Localized Haar Extensions on the Dyadic Plane

We prove that every $L^2$ function on a set $E\subset\mathbb R^2$ of finite measure can be extended linearly to the whole plane using Haar functions. The extension agrees with the original function on $E$, with uniform bounds on its norm and the total area of the Haar rectangles used. In particular, we construct a function equal to $1$ on any finite-measure open set, satisfying the dyadic Hardy norm estimates for all $0