We construct uniformly bounded solutions of the equations div(U)=f and curl(U)=f, for general f’s in the critical regularity spaces L^d(R^d) and, respectively, L^3(R^3). The study of these equations was motivated by recent results of Bourgain & Brezis. The equations are linear, but construction of their solutions is not. Our constructions are, in fact, special cases of a rather general framework for solving linear equations, L(U)=f, covered by the closed range theorem. The solutions are realized in terms of nonlinear hierarchical representations, U=sum(u_j), which we introduced earlier in the context of image processing. The u_j's are constructed recursively as proper minimizers, yielding a multi-scale decomposition of the solutions U.
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