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Riesz Transform Characterizations for Multidimensional Hardy Spaces

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Abstract

We study Hardy space \(H^1_L(X)\) related to a self-adjoint operator L defined on an Euclidean subspace X of \({{\mathbb {R}}^d}\). We continue study from [27], where, under certain assumptions on the heat semigroup \(\exp (-tL)\), the atomic characterization of local type for \(H^1_L(X)\) was proved. In this paper we provide additional assumptions that lead to another characterization of \(H^1_L(X)\) by the Riesz transforms related to L. As an application, we prove the Riesz transform characterization of \(H^1_L(X)\) for multidimensional Bessel and Laguerre operators, and the Dirichlet Laplacian on \({\mathbb {R}}^d_+\).

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References

  1. Auscher, P., Russ, E.: Hardy spaces and divergence operators on strongly Lipschitz domains of \(\mathbb{R}^n\). J. Funct. Anal. 201(1), 148–184 (2003)

    Article  MathSciNet  Google Scholar 

  2. Bernicot, F., Zhao, J.: New abstract Hardy spaces. J. Funct. Anal. 255(7), 1761–1796 (2008)

    Article  MathSciNet  Google Scholar 

  3. Betancor, J., Dziubański, J., Garrigós, G.: Riesz transform characterization of Hardy spaces associated with certain Laguerre expansions. Tohoku Math. J. 62(2), 215–231 (2010)

    Article  MathSciNet  Google Scholar 

  4. Betancor, J., Fariña, J.C., Rodríguez-Mesa, L., Sanabria, A., Torrea, J.L.: Transference between Laguerre and Hermite settings. J. Funct. Anal. 254(3), 826–850 (2008)

    Article  MathSciNet  Google Scholar 

  5. Betancor, J.J., Castro, A.J., Nowak, A.: Calderón–Zygmund operators in the Bessel setting. Monatsh. Math. 167(3–4), 375–403 (2012)

    Article  MathSciNet  Google Scholar 

  6. Betancor, J.J., Dziubański, J., Torrea, J.L.: On Hardy spaces associated with Bessel operators. J. Anal. Math. 107, 195–219 (2009)

    Article  MathSciNet  Google Scholar 

  7. Betancor, J.J., Fariña, J.C., Buraczewski, D., Martínez, T., Torrea, J.L.: Riesz transforms related to Bessel operators. Proc. R. Soc. Edinb. Sect. A 137(4), 701–725 (2007)

    Article  MathSciNet  Google Scholar 

  8. Betancor, J.J., Harboure, E., Nowak, A., Viviani, B.: Mapping properties of fundamental operators in harmonic analysis related to Bessel operators. Studia Math. 197(2), 101–140 (2010)

    Article  MathSciNet  Google Scholar 

  9. Betancor, J.J., Molina, S.M., Rodríguez-Mesa, L.: Area Littlewood–Paley functions associated with Hermite and Laguerre operators. Potential Anal. 34(4), 345–369 (2011)

    Article  MathSciNet  Google Scholar 

  10. Betancor, J.J., Stempak, K.: Relating multipliers and transplantation for Fourier-Bessel expansions and Hankel transform. Tohoku Math. J. 53(1), 109–129 (2001)

    Article  MathSciNet  Google Scholar 

  11. Bownik, M.: Boundedness of operators on Hardy spaces via atomic decompositions. Proc. Am. Math. Soc. 133(12), 3535–3542 (2005)

    Article  MathSciNet  Google Scholar 

  12. Chang, D.-C., Krantz, S.G., Stein, E.M.: \(H^p\) theory on a smooth domain in \({ R}^N\) and elliptic boundary value problems. J. Funct. Anal. 114(2), 286–347 (1993)

    Article  MathSciNet  Google Scholar 

  13. Ciaurri, Ó., Stempak, K.: Conjugacy for Fourier-Bessel expansions. Studia Math. 176(3), 215–247 (2006)

  14. Coifman, R.R.: A real variable characterization of \(H^{p}\). Studia Math. 51, 269–274 (1974)

    Article  MathSciNet  Google Scholar 

  15. Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83(4), 569–645 (1977)

    Article  MathSciNet  Google Scholar 

  16. Dziubański, J.: Hardy spaces for Laguerre expansions. Constr. Approx. 27(3), 269–287 (2008)

    Article  MathSciNet  Google Scholar 

  17. Dziubański, J.: Riesz transforms characterizations of Hardy spaces \(H^1\) for the rational Dunkl setting and multidimensional Bessel operators. J. Geom. Anal. 26(4), 2639–2663 (2016)

    Article  MathSciNet  Google Scholar 

  18. Dziubański, J., Hejna, A.: Remark on atomic decompositions for the Hardy space \(H^1\) in the rational Dunkl setting. Studia Math. 251(1), 89–110 (2020)

    Article  MathSciNet  Google Scholar 

  19. Dziubański, J., Preisner, M.: On Riesz transforms characterization of \(H^1\) spaces associated with some Schrödinger operators. Potential Anal. 35(1), 39–50 (2011)

    Article  MathSciNet  Google Scholar 

  20. Dziubański, J., Preisner, M.: Hardy spaces for semigroups with Gaussian bounds. Ann. Mat. Pura Appl. 197(3), 965–987 (2018)

    Article  MathSciNet  Google Scholar 

  21. Dziubański, J., Zienkiewicz, J.: A characterization of Hardy spaces associated with certain Schrödinger operators. Potential Anal. 41(3), 917–930 (2014)

    Article  MathSciNet  Google Scholar 

  22. Fefferman, C., Stein, E.M.: \(H^{p}\) spaces of several variables. Acta Math. 129(3–4), 137–193 (1972)

    Article  MathSciNet  Google Scholar 

  23. Goldberg, D.: A local version of real Hardy spaces. Duke Math. J. 46(1), 27–42 (1979)

    Article  MathSciNet  Google Scholar 

  24. Hejna, A.: Hardy spaces for the Dunkl harmonic oscillator. Math. Nachr. 293(11), 2112–2139 (2020)

    Article  MathSciNet  Google Scholar 

  25. Hofmann, S., Lu, G., Mitrea, D., Mitrea, M., Yan, L.: Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates. Mem. Am. Math. Soc. 214(1007), vi+78 (2011)

    MathSciNet  MATH  Google Scholar 

  26. Kania, E., Preisner, M.: Hardy spaces for Bessel-Schrödinger operators. Math. Nachr. 291(5–6), 908–927 (2018)

    Article  MathSciNet  Google Scholar 

  27. Kania-Strojec, E., Plewa, P., Preisner, M.: Local atomic decompositions for multidimensional Hardy spaces. Rev. Mat. Complut. 34, 409–434 (2021)

    Article  MathSciNet  Google Scholar 

  28. Latter, R.H.: A characterization of \(H^{p}({ R}^{n})\) in terms of atoms. Studia Math. 62(1), 93–101 (1978)

    Article  MathSciNet  Google Scholar 

  29. Li, J., Wick, B.D.: Characterizations of \(H_{\Delta _N}^1(\mathbb{R}^n)\) and \({{\rm BMO}}_{\Delta _N}(\mathbb{R}^n)\) via weak factorizations and commutators. J. Funct. Anal. 272(12), 5384–5416 (2017)

    Article  MathSciNet  Google Scholar 

  30. Nowak, A., Stempak, K.: Riesz transforms and conjugacy for Laguerre function expansions of Hermite type. J. Funct. Anal. 244(2), 399–443 (2007)

    Article  MathSciNet  Google Scholar 

  31. Nowak, A., Stempak, K.: Riesz transforms for multi-dimensional laguerre function expansions. Adv. Math. 215(2), 642–678 (2007)

    Article  MathSciNet  Google Scholar 

  32. Peloso, M.M., Secco, S.: Local Riesz transforms characterization of local Hardy spaces. Collect. Math. 59(3), 299–320 (2008)

    Article  MathSciNet  Google Scholar 

  33. Preisner, M.: Riesz transform characterization of \(H^1\) spaces associated with certain Laguerre expansions. J. Approx. Theory 164(2), 229–252 (2012)

    Article  MathSciNet  Google Scholar 

  34. Preisner, M., Sikora, A., Yan, L.: Hardy spaces meet harmonic weights, arXiv e-prints (2019), arXiv:1912.00734

  35. Schmüdgen, K.: Unbounded Self-adjoint Operators on Hilbert Space. Graduate Texts in Mathematics, vol. 265. Springer, Dordrecht (2012)

    Book  Google Scholar 

  36. Song, L., Yan, L.: A maximal function characterization for Hardy spaces associated to nonnegative self-adjoint operators satisfying Gaussian estimates. Adv. Math. 287, 463–484 (2016)

    Article  MathSciNet  Google Scholar 

  37. Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III (1993)

  38. Uchiyama, A.: A maximal function characterization of \(H^{p}\) on the space of homogeneous type. Trans. Am. Math. Soc. 262(2), 579–592 (1980)

    MATH  Google Scholar 

  39. Watson, G.N.: A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995, Reprint of the second edition (1944)

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  1. Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384, Wrocław, Poland

    Edyta Kania-Strojec & Marcin Preisner

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  1. Edyta Kania-Strojec
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  2. Marcin Preisner
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Correspondence to Marcin Preisner.

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The authors are supported by the Grant No. 2017/25/B/ST1/00599 from National Science Centre (Narodowe Centrum Nauki), Poland.

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Kania-Strojec, E., Preisner, M. Riesz Transform Characterizations for Multidimensional Hardy Spaces. J Geom Anal 32, 163 (2022). https://doi.org/10.1007/s12220-022-00896-1

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