Lᵢ-α2k is bounded from WHₗᵢᵖ(Rⁿ) to WHₗᵢʳ(Rⁿ). Furthermore, the authors find the dual space of WHₗᵖ(Rⁿ) for p∈(0,1], which can be defined via mean oscillations based on some subtle coverings of bounded open sets and, even when L:=-Δ, are also previously unknown. In particular, if L is a nonnegative self-adjoint operator in L²(Rⁿ) satisfying the Davies-Gaffney estimates, the authors further establish the weak atomic characterization of WHₗᵖ(Rⁿ).