The development of square functions in the work of A. Zygmund
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- by E. M. Stein PDF
- Bull. Amer. Math. Soc. 7 (1982), 359-376
References
- Hans Rademacher, Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen, Math. Ann. 87 (1922), no. 1-2, 112–138 (German). MR 1512104, DOI 10.1007/BF01458040 [1923] A. S. Besicovitch, Sur la nature des fonctions à carré sommable et des ensembles mesurables, Fund. Math. 4, 172-195. [1923] A. N. Kolmogorov, Une série de Fourier-Lebesgue divergente presque-partout, Fund. Math. 4, 324-328. [1923] D. Menshov, Sur les séries de functions orthogonales. I, Fund. Math. 4, 82-105; II, 8 (1926), 56-108; III, 10 (1927), 375-420.
- Stefan Kaczmarz, Über die Konvergenz der Reihen von Orthogonalfunktionen, Math. Z. 23 (1925), no. 1, 263–270 (German). MR 1544743, DOI 10.1007/BF01506233 [1925] A. N. Kolmogorov, Sur les fonctions harmoniques conjuguées et les séries de Fourier, Fund. Math. 7, 23-28. [1926] A. S. Besicovitch, On a general metric property of summable functions, J. London Math. Soc. 1, 120-128. [1926] A. N. Kolmogorov, Une série de Fourier-Lebesgue divergente partout, Comptes Rend. Acad. Sci. 178, 303-305. [1926] A. N. Kolmogorov and G. Seliverstov, Sur la convergence des séries de Fourier, Rend. Acc. Linca. 3, 307-310. [1926] A. Plessner, Über die Konvergenz von trigonometrischen Reihen, J. F. Math. 155, 15-25.
- A. Zygmund, Une remarque sur un théorème de M. Kaczmarz, Math. Z. 25 (1926), no. 1, 297–298 (French). MR 1544811, DOI 10.1007/BF01283839
- Marcel Riesz, Sur les fonctions conjuguées, Math. Z. 27 (1928), no. 1, 218–244 (French). MR 1544909, DOI 10.1007/BF01171098 [1930] N. Lusin, Sur une propriété des fonctions à carré sommable, Bull. Calcutta Math. Soc. 20, 139-154. [1931] J. E. Littlewood and R. E. A. C. Paley, Theorems on Fourier series and power series. I, J. London Math. Soc. 6, 230-233; II, Proc. London Math. Soc. 42 (1936), 52-89; III, 43 (1937), 105-126. [1932] R. E. A. C. Paley, A remarkable system of orthogonal functions. I, Proc. London Math. Soc. 34, 241-264; II, 265-279. [1938] J. Marcinkiewicz, Sur quelques intégrales du type de Dini, Ann. Soc. Math. Polon. 17, 42-50.
- J. Marcinkiewicz and A. Zygmund, A theorem of Lusin. Part I, Duke Math. J. 4 (1938), no. 3, 473–485. MR 1546069, DOI 10.1215/S0012-7094-38-00440-5 [1938] A. Zygmund, On the convergence and summability of power series on the circle of convergence. I, Fund. Math. 30, 170-196. [1939] J. Marcinkiewicz, Sur les multiplicateurs des séries de Fourier, Studia Math. 8, 78-91.
- D. C. Spencer, A function-theoretic identity, Amer. J. Math. 65 (1943), 147–160. MR 7437, DOI 10.2307/2371778
- A. Zygmund, On certain integrals, Trans. Amer. Math. Soc. 55 (1944), 170–204. MR 9966, DOI 10.1090/S0002-9947-1944-0009966-5
- A. Zygmund, Proof of a theorem of Littlewood and Paley, Bull. Amer. Math. Soc. 51 (1945), 439–446. MR 12306, DOI 10.1090/S0002-9904-1945-08374-8
- A. P. Calderón, On the behaviour of harmonic functions at the boundary, Trans. Amer. Math. Soc. 68 (1950), 47–54. MR 32863, DOI 10.1090/S0002-9947-1950-0032863-9
- A. P. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85–139. MR 52553, DOI 10.1007/BF02392130
- S. G. Mihlin, On the multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR (N.S.) 109 (1956), 701–703 (Russian). MR 0080799
- Gen-ichirô Sunouchi, Theorems on power series of the class $H^p$, Tohoku Math. J. (2) 8 (1956), 125–146. MR 84607, DOI 10.2748/tmj/1178245015
- A. Zygmund, On the Littlewood-Paley function $g^*(\theta )$, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 208–212. MR 77700, DOI 10.1073/pnas.42.4.208
- E. M. Stein, On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 430–466. MR 112932, DOI 10.1090/S0002-9947-1958-0112932-2
- Mary Weiss and Antoni Zygmund, A note on smooth functions, Nederl. Akad. Wetensch. Proc. Ser. A 62 = Indag. Math. 21 (1959), 52–58. MR 0107122, DOI 10.1016/S1385-7258(59)50007-4 [1959] A. Zygmund, Trigonometric series (2nd ed.), 2 vols., Cambridge, England.
- 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 121579, DOI 10.1007/BF02546524
- A.-P. Calderón and A. Zygmund, Local properties of solutions of elliptic partial differential equations, Studia Math. 20 (1961), 171–225. MR 136849, DOI 10.4064/sm-20-2-181-225
- Elias M. Stein, On the theory of harmonic functions of several variables. II. Behavior near the boundary, Acta Math. 106 (1961), 137–174. MR 173019, DOI 10.1007/BF02545785
- 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 133653, DOI 10.1073/pnas.48.3.356
- E. M. Stein and A. Zygmund, On the differentiability of functions, Studia Math. 23 (1963/64), 247–283. MR 158955, DOI 10.4064/sm-23-3-247-283
- A.-P. Calderón, Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1092–1099. MR 177312, DOI 10.1073/pnas.53.5.1092
- D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494–1504. MR 208647, DOI 10.1214/aoms/1177699141
- Lennart Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135–157. MR 199631, DOI 10.1007/BF02392815 [1966] E. M. Stein, Classes H, C. R. Acad. Sci. Paris Sér. A 263, 716-719; 780-781; and 264 (1967), 107-108.
- D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249–304. MR 440695, DOI 10.1007/BF02394573
- Elias M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory. , Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. MR 0252961
- 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 274767, DOI 10.1090/S0002-9947-1971-0274767-6
- C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
- Elias M. Stein, Maximal functions. I. Spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 7, 2174–2175. MR 420116, DOI 10.1073/pnas.73.7.2174
- A. Córdoba and R. Fefferman, On differentiation of integrals, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 6, 2211–2213. MR 476977, DOI 10.1073/pnas.74.6.2211
- Björn E. J. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (1977), no. 3, 275–288. MR 466593, DOI 10.1007/BF00280445
- 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 473204
- Jan-Olov Strömberg, Weak estimates on maximal functions with rectangles in certain directions, Ark. Mat. 15 (1977), no. 2, 229–240. MR 487260, DOI 10.1007/BF02386043
- Elias M. Stein and Stephen Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1239–1295. MR 508453, DOI 10.1090/S0002-9904-1978-14554-6
- A. Nagel, E. M. Stein, and S. Wainger, Differentiation in lacunary directions, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), no. 3, 1060–1062. MR 466470, DOI 10.1073/pnas.75.3.1060
- Jean Brossard, Comportement des fonctions biharmoniques là où l’intégrale d’aire est finie, Bull. Sci. Math. (2) 103 (1979), no. 1, 77–95 (French, with English summary). MR 528163
- 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, DOI 10.1073/pnas.76.3.1026 [1982] R. R. Coifman, A. McIntosh, and Y. Meyer, L’Intégrale de Cauchy definit un operateur borné sur L, Ann. of Math. (2) (to appear).
Additional Information
- Journal: Bull. Amer. Math. Soc. 7 (1982), 359-376
- MSC (1980): Primary 42B25, 42A99; Secondary 30D55, 60G46
- DOI: https://doi.org/10.1090/S0273-0979-1982-15040-6
- MathSciNet review: 663787