Dyadic microlocal partitions for position-dependent fiber metrics and Weyl quantization

Abstract

We construct a dyadic microlocal partition adapted to a position-dependent fiber metric on phase space and quantify its interaction with Weyl quantization. Under uniform ellipticity, the normalized fiber variable $\zeta=T_x\xi$ is uniformly comparable with the Euclidean frequency, so the construction does not introduce a new global symbolic order. The essential feature is instead a mesoscopic decomposition on the dyadic annulus $|\zeta|\asymp 2^k$, with block radius $R_k=2^{k/2}$. The corresponding microlocalizers satisfy explicit packing, support and derivative estimates, including the compact-local balance $|\partial_x^\alpha\partial_\xi^\beta\Lambda_{j,k}|\leq C_{\alpha,\beta,K}R_k^{|\alpha|-|\beta|}$ on each fixed spatial compact set $K$. For the spatially localized Sobolev-conjugated Weyl blocks we prove quantitative off-diagonal decay in the normalized mesoscopic distance and verify the Cotlar--Stein summability conditions, yielding strong recombination from $H^s$ to $H^{s-m_2}$. Separately, for the full product-type symbol class $S^{m_1,m_2}$ we prove the global weighted estimate $\operatorname{Op}^w(a):H^{s,\sigma}\to H^{s-m_2,\sigma-m_1}$ with an explicit finite-seminorm budget. We also record finite-order Weyl--Moyal bookkeeping and illustrate the construction through a patchwise parametrix model and the Radon transform as a model Fourier integral operator. For the localized Radon transform blocks in dimensions n=2,3, the mesoscopic construction introduces no additional dyadic localization loss beyond the standard Fourier integral operator Sobolev shift.