On the Product of Functions in BMO and H
Aline Bonami[1]; Tadeusz Iwaniec[2]; Peter Jones[3]; Michel Zinsmeister[4]
- [1] Université d’Orléans MAPMO BP 6759 45067 Orléans cedex
- [2] Syracuse University 215 Carnegie Hall Syracuse NY 13244-1150 (USA)
- [3] Yale University Mathematics Dept. PO Box 208 283 New Haven CT 06520-8283 (USA)
- [4] Université d’Orléans MAPMO BP 6759 45067 Orléans cedex et Ecole Polytechnique PMC 91128 Palaiseau
Annales de l’institut Fourier (2007)
- Volume: 57, Issue: 5, page 1405-1439
- ISSN: 0373-0956
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topBonami, Aline, et al. "On the Product of Functions in BMO and H$^\text{1}$." Annales de l’institut Fourier 57.5 (2007): 1405-1439. <http://eudml.org/doc/10263>.
@article{Bonami2007,
abstract = {The point-wise product of a function of bounded mean oscillation with a function of the Hardy space $H^1$ is not locally integrable in general. However, in view of the duality between $H^1$ and $BMO$, we are able to give a meaning to the product as a Schwartz distribution. Moreover, this distribution can be written as the sum of an integrable function and a distribution in some adapted Hardy-Orlicz space. When dealing with holomorphic functions in the unit disc, the converse is also valid: every holomorphic of the corresponding Hardy-Orlicz space can be written as a product of a function in the holomorphic Hardy space $H^1$ and a holomorphic function with boundary values of bounded mean oscillation.},
affiliation = {Université d’Orléans MAPMO BP 6759 45067 Orléans cedex; Syracuse University 215 Carnegie Hall Syracuse NY 13244-1150 (USA); Yale University Mathematics Dept. PO Box 208 283 New Haven CT 06520-8283 (USA); Université d’Orléans MAPMO BP 6759 45067 Orléans cedex et Ecole Polytechnique PMC 91128 Palaiseau},
author = {Bonami, Aline, Iwaniec, Tadeusz, Jones, Peter, Zinsmeister, Michel},
journal = {Annales de l’institut Fourier},
keywords = {Hardy spaces; bounded mean oscillation; Jacobian lemma; Jacobian equation; Hardy-Orlicz spaces; div-curl lemma; factorization in Hardy spaces; weak Jacobian; Hardy-Orlizc spaces},
language = {eng},
number = {5},
pages = {1405-1439},
publisher = {Association des Annales de l’institut Fourier},
title = {On the Product of Functions in BMO and H$^\text\{1\}$},
url = {http://eudml.org/doc/10263},
volume = {57},
year = {2007},
}
TY - JOUR
AU - Bonami, Aline
AU - Iwaniec, Tadeusz
AU - Jones, Peter
AU - Zinsmeister, Michel
TI - On the Product of Functions in BMO and H$^\text{1}$
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 5
SP - 1405
EP - 1439
AB - The point-wise product of a function of bounded mean oscillation with a function of the Hardy space $H^1$ is not locally integrable in general. However, in view of the duality between $H^1$ and $BMO$, we are able to give a meaning to the product as a Schwartz distribution. Moreover, this distribution can be written as the sum of an integrable function and a distribution in some adapted Hardy-Orlicz space. When dealing with holomorphic functions in the unit disc, the converse is also valid: every holomorphic of the corresponding Hardy-Orlicz space can be written as a product of a function in the holomorphic Hardy space $H^1$ and a holomorphic function with boundary values of bounded mean oscillation.
LA - eng
KW - Hardy spaces; bounded mean oscillation; Jacobian lemma; Jacobian equation; Hardy-Orlicz spaces; div-curl lemma; factorization in Hardy spaces; weak Jacobian; Hardy-Orlizc spaces
UR - http://eudml.org/doc/10263
ER -
References
top- K. Astala, T. Iwaniec, P. Koskela, G. Martin, Mappings of -bounded distortion, Math. Ann. 317 (2000), 703-726 Zbl0954.30009MR1777116
- K. Astala, M. Zinsmeister, Teichmüller spaces and BMOA, Math. Ann. 289 (1991), 613-625 Zbl0896.30028MR1103039
- J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1977), 703-726 Zbl0368.73040MR475169
- J. M. Ball, F. Murat, Remarks on Chacon’s Biting Lemma, Proc. Amer. Math. Soc. 107 (1989), 655-663 Zbl0678.46023
- J. M. Ball, K. Zhang, Lower semicontinuity of multiple integrals and the biting lemma, Proc. Royal Soc. Edinburgh. Sec. A 114 (1990), 367-379 Zbl0716.49011MR1055554
- A. Bonami, S. Madan, Balayage of Carleson measures and Hankel operators on generalized Hardy spaces, Math. Nachr. 193 (1991), 237-245 Zbl0797.31011MR1131946
- J. K. Brooks, R.C. Chacon, Continuity and compactness of measures, Advances Math. 107 (1980), 16-26 Zbl0463.28003MR585896
- D.-C. Chang, G. Dafni, C. Sadosky, A lemma in on a domain, 238 (2005), Birkhäuser, Boston MR2174309
- D. C. Chang, S. G. Krantz, E. M. Stein, -theory on a smooth domain in and elliptic boundary value problems, J. Funct. Anal. 114 (1993), 286-347 Zbl0804.35027MR1223705
- R. R. Coifman, P. L. Lions, Y. Meyer, S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. 72 (1993), 247-286 Zbl0864.42009MR1225511
- R. R. Coifman, R. Rochberg, Another characterization of , Proc. Amer. Math. Soc. 79 (1980), 249-254 Zbl0432.42016MR565349
- R. R. Coifman, R. Rochberg, G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611-635 Zbl0326.32011MR412721
- R. R. Coifman, G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645 Zbl0358.30023MR447954
- G. Dafni, Local and weak convergence in , Canad. Math. Bull. 45 (2002), 46-59 Zbl1004.42020MR1884133
- G. Dafni, Nonhomogeneous lemmas and local Hardy spaces, Adv. Differential Equations 10 (2005), 505-526 Zbl1129.42396MR2134048
- L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, 74 (1990), American Mathematical Society, Providence Zbl0698.35004MR1034481
- L. C. Evans, S. Müller, Hardy spaces and the two-dimensional Euler equations with nonnegative vorticity, Journ. Amer. Math. Soc. 7 (1994), 199-219 Zbl0802.35120MR1220787
- C. Fefferman, Characterization of bounded mean oscillations, Bull. Amer. Math. Soc. 77 (1971), 587-588 Zbl0229.46051MR280994
- C. Fefferman, E. M. Stein, -spaces of several variables, Acta Math. 129 (1972), 137-193 Zbl0257.46078MR447953
- J. B. Garnett, Bounded Analytic Functions, 96 (1981), Academic Press, New York Zbl0469.30024MR628971
- F. Giannetti, T. Iwaniec, J. Onninen, A. Verde, Estimates of Jacobians by subdeterminants, Journ. of Geometric Anal. 12 (2002), 223-254 Zbl1053.42024MR1888516
- Y. Gotoh, Remarks on multipliers for on general domains, Kodai Math. J. 16 (1993), 79-89 Zbl0783.42014MR1207993
- Y. Gotoh, On multipliers for on general domains, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 19 (1994), 143-161 Zbl0809.42003MR1274086
- L. Greco, T. Iwaniec, C. Sbordone, Inverting the -harmonic operator, Manuscripta Math. 92 (1997), 249-258 Zbl0869.35037MR1428651
- F. Hélein, Regularity of weakly harmonic maps from a surface into a manifold with symmetries, Manuscripta Math. 70 (1991), 203-218 Zbl0718.58019MR1085633
- T. Iwaniec, P. Koskela, G. Martin, C. Sbordone, Mappings of finite distortion: - integrability, J. London Math. Soc. 67 (2003), 123-136 Zbl1047.30010MR1942415
- T. Iwaniec, G. Martin, Geometric Function Theory and Nonlinear Analysis, (2001), Oxford University Press, New-York Zbl1045.30011
- T. Iwaniec, J. Onninen, - estimates of Jacobians by subdeterminants, Mathematische Annalen 324 (2002), 341-358 Zbl1055.42011MR1933861
- T. Iwaniec, C. Sbordone, On the integrability of the Jacobian under minimal hypothesis, Arch. Rational Mech. Anal. 119 (1992), 129-143 Zbl0766.46016MR1176362
- T. Iwaniec, C. Sbordone, Weak minima of variational integrals, J. Reine Angew. Math. 454 (1994), 143-161 Zbl0802.35016MR1288682
- T. Iwaniec, C. Sbordone, Quasiharmonic fields, Ann. I.H. Poincaré Anal. Non Lin. 18 (2001), 519-572 Zbl1068.30011MR1849688
- T. Iwaniec, C. Sbordone, New and old function spaces in the theory of PDEs and nonlinear analysis, 64 (2004), Polish Acad. Sci., Warsaw Zbl1061.46027MR2099461
- T. Iwaniec, A. Verde, A study of Jacobians in Hardy-Orlicz Spaces, Proc. Royal Soc. Edinburgh 129A (1999), 539-570 Zbl0954.46018MR1693625
- S. Janson, On functions with conditions on the mean oscillation, Ark. Math. 14 (1976), 189-196 Zbl0341.43005MR438030
- S. Janson, Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation, Duke Math. J. 47 (1980), 959-982 Zbl0453.46027MR596123
- S. Janson, P. W. Jones, Interpolation between spaces; The complex method, Journ. of Funct. Anal. 48 (1982), 58-80 Zbl0507.46047MR671315
- F. John, L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426 Zbl0102.04302MR131498
- P. W. Jones, Carleson measures and the Fefferman-Stein decomposition of , Ann. of Math. 111 (1980), 197-208 Zbl0393.30029MR558401
- P. W. Jones, Extension theorems for , Indiana Univ. Math. J. 29 (1980), 41-66 Zbl0432.42017MR554817
- P. W. Jones, Interpolation between Hardy spaces, I, II (Chicago, III, 1981) (1983), Wadsworth, Belmont, CA Zbl0523.46048MR730083
- P. W. Jones, J. L. Journé, On weak convergence in , Proc. Amer. Math. Soc. 120 (1994), 137-138 Zbl0814.42011MR1159172
- A. Jonsson, P. Sjögren, H. Wallin, Hardy and Lipschitz spaces on subsets of , Studia Math. 80 (1984), 141-166 Zbl0513.42020MR781332
- Z. Lou, A. McIntosh, Hardy spaces of exact forms on Lipschitz domains in , Indiana Univ. Math. J. 53 (2004), 583-611 Zbl1052.42021MR2060046
- M. Milman, T. Schonbek, Second order estimates in interpolation theory and applications, Proc. Amer. Math. Soc. 110 (1990), 961-969 Zbl0717.46066MR1075187
- A. Miyachi, spaces over open subsets of , Studia Math. 95 (1990), 205-228 Zbl0716.42017MR1060724
- S. Müller, A surprising higher integrability property of mappings with positive determinant, Bull. Amer. Math. Soc. 21 (1989), 245-248 Zbl0689.49006MR999618
- S. Müller, Weak continuity of determinants and nonlinear elasticity, C.R. Acad. Sci. Paris Ser. I Math. 311 (1990), 13-17 Zbl0679.34051MR964116
- S. Müller, T. Qi, B.S. Yan, On a new class of elastic deformations not allowing for cavitation, Ann. Inst. H. Poincaré, Anal. Non Lin. 11 (1994), 217-243 Zbl0863.49002MR1267368
- F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5 (1978), 489-507 Zbl0399.46022MR506997
- E. Nakai, Pointwise multipliers for functions of weighted bounded mean oscillation, Studia Math. 105 (1993), 105-119 Zbl0812.42008MR1226621
- E. Nakai, K. Yabuta, Pointwise multipliers for functions of bounded mean oscillation, J. Math. Soc. Japan 37 (1985), 207-218 Zbl0546.42019MR780660
- M. M. Rao, Z. D. Ren, Theory of Orlicz Spaces, 146 (1991), Dekker, New York Zbl0724.46032MR1113700
- C. Sbordone, Grand Sobolev spaces and their applications to variational problems, Le Matematiche (Catania) 51 (1996(1997)), 335-347 Zbl0915.46030MR1488076
- D. A. Stegenga, Bounded Toeplitz operators on and applications of the duality between and the functions of bounded mean oscillation, Amer. J. Math. 98 (1976), 573-589 Zbl0335.47018MR420326
- E. M. Stein, Note on the class , Studia Math. 32 (1969), 305-310 Zbl0182.47803MR247534
- E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, 43 (1993), Princeton University Press, Princeton Zbl0821.42001MR1232192
- J. O. Stromberg, Bounded mean oscillations with Orlicz norms and duality of Hardy spaces, Indiana Univ. Math. J. 28 (1979), 511-544 Zbl0429.46016MR529683
- V. Sverak, Regularity properties of deformations with finite energy, Arch. Rational Mech. Anal. 100 (1988), 105-127 Zbl0659.73038MR913960
- L. Tartar, Compensated compactness and applications to partial differential equations, 39 (1979), Pitman, Boston Zbl0437.35004MR584398
- A. Uchiyama, A constructive proof of the Fefferman-Stein decomposition of , Acta. Math. 148 (1982), 215-241 Zbl0514.46018MR666111
- A. Uchiyama, Hardy spaces on the Euclidean space, (2001), Springer-Verlag, Tokyo Zbl0984.42015MR1845883
- K. Zhang, Biting theorems for Jacobians and their applications, Ann. I. H. P. Anal. Non Lin. 7 (1990), 345-365 Zbl0717.49012MR1067780
- M. Zinsmeister, Espaces de Hardy et domaines de Denjoy, Ark. Mat. 27 (1989), 363-378 Zbl0682.30030MR1022286
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