Adaptive Hybrid Norms in Vector-Valued Function Spaces: Compactness, Duality, and Applications to Nonlinear PDEs

This paper develops a comprehensive theory of generalized norm structures in vector-valued function spaces, introducing three fundamental advances: (1) (adaptive hybrid)- norms that unify variable-exponent Lebesgue spaces with Banach lattice operations, enabling precise control of anisotropic singularities in nonlinear PDEs; (2)(non-iterated compactness criteria)for non-separable ranges, extending classical Aubin-Lions theory; and (3) a (hybrid Radon-Nikodym property)- that resolves duality gaps in variable-exponent spaces. Applications include existence theorems for fractional quasilinear PDEs, optimal convergence rates for coupled discontinuous Galerkin systems, and rigorous error bounds for neural operators. The framework bridges harmonic analysis with data-driven modeling, offering a unified toolkit for multiscale nonlinear phenomena.