### Restrictions from ${\mathbb{R}}^{n}$ to ${\mathbb{Z}}^{n}$ of weak type (1,1) multipliers

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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 ${L}^{p}\left(G\right)$, 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...

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.

We define a new type of multiplier operators on ${L}^{p}{(}^{N})$, where ${}^{N}$ is the N-dimensional torus, and use tangent sequences from probability theory to prove that the operator norms of these multipliers are independent of the dimension N. Our construction is motivated by the conjugate function operator on ${L}^{p}{(}^{N})$, to which the theorem applies as a particular example.

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...

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