Haar projection error for a Lipschitz function ($\Vert K_{j}f-f{\Vert }_{\infty }\le L{2}^{-j}$)

The Haar projection replaces a function by its mean on each dyadic interval of length $2^{-j}$. For a function satisfying Lipschitz continuity with constant $L$, every value in a cell differs from its cell mean by at most $L{2}^{-j}$. This bound holds for all points with a consistent endpoint convention, and requires no global L2 norm assumption when the projection is defined by local averages.