Weights arising from parabolic partial differential equations

Olli Saari

Research output: ThesisDoctoral ThesisCollection of Articles

Abstract

This thesis is devoted to the study of one-sided weights and parabolic partial differential equations in the Euclidean n-space. We define a tailored maximal operator, whose weighted theory has the ideal connection to the regularity theory of parabolic partial differential equations. It can also be regarded as the multidimensional version of the one-sided maximal function. We give a Muckenhoupt type characterization for the good weights of the parabolic forward-in-time maximal operator. This applies to both weak and strong type weighted norm inequalities. Moreover, we combine the characterization with the classical Rubio de Francia algorithm to prove a factorization result. There is a related class of functions, those with parabolic bounded mean oscillation (parabolic BMO). We prove local-to-global results for their definition and John-Nirenberg inequality. As an application, global integrability of supersolutions to a wide class of parabolic partial differential equations is established. In addition, we give a characterization of the parabolic BMO through maximal functions of Borel measures. Finally, we study related topics in the context of functions of bounded mean oscillation, in the classical sense as defined by John and Nirenberg. We study the relation between quasiconformal and BMO-preserving coordinate changes in the Heisenberg group. In addition, we generalize local-to-global results for a wide scale of BMO type spaces in the context of general metric measure spaces.
Translated title of the contributionParabolisten osittaisdifferentiaaliyhtälöiden painoteoria
Original languageEnglish
QualificationDoctor's degree
Awarding Institution
  • Aalto University
Supervisors/Advisors
  • Kinnunen, Juha, Supervising Professor
  • Kinnunen, Juha, Thesis Advisor
Publisher
Print ISBNs978-952-60-6677-6
Electronic ISBNs978-952-60-6678-3
Publication statusPublished - 2016
MoE publication typeG5 Doctoral dissertation (article)

Keywords

  • parabolic partial differential equation
  • heat equation
  • weighted norm inequality
  • maximal operator
  • Muckenhoupt weight
  • BMO
  • John-Nirenberg inequality
  • metric space

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