Anisotropic Higher-Order Semiregularity of Degenerate Generalized Equations

Abstract

We give a self-contained triangular formulation of anisotropic higher-order covering and inverse estimates for smooth mappings and generalized equations. For a $C^p$ mapping into a finite-dimensional target, an indexed target decomposition is fixed, the corresponding triangular factor condition is imposed, and a direction satisfying the associated triangular $p$-kernel condition is chosen. Surjectivity of the resulting triangular $p$-factor operator yields a fixed-scale inclusion in which each nonzero block $Y_i$ is covered at order $t^i$, and hence an inverse estimate with exponent $1/i$ on that block. Zero blocks are allowed, and $p$ denotes the highest active grade. The smooth result is a derivative-level graded formulation: it verifies the model directly from the $C^p$ derivatives and records the blockwise fixed-scale inclusion needed for the set-valued analysis. For generalized equations, an auxiliary exact-model theorem isolates the range-transfer mechanism with acyclic correction fibres. The main sufficient criterion uses a closed convex-process inner approximation whose sum with the smooth triangular factor operator is surjective, allowing the set-valued term to supply missing directions.