The L^p-to-L^q boundedness of commutators with applications to the Jacobian operator

Tuomas P. Hytönen

paper

The L^p-to-L^q boundedness of commutators with applications to the Jacobian operator

arXiv:1804.11167

Abstract

Supplying the missing necessary conditions, we complete the characterisation of the $L^{p} \to L^{q}$ boundedness of commutators $[b, T]$ of pointwise multiplication and Calderón-Zygmund operators, for arbitrary pairs of $1 < p, q < \infty$ and under minimal non-degeneracy hypotheses on $T$ . For $p \leq q$ (and especially $p = q$ ), this extends a long line of results under more restrictive assumptions on $T$ . In particular, we answer a recent question of Lerner, Ombrosi, and Rivera-Ríos by showing that $b \in B M O$ is necessary for the $L^{p}$ -boundedness of $[b, T]$ for any non-zero homogeneous singular integral $T$ . We also deal with iterated commutators and weighted spaces. For $p > q$ , our results are new even for special classical operators with smooth kernels. As an application, we show that every $f \in L^{p} (R^{d})$ can be represented as a convergent series of normalised Jacobians $J u = det \nabla u$ of $u \in W ˙^{1, d p} (R^{d})^{d}$ . This extends, from $p = 1$ to $p > 1$ , a result of Coifman, Lions, Meyer and Semmes about $J : W ˙^{1, d} (R^{d})^{d} \to H^{1} (R^{d})$ , and supports a conjecture of Iwaniec about the solvability of the equation $J u = f \in L^{p} (R^{d})$ .

V4: 40 pages, final author version to appear in J. Math. Pures Appl. New examples on the non-degeneracy condition and some new references. V3: Minor revision: a couple of new references and short related discussion; numbering scheme changed. V2: Corrected a detail in the proof of Theorem 3.7 on page 22 (random d-th roots of unity are needed here rather than random signs)

The L^p-to-L^q boundedness of commutators with applications to the Jacobian operator

Abstract

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