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An interpolatory estimate for the UMD-valued directional Haar projection

Richard Lechner — 2014

We prove an interpolatory estimate linking the directional Haar projection P ( ε ) to the Riesz transform in the context of Bochner-Lebesgue spaces L p ( ; X ) , 1 < p < ∞, provided X is a UMD-space. If ε i = 1 , the result is the inequality | | P ( ε ) u | | L p ( ; X ) C | | u | | L p ( ; X ) 1 / | | R i u | | L p ( ; X ) 1 - 1 / , (1) where the constant C depends only on n, p, the UMD-constant of X and the Rademacher type of L p ( ; X ) . In order to obtain the interpolatory result (1) we analyze stripe operators S λ , λ ≥ 0, which are used as basic building blocks to dominate the directional Haar projection. The...

An introduction to Cartan Geometries

Sharpe, Richard — 2002

Proceedings of the 21st Winter School "Geometry and Physics"

A principal bundle with a Lie group H consists of a manifold P and a free proper smooth H -action P × H P . There is a unique smooth manifold structure on the quotient space M = P / H such that the canonical map π : P M is smooth. M is called a base manifold and H P M stands for the bundle. The most fundamental examples of principal bundles are the homogeneous spaces H G G / H , where H is a closed subgroup of G . The pair ( 𝔤 , 𝔥 ) is a Klein pair. A model geometry consists of a Klein pair ( 𝔤 , 𝔥 ) and a Lie group H with Lie algebra 𝔥 . In this...

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