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Summability in anisotropic mixed-norm Hardy spaces


  • Received: 14 May 2022 Revised: 01 July 2022 Accepted: 08 July 2022 Published: 15 July 2022
  • Let $ H_A^{\vec{p}}(\mathbb{R}^n) $ be the anisotropic mixed-norm Hardy space, where $ \vec{p}\in(0, \infty)^n $ and $ A $ is a general expansive matrix on $ \mathbb{R}^n $. In this paper, a general summability method, the so-called $ \theta $-summability is considered for multi-dimensional Fourier transforms in $ H_A^{\vec{p}}(\mathbb{R}^n) $. Precisely, the author establishes the boundedness of maximal operators, induced by the so-called $ \theta $-means, from $ H_A^{\vec{p}}(\mathbb{R}^n) $ to the mixed-norm Lebesgue space $ L^{\vec{p}}(\mathbb{R}^n) $. As applications, some norm and almost everywhere convergence results of the $ \theta $-means are presented. Finally, the corresponding conclusions of two well-known specific summability methods, namely, Bochner–Riesz and Weierstrass means, are also obtained.

    Citation: Nan Li. Summability in anisotropic mixed-norm Hardy spaces[J]. Electronic Research Archive, 2022, 30(9): 3362-3376. doi: 10.3934/era.2022171

    Related Papers:

  • Let $ H_A^{\vec{p}}(\mathbb{R}^n) $ be the anisotropic mixed-norm Hardy space, where $ \vec{p}\in(0, \infty)^n $ and $ A $ is a general expansive matrix on $ \mathbb{R}^n $. In this paper, a general summability method, the so-called $ \theta $-summability is considered for multi-dimensional Fourier transforms in $ H_A^{\vec{p}}(\mathbb{R}^n) $. Precisely, the author establishes the boundedness of maximal operators, induced by the so-called $ \theta $-means, from $ H_A^{\vec{p}}(\mathbb{R}^n) $ to the mixed-norm Lebesgue space $ L^{\vec{p}}(\mathbb{R}^n) $. As applications, some norm and almost everywhere convergence results of the $ \theta $-means are presented. Finally, the corresponding conclusions of two well-known specific summability methods, namely, Bochner–Riesz and Weierstrass means, are also obtained.



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    [1] M. Bownik, Anisotropic Hardy Spaces and Wavelets, American Mathematical Society, Rhode Island, 2003. https://doi.org/http://dx.doi.org/10.1090/memo/0781
    [2] A. P. Calderón, A. Torchinsky, Parabolic maximal functions associated with a distribution, Adv. Math., 16 (1975), 1–64. https://doi.org/10.1016/0001-8708(75)90099-7 doi: 10.1016/0001-8708(75)90099-7
    [3] L. Huang, J. Liu, D. Yang, W. Yuan, Real-variable characterizations of new anisotropic mixed-norm Hardy spaces, Commun. Pure Appl. Anal., 19 (2020), 3033–3082. https://doi.org/10.3934/cpaa.2020132 doi: 10.3934/cpaa.2020132
    [4] T. Chen, W. Sun, Iterated and mixed weak norms with applications to geometric inequalities, J. Geom. Anal., 30 (2020), 4268–4323. https://doi.org/10.1007/s12220-019-00243-x doi: 10.1007/s12220-019-00243-x
    [5] T. Chen, W. Sun, Extension of multilinear fractional integral operators to linear operators on mixed-norm Lebesgue spaces, Math. Ann., 379 (2021), 1089–1172. https://doi.org/10.1007/s00208-020-02105-2 doi: 10.1007/s00208-020-02105-2
    [6] T. Chen, W. Sun, Hardy–Littlewood–Sobolev inequality on mixed-norm Lebesgue spaces, J. Geom. Anal., 32 (2022), 43. https://doi.org/10.1007/s12220-021-00855-2 doi: 10.1007/s12220-021-00855-2
    [7] G. Cleanthous, A. G. Georgiadis, Mixed-norm $\alpha$-modulation spaces, Trans. Amer. Math. Soc., 373 (2020), 3323–3356. https://doi.org/10.1090/tran/8023 doi: 10.1090/tran/8023
    [8] G. Cleanthous, A. G. Georgiadis, M. Nielsen, Molecular decomposition of anisotropic homogeneous mixed-norm spaces with applications to the boundedness of operators, Appl. Comput. Harmon. Anal., 47 (2019), 447–480. https://doi.org/10.1016/j.acha.2017.10.001 doi: 10.1016/j.acha.2017.10.001
    [9] J. Johnsen, S. Munch Hansen, W. Sickel, Characterisation by local means of anisotropic Lizorkin–Triebel spaces with mixed norms, Z. Anal. Anwend., 32 (2013), 257–277. https://doi.org/10.4171/ZAA/1484 doi: 10.4171/ZAA/1484
    [10] J. Johnsen, S. Munch Hansen, W. Sickel, Anisotropic Lizorkin–Triebel spaces with mixed norms–traces on smooth boundaries, Math. Nachr., 288 (2015), 1327–1359. https://doi.org/10.1002/mana.201300313 doi: 10.1002/mana.201300313
    [11] T. Nogayama, T. Ono, D. Salim, Y. Sawano, Atomic decomposition for mixed Morrey spaces, J. Geom. Anal., 31 (2021), 9338–9365. https://doi.org/10.1007/s12220-020-00513-z doi: 10.1007/s12220-020-00513-z
    [12] E. M. Stein, M. H. Taibleson, G. Weiss, Weak type estimates for maximal operators on certain $H^p$ classes, Rend. Circ. Mat. Palermo, 1 (1981), 81–97.
    [13] J. Liu, F. Weisz, D. Yang, W. Yuan, Variable anisotropic Hardy spaces and their applications, Taiwanese J. Math., 22 (2018), 1173–1216. https://doi.org/10.11650/tjm/171101 doi: 10.11650/tjm/171101
    [14] J. Liu, F. Weisz, D. Yang, W. Yuan, Littlewood-Paley and finite atomic characterizations of anisotropic variable Hardy-Lorentz spaces and their applications, J. Fourier Anal. Appl., 25 (2019), 874–922. https://doi.org/10.1007/s00041-018-9609-3 doi: 10.1007/s00041-018-9609-3
    [15] F. Weisz, Summability of Multi-dimensional Fourier Series and Hardy Spaces, Dordrecht: Kluwer Academic Publishers, 2002. https://doi.org/10.1007/978-94-017-3183-6
    [16] F. Weisz, Convergence and Summability of Fourier Transforms and Hardy Spaces, Basel, Birkhäuser, 2017.
    [17] F. Weisz, Summability of Fourier transforms in variable Hardy and Hardy–Lorentz spaces, Jaen J. Approx., 10 (2018), 101–131.
    [18] F. Weisz, Summability in mixed-norm Hardy spaces, Ann. Univ. Sci. Budapest. Sect. Comput., 48 (2018), 233–246.
    [19] A. Benedek, R. Panzone, The space $L^p$, with mixed norm, Duke Math. J., 28 (1961), 301–324. https://doi.org/10.1215/S0012-7094-61-02828-9 doi: 10.1215/S0012-7094-61-02828-9
    [20] G. Cleanthous, A. G. Georgiadis, M. Nielsen, Anisotropic mixed-norm Hardy spaces, J. Geom. Anal., 27 (2017), 2758–2787. https://doi.org/10.1007/s12220-017-9781-8 doi: 10.1007/s12220-017-9781-8
    [21] F. Weisz, Summability of multi-dimensional trigonometric Fourier series, Surv. Approx. Theory, 7 (2012), 1–179.
    [22] J. Liu, L. Huang, C. Yue, Molecular characterizations of anisotropic mixed-norm Hardy spaces and their applications, Mathematics, 9 (2021), 2216. https://doi.org/10.3390/math9182216 doi: 10.3390/math9182216
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