Home Carleson measure characterizations of the Campanato type space associated with Schrödinger operators on stratified Lie groups
Article
Licensed
Unlicensed Requires Authentication

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).

References

[1] C. Bennett, R. A. DeVore and R. Sharpley, Weak-L∞ and BMO, Ann. of Math. (2) 113 (1981), no. 3, 601–611. 10.2307/2006999Search in Google Scholar

[2] R. R. Coifman, Y. Meyer and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), no. 2, 304–335. 10.1016/0022-1236(85)90007-2Search in Google Scholar

[3] X. T. Duong, L. Yan and C. Zhang, On characterization of Poisson integrals of Schrödinger operators with BMO traces, J. Funct. Anal. 266 (2014), no. 4, 2053–2085. 10.1016/j.jfa.2013.09.008Search in Google Scholar

[4] J. Dziubański, Note on H1 spaces related to degenerate Schrödinger operators, Illinois J. Math. 49 (2005), no. 4, 1271–1297. 10.1215/ijm/1258138138Search in Google Scholar

[5] J. Dziubański, G. Garrigós, T. Martínez, J. L. Torrea and J. Zienkiewicz, BMO spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality, Math. Z. 249 (2005), no. 2, 329–356. 10.1007/s00209-004-0701-9Search in Google Scholar

[6] J. Dziubański and J. Zienkiewicz, Hardy space H1 associated to Schrödinger operator with potential satisfying reverse Hölder inequality, Rev. Mat. Iberoam. 15 (1999), no. 2, 279–296. 10.4171/RMI/257Search in Google Scholar

[7] J. Dziubański and J. Zienkiewicz, Hp spaces for Schrödinger operators, Fourier Analysis and Related Topics (Bȩdlewo 2000), Banach Center Publ. 56, Polish Acadademy of Sciences, Warsaw (2002), 45–53. 10.4064/bc56-0-4Search in Google Scholar

[8] G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Math. Notes 28, Princeton University, Princeton, 1982. 10.1515/9780691222455Search in Google Scholar

[9] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Math. Monogr., Oxford University, New York, 1985. Search in Google Scholar

[10] R. Gong, J. Li and L. Song, Besov and Hardy spaces associated with the Schrödinger operator on the Heisenberg group, J. Geom. Anal. 24 (2014), no. 1, 144–168. 10.1007/s12220-012-9331-3Search in Google Scholar

[11] E. Harboure, O. Salinas and B. Viviani, A look at BMOϕ⁢(ω) through Carleson measures, J. Fourier Anal. Appl. 13 (2007), no. 3, 267–284. 10.1007/s00041-005-5044-3Search in Google Scholar

[12] D. S. Jerison and A. Sánchez-Calle, Estimates for the heat kernel for a sum of squares of vector fields, Indiana Univ. Math. J. 35 (1986), no. 4, 835–854. 10.1512/iumj.1986.35.35043Search in Google Scholar

[13] T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Classics Math., Springer, Berlin, 1984. Search in Google Scholar

[14] H.-Q. Li, Estimations Lp des opérateurs de Schrödinger sur les groupes nilpotents, J. Funct. Anal. 161 (1999), no. 1, 152–218. 10.1006/jfan.1998.3347Search in Google Scholar

[15] C.-C. Lin and H. Liu, BMOL⁢(ℍn) spaces and Carleson measures for Schrödinger operators, Adv. Math. 228 (2011), no. 3, 1631–1688. 10.1016/j.aim.2011.06.024Search in Google Scholar

[16] C.-C. Lin, H. Liu and Y. Liu, Hardy spaces associated with Schrödinger operators on the Heisenberg group, preprint (2011), https://arxiv.org/abs/1106.4960. Search in Google Scholar

[17] G. Lu, A Fefferman–Phong type inequality for degenerate vector fields and applications, Panamer. Math. J. 6 (1996), no. 4, 37–57. Search in Google Scholar

[18] T. Ma, P. R. Stinga, J. L. Torrea and C. Zhang, Regularity properties of Schrödinger operators, J. Math. Anal. Appl. 388 (2012), no. 2, 817–837. 10.1016/j.jmaa.2011.10.006Search in Google Scholar

[19] C. Segovia and R. L. Wheeden, On certain fractional area integrals, J. Math. Mech. 19 (1969/1970), 247–262. 10.1512/iumj.1970.19.19023Search in Google Scholar

[20] Z. W. Shen, Lp estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 2, 513–546. 10.5802/aif.1463Search in Google Scholar

[21] M. H. Taibleson and G. Weiss, The molecular characterization of certain Hardy spaces, Representation Theorems for Hardy Spaces, Astérisque 77, Société Mathématique de France, Paris (1980), 67–149. Search in Google Scholar

[22] N. T. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, Cambridge Tracts in Math. 100, Cambridge University, Cambridge, 1992. 10.1017/CBO9780511662485Search in Google Scholar

[23] L. Wu and L. Yan, Heat kernels, upper bounds and Hardy spaces associated to the generalized Schrödinger operators, J. Funct. Anal. 270 (2016), no. 10, 3709–3749. 10.1016/j.jfa.2015.12.016Search in Google Scholar

[24] D. Yang, D. Yang and Y. Zhou, Localized Morrey–Campanato spaces on metric measure spaces and applications to Schrödinger operators, Nagoya Math. J. 198 (2010), 77–119. 10.1215/00277630-2009-008Search in Google Scholar

[25] D. Yang and Y. Zhou, Localized Hardy spaces H1 related to admissible functions on RD-spaces and applications to Schrödinger operators, Trans. Amer. Math. Soc. 363 (2011), no. 3, 1197–1239. 10.1090/S0002-9947-2010-05201-8Search in Google Scholar

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

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
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
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 23.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2019-0224/html
Scroll to top button