Hierarchical Solutions for Linear Equations: a Constructive Proof of the Closed Range Theorem

We construct uniformly bounded solutions for the equations divU = f and curlU = f in the critical cases f ∈ L # (T ,R) and f∈ L # (R,R). Bourgain& Brezis, [BB03, BB07], have shown that there exists no linear construction for such solutions. Our constructions are special cases of a general framework for solving linear equations of the form LU = f , where L is a linear operator densely defined in Banach space Bwith a closed range in a (proper subspace) of Lebesgue space L p # (Ω), and with an injective dual L ∗. The solutions are realized in terms of a multiscale hierarchical representation, U =∑∞j=1u j, interesting for its own sake. Here, the u j’s are constructed recursively as minimizers of the iterative refinement step, u j+1= argminu { ‖u‖B+λ j+1‖r j−L u‖ L(Ω) } , where r j := f −L (∑ j k=1uk), are resolved in terms of an exponentially increasing sequence of scales λ j+1 := λ1ζ . The resulting hierarchical decompositions,U = ∑∞j=1u j, are nonlinear.