Global L^p Second Commutation Lemma

2026/08/02 by Marin Mišur
Mathematics · #math.AP

paper · pdf

Work in progress

arxiv created 2026/08/02 · arxiv updated 2026/08/04

Abstract

We prove the second commutation lemma for Lebesgue spaces L^p(\R^d) (1 < p < ∞) on the global unbounded domain, extending the L² theory of Tartar \citeTartar1990 to the Banach space framework of H-distributions. Unlike the L² setting, where the Plancherel isometry and the compactness tools of Hilbert space are available (as in the parabolic variants of Antonić and Lazar \citeAntonicLazar2013), the global L^p setting lacks both. We resolve the non-local tail problem through elementary Calderón--Zygmund kernel estimates, an explicit Taylor-remainder identity, and spatial truncations. Furthermore, by pairing macroscopic sequences with their canonical Nemyckij duals, we use the lemma to derive L^p transport equations. We establish the phase-space bicharacteristic flow (Vlasov equation) for first-order scalar PDEs (for p ≥ 2), and for the quasilinear p-Wave system we lift the local energy identity to the microlocal level, obtaining a Poynting-flux transport of microlocal energy in the linear core (p = 2) and isolating the structural obstruction to its closure when p ≠ 2. These results provide functional analytic tools for tracking the propagation of singularities in degenerate nonlinear and fractional partial differential equations.

Citations

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