Login | Register

Geometric Inequalities and Bounded Mean Oscillation


Geometric Inequalities and Bounded Mean Oscillation

Gibara, Ryan (2020) Geometric Inequalities and Bounded Mean Oscillation. PhD thesis, Concordia University.

[thumbnail of Gibara_PhD_F2020.pdf]
Text (application/pdf)
Gibara_PhD_F2020.pdf - Accepted Version
Available under License Spectrum Terms of Access.


In this thesis, we study the space of functions of bounded mean oscillation (BMO) on shapes. We prove the boundedness of important nonlinear operators, such as maximal functions and rearrangements, on this space and analyse how the bounds are affected by the underlying geometry of the shapes.

We provide a general definition of BMO on a domain in R^n, where mean oscillation is taken with respect to a basis of shapes, i.e. a collection of open sets covering the domain. We prove many properties inherent to BMO that are valid for any choice of basis; in particular, BMO is shown to be complete. Many shapewise inequalities, which hold for every shape in a given basis, are proven with sharp constants. Moreover, a sharp norm inequality, which holds for the BMO norm that involves taking a supremum over all shapes in a given basis, is obtained for the truncation of a BMO function. When the shapes exhibit some product structure, a product decomposition is obtained for BMO.

We consider the boundedness of maximal functions on BMO on shapes in R^n. We prove that for bases of shapes with an engulfing property, the corresponding maximal function is bounded from BMO to BLO, the collection of functions of bounded lower oscillation. When the basis of shapes does not possess an engulfing property but exhibits a product structure with respect to lower-dimensional shapes coming from bases that do possess an engulfing property, we show that the corresponding maximal function is bounded from BMO to a space we define and call rectangular BLO.

We obtain boundedness and continuity results for rearrangements on BMO. This allows for an improvement of the known bound for the basis of cubes. We show, by example, that the decreasing rearrangement is not continuous on BMO, but that it is both bounded and continuous on VMO, the subspace of functions of vanishing mean oscillation. Boundedness for the symmetric decreasing rearrangement is then established for the basis of balls in R^n.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Thesis (PhD)
Authors:Gibara, Ryan
Institution:Concordia University
Degree Name:Ph. D.
Date:3 July 2020
Thesis Supervisor(s):Dafni, Galia and Burchard, Almut
ID Code:987220
Deposited By: RYAN GIBARA
Deposited On:25 Nov 2020 16:16
Last Modified:25 Nov 2020 16:16
All items in Spectrum are protected by copyright, with all rights reserved. The use of items is governed by Spectrum's terms of access.

Repository Staff Only: item control page

Downloads per month over past year

Research related to the current document (at the CORE website)
- Research related to the current document (at the CORE website)
Back to top Back to top