Abstract
In this paper, carrying on with our study of the Hardy-amalgam spaces \({{\cal H}^{(q,p)}}\) and \({\cal H}_{{\rm{loc}}}^{(q,p)}\) (0 < q, p < ∞), we give a characterization of their dual spaces whenever 0 < q ≤ 1 and q ≤ p < ∞. Moreover, when 0 < q ≤ p ≤ 1, these characterizations coincide with those obtained in our earlier papers.
Similar content being viewed by others
References
Ablé, Z. V. de P., Feuto, J.: Atomic decomposition of Hardy-amalgam spaces. J. Math. Anal. Appl., 455, 1899–1936 (2017)
Ablé, Z. V. de P., Feuto, J.: Dual of Hardy-amalgam spaces and norms inequalities. Anal. Math., 45(4), 647–686 (2019)
Ablé, Z. V. de P., Feuto, J.: Dual of Hardy-amalgam spaces \({\cal H}_{{\rm{loc}}} {(q,p)}\) and pseudo-differential operators. arXiv: 1803.03595
Abu-Shammala, W.: The Hardy-Lorentz spaces, Ph.D. Thesis Ińdiana University, 2007
Benedek, A., Panzone, R.: The space Lp, with mixed norm. Duke Math. J., 28, 301–324 (1961)
Bertrandias, J. P., Datry, C., Dupuis, C.: Unions et intersections d’espaces Lp invariantes par translation ou convolution. Ann. Inst. Fourier (Grenoble), 28, 53–84 (1978)
Bownik, M.: Anisotropic Hardy spaces and wavelets. Mem. Amer. Math. Soc., 164 (2003)
Busby, R. C., Smith, H. A.: Product-convolution operators and mixed-norm spaces. Trans. Amer. Math. Soc., 263, 309–341 (1981)
Carton-Lebrun, C., Heinig, H. P., Hofmann, S. C.: Integral operators on weighted amalgams. Studia Math., 109, 133–157 (1994)
Chang, D. C., Wang, S., Yang, D., et al.: Littlewood-Paley characterizations of Hardy-type spaces associated with ball quasi-banach function spaces. Complex Anal. Oper. Theory, 14 (2020), Paper No. 40, 33 pp.
Cleanthous, G., Georgiadis, A. G., Nielsen, M.: Anisotropic mixed-norm Hardy spaces. J. Geom. Anal., 27, 2758–2787 (2017)
Fournier, J. J. F., Stewart, J.: Amalgams of Lp and lp. Bull. Amer. Math. Soc., 13, 1–21 (1985)
García-Cuerva, J., Rubio de Francia, J. L.: Weighted Norm Inequalities and Related Topics. North-Holland, 1985
Goldberg, D.: A local version of real Hardy spaces. Duke Math. J., 46, 27–42 (1979)
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., 370, 8581–8612 (2018)
Holland, F.: Harmonic analysis on amalgams of Lp and lq. J. London Math. Soc., 10, 295–305 (1975).
Huang, L., Liu, J., Yang, D., et al.: Atomic and Littlewood-Paley characterizations of anisotropic mixed-norm Hardy spaces and their applications. J. Geom. Anal., 29, 1991–2067 (2019)
Huang, L., Liu, J., Yang, D., et al.: Dual spaces of anisotropic mixed-norm Hardy spaces. Proc. Amer. Math. Soc., 147, 1201–1215 (2019)
Huang, L., Liu, J., Yang, D., et al.: Identification of anisotropic mixed-norm Hardy spaces and certain homogeneous Triebel-Lizorkin spaces. J. Approx. Theory, 258, 105459 (2020)
Huang, L., Yang, D.: On function spaces with mixed norms-a survey. J. Math. Study, 54(3), 262–336 (2021)
Liang, Y., Sawano, Y., Ullrich, T., et al.: A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces. Dissertationes Math. (Rozprawy Mat.), 489, 1–114 (2013)
Lu, S. Z.: Four Lectures on Real Hp Spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 1995
Nakai, E., Sawano, Y.: Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal., 262, 3665–3748 (2012)
Ruzhansky, M., Turunen, V.: Pseudo-differential operators and symmetries: Background Analysis and Advanced Topics. Vol. 2 of Pseudo-Differential Operators Theory and Applications, Birkhauser Verlag AG, Basel, 2010
Sawano, Y., Ho, K. P., Yang, D., et al.: Hardy spaces for ball quasi-Banach function spaces. Dissertationes Math. (Rozprawy Mat.), 525, 102 pp. (2017)
Stein, E. M.: Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integral. Princeton Mathematical Series, Vol. 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993
Stewart, J.: Fourier transforms of unbounded measures. Canad. J. Math., 31, 1281–1292 (1979)
Taylor, M. E.: Pseudodifferential operators and nonlinear PDE. Progress in Mathematics, Vol. 100, Birkhaäser, Boston, 1991
Triebel, H.: Theory of Function Spaces II, Birkhäuser Verlag, Basel, 1992
Wang, F., Yang, D., Yang, S.: Applications of Hardy spaces associated with ball quasi-Banach function spaces. Results Math., 75(1), Art. 26, 58 pp. (2020)
Wang, S., Yang, D., Yuan, W., et al.: Weak Hardy-type spaces associated with ball quasi-Banach function spaces II: Littlewood-Paley characterizations and real interpolation. J. Geom. Anal., 31(1), 631–696 (2021)
Wong, M. W.: An Introduction to Pseudo-Differential Operators, Second Edition, World Scientific, 1999
Yan, X., Yang, D., Yuan, W.: Intrinsic square function characterizations of Hardy spaces associated with ball quasi-Banach function spaces. Front. Math. China, 15, 769–806 (2020)
Zhang, Y., Yang, D., Yuan, W., et al.: Weak Hardy-type spaces associated with ball quasi-Banach function spaces I: Decompositions with applications to boundedness of Caldernon-Zygmund operators. Sci. China Math., 64(9), 2007–2064 (2021)
Zhang, Y., Yang, D., Yuan, W., et al.: Real-variable characterizations of Orlicz-slice Hardy spaces. Anal. Appl. (Singap.), 17, 597–664 (2019)
Acknowledgements
We thank the referees for their time and valuable comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ablé, Z.V.d.P., Feuto, J. New Characterizations of the Dual Spaces of Hardy-amalgam Spaces. Acta. Math. Sin.-English Ser. 38, 519–546 (2022). https://doi.org/10.1007/s10114-022-0572-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-022-0572-1