# Linear PDEs in critical regularity spaces: Hierarchical construction of their nonlinear solutions

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We construct uniformly bounded solutions of the equations $div(U)=f$ and $curl(U)=f$, for general $f$ in the critical regularity spaces $L^d(R^d)$ and, respectively, $L3(R3)$. 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 $U$.