Riesz means of Fourier series and integrals: Strong summability at the critical index

Kim, Jongchon; Seeger, Andreas

doi:10.1090/tran/7818

Riesz means of Fourier series and integrals: Strong summability at the critical index
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by Jongchon Kim and Andreas Seeger PDF

Trans. Amer. Math. Soc. 372 (2019), 2959-2999 Request permission

Abstract:

We consider spherical Riesz means of multiple Fourier series and some generalizations. While almost everywhere convergence of Riesz means at the critical index $(d-1)/2$ may fail for functions in the Hardy space $h^1(\mathbb T^d)$, we prove sharp positive results for strong summability almost everywhere. For functions in $L^p(\mathbb T^d)$, $1<p<2$, we consider Riesz means at the critical index $d(1/p-1/2)-1/2$ and prove an almost sharp theorem on strong summability. The results follow via transference from corresponding results for Fourier integrals. We include an endpoint bound on maximal operators associated with generalized Riesz means on Hardy spaces $H^p(\mathbb {R}^d)$ for $0<p<1$.

References

Nakhlé Asmar, Earl Berkson, and Jean Bourgain, Restrictions from $\mathbf R^n$ to $\mathbf Z^n$ of weak type $(1,1)$ multipliers, Studia Math. 108 (1994), no. 3, 291–299. MR 1259281, DOI 10.4064/sm-108-3-291-299
Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275, DOI 10.1007/978-3-642-66451-9
Salomon Bochner, Summation of multiple Fourier series by spherical means, Trans. Amer. Math. Soc. 40 (1936), no. 2, 175–207. MR 1501870, DOI 10.1090/S0002-9947-1936-1501870-1
Jean Bourgain, Estimations de certaines fonctions maximales, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 10, 499–502 (French, with English summary). MR 812567
Jean Bourgain and Larry Guth, Bounds on oscillatory integral operators based on multilinear estimates, Geom. Funct. Anal. 21 (2011), no. 6, 1239–1295. MR 2860188, DOI 10.1007/s00039-011-0140-9
Anthony Carbery and Fernando Soria, Pointwise Fourier inversion and localisation in $\textbf {R}^n$, Proceedings of the conference dedicated to Professor Miguel de Guzmán (El Escorial, 1996), 1997, pp. 847–858. MR 1600203, DOI 10.1007/BF02656490
Sun-Yung A. Chang and Robert Fefferman, A continuous version of duality of $H^{1}$ with BMO on the bidisc, Ann. of Math. (2) 112 (1980), no. 1, 179–201. MR 584078, DOI 10.2307/1971324
Michael Christ, Weak type $(1,1)$ bounds for rough operators, Ann. of Math. (2) 128 (1988), no. 1, 19–42. MR 951506, DOI 10.2307/1971461
Michael Christ, Weak type endpoint bounds for Bochner-Riesz multipliers, Rev. Mat. Iberoamericana 3 (1987), no. 1, 25–31. MR 1008443, DOI 10.4171/RMI/44
F. M. Christ and C. D. Sogge, The weak type $L^1$ convergence of eigenfunction expansions for pseudodifferential operators, Invent. Math. 94 (1988), no. 2, 421–453. MR 958838, DOI 10.1007/BF01394331
Michael Christ and Christopher D. Sogge, On the $L^1$ behavior of eigenfunction expansions and singular integral operators, Miniconferences on harmonic analysis and operator algebras (Canberra, 1987) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 16, Austral. Nat. Univ., Canberra, 1988, pp. 29–50. MR 953981
A. Córdoba, A note on Bochner-Riesz operators, Duke Math. J. 46 (1979), no. 3, 505–511. MR 544242, DOI 10.1215/S0012-7094-79-04625-8
A. Erdélyi, Asymptotic expansions, Dover Publications, Inc., New York, 1956. MR 0078494
Da Shan Fan and Zhi Jian Wu, Transference of maximal multipliers on Hardy spaces, Proc. Amer. Math. Soc. 123 (1995), no. 10, 3169–3174. MR 1273491, DOI 10.1090/S0002-9939-1995-1273491-2
Charles Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9–36. MR 257819, DOI 10.1007/BF02394567
Gustavo Garrigós and Andreas Seeger, A note on maximal operators associated with Hankel multipliers, Rev. Un. Mat. Argentina 50 (2009), no. 2, 137–148. MR 2656531
David Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979), no. 1, 27–42. MR 523600
G. H. Hardy and J. E. Littlewood, Sur la séries de Fourier d’une fonction à carré sommable, Comptes Rendus (Paris) 156 (1913), 1307–1309.
Sunggeum Hong, Joonil Kim, and Chan Woo Yang, Riesz means associated with certain product type convex domain, J. Math. Anal. Appl. 380 (2011), no. 2, 585–606. MR 2794416, DOI 10.1016/j.jmaa.2011.02.070
Carlos E. Kenig and Peter A. Tomas, Maximal operators defined by Fourier multipliers, Studia Math. 68 (1980), no. 1, 79–83. MR 583403, DOI 10.4064/sm-68-1-79-83
N. J. Kalton, Linear operators on $L_{p}$ for $0<p<1$, Trans. Amer. Math. Soc. 259 (1980), no. 2, 319–355. MR 567084, DOI 10.1090/S0002-9947-1980-0567084-3
Y. Kim and A. Seeger, A note on pointwise convergence of quasiradial Riesz means, Acta Sci. Math. (Szeged) 62 (1996), no. 1-2, 187–199. MR 1412927
A. N. Kolmogorov, Une série de Fourier-Lebesgue divergente presque partout, Fund. Math. 4 (1923), 324–328 (French).
Sergey V. Konyagin, Almost everywhere convergence and divergence of Fourier series, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1393–1403. MR 2275651
Karel de Leeuw, On $L_{p}$ multipliers, Ann. of Math. (2) 81 (1965), 364–379. MR 174937, DOI 10.2307/1970621
Zhi Xin Liu and Shan Zhen Lu, Transference and restriction of maximal multiplier operators on Hardy spaces, Studia Math. 105 (1993), no. 2, 121–134. MR 1226622, DOI 10.4064/sm-105-2-121-134
Shanzhen Lu, Conjectures and problems on Bochner-Riesz means, Front. Math. China 8 (2013), no. 6, 1237–1251. MR 3122308, DOI 10.1007/s11464-013-0323-9
J. Marcinkiewicz, Sur la sommabilité forte de séries de Fourier, J. London Math. Soc. 14 (1939), 162–168 (French). MR 66, DOI 10.1112/jlms/s1-14.3.162
Jaak Peetre, On spaces of Triebel-Lizorkin type, Ark. Mat. 13 (1975), 123–130. MR 380394, DOI 10.1007/BF02386201
V. A. Rodin, The space BMO and strong means of Fourier series, Anal. Math. 16 (1990), no. 4, 291–302 (English, with Russian summary). MR 1094184, DOI 10.1007/BF02630362
Andreas Seeger, Estimates near $L^1$ for Fourier multipliers and maximal functions, Arch. Math. (Basel) 53 (1989), no. 2, 188–193. MR 1004277, DOI 10.1007/BF01198570
Andreas Seeger, Endpoint estimates for multiplier transformations on compact manifolds, Indiana Univ. Math. J. 40 (1991), no. 2, 471–533. MR 1119186, DOI 10.1512/iumj.1991.40.40024
Andreas Seeger, Endpoint inequalities for Bochner-Riesz multipliers in the plane, Pacific J. Math. 174 (1996), no. 2, 543–553. MR 1405600, DOI 10.2140/pjm.1996.174.543
Elias M. Stein, Localization and summability of multiple Fourier series, Acta Math. 100 (1958), 93–147. MR 105592, DOI 10.1007/BF02559603
E. M. Stein, On limits of seqences of operators, Ann. of Math. (2) 74 (1961), 140–170. MR 125392, DOI 10.2307/1970308
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
E. M. Stein, An $H^{1}$ function with nonsummable Fourier expansion, Harmonic analysis (Cortona, 1982) Lecture Notes in Math., vol. 992, Springer, Berlin, 1983, pp. 193–200. MR 729354, DOI 10.1007/BFb0069159
E. M. Stein, Oscillatory integrals in Fourier analysis, Beijing lectures in harmonic analysis (Beijing, 1984) Ann. of Math. Stud., vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 307–355. MR 864375
Elias M. Stein, Mitchell H. Taibleson, and Guido Weiss, Weak type estimates for maximal operators on certain $H^{p}$ classes, Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980), 1981, pp. 81–97. MR 639468
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
E. M. Stein and N. J. Weiss, On the convergence of Poisson integrals, Trans. Amer. Math. Soc. 140 (1969), 35–54. MR 241685, DOI 10.1090/S0002-9947-1969-0241685-X
Gen-ichirô Sunouchi, On the Littlewood-Paley function $g^*$ of multiple Fourier integrals and Hankel multiplier transformations, Tohoku Math. J. (2) 19 (1967), 496–511. MR 225097, DOI 10.2748/tmj/1178243254
Terence Tao, The weak-type endpoint Bochner-Riesz conjecture and related topics, Indiana Univ. Math. J. 47 (1998), no. 3, 1097–1124. MR 1665753, DOI 10.1512/iumj.1998.47.1544
Terence Tao, On the maximal Bochner-Riesz conjecture in the plane for $p<2$, Trans. Amer. Math. Soc. 354 (2002), no. 5, 1947–1959. MR 1881025, DOI 10.1090/S0002-9947-02-02942-2
Walter Trebels, Some Fourier multiplier criteria and the spherical Bochner-Riesz kernel, Rev. Roumaine Math. Pures Appl. 20 (1975), no. 10, 1173–1185. MR 404967
Bobby Wilson, On almost everywhere convergence of strong arithmetic means of Fourier series, Trans. Amer. Math. Soc. 367 (2015), no. 2, 1467–1500. MR 3280051, DOI 10.1090/S0002-9947-2014-06297-1
A. Zygmund, On the convergence and summability of power series on the circle of convergence. II, Proc. London Math. Soc. (2) 47 (1942), 326–350. MR 7042, DOI 10.1112/plms/s2-47.1.326
A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587