Dimension of divergence set of the wave equation. (February 2022)

• Record Type: Journal Article
• Authors: Ham, Seheon; Ko, Hyerim; Lee, Sanghyuk
• Is Part Of: Nonlinear analysis. Volume 215(2022)
• Publication Date: 2022-02
• Subjects: 42B37; Divergence set -- Fractal Strichartz estimate -- Spherical average; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112631
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider the Hausdorff dimension of the divergence set on which the pointwise convergence lim t → 0 e i t − Δ f ( x ) = f ( x ) fails when f ∈ H s ( R d ) . We especially prove the conjecture raised by Barceló et al. (2011) for d = 3, and improve the previous results in higher dimensions d ≥ 4 . We also show that a Strichartz type estimate for f → e i t − Δ f with the measure d t d μ ( x ) is essentially equivalent to the estimate for the spherical average of μ ̂ which has been extensively studied for the Falconer distance set problem. The equivalence provides shortcuts to the recent results due to Liu (2019) and Rogers (2018).