When Kernel Ridge Regression Meets the H\"older-Zygmund Class: Minimax Optimality and Failure of Properness

We study kernel ridge regression for nonparametric regression over the H"older-Zygmund class. Using an RKHS equivalent to a Sobolev space of smoothness s+d/2, we prove that misspecified KRR attains the minimax L2 rate n^{-2s/(2s+d)}. We also show that properness fails in the H"older-Zygmund norm: even for the zero regression function with Gaussian noise, the expected squared H"older-Zygmund norm of the KRR noise component grows as log n.