Local Hardy spaces associated with inhomogeneous higher order elliptic operators

Jun Cao, Svitlana Mayboroda, Dachun Yang

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

Let L be a divergence form inhomogeneous higher order elliptic operator with complex bounded measurable coefficients. In this paper, for all p∈(0,∞) and L satisfying a weak ellipticity condition, the authors introduce the local Hardy spaces hLp(Rn) associated with L, which coincide with Goldberg's local Hardy spaces hLp(Rn) for all p∈(0,∞) when ≡-Δ (the Laplace operator). The authors also establish a real-variable theory of hLp(Rn), which includes their characterizations in terms of the local molecules, the square functions or the maximal functions, the complex interpolation and dual spaces. These real-variable characterizations on the local Hardy spaces are new even when ≡-div(A∇) (the divergence form homogeneous second-order elliptic operator). Moreover, the authors show that hLp(Rn) coincides with the Hardy space HpL(Rn) associated with the operator L + δ for all p∈(0,∞), where δ is some positive constant depending on the ellipticity and the off-diagonal estimates of L. As an application, the authors establish some mapping properties for the local Riesz transforms ∇k(L+δ))-1/2 on HL+δp(Rn), where k {0,...,m} and p∈(0, 2].

Original languageEnglish (US)
Pages (from-to)137-224
Number of pages88
JournalAnalysis and Applications
Volume15
Issue number2
DOIs
StatePublished - Mar 1 2017

Keywords

  • Higher order elliptic operator
  • Lipschitz space
  • Riesz transform
  • complex interpolation
  • ellipticity condition
  • local Hardy space
  • maximal function
  • molecule
  • off-diagonal estimate
  • parabolic Caccioppoli inequality
  • square function
  • tent space

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