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Hahn's Embedding Theorem for orders and harmonic analysis on groups with ordered duals

Nakhlé AsmarStephen Montgomery-Smith — 1996

Colloquium Mathematicae

Let G be a locally compact abelian group whose dual group Γ contains a Haar measurable order P. Using the order P we define the conjugate function operator on L p ( G ) , 1 ≤ p < ∞, as was done by Helson [7]. We will show how to use Hahn’s Embedding Theorem for orders and the ergodic Hilbert transform to study the conjugate function. Our approach enables us to define a filtration of the Borel σ-algebra on G, which in turn will allow us to introduce tools from martingale theory into the analysis on groups...

Decomposition of analytic measures on groups and measure spaces

Nakhlé AsmarStephen Montgomery-Smith — 2001

Studia Mathematica

We consider an arbitrary locally compact abelian group G, with an ordered dual group Γ, acting on a space of measures. Under suitable conditions, we define the notion of analytic measures using the representation of G and the order on Γ. Our goal is to study analytic measures by applying a new transference principle for subspaces of measures, along with results from probability and Littlewood-Paley theory. As a consequence, we derive new properties of analytic measures as well as extensions of previous...

A counterexample to the smoothness of the solution to an equation arising in fluid mechanics

Stephen Montgomery-SmithMilan Pokorný — 2002

Commentationes Mathematicae Universitatis Carolinae

We analyze the equation coming from the Eulerian-Lagrangian description of fluids. We discuss a couple of ways to extend this notion to viscous fluids. The main focus of this paper is to discuss the first way, due to Constantin. We show that this description can only work for short times, after which the ``back to coordinates map'' may have no smooth inverse. Then we briefly discuss a second way that uses Brownian motion. We use this to provide a plausibility argument for the global regularity for...

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