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Vector-valued Hilbert transforms along curves. (English) Zbl 1337.43003
Summary: We show that Hilbert transforms along some curves are bounded on $$L^{p}({\mathbb{R}}^{n};X)$$ for some $$1< p< \infty$$ and some UMD spaces $$X$$. In particular, we prove that Hilbert transforms along some curves are completely $$L^{p}$$-bounded in the terminology from operator space theory. Moreover, we obtain the $$L^{p}(\mathbb{R}^{n};X)$$-boundedness of anisotropic singular integrals by using the “method of rotations” of Calderón-Zygmund. All these results extend preexisting related ones.

##### MSC:
 43A32 Other transforms and operators of Fourier type 46B99 Normed linear spaces and Banach spaces; Banach lattices 44A15 Special integral transforms (Legendre, Hilbert, etc.)
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##### References:
 [1] A. Benedek, A. P. Calderón, and R. Panzone, Convolution operators on Banach space valued functions , Proc. Natl. Acad. Sci. USA 48 (1962), no. 3, 356-365. · Zbl 0103.33402 [2] J. Bourgain, “Vector-valued singular integrals and the $$H^{1}$$-BMO duality” in Probability Theory and Harmonic Analysis (Cleveland, 1983) , Pure Appl. Math. 98 Dekker, New York, 1986, 1-19. [3] D. L. Burkholder, “A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions” in Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. I, II (Chicago, 1981) , Wadsworth, Belmont, CA, 1983, 270-286. [4] A. P. Calderón and A. Zygmund, On the existence of certain singular integrals , Acta Math. 88 (1952), no. 1, 85-139. · Zbl 0047.10201 [5] H. Carlsson, M. Christ, A. Córdoba, J. Duoandikoetxea, J. L. Rubio de Francia, J. Vance, S. Wainger, and D. Weinberg, $$L^{p}$$ estimates for maximal functions and Hilbert transforms along flat convex curves in $$\mathbb{R}^{2}$$ , Bull. Amer. Math. Soc. (N.S.) 14 (1986), no. 2, 263-267. · Zbl 0588.44007 [6] M. Christ, A. Nagel, E. M. Stein, and S. Wainger, Singular and maximal radon transforms: Analysis and geometry , Ann. of Math. (2) 150 (1999), no. 2, 489-577. · Zbl 0960.44001 [7] E. B. Fabes, Singular integrals and partial differential equations of parabolic type , Studia Math. 28 (1966), no. 1, 81-131. · Zbl 0144.35002 [8] V. S. Guliev, Imbedding theorems for spaces of UMD-valued functions , Dokl. Akad. Nauk 329 (1993), no. 4, 408-410; English translation in Russian Acad. Sci. Dokl. Math. 47 (1993), no. 2, 274-277. · Zbl 0836.46022 [9] G. Hong, L. D. López-Sánchez, J. M. Martell, and J. Parcet, Calderón-Zygmund operators associated to matrix-valued kernels , Int. Math. Res. Not. IMRN 2014 (2014), no. 5, 1221-1252. · Zbl 1296.42008 [10] G. Hong and J. Parcet, Necessity of property $$(\alpha)$$ for vector-valued Littlewood-Paley sets associated sumsets , in preparation. [11] T. Hytönen, Anisotropic Fourier multipliers and singular integrals for vector-valued functions , Ann. Mat. Pura Appl. (4) 186 (2007), no. 3, 455-468. · Zbl 1223.42007 [12] T. Hytönen and L. Weis, Singular convolution integrals with operator-valued kernel , Math. Z. 255 (2007), no. 2, 393-425. · Zbl 1189.45017 [13] T. Hytönen and L. Weis, On the necessity of property $$(\alpha)$$ for some vector-valued multiplier theorems , Arch. Math. (Basel) 90 (2008), no. 1, 44-52. · Zbl 1151.42003 [14] M. Junge, T. Mei, and J. Parcet. Smooth Fourier multipliers on group von Neumann algebras , Geom. Funct. Anal. 24 (2014), no. 6, 1913-1980. · Zbl 1306.43001 [15] H. Liu, Hilbert transforms along convex curves for valued functions , ISRN Math. Anal. 2014 (2014), art ID 827072. [16] T. R. McConnell, On Fourier multiplier transformations of Banach-valued functions , Trans. Amer. Math. Soc. 285 (1984), no. 2, 739-757. · Zbl 0566.42009 [17] T. Mei, Operator Valued Hardy Spaces , Mem. Amer. Math. Soc. 188 (2007), no. 881. · Zbl 1138.46038 [18] A. Nagel, N. M. Rivière, and S. Wainger, On Hilbert transforms along curves, II , Amer. J. Math. 98 (1976), no. 2, 395-403. · Zbl 0334.44012 [19] A. Nagel and S. Wainger, Hilbert transforms associated with plane curves , Trans. Amer. Math. Soc. 223 (1976), 235-252. · Zbl 0341.44005 [20] J. Parcet, Pseudo-localization of singular integrals and noncommutative Calderón-Zygmund theory , J. Funct. Anal. 256 (2009), no. 2, 509-593. · Zbl 1179.46051 [21] J. L. Rubio de Francia, “Martingale and integral transforms of Banach space valued functions” in Probability and Banach Spaces (Zaragoza, 1985) , Lecture Notes in Math. 1221 , Springer, Berlin, 1986, 195-222. [22] J. L. Rubio de Francia, F. J. Ruiz, and J. L. Torra, Calderón-Zygmund theory for operator-valued kernels , Adv. Math. 62 (1986), no. 1, 7-48. · Zbl 0627.42008 [23] E. M. Stein, Interpolation of linear operators , Trans. Amer. Math. Soc. 83 (1956), no. 2, 482-492. · Zbl 0072.32402 [24] E. M. Stein and S. Wainger, The estimation of an integrals arising in multiplier transformations , Studia Math. 35 (1970), no. 1, 101-104. · Zbl 0202.12401 [25] E. M. Stein and S. Wainger, Problems in harmonic analysis related to curvature , Bull. Amer. Math. Soc. 84 (1978), no. 6, 1239-1295. · Zbl 0393.42010 [26] Ž. Štrkalj and L. Weis, On operator-valued Fourier multiplier theorems , Trans. Amer. Math. Soc. 359 (2007), no. 8, 3529-3547. · Zbl 1209.42005 [27] F. Zimmermann, On vector-valued Fourier multiplier theorems , Studia Math. 93 (1989), no. 3, 201-222. · Zbl 0686.42008
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