Quadratic variation obstruction to Hölder regularity (${\sum }_{k=0}^{N-1}(w((k+1)/N)-w(k/N){)}^2\le C^2{N}^{1-2\alpha }$)

A function with finite Hölder seminorm of exponent $\alpha >1/2$ has squared-increment sum on a uniform $N$-interval partition bounded by $C^2{N}^{1-2\alpha }$, hence tending to zero. This conflicts with a nonzero quadratic-variation limit. In particular dyadic quadratic variation of Brownian motion prevents a Brownian coordinate law supported on such a Hölder space. This argument alone gives no conclusion at the critical exponent $1/2$.