TY - JOUR
T1 - Local Hardy spaces associated with inhomogeneous higher order elliptic operators
AU - Cao, Jun
AU - Mayboroda, Svitlana
AU - Yang, Dachun
N1 - Publisher Copyright:
© 2017 World Scientific Publishing Company.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.
PY - 2017/3/1
Y1 - 2017/3/1
N2 - 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].
AB - 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].
KW - Higher order elliptic operator
KW - Lipschitz space
KW - Riesz transform
KW - complex interpolation
KW - ellipticity condition
KW - local Hardy space
KW - maximal function
KW - molecule
KW - off-diagonal estimate
KW - parabolic Caccioppoli inequality
KW - square function
KW - tent space
UR - http://www.scopus.com/inward/record.url?scp=85010867750&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85010867750&partnerID=8YFLogxK
U2 - 10.1142/S0219530515500189
DO - 10.1142/S0219530515500189
M3 - Article
AN - SCOPUS:85010867750
VL - 15
SP - 137
EP - 224
JO - Analysis and Applications
JF - Analysis and Applications
SN - 0219-5305
IS - 2
ER -