Abstract
I discuss the prescribed Jacobian equation \(Ju=\det \nabla u=f\) for an unknown vector-function u, and the connection of this problem to the boundedness of commutators of multiplication operators with singular integrals in general, and with the Beurling operator in particular. A conjecture of T. Iwaniec regarding the solvability for general datum \(f\in L^p(\mathbb {R}^d)\) remains open, but recent partial results in this direction will be presented. These are based on a complete characterisation of the L p-to-L q boundedness of commutators, where the regime of exponents p > q, unexplored until recently, plays a key role. These results have been proved in general dimension d ≥ 2 elsewhere, but I will here present a simplified approach to the important special case d = 2, using a framework suggested by S. Lindberg.
Dedicated to Professor Fulvio Ricci
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References
Avila, A.: On the regularization of conservative maps. Acta Math. 205(1), 5–18 (2010)
Coifman, R.R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. (2) 103(3), 611–635 (1976)
Coifman, R., Lions, P.-L., Meyer, Y., Semmes, S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. (9) 72(3), 247–286 (1993)
Dacorogna, B., Moser, J.: On a partial differential equation involving the Jacobian determinant. Ann. Inst. H. Poincaré Anal. Non Linéaire 7(1), 1–26 (1990)
Hytönen, T.P.: The L p-to-L q boundedness of commutators with applications to the Jacobian operator. Preprint, arXiv:1804.11167 (2018)
Iwaniec, T.: Nonlinear commutators and Jacobians. J. Fourier Anal. Appl. 3, 775–796 (1997)
Janson, S.: Mean oscillation and commutators of singular integral operators. Ark. Mat. 16(2), 263–270 (1978)
Lerner, A.K.: A pointwise estimate for the local sharp maximal function with applications to singular integrals. Bull. Lond. Math. Soc. 42(5), 843–856 (2010)
Lerner, A.K., Ombrosi, S., Rivera-Ríos, I.P.: Commutators of singular integrals revisited. Bull. Lond. Math. Soc. 51(1), 107–119 (2019)
Lindberg, S.: On the Hardy space theory of compensated compactness quantities. Arch. Ration. Mech. Anal. 224(2), 709–742 (2017)
Uchiyama, A.: On the compactness of operators of Hankel type. Tôhoku Math. J. (2) 30(1), 163–171 (1978)
Ye, D.: Prescribing the Jacobian determinant in Sobolev spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire 11(3), 275–296 (1994)
Acknowledgements
The author is supported by the Academy of Finland via project Nos. 307333 (Centre of Excellence in Analysis and Dynamics Research) and 314829 (Frontiers of singular integrals).
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Hytönen, T.P. (2021). Of Commutators and Jacobians. In: Ciatti, P., Martini, A. (eds) Geometric Aspects of Harmonic Analysis. Springer INdAM Series, vol 45. Springer, Cham. https://doi.org/10.1007/978-3-030-72058-2_13
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