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We study unbounded harmonic functions for a second order differential operator on a homogeneous manifold of negative curvature which is a semidirect product of a nilpotent Lie group N and A = ℝ⁺. We prove that if F is harmonic and satisfies some growth condition then F has an asymptotic expansion as a → 0 with coefficients from 𝓓'(N). Then we single out a set of at most two of these coefficients which determine F. Then using asymptotic expansions we are able to prove some theorems...

Unbounded Lebesgue Constants on Compact Groups.

Christopher Meaney (1981)

Monatshefte für Mathematik

Uncertainty principles for the Schrödinger equation on Riemannian symmetric spaces of the noncompact type

Angela Pasquale, Maddala Sundari (2012)

Annales de l’institut Fourier

Let $X$ be a Riemannian symmetric space of the noncompact type. We prove that the solution of the time-dependent Schrödinger equation on $X$ with square integrable initial condition $f$ is identically zero at all times $t$ whenever $f$ and the solution at a time $t_{0} > 0$ are simultaneously very rapidly decreasing. The stated condition of rapid decrease is of Beurling type. Conditions respectively of Gelfand-Shilov, Cowling-Price and Hardy type are deduced.

Uncertainty principles for the Weinstein transform

Hatem Mejjaoli, Makren Salhi (2011)

Czechoslovak Mathematical Journal

The Weinstein transform satisfies some uncertainty principles similar to the Euclidean Fourier transform. A generalization and a variant of Cowling-Price theorem, Miyachi's theorem, Beurling's theorem, and Donoho-Stark's uncertainty principle are obtained for the Weinstein transform.

Unconditionality, Fourier multipliers and Schur multipliers

Cédric Arhancet (2012)

Colloquium Mathematicae

Let G be an infinite locally compact abelian group and X be a Banach space. We show that if every bounded Fourier multiplier T on L²(G) has the property that $T \otimes I d_{X}$ is bounded on L²(G,X) then X is isomorphic to a Hilbert space. Moreover, we prove that if 1 < p < ∞, p ≠ 2, then there exists a bounded Fourier multiplier on $L^{p} (G)$ which is not completely bounded. Finally, we examine unconditionality from the point of view of Schur multipliers. More precisely, we give several necessary and sufficient conditions...

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