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Fourier transform of anisotropic mixed-norm Hardy spaces


Let \(\overrightarrow a \,: = \,\left( {{a_1}, \ldots,{a_n}} \right) \in {\left[ {1,\infty } \right)^n},\,\overrightarrow {p\,}: = \left( {{p_1}, \ldots,{p_n}} \right) \in {\left( {0,1} \right]^n},H_{\overrightarrow a }^{\overrightarrow p }\left( {{\mathbb{R}^n}} \right)\) be the anisotropic mixed-norm Hardy space associated with \(\overrightarrow a \) defined via the radial maximal function, and let f belong to the Hardy space \(H_{\overrightarrow a }^{\overrightarrow p }\left( {{\mathbb{R}^n}} \right)\). In this article, we show that the Fourier transform \(\widehat{f}\) coincides with a continuous function g on ℝn in the sense of tempered distributions and, moreover, this continuous function g, multiplied by a step function associated with \(\overrightarrow a \), can be pointwisely controlled by a constant multiple of the Hardy space norm of f. These proofs are achieved via the known atomic characterization of \(H_{\overrightarrow a }^{\overrightarrow p }\left( {{\mathbb{R}^n}} \right)\) and the establishment of two uniform estimates on anisotropic mixed-norm atoms. As applications, we also conclude a higher order convergence of the continuous function g at the origin. Finally, a variant of the Hardy-Littlewood inequality in the anisotropic mixed-norm Hardy space setting is also obtained. All these results are a natural generalization of the well-known corresponding conclusions of the classical Hardy spaces Hp(ℝn) with p ∈ (0, 1], and are even new for isotropic mixed-norm Hardy spaces on ∈n.

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  1. 1.

    Benedek A, Panzone R. The space Lp, with mixed norm. Duke Math J, 1961, 28: 301–324

    MathSciNet  Article  Google Scholar 

  2. 2.

    Besov O V, Il’in V P, Lizorkin P I. The Lp-estimates of a certain class of non-isotropically singular integrals. Dokl Akad Nauk SSSR, 1966, 169: 1250–1253 (in Russian)

    MathSciNet  Google Scholar 

  3. 3.

    Besov O V, Il’in V P, Nikol’skiĭ S M. Integral Representations of Functions and Imbedding Theorems. Vol I. New York: Halsted Press (John Wiley and Sons), 1978

    MATH  Google Scholar 

  4. 4.

    Bownik M. Anisotropic Hardy Spaces and Wavelets. Mem Amer Math Soc, Vol 164, No 781. Providence: Amer Math Soc, 2003

    MATH  Google Scholar 

  5. 5.

    Bownik M, Wang L-A D. Fourier transform of anisotropic Hardy spaces. Proc Amer Math Soc, 2013, 141: 2299–2308

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bu R, Fu Z, Zhang Y. Weighted estimates for bilinear square functions with non-smooth kernels and commutators. Front Math China, 2020, 15: 1–20

    MathSciNet  Article  Google Scholar 

  7. 7.

    Chen T, Sun W. Hardy-Littlewood-Sobolev inequality on mixed-norm Lebesgue spaces. arXiv: 1912.03712

  8. 8.

    Chen T, Sun W. Iterated and mixed weak norms with applications to geometric inequalities. J Geom Anal, 2020, 30: 4268–4323

    MathSciNet  Article  Google Scholar 

  9. 9.

    Chen T, Sun W. Extension of multilinear fractional integral operators to linear operators on mixed-norm Lebesgue spaces. Math Ann, 2020, doi.

  10. 10.

    Cleanthous G, Georgiadis A G. Mixed-norm α-modulation spaces. Trans Amer Math Soc, 2020, 373: 3323–3356

    MathSciNet  Article  Google Scholar 

  11. 11.

    Cleanthous G, Georgiadis A G, Nielsen M. Anisotropic mixed-norm Hardy spaces. J Geom Anal, 2017, 27: 2758–2787

    MathSciNet  Article  Google Scholar 

  12. 12.

    Cleanthous G, Georgiadis A G, Nielsen M. Molecular decomposition of anisotropic homogeneous mixed-norm spaces with applications to the boundedness of operators. Appl Comput Harmon Anal, 2019, 47: 447–480

    MathSciNet  Article  Google Scholar 

  13. 13.

    Coifman R R. Characterization of Fourier transforms of Hardy spaces. Proc Natl Acad Sci USA, 1974, 71: 4133–4134

    MathSciNet  Article  Google Scholar 

  14. 14.

    Colzani L. Fourier transform of distributions in Hardy spaces. Boll Unione Mat Ital A (6), 1982, 1: 403–410

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Deng Q, Guedjiba D E. Weighted product Hardy space associated with operators. Front Math China, 2020, 15: 649–683

    MathSciNet  Article  Google Scholar 

  16. 16.

    Fabes E B, Rivière N M. Singular integrals with mixed homogeneity. Studia Math, 1966, 27: 19–38

    MathSciNet  Article  Google Scholar 

  17. 17.

    Fefferman C, Stein E M. Hp spaces of several variables. Acta Math, 1972, 129: 137–193

    MathSciNet  Article  Google Scholar 

  18. 18.

    García-Cuerva J, Kolyada V I. Rearrangement estimates for Fourier transforms in Lp and Hp in terms of moduli of continuity. Math Nachr, 2001, 228: 123–144

    MathSciNet  Article  Google Scholar 

  19. 19.

    Georgiadis A G, Johnsen J, Nielsen M. Wavelet transforms for homogeneous mixed-norm Triebel-Lizorkin spaces. Monatsh Math, 2017, 183: 587–624

    MathSciNet  Article  Google Scholar 

  20. 20.

    Hart J, Torres R H, Wu X. Smoothing properties of bilinear operators and Leibniz-type rules in Lebesgue and mixed Lebesgue spaces. Trans Amer Math Soc, 2018, 370: 8581–8612

    MathSciNet  Article  Google Scholar 

  21. 21.

    Hörmander L. Estimates for translation invariant operators in Lp spaces. Acta Math, 1960, 104: 93–140

    MathSciNet  Article  Google Scholar 

  22. 22.

    Huang L, Liu J, Yang D, Yuan W. Atomic and Littlewood-Paley characterizations of anisotropic mixed-norm Hardy spaces and their applications. J Geom Anal, 2019, 29: 1991–2067

    MathSciNet  Article  Google Scholar 

  23. 23.

    Huang L, Liu J, Yang D, Yuan W. Dual spaces of anisotropic mixed-norm Hardy spaces. Proc Amer Math Soc, 2019, 147: 1201–1215

    MathSciNet  Article  Google Scholar 

  24. 24.

    Huang L, Liu J, Yang D, Yuan W. Identification of anisotropic mixed-norm Hardy spaces and certain homogeneous Triebel-Lizorkin spaces. J Approx Theory, 2020, 258: 1–27

    MathSciNet  Article  Google Scholar 

  25. 25.

    Huang L, Liu J, Yang D, Yuan W. Real-variable characterizations of new anisotropic mixed-norm Hardy spaces. Comm Pure Appl Anal, 2020, 19: 3033–3082

    MathSciNet  Article  Google Scholar 

  26. 26.

    Huang L, Weisz F, Yang D, Yuan W. Summability of Fourier transforms on mixed-norm Lebesgue spaces via associated Herz spaces. Preprint

  27. 27.

    Huang L, Yang D. On function spaces with mixed norms—a survey. J Math Study, 2019,

  28. 28.

    Huang L, Yang D, Yuan W, Zhang Y. New ball Campanato-type function spaces and their applications. Preprint

  29. 29.

    Jawerth B. Some observations on Besov and Lizorkin-Triebel spaces. Math Scand, 1977, 40: 94–104

    MathSciNet  Article  Google Scholar 

  30. 30.

    Johnsen J, Munch Hansen S, Sickel W. Characterisation by local means of anisotropic Lizorkin-Triebel spaces with mixed norms. Z Anal Anwend, 2013, 32: 257–277

    MathSciNet  Article  Google Scholar 

  31. 31.

    Johnsen J, Munch Hansen S, Sickel W. Anisotropic Lizorkin-Triebel spaces with mixed norms—traces on smooth boundaries. Math Nachr, 2015, 288: 1327–1359

    MathSciNet  Article  Google Scholar 

  32. 32.

    Johnsen J, Sickel W. A direct proof of Sobolev embeddings for quasi-homogeneous Lizorkin-Triebel spaces with mixed norms. J Funct Spaces Appl, 2007, 5: 183–198

    MathSciNet  Article  Google Scholar 

  33. 33.

    Johnsen J, Sickel W. On the trace problem for Lizorkin-Triebel spaces with mixed norms. Math Nachr, 2008, 281: 669–696

    MathSciNet  Article  Google Scholar 

  34. 34.

    Lizorkin P I. Multipliers of Fourier integrals and estimates of convolutions in spaces with mixed norm. Applications. Izv Akad Nauk SSSR Ser Mat, 1970, 34: 218–247

    MathSciNet  Google Scholar 

  35. 35.

    Peetre J. New Thoughts on Besov Spaces. Duke Univ Math Ser, No 1. Durham: Math Department, Duke Univ, 1976

    MATH  Google Scholar 

  36. 36.

    Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton: Princeton Univ Press, 1993

    MATH  Google Scholar 

  37. 37.

    Stein E M, Wainger S. Problems in harmonic analysis related to curvature. Bull Amer Math Soc, 1978, 84: 1239–1295

    MathSciNet  Article  Google Scholar 

  38. 38.

    Taibleson M H, Weiss G. The molecular characterization of certain Hardy spaces. Astérisque, 1980, 77: 67–149

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Wang H, Xu J, Tan J. Boundedness of multilinear singular integrals on central Morrey spaces with variable exponents. Front Math China, 2020, 15: 1011–1034

    MathSciNet  Article  Google Scholar 

  40. 40.

    Yan X, Yang D, Yuan W. Intrinsic square function characterizations of Hardy spaces associated with ball quasi-Banach function spaces. Front Math China, 2020, 15: 769–806

    MathSciNet  Article  Google Scholar 

  41. 41.

    Zhang X, Wei M, Yan D, He Q. Equivalence of operator norm for Hardy-Littlewood maximal operators and their truncated operators on Morrey spaces. Front Math China, 2020, 15: 215–223

    MathSciNet  Article  Google Scholar 

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The authors would like to express their thanks to Professor Winfried Sickel of Friedrich-Schiller-University Jena for bringing references [29] and [35] to their attention. This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11971058, 12071197) and the National Key Research and Development Program of China (Grant No. 2020YFA0712900), and Der-Chen Chang was partially supported by an NSF grant DMS-1408839 and a McDevitt Endowment Fund at Georgetown University.

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Correspondence to Dachun Yang.

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Huang, L., Chang, DC. & Yang, D. Fourier transform of anisotropic mixed-norm Hardy spaces. Front. Math. China 16, 119–139 (2021).

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  • Anisotropic (mixed-norm) Hardy space
  • Fourier transform
  • Hardy-Littlewood inequality


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  • 42B30
  • 42B10