A note on the weak* and pointwise convergence of BV functions

2020/09/21 by Beck, Lisa, Lahti, Panu
#26B30 #28A20 #28A78 #31C40 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2009.09889

Abstract

We study pointwise convergence properties of weakly* converging sequences \u_i\_i ∈ \mathbb N in BV(\mathbb Rⁿ). We show that, after passage to a suitable subsequence (not relabeled), we have pointwise convergence u_i^*(x)→ u^*(x) of the precise representatives for all x∈ \mathbb Rⁿ ∖ E, where the exceptional set E ⊂ \mathbb Rⁿ has on the one hand Hausdorff dimension at most n-1, and is on the other hand also negligible with respect to the Cantor part of |D u|. Furthermore, we discuss the optimality of these results.

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