• DOI:10.1002/cpa.21575
• Corpus ID: 959977

Hierarchical Construction of Bounded Solutions in Critical Regularity Spaces

@article{Tadmor2010HierarchicalCO,
  title={Hierarchical Construction of Bounded Solutions in Critical Regularity Spaces},
  author={Eitan Tadmor},
  journal={Communications on Pure and Applied Mathematics},
  year={2010},
  volume={69}
}

• E. Tadmor
• Published 7 March 2010
• Mathematics
• Communications on Pure and Applied Mathematics

We construct uniformly bounded solutions for the equations div U = f and U = f in the critical cases f∈L#d(Td,ℝ) and f∈L#3(T3,ℝ3) , respectively. Criticality in this context manifests itself by the lack of a linear solution operator mapping L#d↦ L∞(Td) . Thus, the intriguing aspect here is that although the problems are linear, construction of their solutions is not. 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 B with 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=1∞uj , interesting for its own sake. Here, uj's are constructed recursively as minimizers of uj+1=arg minn{ ‖ u ‖B+λj+1‖ rj−ℒu ‖Lp(Ω)p }, where the residuals rj:=f−ℒ(∑k=1juk) are resolved in terms of a dyadic sequence of scales λj+1:=λ12j with large enough λ1≳| f |Lp1−p . The nonlinear aspect of this construction is a counterpart of the fact that one cannot linearly solve LU=f in critical regularity spaces.© 2016 Wiley Periodicals, Inc.