Parabolic bounded mean oscillation and Muckenhoupt weights

Kim Myyryläinen

Parabolic bounded mean oscillation and Muckenhoupt weights

Julkaisun otsikon käännös: Parabolinen rajoitettu keskivärähtely ja Muckenhouptin painot

Kim Myyryläinen

Tutkimustuotos: Doctoral Thesis › Collection of Articles

Abstrakti

Väitöskirjassa tutkitaan parabolista rajoitetun keskivärähtelyn funktioluokkaa (BMO) ja parabolisia Muckenhouptin painoja, joiden teoria on motivoitu yksipuolisilla maksimaalifunktioilla ja kaksinkertaisesti epälineaarisilla p-Laplacen tyyppisillä osittaisdifferentiaaliyhtälöillä. Parabolinen BMO koostuu kahdesta funktion keskivärähtelyyn liittyvästä ehdosta, joiden välillä on aikaviive. Ensimmäinen ehto arvioi värähtelyä menneisyydessä ja toinen ehto puolestaan tulevaisuudessa. Useita eri parabolisia John–Nirenbergin epäyhtälöitä näytetään osittaisdifferentiaaliyhtälöiden luontaisessa geometriassa. Epäyhtälöt antavat eksponentiaalisen vaimenemisarvion funktioiden värähtelylle. Työssä laajennetaan ja täydennetään parabolisten Muckenhouptin Aq painojen nykyteoriaa sekä luodaan täydellinen teoria paraboliselle Muckenhouptin A1 ääriluokalle sisältäen faktorointi- ja karakterisaatiotulokset. Teorian kehittämiseen sovelletaan epäkeskitettyä parabolista maksimaalifunktiota aikaviiveellä sekä sen painotettuja normiepäyhtälöitä. Näiden avulla johdetaan paraboliset versiot Jonesin faktoroinnista ja Coifman–Rochbergin karakterisaatiosta. Lisäksi tutkitaan toista parabolisten Muckenhouptin painojen A∞ ääriluokkaa. Luokalle osoitetaan uusia karakterisaatiotuloksia perustuen kvantitatiiviseen absoluuttiseen jatkuvuuteen aikaviiveellä. Teoriaa lähestytään myös parabolisten käänteisten Hölderin epäyhtälöiden näkökulmasta, joiden luokalle todistetaan useita karakterisaatiotuloksia ja itseparantuvuusominaisuuksia. Sen lisäksi selvitetään kyseisen luokan yhteyttä parabolisiin Muckenhouptin painoihin. Paraboliset Muckenhouptin painot toteuttavat parabolisen käänteisen Hölderin epäyhtälön, kun taas käänteistä suuntaa tarkastellaan aikaviiveellisen parabolisen tuplausehdon avulla. Olennaisia työkaluja parabolisessa teoriassa ovat muun muassa paraboliset Calderón–Zygmundin hajotelmat, hyvä-lambdan estimaatit, peite- ja ketjutusargumentit. Parabolisen BMO:n ohella tutkitaan erilaisia BMO-tyyppisiä funktioavaruuksia metrisissä mitta-avaruuksissa tuplaavalla mitalla. Erityisesti käsitellään John–Nirenbergin avaruutta mediaaneilla ja heikkoa versiota Gurov–Reshetnyakin luokasta sekä vastaavia John–Nirenbergin epäyhtälöitä ja niiden seurauksia. Mediaanityyppisen John–Nirenbergin avaruuden John–Nirenbergin epäyhtälö antaa funktioiden värähtelylle polynomisen vaimenenisarvion. Sen sijaan heikon Gurov–Reshetnyakin luokan vastaava lause antaa spesifisen vaimenemisarvion värähtelylle.

Julkaisun otsikon käännös	Parabolinen rajoitettu keskivärähtely ja Muckenhouptin painot
Alkuperäiskieli	Englanti
Pätevyys	Tohtorintutkinto
Myöntävä instituutio	Aalto-yliopisto
Valvoja/neuvonantaja	Kinnunen, Juha, Vastuuprofessori Kinnunen, Juha, Ohjaaja
Kustantaja	Aalto University
Painoksen ISBN	978-952-64-1842-1
Sähköinen ISBN	978-952-64-1843-8
Tila	Julkaistu - 2024
OKM-julkaisutyyppi	G5 Artikkeliväitöskirja

Tutkimusalat

Parabolinen BMO
paraboliset Muckenhouptin painot
John–Nirenbergin epäyhtälö
parabolinen maksimaalifunktio
käänteinen Hölderin epäyhtälö
yksipuoliset painot
kaksinkertaisesti epälineaarinen yhtälö
Gurov–Reshetnyakin luokka
John–Nirenbergin avaruus
mediaani
tuplaava mitta
metrinen avaruus

Pääsy asiakirjaan

http://urn.fi/URN:ISBN:978-952-64-1843-8

Siteeraa tätä

@phdthesis{e4072a5ba7274829ade8f59ede63bb31,

title = "Parabolic bounded mean oscillation and Muckenhoupt weights",

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 H{\"o}lder inequalities. We obtain several characterizations and self-improving properties for the weights satisfying a parabolic reverse H{\"o}lder inequality and study their connection to parabolic Muckenhoupt weights. Parabolic Muckenhoupt weights satisfy the parabolic reverse H{\"o}lder 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 Calder{\'o}n–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.",

keywords = "parabolic BMO, parabolic Muckenhoupt weights, John–Nirenberg inequality, parabolic maximal function, reverse H{\"o}lder inequality, one-sided weights, doubly nonlinear equation, Gurov–Reshetnyak class, John–Nirenberg space, median, doubling measure, metric space, Parabolinen BMO, paraboliset Muckenhouptin painot, John–Nirenbergin ep{\"a}yht{\"a}l{\"o}, parabolinen maksimaalifunktio, k{\"a}{\"a}nteinen H{\"o}lderin ep{\"a}yht{\"a}l{\"o}, yksipuoliset painot, kaksinkertaisesti ep{\"a}lineaarinen yht{\"a}l{\"o}, Gurov–Reshetnyakin luokka, John–Nirenbergin avaruus, mediaani, tuplaava mitta, metrinen avaruus, parabolic BMO, parabolic Muckenhoupt weights, John–Nirenberg inequality, parabolic maximal function, reverse H{\"o}lder inequality, one-sided weights, doubly nonlinear equation, Gurov–Reshetnyak class, John–Nirenberg space, median, doubling measure, metric space",

author = "Kim Myyryl{\"a}inen",

year = "2024",

language = "English",

isbn = "978-952-64-1842-1",

series = "Aalto University publication series DOCTORAL THESES",

publisher = "Aalto University",

number = "112",

address = "Finland",

school = "Aalto University",

}

TY - BOOK

T1 - Parabolic bounded mean oscillation and Muckenhoupt weights

AU - Myyryläinen, Kim

PY - 2024

Y1 - 2024

N2 - 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.

AB - 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.

KW - parabolic BMO

KW - parabolic Muckenhoupt weights

KW - John–Nirenberg inequality

KW - parabolic maximal function

KW - reverse Hölder inequality

KW - one-sided weights

KW - doubly nonlinear equation

KW - Gurov–Reshetnyak class

KW - John–Nirenberg space

KW - median

KW - doubling measure

KW - metric space

KW - Parabolinen BMO

KW - paraboliset Muckenhouptin painot

KW - John–Nirenbergin epäyhtälö

KW - parabolinen maksimaalifunktio

KW - käänteinen Hölderin epäyhtälö

KW - yksipuoliset painot

KW - kaksinkertaisesti epälineaarinen yhtälö

KW - Gurov–Reshetnyakin luokka

KW - John–Nirenbergin avaruus

KW - mediaani

KW - tuplaava mitta

KW - metrinen avaruus

KW - parabolic BMO

KW - parabolic Muckenhoupt weights

KW - John–Nirenberg inequality

KW - parabolic maximal function

KW - reverse Hölder inequality

KW - one-sided weights

KW - doubly nonlinear equation

KW - Gurov–Reshetnyak class

KW - John–Nirenberg space

KW - median

KW - doubling measure

KW - metric space

M3 - Doctoral Thesis

SN - 978-952-64-1842-1

T3 - Aalto University publication series DOCTORAL THESES

PB - Aalto University

ER -

Parabolic bounded mean oscillation and Muckenhoupt weights

Abstrakti

Tutkimusalat

Pääsy asiakirjaan

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