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

We construct a dyadic microlocal partition adapted to a position-dependent fiber metric on phase space. Under uniform ellipticity, the associated fiber norm is equivalent to the Euclidean one; the main effect of the construction is therefore not a new global symbolic order, but the $x$-dependent deformation of the microlocal patches and the derivative losses produced by differentiating the moving normalization. We prove finite-seminorm estimates for the localized symbols, with explicit losses depending on the number of controlled derivatives, and derive corresponding local Weyl quantization bounds through Calderón–Vaillancourt estimates. We also record finite-order Moyal truncation estimates and a semiclassical band normalization. Global recombination is formulated as a conditional Cotlar–Stein criterion with explicit almost-orthogonality hypotheses. Finally, we present two model uses: a patchwise parametrix construction and a compatibility discussion for the Radon transform as a model Fourier integral operator.