Tehnika Bellmanovih funkcija u matematičkoj analizi

master's thesis

• Author: Marin Bužančić
• Mentor: Vjekoslav Kovač (mentor)
• Committee member: Vjekoslav Kovač (predsjednik povjerenstva)
• Committee member: Marko Vrdoljak (član povjerenstva)
• Committee member: Ljiljana Arambašić (član povjerenstva)
• Committee member: Matija Kazalicki (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-09-23, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu obrazlaže se osnovni princip tehnike Bellmanovih funkcija primjenama na nekoliko konkretnih problema u harmonijskoj analizi. Najprije se definiraju osnovni pojmovi i dokazuju pomoćni rezultati potrebni u tehnici Bellmanovih funkcija. Potom se detaljno opisuje navedena tehnika na primjerima dokaza dijadske \(L^p\) verzije Carlesonog teorema ulaganja i dokaza dijadske \(L^p\) verzije Hardy-Littlewoodovog maksimalnog teorema, gdje se konstruira Bellmanova funkcija kao rješenje homogene Monge-Ampèreove jednadžbe s rubnim uvjetima. Na kraju se dokazuje nužnost sumacijskih uvjeta za nekoliko klasa dijadskih težina.

Abstract (english)

In this thesis the basic principle of the Bellman function technique is explained on a few specific problems in harmonic analysis. Firstly, basic concepts are defined and additional results needed in the Bellman function technique are proven. Then, the mentioned technique is described in detail by examples of proof of the dyadic \(L^p\) version of Carleson embedding theorem and the proof of the dyadic \(L^p\) version of Hardy-Littlewood maximal theorem, where the Bellman function is constructed as a solution to the homogeneous Monge-Amp èreove equation with boundary conditions. Lastly, the necessity of summation conditions is proven for a few classes of dyadic weights.

• Keywords: tehnike Bellmanovih funkcija; harmonijska analiza; Carlesonov teorem ulaganja; Hardy-Littlewoodov maksimalni teorem
• Keywords (english): Bellman funcions technique; harmonic analysis; Carleson embedding theorem; Hardy-Littlewood maximal theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:313165
• Study programme: Title: Pure Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-01-07 09:45:59