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

100%

Asmar N. H., Montgomery-Smith S. J.

Colloquium Mathematicum

1996

tom 70

nr 2

235-252

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 with ordered duals. We illustrate our methods by describing a concrete way to construct the conjugate function in $L^p(G)$. This construction is in terms of an unconditionally convergent conjugate series whose individual terms are constructed from specific ergodic Hilbert transforms. We also present a study of the square function associated with the conjugate series.

On a weak type (1,1) inequality for a maximal conjugate function

100%

Asmar N. H., Montgomery-Smith S. J.

Studia Mathematica

1997

tom 125

nr 1

13-21

In their celebrated paper [3], Burkholder, Gundy, and Silverstein used Brownian motion to derive a maximal function characterization of $H^p$ spaces for 0 < p < ∞. In the present paper, we show that the methods in [3] extend to higher dimensions and yield a dimension-free weak type (1,1) estimate for a conjugate function on the N-dimensional torus.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Studia Mathematica

2 Asmar N. H.

2 Montgomery-Smith S. J.

1 1997

1 1996

Wyniki wyszukiwania

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

On a weak type (1,1) inequality for a maximal conjugate function