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Carleson measure characterizations of the Campanato type space associated with Schrödinger operators on stratified Lie groups

  • Yixin Wang , Yu Liu , Chuanhong Sun and Pengtao Li EMAIL logo
Published/Copyright: April 17, 2020
Forum Mathematicum
From the journal Forum Mathematicum Volume 32 Issue 5

Abstract

Let ℒ=-Δ𝔾+V be a Schrödinger operator on the stratified Lie group 𝔾, where Δ𝔾 is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class Bq0 with q0>𝒬/2 and 𝒬 is the homogeneous dimension of 𝔾. In this article, by Campanato type spaces Λℒα⁢(𝔾), we introduce Hardy type spaces associated with ℒ denoted by Hℒp⁢(𝔾) and prove the atomic characterization of Hℒp⁢(𝔾). Further, we obtain the following duality relation:

Λℒ𝒬⁢(1/p-1)⁢(𝔾)=(Hℒp⁢(𝔾))∗,𝒬/(𝒬+δ)<p<1 for⁢δ=min⁡{1,2-𝒬/q0}.

The above relation enables us to characterize Λℒα⁢(𝔾) via two families of Carleson measures generated by the heat semigroup and the Poisson semigroup, respectively. Also, we obtain two classes of perturbation formulas associated with the semigroups related to ℒ. As applications, we obtain the boundedness of the Littlewood–Paley function and the Lusin area function on Λℒα⁢(𝔾).

Keywords: Campanato type spaces; stratified Lie groups; Schrödinger operator; Carleson measures; square functions
MSC 2010: 22E30; 42B35; 35J10; 47B38

Communicated by Christopher D. Sogge

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11871293

Award Identifier / Grant number: 11571217

Award Identifier / Grant number: 11671031

Funding source: Natural Science Foundation of Shandong Province

Award Identifier / Grant number: ZR2017JL008

Funding source: Beijing Municipal Science and Technology Commission

Award Identifier / Grant number: Z17111000220000

Funding statement: Pengtao Li was in part supported by National Natural Science Foundation of China (#11871293 and #11571217) and Shandong Natural Science Foundation of China (#ZR2017JL008). Yu Liu was supported by National Natural Science Foundation of China (#11671031) and Beijing Municipal Science and Technology Project (#Z17111000220000).

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Received: 2019-08-20
Revised: 2019-11-06
Published Online: 2020-04-17
Published in Print: 2020-09-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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  2. Conjugacy classes and automorphisms of twin groups
  3. Decomposition and classification of length functions
  4. Successive coefficients of close-to-convex functions
  5. Frobenius cowreaths and Morita contexts
  6. The non-linear sewing lemma III: Stability and generic properties
  7. On hyperquadrics containing projective varieties
  8. Explicit Burgess-like subconvex bounds for GL2 × GL1
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  10. Solvable Lie and Leibniz superalgebras with a given nilradical
  11. Alvis–Curtis duality for representations of reductive groups with Frobenius maps
  12. The third partial cohomology group and existence of extensions of semilattices of groups by groups
  13. Instanton bundles on two Fano threefolds of index 1
  14. Carleson measure characterizations of the Campanato type space associated with Schrödinger operators on stratified Lie groups
https://doi.org/10.1515/forum-2019-0224
Keywords for this article
Campanato type spaces; stratified Lie groups; Schrödinger operator; Carleson measures; square functions

Articles in the same Issue

  1. Frontmatter
  2. Conjugacy classes and automorphisms of twin groups
  3. Decomposition and classification of length functions
  4. Successive coefficients of close-to-convex functions
  5. Frobenius cowreaths and Morita contexts
  6. The non-linear sewing lemma III: Stability and generic properties
  7. On hyperquadrics containing projective varieties
  8. Explicit Burgess-like subconvex bounds for GL2 × GL1
  9. Characterisation of polyhedral products with finite generalised Postnikov decomposition
  10. Solvable Lie and Leibniz superalgebras with a given nilradical
  11. Alvis–Curtis duality for representations of reductive groups with Frobenius maps
  12. The third partial cohomology group and existence of extensions of semilattices of groups by groups
  13. Instanton bundles on two Fano threefolds of index 1
  14. Carleson measure characterizations of the Campanato type space associated with Schrödinger operators on stratified Lie groups
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