Some recent developments in Fourier analysis and 𝐻^{𝑝}-theory on product domains

Chang, Sun-Yung A.; Fefferman, Robert

doi:10.1090/S0273-0979-1985-15291-7

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-9485(online) ISSN 0273-0979(print)

Some recent developments in Fourier analysis and $H^p$-theory on product domains

Authors: Sun-Yung A. Chang and Robert Fefferman
Journal: Bull. Amer. Math. Soc. 12 (1985), 1-43
MSC (1980): Primary 42B20, 42B25, 32A35
MathSciNet review: 766959
Full-text PDF

1. Albert Baernstein II, Univalence and bounded mean oscillation, Michigan Math. J. 23 (1976), no. 3, 217–223 (1977). MR 0444935 (56 #3281)
2. A. Benedek, A.-P. Calderón, and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 356–365. MR 0133653 (24 #A3479)
3. Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. MR 0482275 (58 #2349)
4. Alain Bernard, Espaces 𝐻¹ de martingales à deux indices. Dualité avec les martingales de type “BMO”, Bull. Sci. Math. (2) 103 (1979), no. 3, 297–303 (French, with English summary). MR 539351 (82d:60092)
5. S. V. Bockarev, On a basis in the space of functions continuous in the closed disc and analytic inside it, Soviet Math. Dokl. 15 (1974), 1195-1216.
6. D. L. Burkholder, Martingale theory and harmonic analysis in Euclidean spaces, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 2, Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 283–301. MR 545315 (82c:60089)
7. D. L. Burkholder, R. F. Gundy, and M. L. Silverstein, A maximal function characterization of the class 𝐻^{𝑝}, Trans. Amer. Math. Soc. 157 (1971), 137–153. MR 0274767 (43 #527), http://dx.doi.org/10.1090/S0002-9947-1971-0274767-6
8. D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249–304. MR 0440695 (55 #13567)
9. A.-P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190. MR 0167830 (29 #5097)
10. A. P. Calderón, On the behaviour of harmonic functions at the boundary, Trans. Amer. Math. Soc. 68 (1950), 47–54. MR 0032863 (11,357e), http://dx.doi.org/10.1090/S0002-9947-1950-0032863-9
11. A. P. Calderón, On a theorem of Marcinkiewicz and Zygmund, Trans. Amer. Math. Soc. 68 (1950), 55–61. MR 0032864 (11,357f), http://dx.doi.org/10.1090/S0002-9947-1950-0032864-0
12. Alberto-P. Calderón, An atomic decomposition of distributions in parabolic 𝐻^{𝑝}\ spaces, Advances in Math. 25 (1977), no. 3, 216–225. MR 0448066 (56 #6376)
13. A.-P. Calderón and A. Torchinsky, Parabolic maximal functions associated with a distribution. II, Advances in Math. 24 (1977), no. 2, 101–171. MR 0450888 (56 #9180)
A. P. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85–139. MR 0052553 (14,637f)
15. Lennart Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547–559. MR 0141789 (25 #5186)
16. L. Carleson, The Corona problem (Proc. 15th Scandinavian Congress, 968 (1970). Lecture Notes in Math., Springer, 118.
17. L. Carleson, A counterexample for measures bounded on H, Mittag-Leffler Report No. 7, 1974.
18. Lennart Carleson, An explicit unconditional basis in 𝐻¹, Bull. Sci. Math. (2) 104 (1980), no. 4, 405–416 (English, with French summary). MR 602408 (82b:46028)
19. L. Carleson, BMO in 10 years, Proc. Scandinavian Congress Lecture Notes, 1981.
20. Sun Yung A. Chang, A characterization of Douglas subalgebras, Acta Math. 137 (1976), no. 2, 82–89. MR 0428044 (55 #1074a)
21. Sun-Yung A. Chang, Carleson measure on the bi-disc, Ann. of Math. (2) 109 (1979), no. 3, 613–620. MR 534766 (80j:32009), http://dx.doi.org/10.2307/1971229
22. Sun-Yung A. Chang, Two remarks about 𝐻¹ and BMO on the bidisc, (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 373–393. MR 730080 (85e:32007)
23. Sun-Yung A. Chang and Z. Ciesielski, Spline characterizations of 𝐻¹, Studia Math. 75 (1983), no. 2, 183–192. MR 722263 (85f:46051)
24. Sun-Yung A. Chang and Robert Fefferman, A continuous version of duality of 𝐻¹ with BMO on the bidisc, Ann. of Math. (2) 112 (1980), no. 1, 179–201. MR 584078 (82a:32009), http://dx.doi.org/10.2307/1971324
25. Sun-Yung A. Chang and Robert Fefferman, The Calderón-Zygmund decomposition on product domains, Amer. J. Math. 104 (1982), no. 3, 455–468. MR 658542 (84a:42028), http://dx.doi.org/10.2307/2374150
26. Z. Ciesielski, Properties of the orthonormal Franklin system. II, Studia Math. 27 (1966), 289–323. MR 0203349 (34 #3202)
27. Ronald R. Coifman, A real variable characterization of 𝐻^{𝑝}, Studia Math. 51 (1974), 269–274. MR 0358318 (50 #10784)
28. R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 0358205 (50 #10670)
29. R. R. Coifman, Y. Meyer and E. M. Stein, Un nouvel espace functionnel adapte a l'etude des operateurs definis par des integrales singulieres (preprint).
30. 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 0447954 (56 #6264), http://dx.doi.org/10.1090/S0002-9904-1977-14325-5
31. Antonio Córdoba, Maximal functions, covering lemmas and Fourier multipliers, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 1, Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 29–50. MR 545237 (81f:42017)
32. A. Cordoba and R. Fefferman, A geometric proof of the strong maximal theorem, Ann. of Math. (2) 102 (1975), no. 1, 95–100. MR 0379785 (52 #690)
33. A. Córdoba and R. Fefferman, On differentiation of integrals, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 6, 2211–2213. MR 0476977 (57 #16522)
34. Burgess Davis, Hardy spaces and rearrangements, Trans. Amer. Math. Soc. 261 (1980), no. 1, 211–233. MR 576872 (81h:42011), http://dx.doi.org/10.1090/S0002-9947-1980-0576872-9
35. Charles Fefferman, Characterizations of bounded mean oscillation, Bull. Amer. Math. Soc. 77 (1971), 587–588. MR 0280994 (43 #6713), http://dx.doi.org/10.1090/S0002-9904-1971-12763-5
36. Charles Fefferman, Estimates for double Hilbert transforms, Studia Math. 44 (1972), 1–15. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I. MR 0312166 (47 #728)
37. C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115. MR 0284802 (44 #2026)
38. C. Fefferman and E. M. Stein, 𝐻^{𝑝} spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 0447953 (56 #6263)
39. C. Fefferman, N. M. Rivière, and Y. Sagher, Interpolation between 𝐻^{𝑝} spaces: the real method, Trans. Amer. Math. Soc. 191 (1974), 75–81. MR 0388072 (52 #8909), http://dx.doi.org/10.1090/S0002-9947-1974-0388072-3
40. R. Fefferman, Bounded mean oscillation on the polydisk, Ann. of Math. (2) 110 (1979), no. 2, 395–406. MR 549492 (81c:32016), http://dx.doi.org/10.2307/1971231
41. R. Fefferman, Strong differentiation with respect to measures, Amer. J. Math. 103 (1981), no. 1, 33–40. MR 601461 (83g:42009), http://dx.doi.org/10.2307/2374188
42. R. Fefferman, The atomic decomposition of 𝐻¹ in product spaces, Adv. in Math. 55 (1985), no. 1, 90–100. MR 772072 (86e:42034), http://dx.doi.org/10.1016/0001-8708(85)90006-4
43. R. Fefferman, Some weighted norm inequalities for Córdoba’s maximal function, Amer. J. Math. 106 (1984), no. 5, 1261–1264. MR 761586 (86a:42024), http://dx.doi.org/10.2307/2374280
44. Robert Fefferman and Elias M. Stein, Singular integrals on product spaces, Adv. in Math. 45 (1982), no. 2, 117–143. MR 664621 (84d:42023), http://dx.doi.org/10.1016/S0001-8708(82)80001-7
45. G. B. Folland and Elias M. Stein, Hardy spaces on homogeneous groups, Mathematical Notes, vol. 28, Princeton University Press, Princeton, N.J., 1982. MR 657581 (84h:43027)
46. John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1981. MR 628971 (83g:30037)
47. John B. Garnett and Peter W. Jones, BMO from dyadic BMO, Pacific J. Math. 99 (1982), no. 2, 351–371. MR 658065 (85d:42021)
48. R. Gundy, Maximal function characterizations of H, Lecture Notes in Math., Springer-Verlag, 1978.
49. R. Gundy, Inégalités pour martingales à un et deux indices l'espace H, L'École d'ete de St. Flour, 1978.
50. R. F. Gundy and E. M. Stein, 𝐻^{𝑝} theory for the poly-disc, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 3, 1026–1029. MR 524328 (80j:32012)
51. Richard Hunt, Benjamin Muckenhoupt, and Richard Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251. MR 0312139 (47 #701), http://dx.doi.org/10.1090/S0002-9947-1973-0312139-8
52. Satoru Igari, An extension of the interpolation theorem of Marcinkiewicz, Proc. Japan Acad. 38 (1962), 731–734. MR 0147815 (26 #5328)
53. Björn Jawerth and Alberto Torchinsky, The strong maximal function with respect to measures, Studia Math. 80 (1984), no. 3, 261–285. MR 783994 (87b:42024)
54. B. Jessen, J. Marcinkiewicz and A. Zygmund, Note on the differentiability of multiple integrals, Fund. Math. 24 (1953), 217-234.
55. Fritz John, Rotation and strain, Comm. Pure Appl. Math. 14 (1961), 391–413. MR 0138225 (25 #1672)
56. F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426. MR 0131498 (24 #A1348)
57. Peter W. Jones, Carleson measures and the Fefferman-Stein decomposition of 𝐵𝑀𝑂(𝑅), Ann. of Math. (2) 111 (1980), no. 1, 197–208. MR 558401 (81d:30059), http://dx.doi.org/10.2307/1971197
58. P. Jones, L, Acta Math. 150 (1982), 136-152.
59. Peter W. Jones, Interpolation between Hardy spaces, (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 437–451. MR 730083 (85a:42029)
60. K. C. Lin, Thesis, Univ. of California, Los Angeles, 1984.
61. K. C. Lin, H (to appear).
62. Marie-Paule Malliavin and Paul Malliavin, Intégrales de Lusin-Calderon pour les fonctions biharmoniques, Bull. Sci. Math. (2) 101 (1977), no. 4, 357–384 (French, with English summary). MR 0473204 (57 #12879)
63. Donald E. Marshall, Subalgebras of 𝐿^{∞} containing 𝐻^{∞}, Acta Math. 137 (1976), no. 2, 91–98. MR 0428045 (55 #1074b)
64. Bernard Maurey, Isomorphismes entre espaces 𝐻₁, Acta Math. 145 (1980), no. 1-2, 79–120 (French). MR 586594 (84b:46027), http://dx.doi.org/10.1007/BF02414186
65. Jürgen Moser, On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577–591. MR 0159138 (28 #2356)
66. Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. MR 0293384 (45 #2461), http://dx.doi.org/10.1090/S0002-9947-1972-0293384-6
67. Ch. Pommerenke, Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation, Comment. Math. Helv. 52 (1977), no. 4, 591–602 (German). MR 0454017 (56 #12268)
68. J. Privalov, Sur les fonctions conjuguées, Bull. Soc. Math. France 44 (1916), 100-103.
69. H. M. Reimann, Functions of bounded mean oscillation and quasiconformal mappings, Comment. Math. Helv. 49 (1974), 260–276. MR 0361067 (50 #13513)
70. N. M. Rivière and Y. Sagher, Interpolation between 𝐿^{∞} and 𝐻¹, the real method, J. Functional Analysis 14 (1973), 401–409. MR 0361759 (50 #14204)
71. Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York, 1966. MR 0210528 (35 #1420)
72. Donald Sarason, Function theory on the unit circle, Virginia Polytechnic Institute and State University Department of Mathematics, Blacksburg, Va., 1978. Notes for lectures given at a Conference at Virginia Polytechnic Institute and State University, Blacksburg, Va., June 19–23, 1978. MR 521811 (80d:30035)
73. Donald Sarason, Algebras between 𝐿^{∞} and 𝐻^{∞}, Spaces of analytic functions (Sem. Functional Anal. and Function Theory, Kristiansand, 1975), Springer, Berlin, 1976, pp. 117–130. Lecture Notes in Math., Vol. 512. MR 0493361 (58 #12386)
74. Elias M. Stein, On the theory of harmonic functions of several variables. II. Behavior near the boundary, Acta Math. 106 (1961), 137–174. MR 0173019 (30 #3234)
75. Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095 (44 #7280)
76. Elias M. Stein, A variant of the area integral, Bull. Sci. Math. (2) 103 (1979), no. 4, 449–461 (English, with French summary). MR 548920 (81a:31008)
77. Elias M. Stein and Guido Weiss, On the theory of harmonic functions of several variables. I. The theory of 𝐻^{𝑝}-spaces, Acta Math. 103 (1960), 25–62. MR 0121579 (22 #12315)
78. Jan-Olov Strömberg, Weak estimates on maximal functions with rectangles in certain directions, Ark. Mat. 15 (1977), no. 2, 229–240. MR 0487260 (58 #6911)
79. Jan-Olov Strömberg, A modified Franklin system and higher-order spline systems on 𝑅ⁿ as unconditional bases for Hardy spaces, (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 475–494. MR 730086 (85d:42027)
80. Akihito Uchiyama, A constructive proof of the Fefferman-Stein decomposition of BMO (𝑅ⁿ), Acta Math. 148 (1982), 215–241. MR 666111 (84h:42037), http://dx.doi.org/10.1007/BF02392729
81. N. Th. Varopoulos, BMO functions and the \𝑜𝑣𝑒𝑟𝑙𝑖𝑛𝑒∂-equation, Pacific J. Math. 71 (1977), no. 1, 221–273. MR 0508035 (58 #22639a)
82. Nicholas Th. Varopoulos, Probabilistic approach to some problems in complex analysis, Bull. Sci. Math. (2) 105 (1981), no. 2, 181–224 (English, with French summary). MR 618877 (82k:32025)
83. P. Wojtaszczyk, The Franklin system is an unconditional basis in 𝐻₁, Ark. Mat. 20 (1982), no. 2, 293–300. MR 686177 (84f:46047), http://dx.doi.org/10.1007/BF02390514
84. Thomas H. Wolff, A note on interpolation spaces, Harmonic analysis (Minneapolis, Minn., 1981) Lecture Notes in Math., vol. 908, Springer, Berlin, 1982, pp. 199–204. MR 654187 (83f:46082)
85. A. Zygmund, Trigonometric series: Vols. I, II, Second edition, reprinted with corrections and some additions, Cambridge University Press, London, 1968. MR 0236587 (38 #4882)
86. A.-P. Calderón, Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1092–1099. MR 0177312 (31 #1575)
87. R. Coifman and Y. Meyer, Commutateurs d’intégrales singulières et opérateurs multilinéaires, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 3, xi, 177–202 (French, with English summary). MR 511821 (80a:47076)
88. Guy David and Jean-Lin Journé, A boundedness criterion for generalized Calderón-Zygmund operators, Ann. of Math. (2) 120 (1984), no. 2, 371–397. MR 763911 (85k:42041), http://dx.doi.org/10.2307/2006946
89. J. L. Journé, Vector valued singular integral operators and singular integral operators on product spaces (preprint).

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|