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We develop a strategy making extensive use of tent spaces to study parabolic equations with quadratic nonlinearities as for the Navier–Stokes system. We begin with a new proof of the well-known result of Koch and Tataru on the well-posedness of Navier–Stokes equations in $\mathbb{R}^{n}$ with small initial data in $\mathit{BMO}^{-1}(\mathbb{R}^{n})$. We then study another model where neither pointwise kernel bounds nor self-adjointness are available.

In this paper, we study the boundedness of є-families of operators on Triebel-Lizorkin with wide range of parameters. We also prove that є -families of operators are bounded from Triebel-Lizorkin spaces into (generalized) tent spaces, and obtain a characterization of certain Triebel-Lizorkin spaces in terms of tent spaces. In particular, the boundedness of fractional operators in Triebel-Lizorkin, and a sharp version of T\theorem for generalized Calderón-Zygmund operators on Triebel-Lizorkin spaces can be considered as applications of (proofs of) these results.

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ON THE WELL-POSEDNESS OF PARABOLIC EQUATIONS OF NAVIER–STOKES TYPE WITH $\mathit{BMO}^{-1}$ DATA

є-Families of Operators in Triebel-Lizorkin and Tent Spaces

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ON THE WELL-POSEDNESS OF PARABOLIC EQUATIONS OF NAVIER–STOKES TYPE WITH $\mathit{BMO}^{-1}$ DATA

є-Families of Operators in Triebel-Lizorkin and Tent Spaces