Skip to main content

Part of the book series: Progress in Mathematics ((PM,volume 346))

  • 215 Accesses


This chapter is concerned with identifying three pre-Hardy spaces, \(\mathbb {H}_L^p\), \(\mathbb {H}_L^{1,p}\), and \(\mathbb {H}_{DB}^p\), that play a crucial role for Dirichlet and regularity problems, with classical smoothness spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions


  1. A. Amenta, P. Auscher, Elliptic Boundary Value Problems with Fractional Regularity Data, volume 37 of CRM Monograph Series (American Mathematical Society, Providence, 2018). The first order approach

    Google Scholar 

  2. P. Auscher, On the Calderón-Zygmund lemma for Sobolev functions.

  3. P. Auscher, On necessary and sufficient conditions for Lp-estimates of Riesz transforms associated to elliptic operators on \(\mathbb {R}^n\) and related estimates. Mem. Am. Math. Soc. 186(871), xviii+ 75 (2007)

    Google Scholar 

  4. P. Auscher, S. Hofmann, J.-M. Martell, Vertical versus conical square functions. Trans. Am. Math. Soc. 364(10), 5469–5489 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Auscher, M. Mourgoglou, Representation and uniqueness for boundary value elliptic problems via first order systems. Rev. Mat. Iberoam. 35(1), 241–315 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  6. P. Auscher, S. Stahlhut, Functional calculus for first order systems of Dirac type and boundary value problems. Mém. Soc. Math. Fr. (N.S.) (144), vii+ 164 (2016)

    Google Scholar 

  7. M. Cowling, I. Doust, A. McIntosh, A. Yagi, Banach space operators with a bounded H functional calculus. J. Austral. Math. Soc. Ser. A 60(1), 51–89 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Hofmann, S. Mayboroda, A. McIntosh, Second order elliptic operators with complex bounded measurable coefficients in Lp, Sobolev and Hardy spaces. Ann. Sci. Éc. Norm. Supér. (4) 44(5), 723–800 (2011)

    Google Scholar 

  9. T. Hytönen, J. van Neerven, M. Veraar, L. Weis, Analysis in Banach Spaces. Vol. II, volume 67 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics (Springer, Cham, 2017). Probabilistic methods and operator theory

    Google Scholar 

  10. W.P. Ziemer, Weakly Differentiable Functions, volume 120 of Graduate Texts in Mathematics. (Springer, New York, 1989). Sobolev spaces and functions of bounded variation

    Google Scholar 

Download references

Author information

Authors and Affiliations


Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Auscher, P., Egert, M. (2023). Identification of Adapted Hardy Spaces. In: Boundary Value Problems and Hardy Spaces for Elliptic Systems with Block Structure. Progress in Mathematics, vol 346. Birkhäuser, Cham.

Download citation

Publish with us

Policies and ethics