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MiS Preprint

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

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


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.

MSC Codes:
49J45, 42C15, 35B35

Related publications

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