$H^1$ boundedness of determinants of vector fields
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Abstract:
We consider multilinear operators $T(f_{1},\dots , f_{l})$ given by determinants of matrices of the form $(X_{k}f_{j})_{1\le j,k\le l}$, where the $X_{k}$’s are $C^{\infty }$ vector fields on $\mathbb {R}^{n}$. We give conditions on the $X_{k}$’s so that the corresponding operator $T$ maps products of Lebesgue spaces $L^{p_{1}}\times \dots \times L^{p_{l}}$ into some anisotropic space $H^{1}$, when ${\frac {1}{p_{1}}}+\dots +{\frac {1}{p_{l}}}=1$.References
- Ronald R. Coifman and Loukas Grafakos, Hardy space estimates for multilinear operators. I, Rev. Mat. Iberoamericana 8 (1992), no. 1, 45–67. MR 1178448, DOI 10.4171/RMI/116
- Ronald R. Coifman, Pierre-Louis Lions, Yves Meyer, and Stephen Semmes, Compacité par compensation et espaces de Hardy, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 18, 945–949 (French, with English summary). MR 1054740
- R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9) 72 (1993), no. 3, 247–286 (English, with English and French summaries). MR 1225511
- R. R. Coifman, R. Rochberg, and Guido Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (1976), no. 3, 611–635. MR 412721, DOI 10.2307/1970954
- Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. MR 447954, DOI 10.1090/S0002-9904-1977-14325-5
- G. Folland and E. M. Stein, Harmonic Analysis on Homogeneous groups, Mathematical Notes, Princeton Univ. Press, Princeton NJ (1982).
- Loukas Grafakos, Hardy space estimates for multilinear operators. II, Rev. Mat. Iberoamericana 8 (1992), no. 1, 69–92. MR 1178449, DOI 10.4171/RMI/117
- Loukas Grafakos and Richard Rochberg, Compensated compactness and the Heisenberg group, Math. Ann. 301 (1995), no. 3, 601–611. MR 1324529, DOI 10.1007/BF01446650
- Tadeusz Iwaniec and Adam Lutoborski, Integral estimates for null Lagrangians, Arch. Rational Mech. Anal. 125 (1993), no. 1, 25–79. MR 1241286, DOI 10.1007/BF00411477
- Tadeusz Iwaniec and Carlo Sbordone, On the integrability of the Jacobian under minimal hypotheses, Arch. Rational Mech. Anal. 119 (1992), no. 2, 129–143. MR 1176362, DOI 10.1007/BF00375119
- David Jerison, The Poincaré inequality for vector fields satisfying Hörmander’s condition, Duke Math. J. 53 (1986), no. 2, 503–523. MR 850547, DOI 10.1215/S0012-7094-86-05329-9
- Guozhen Lu, Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition and applications, Rev. Mat. Iberoamericana 8 (1992), no. 3, 367–439. MR 1202416, DOI 10.4171/RMI/129
- Guozhen Lu, The sharp Poincaré inequality for free vector fields: an endpoint result, Rev. Mat. Iberoamericana 10 (1994), no. 2, 453–466. MR 1286482, DOI 10.4171/RMI/158
- Stefan Müller, Higher integrability of determinants and weak convergence in $L^1$, J. Reine Angew. Math. 412 (1990), 20–34. MR 1078998, DOI 10.1515/crll.1990.412.20
- Alexander Nagel, Elias M. Stein, and Stephen Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), no. 1-2, 103–147. MR 793239, DOI 10.1007/BF02392539
- Linda Preiss Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3-4, 247–320. MR 436223, DOI 10.1007/BF02392419
- Stephen Semmes, A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller, Comm. Partial Differential Equations 19 (1994), no. 1-2, 277–319. MR 1257006, DOI 10.1080/03605309408821017
- E. M. Stein, A note on the class $LlogL$, Studia Math 32 (1968), 305-310.
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
Additional Information
- Loukas Grafakos
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211-0001
- MR Author ID: 288678
- ORCID: 0000-0001-7094-9201
- Email: loukas@math.missouri.edu
- Received by editor(s): December 11, 1995
- Received by editor(s) in revised form: May 20, 1996
- Additional Notes: Research partially supported by the NSF and the University of Missouri Research Board
- Communicated by: J. Marshall Ash
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 3279-3288
- MSC (1991): Primary 42B30
- DOI: https://doi.org/10.1090/S0002-9939-97-03958-0
- MathSciNet review: 1403130