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theorem is an
boundedness criterion by which the question of the global behavior of an operator is reduced to its local behavior, acting on a family of test functions
indexed by the dyadic cubes. We present two versions of such results, in particular, treating square function operators whose kernels do not satisfy the standard Littlewood–Paley pointwise estimates. As an application of one version of the local
theorem, we show how the solvability of the Kato problem (which was implicitly based on local
theory) may be deduced from this general criterion.
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