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Radial averaging operator acting on Bergman and Lebesgue spaces

  • Taneli Korhonen EMAIL logo , José Ángel Peláez and Jouni Rättyä
Published/Copyright: May 11, 2019
Forum Mathematicum
From the journal Forum Mathematicum Volume 31 Issue 4

Abstract

It is shown that the radial averaging operator

Tω⁢(f)⁢(z)=∫|z|1f⁢(s⁢z|z|)⁢ω⁢(s)⁢𝑑sω^⁢(z),ω^⁢(z)=∫|z|1ω⁢(s)⁢𝑑s,

induced by a radial weight ω on the unit disc 𝔻, is bounded from the weighted Bergman space Aνp, where 0<p<∞ and the radial weight ν satisfies ν^⁢(r)≤C⁢ν^⁢(1+r2) for all 0≤r<1, to Lνp if and only if the self-improving condition

sup0≤r<1⁡ω^⁢(r)p∫r1s⁢ν⁢(s)⁢𝑑s⁢∫0rt⁢ν⁢(t)ω^⁢(t)p⁢𝑑t<∞

is satisfied. Further, two characterizations of the weak-type inequality

η⁢({z∈𝔻:|Tω⁢(f)⁢(z)|≥λ})≲λ-p⁢∥f∥Lνpp,λ>0,

are established for arbitrary radial weights ω, ν and η. Moreover, differences and interrelationships between the cases Aνp→Lνp, Lνp→Lνp and Lνp→Lνp,∞ are analyzed.

Keywords: Inner function; Bergman space; Hardy space; radial averaging operator; Schwarz–Pick lemma; doubling weight; Muckenhoupt weight; maximal function; Carleson measure
MSC 2010: 30H20; 47G10

Communicated by Siegfried Echterhoff

Funding source: Ministerio de Economía y Competitividad

Award Identifier / Grant number: MTM2014-52865-P

Award Identifier / Grant number: MTM2017-90584-REDT

Funding source: Junta de Andalucía

Award Identifier / Grant number: FQM210

Funding source: Suomen Akatemia

Award Identifier / Grant number: 268009

Funding statement: This research was supported in part by Ministerio de Economía y Competitividad, Spain, projects MTM2014-52865-P and MTM2017-90584-REDT; Junta de Andalucía, project FQM210; Academy of Finland project no. 268009, and by Faculty of Science and Forestry of University of Eastern Finland.

Acknowledgements

We would like to thank professor Francisco J. Martín-Reyes for helpful conversations about Hardy operators and pointing out relevant references on the topic.

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Received: 2018-11-29
Revised: 2019-04-09
Published Online: 2019-05-11
Published in Print: 2019-07-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Characteristic functions of semigroups in semi-simple Lie groups
  3. Damping estimates for oscillatory integral operators with real-analytic phases and its applications
  4. Formality properties of finitely generated groups and Lie algebras
  5. Toric actions in cosymplectic geometry
  6. On classification of typical representations for GL3(F)
  7. Integrable generators of Lie algebras of vector fields on ℂn
  8. Expanding phenomena over matrix rings
  9. On sup-norm bounds part II: GL(2) Eisenstein series
  10. Sublinear elliptic problems under radiality. Harmonic NA groups and Euclidean spaces
  11. Partial and full boundary regularity for non-autonomous functionals with Φ-growth conditions
  12. Radial averaging operator acting on Bergman and Lebesgue spaces
  13. The 𝑝-adic variation of the Gross–Kohnen–Zagier theorem
https://doi.org/10.1515/forum-2018-0293
Keywords for this article
Inner function; Bergman space; Hardy space; radial averaging operator; Schwarz–Pick lemma; doubling weight; Muckenhoupt weight; maximal function; Carleson measure

Articles in the same Issue

  1. Frontmatter
  2. Characteristic functions of semigroups in semi-simple Lie groups
  3. Damping estimates for oscillatory integral operators with real-analytic phases and its applications
  4. Formality properties of finitely generated groups and Lie algebras
  5. Toric actions in cosymplectic geometry
  6. On classification of typical representations for GL3(F)
  7. Integrable generators of Lie algebras of vector fields on ℂn
  8. Expanding phenomena over matrix rings
  9. On sup-norm bounds part II: GL(2) Eisenstein series
  10. Sublinear elliptic problems under radiality. Harmonic NA groups and Euclidean spaces
  11. Partial and full boundary regularity for non-autonomous functionals with Φ-growth conditions
  12. Radial averaging operator acting on Bergman and Lebesgue spaces
  13. The 𝑝-adic variation of the Gross–Kohnen–Zagier theorem
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