Parabolic bounded mean oscillation and Muckenhoupt weights

Research output: ThesisDoctoral ThesisCollection of Articles

Abstract

This thesis further develops the parabolic theory of functions of bounded mean oscillation (BMO) and Muckenhoupt weights motivated by one-sided maximal functions and a doubly nonlinear parabolic partial differential equation of p-Laplace type. The definition of parabolic BMO consists of two conditions on the mean oscillation of a function, one in the past and the other one in the future with a time lag between the estimates. Various parabolic John–Nirenberg inequalities, which give exponential decay estimates for the oscillation of a function, are shown in the natural geometry of the partial differential equation. We extend and complement the existing theory for the parabolic Muckenhoupt Aq weights and obtain a complete theory for the limiting parabolic Muckenhoupt A1 class including factorization and characterization results. In particular, an uncentered parabolic maximal function with a time lag is applied leading to a more streamlined theory. Weighted norm inequalities are shown for the parabolic maximal function which allows us to establish parabolic versions of the Jones factorization and the Coifman–Rochberg characterization. The other endpoint class of parabolic Muckenhoupt A∞ weights is also discussed and new characterizations are discovered in terms of quantitative absolute continuity with a time lag. Furthermore, this is considered from the perspective of parabolic reverse Hölder inequalities. We obtain several characterizations and self-improving properties for the weights satisfying a parabolic reverse Hölder inequality and study their connection to parabolic Muckenhoupt weights. Parabolic Muckenhoupt weights satisfy the parabolic reverse Hölder inequality, whereas the reverse direction is investigated in terms of a parabolic doubling condition with a time lag. Essential tools in the parabolic theory include delicate parabolic Calderón–Zygmund decompositions, good lambda estimates, covering and chaining arguments. In addition to parabolic BMO, different function spaces of BMO type are studied in the setting of metric measure spaces with a doubling measure. We consider the John–Nirenberg space defined via medians and a weak version of the Gurov–Reshetnyak class. Moreover, we show the corresponding John–Nirenberg inequalities and discuss their consequences. The John–Nirenberg lemma for the median-type John–Nirenberg space gives a polynomial decay estimate for the oscillation of a function. On the other hand, the John–Nirenberg lemma for the weak Gurov–Reshetnyak class provides a specific decay estimate.
Translated title of the contributionParabolinen rajoitettu keskivärähtely ja Muckenhouptin painot
Original languageEnglish
QualificationDoctor's degree
Awarding Institution
  • Aalto University
Supervisors/Advisors
  • Kinnunen, Juha, Supervising Professor
  • Kinnunen, Juha, Thesis Advisor
Publisher
Print ISBNs978-952-64-1842-1
Electronic ISBNs978-952-64-1843-8
Publication statusPublished - 2024
MoE publication typeG5 Doctoral dissertation (article)

Keywords

  • parabolic BMO
  • parabolic Muckenhoupt weights
  • John–Nirenberg inequality
  • parabolic maximal function
  • reverse Hölder inequality
  • one-sided weights
  • doubly nonlinear equation
  • Gurov–Reshetnyak class
  • John–Nirenberg space
  • median
  • doubling measure
  • metric space

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