7 papers
Structural aspects of extremal functions in the Krzyż conjecture
Sullivan F. MacDonald
Extremal functions for the th coefficient in the Krzyż conjecture are atomic singular inner functions with at most atoms. This paper gives a lower bound on the number of ato…
Bounded solutions of degenerate elliptic equations with an Orlicz-gain Sobolev inequality
David Cruz-Uribe, Sullivan F. MacDonald, Scott Rodney
We consider the boundedness and exponential integrability of solutions to the Dirichlet problem for the degenerate elliptic equation \[ -v^{-1}\mathrm{Div}(|\sqrt{Q}\nabla u|^{p-2}…
Sum of Squares Decompositions in Hölder Spaces
Sullivan Francis MacDonald
We investigate the number of half-regular squares required to decompose a non-negative function into a sum of squares. Each non-negative $C^{3,1}(\mathbb{R}…
Regularity of Singular Solutions to -Poisson Equations
Sullivan Francis MacDonald
This work showcases level set estimates for weak solutions to the -Poisson equation on a bounded domain, which we use to establish Lebesgue space inclusions for weak solutions.…
Regularity Preserving Sum of Squares Decompositions
Sullivan F. MacDonald
In the process of proving a sharpened form of Gårding's inequality, Fefferman & Phong demonstrated that every non-negative function can be written as a…
Bounded Weak Solutions of Degenerate -Poisson Equations
Sullivan Francis MacDonald, Scott Rodney
In this work we study global boundedness and exponential integrability of weak solutions to degenerate -Poisson equations using an iterative method of De Giorgi type. Given a sy…
Families of Young Functions and Limits of Orlicz Norms
Scott Rodney, Sullivan F. MacDonald
Given a -finite measure space , a Young function , and a one-parameter family of Young functions , we find necessary and sufficient conditions for the associa…