Category Archives: math.CA
Open Question on Sufficiency To Be a Density Basis
This post is adapted from a question I posted on Math.StackExchange approximately two weeks ago, which has not received a satisfactory answer. If you have a solution or a counterexample, please add it to Math.SE instead of here. Let be … Continue reading
A Differentiation Basis without the Vitali Covering Property
— 1. Introduction — A differentiation basis is a fine collection of bounded measurable sets ; i.e. for every there exists a sequence with diameter tending to zero. Classical examples of differentiation bases are the collections of balls and cubes. … Continue reading
Kuran’s Theorem
It is well-known that a harmonic function satisfies the mean value property for all balls and spheres contained in . (We abuse notation and use to denote both the -dimensional Lebesgue measure and the -dimensional surface measure.) This mean value … Continue reading
Heat Ball and Heat Sphere Mean Value Property
Like solutions to the Laplace equation, (classical) solutions to the heat equation satisfy a mean value property. But instead of integrals over balls or spheres, the heat mean value property involves integrals over a heat ball. The heat ball of … Continue reading
Harmonic Functions: Weyl Lemma
In this post, we prove a generalization of the result that any (clasically) harmonic function defined on a bounded, open set is analytic. If is weakly harmonic, i.e. for any compactly supported function (the space of such functions is denoted … Continue reading
Harmonic Functions: Regularity
I have a confession: I have never actually taken a PDE course. I took an ODE and PDE class offered by the Applied Math department my freshman year, but it wasn’t really a mathematics class–most of the students were physics … Continue reading
Kolmogorov Normability Criterion
This past week I have been refreshing my knowledge of topological vector spaces (tvs) while reading some papers on generalizations of the Mazur-Ulam theorem to metrizable tvs. I intend to upload a typed set of notes on the subject which … Continue reading
Intermediate Value Property of Measures
It is well known that the Lebesgue measure has an intermediate value property like that of real-valued functions defined on some bounded interval. For any Lebesgue measurable set , the function is continuous (continuity of measure) and bounded from above … Continue reading
Discontinuous Midpoint Convex Functions
I’ve have provided a cleaner proof of Lemma 3 than the one presented in my previous post. I have also included a stronger result, which shows that a midpoint convex function is either continuous everywhere or discontinuous everywhere. In what … Continue reading
Convexity, Continuity, and Jensen’s Inequality Revision
A comment to my post Convexity, Continuity, and Jensen’s Inequality asks how we know that , as , in the proof of Lemma 3. It’s actually false that the sequence converges to . What I believe to be a correct proof … Continue reading
Math by Matt 