Hardyjevi prostori

master's thesis

• Author: Fran Domagoj Tomac
• Mentor: Hrvoje Šikić (mentor)
• Committee member: Hrvoje Šikić (predsjednik povjerenstva)
• Committee member: Tomislav Pejković (član povjerenstva)
• Committee member: Marko Erceg (član povjerenstva)
• Committee member: Siniša Slijepčević (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-07-20, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Hardyjevim se prostorima klasično pristupa u kontekstu kompleksne analize gdje ih se promatra na nekom dijelu kompleksne ravnine. U ovom je tekstu, međutim, predstavljen moderan tretman prostora \(H^p(\mathbb{R}^n)\), uz \(0<p\leq \infty\), kao klase onih temperiranih distribucija kojima je prikladno definirani maksimalni operator sadržan u \(L^p(\mathbb{R}^n)\) prostoru. Prvi važni dokazani rezultat tvrdi da maksimalni operator koji korišten u definiciji može biti izabran na nekoliko ekvivalentnih načina. Osim toga, dokazali smo i da za proizvoljni \(f \in H^p\) možemo pronaći familiju \(\{a_k\}\) jednostavnijih elemenata te kompleksni niz koeficijenata \(\lambda_k\) tako da je \(f=\Sigma_k\lambda_ka_k\) u \(H^p\) normi; elemente \(a_k\) nazivamo atomima, a rezultirajuću dekompoziciju atomarnom. Konačno, dan je i detaljniji pregled posebno zanimljivog prostora \(H^1\) te demonstracija činjenice da je njegov dual prostor funkcija ograničene srednje oscilacije (BMO).

Abstract (english)

One classically addresses Hardy spaces in the context of complex analysis where they are observed on some section of complex plane. In this text, however, a modern treatment of the \(H^p(\mathbb{R}^n)\) space, with \(0<p\leq \infty\), as the class of those tempered distributions whose appropriately defined maximal operator is contained in the \(L^p(\mathbb{R}^n)\) space is laid out. The first major result that is proved claims that the maximal operator used in the definition may be chosen in a couple of equivalent ways. Moreover, we proved that for an arbitrary \(f \in H^p\) a family \(\{a_k\}\) of simpler elements and a sequence of complex coefficients \(\lambda_k\) can be obtained such that \(f=\Sigma_k\lambda_ka_k\) in the \(H^p\) norm; we call the elements \(a_k\) atoms, and the resulting decomposition atomic. Finally, a thorough overview of a particularly interesting space \(H^1\) is given along with the demonstration of the fact that its dual is the space of functions with bounded mean oscillation (BMO).

• Keywords: Hardyjevi prostori; atomarna dekompozicija
• Keywords (english): Hardy spaces; atomic decomposition
• Language: croatian
• URN:NBN: urn:nbn:hr:217:832846
• Study programme: Title: Applied 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: 2021-03-10 11:53:28