On Riesz transforms and maximal functions in the context of Gaussian harmonic analysis. (English) Zbl 1128.42006
In the paper the weak type \((1,1)\) inequality with respect to Gaussian measure \(\gamma\) for the maximal operator
\[ (M_{\Phi}f)(x) =\sup_{0<r<1} \frac{1}{\gamma((1+\delta)B(\frac{x}{r}, \frac{| x| }{r}(1-r)))}\int_{\mathbb R^n} \Phi\Big( \frac{| x-ry| }{(1-r^2)^{1/2}} \Big) | f(y)| \,d\gamma(y) \]
has been proved. This result, in particular, gives the same conclusion for the operator \[ T^{*}f(y)=\sup_{0<r<1} \bigg| \frac{e^{| x| ^2}}{\pi^{1/2} (1-r^2)^{n/2}} \int_{\mathbb R^n} e^{-\frac{| x-ry| ^2}{1-r^2}}f(y) \,d\gamma(y)\bigg| , \] which was proved in [P. Sjögren, Harmonic analysis, Proc. Conf., Cortona/Italy 1982, Lect. Notes Math. 992, 73–82 (1983; Zbl 0515.42022)] for \(n>1\). The authors introduce a new class of Gaussian Riesz transforms and using the maximal function \(M_{\Phi}\), they prove the weak type \((1,1)\) inequality for new Gaussian Riesz transforms.
\[ (M_{\Phi}f)(x) =\sup_{0<r<1} \frac{1}{\gamma((1+\delta)B(\frac{x}{r}, \frac{| x| }{r}(1-r)))}\int_{\mathbb R^n} \Phi\Big( \frac{| x-ry| }{(1-r^2)^{1/2}} \Big) | f(y)| \,d\gamma(y) \]
has been proved. This result, in particular, gives the same conclusion for the operator \[ T^{*}f(y)=\sup_{0<r<1} \bigg| \frac{e^{| x| ^2}}{\pi^{1/2} (1-r^2)^{n/2}} \int_{\mathbb R^n} e^{-\frac{| x-ry| ^2}{1-r^2}}f(y) \,d\gamma(y)\bigg| , \] which was proved in [P. Sjögren, Harmonic analysis, Proc. Conf., Cortona/Italy 1982, Lect. Notes Math. 992, 73–82 (1983; Zbl 0515.42022)] for \(n>1\). The authors introduce a new class of Gaussian Riesz transforms and using the maximal function \(M_{\Phi}\), they prove the weak type \((1,1)\) inequality for new Gaussian Riesz transforms.
Reviewer: Alexander Meskhi (Lahore)
MSC:
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |
| 47D06 | One-parameter semigroups and linear evolution equations |
| 42A50 | Conjugate functions, conjugate series, singular integrals |
| 60H07 | Stochastic calculus of variations and the Malliavin calculus |
Keywords:
Gaussian measure; singular integrals; maximal functions; Fourier analysis in several variables; Hermite expansionsCitations:
Zbl 0515.42022References:
| [1] | Calixto P. Calderón, Some remarks on the multiple Weierstrass transform and Abel summability of multiple Fourier-Hermite series, Studia Math. 32 (1969), 119 – 148. · Zbl 0182.15901 |
| [2] | Eugene B. Fabes, Cristian E. Gutiérrez, and Roberto Scotto, Weak-type estimates for the Riesz transforms associated with the Gaussian measure, Rev. Mat. Iberoamericana 10 (1994), no. 2, 229 – 281. · Zbl 0810.42006 · doi:10.4171/RMI/152 |
| [3] | Forzani L. Lemas de cubrimiento de tipo Besicovitch y su aplicación al estudio del operador maximal de Ornstein-Uhlenbeck. (Besicovitch type covering lemmas and their applications to the study of the Ornstein-Uhlenbeck maximal operator). Ph.D. thesis disertation. Universidad Nacional de San Luis, Argentina. |
| [4] | Forzani L., Harboure E., and Scotto R. Weak type inequality for a family of singular integral operators related with the gaussian measure. Preprint. · Zbl 1205.42015 |
| [5] | Liliana Forzani and Roberto Scotto, The higher order Riesz transform for Gaussian measure need not be of weak type (1,1), Studia Math. 131 (1998), no. 3, 205 – 214. · Zbl 0954.42009 |
| [6] | Liliana Forzani, Roberto Scotto, and Wilfredo Urbina, Riesz and Bessel potentials, the \?^{\?} functions and an area function for the Gaussian measure \?, Rev. Un. Mat. Argentina 42 (2000), no. 1, 17 – 37 (2001). · Zbl 0996.42011 |
| [7] | Liliana Forzani, Roberto Scotto, and Wilfredo Urbina, A simple proof of the \?^{\?} continuity of the higher order Riesz transforms with respect to the Gaussian measure \?_{\?}, Séminaire de Probabilités, XXXV, Lecture Notes in Math., vol. 1755, Springer, Berlin, 2001, pp. 162 – 166 (English, with English and French summaries). · Zbl 1006.47030 · doi:10.1007/978-3-540-44671-2_12 |
| [8] | José García-Cuerva, Giancarlo Mauceri, Peter Sjögren, and José Luis Torrea, Spectral multipliers for the Ornstein-Uhlenbeck semigroup, J. Anal. Math. 78 (1999), 281 – 305. · Zbl 0939.42007 · doi:10.1007/BF02791138 |
| [9] | José García-Cuerva, Giancarlo Mauceri, Peter Sjögren, and José-Luis Torrea, Higher-order Riesz operators for the Ornstein-Uhlenbeck semigroup, Potential Anal. 10 (1999), no. 4, 379 – 407. · Zbl 0954.42010 · doi:10.1023/A:1008685801945 |
| [10] | Richard F. Gundy, Sur les transformations de Riesz pour le semi-groupe d’Ornstein-Uhlenbeck, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 19, 967 – 970 (French, with English summary). · Zbl 0606.60063 |
| [11] | Cristian E. Gutiérrez, On the Riesz transforms for Gaussian measures, J. Funct. Anal. 120 (1994), no. 1, 107 – 134. · Zbl 0807.46030 · doi:10.1006/jfan.1994.1026 |
| [12] | Cristian E. Gutiérrez, Carlos Segovia, and José Luis Torrea, On higher Riesz transforms for Gaussian measures, J. Fourier Anal. Appl. 2 (1996), no. 6, 583 – 596. · Zbl 0893.42007 · doi:10.1007/s00041-001-4044-1 |
| [13] | Cristian E. Gutiérrez and Wilfredo O. Urbina, Estimates for the maximal operator of the Ornstein-Uhlenbeck semigroup, Proc. Amer. Math. Soc. 113 (1991), no. 1, 99 – 104. · Zbl 0737.42019 |
| [14] | Einar Hille, A class of reciprocal functions, Ann. of Math. (2) 27 (1926), no. 4, 427 – 464. · JFM 52.0400.02 · doi:10.2307/1967695 |
| [15] | Itô K. Malliavin’s \( C^{\infty}\) Functionals of a Centered Gaussian System. IMA Preprint Series 327, (1987), University of Minnesota. |
| [16] | Albert Messiah, Quantum mechanics. Vol. I, Translated from the French by G. M. Temmer, North-Holland Publishing Co., Amsterdam; Interscience Publishers Inc., New York, 1961. · Zbl 0102.42602 |
| [17] | P.-A. Meyer, Transformations de Riesz pour les lois gaussiennes, Seminar on probability, XVIII, Lecture Notes in Math., vol. 1059, Springer, Berlin, 1984, pp. 179 – 193 (French). · Zbl 0543.60078 · doi:10.1007/BFb0100043 |
| [18] | Benjamin Muckenhoupt, Hermite conjugate expansions, Trans. Amer. Math. Soc. 139 (1969), 243 – 260. · Zbl 0175.12701 |
| [19] | Sonsoles Pérez, Boundedness of Littlewood-Paley \?-functions of higher order associated with the Ornstein-Uhlenbeck semigroup, Indiana Univ. Math. J. 50 (2001), no. 2, 1003 – 1014. · Zbl 1058.42016 · doi:10.1512/iumj.2001.50.1809 |
| [20] | Sonsoles Pérez, The local part and the strong type for operators related to the Gaussian measure, J. Geom. Anal. 11 (2001), no. 3, 491 – 507. · Zbl 1058.42015 · doi:10.1007/BF02922016 |
| [21] | Gilles Pisier, Riesz transforms: a simpler analytic proof of P.-A. Meyer’s inequality, Séminaire de Probabilités, XXII, Lecture Notes in Math., vol. 1321, Springer, Berlin, 1988, pp. 485 – 501. · Zbl 0645.60061 · doi:10.1007/BFb0084154 |
| [22] | Peter Sjögren, On the maximal function for the Mehler kernel, Harmonic analysis (Cortona, 1982) Lecture Notes in Math., vol. 992, Springer, Berlin, 1983, pp. 73 – 82. · doi:10.1007/BFb0069151 |
| [23] | Peter Sjögren, A remark on the maximal function for measures in \?\(^{n}\), Amer. J. Math. 105 (1983), no. 5, 1231 – 1233. · Zbl 0528.42007 · doi:10.2307/2374340 |
| [24] | Sjögren P. Personal communication. |
| [25] | 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. · Zbl 0193.10502 |
| [26] | Wilfredo Urbina, On singular integrals with respect to the Gaussian measure, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), no. 4, 531 – 567. · Zbl 0737.42018 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
