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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
7/2008

Compensated Compactness, Separately convex Functions and Interpolatory Estimates between Riesz Transforms and Haar Projections

Jihoon Lee, Paul F. X. Müller and Stefan Müller

Abstract

In this work we prove sharp interpolatory estimates that exhibit a new link between Riesz transforms and directional projections of the Haar system in ${\mathbb R}^n . $ To a given direction $ \varepsilon \in \{ 0, 1 \}^n , \varepsilon \ne ( 0,\dots , 0 ) , $ we let $P^{(\varepsilon)}$ be the orthogonal projection onto the span of those Haar functions that oscillate along the coordinates $\{ i : \varepsilon_i = 1\} . $ When $ \varepsilon_{i_0} = 1 $ the identity operator and the Riesz transform $ R_{i_0} $ provide a logarithmically convex estimate for the $L^p$ norm of $ P^{(\varepsilon)},$ see Theorem 1.1. Apart from its intrinsic interest Theorem 1.1 has direct applications to variational integrals, the theory of compensated compactness, Young measures, and to the relation between rank one and quasi convex functions. In particular we exploit our Theorem 1.1 in the course of proving a conjecture of L. Tartar on semi-continuity of separately convex integrands; see Theorem 1.5.

Received:
31.01.08
Published:
31.01.08
MSC Codes:
49J45, 42C15, 35B35

Related publications

inJournal
2011 Repository Open Access
Jihoon Lee, Paul F. X. Müller and Stefan Müller

Compensated compactness, separately convex functions and interpolatory estimates between Riesz transforms and Haar projections

In: Communications in partial differential equations, 36 (2011) 4, pp. 547-601