Vertical and horizontal square functions on a class of non-doubling manifolds

Julian Bailey, Adam Sikora*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We consider a class of non-doubling manifolds M that are the connected sum of a finite number of N-dimensional manifolds of the form Rni×Mi. Following on from the work of Hassell and the second author [20], a particular decomposition of the resolvent operators (Δ+k2)−M, for M∈N, will be used to demonstrate that the vertical square function operator [Formula presented] is bounded on Lp(M) for 1<p<nmin=mini⁡ni and weak-type (1,1). In addition, it will be proved that the reverse inequality ‖f‖p≲‖Sf‖p holds for p∈(nmin,nmin) and that S is unbounded for p≥nmin provided 2M<nmin. Similarly, for M>1, this method of proof will also be used to ascertain that the horizontal square function operator [Formula presented] is bounded on Lp(M) for all 1<p<∞ and weak-type (1,1). Hence, for p≥nmin, the vertical and horizontal square function operators are not equivalent and their corresponding Hardy spaces Hp do not coincide.

Original languageEnglish
Pages (from-to)41-102
Number of pages62
JournalJournal of Differential Equations
Volume358
DOIs
Publication statusPublished - 15 Jun 2023

Keywords

  • Non-doubling spaces
  • Resolvent estimates
  • Square functions

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