Intrinsic square function characterizations of Musielak-Orlicz Hardy spaces
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- by Yiyu Liang and Dachun Yang PDF
- Trans. Amer. Math. Soc. 367 (2015), 3225-3256 Request permission
Abstract:
Let $\varphi : \mathbb R^n\times [0,\infty )\to [0,\infty )$ be such that $\varphi (x,\cdot )$ is an Orlicz function and $\varphi (\cdot ,t)$ is a Muckenhoupt $A_\infty (\mathbb R^n)$ weight uniformly in $t$. In this article, for any $\alpha \in (0,1]$ and $s\in \mathbb {Z}_+$, the authors establish the $s$-order intrinsic square function characterizations of $H^{\varphi }(\mathbb R^n)$ in terms of the intrinsic Lusin area function $S_{\alpha ,s}$, the intrinsic $g$-function $g_{\alpha ,s}$ and the intrinsic $g_{\lambda }^*$-function $g^\ast _{\lambda , \alpha ,s}$ with the best known range $\lambda \in (2+2(\alpha +s)/n,\infty )$, which are defined via $\mathrm {Lip}_\alpha ({\mathbb R}^n)$ functions supporting in the unit ball. A $\varphi$-Carleson measure characterization of the Musielak-Orlicz Campanato space ${\mathcal L}_{\varphi ,1,s}({\mathbb R}^n)$ is also established via the intrinsic function. To obtain these characterizations, the authors first show that these $s$-order intrinsic square functions are pointwise comparable with those similar-looking $s$-order intrinsic square functions defined via $\mathrm {Lip}_\alpha ({\mathbb R}^n)$ functions without compact supports, which when $s=0$ was obtained by M. Wilson. All these characterizations of $H^{\varphi }(\mathbb R^n)$, even when $s=0$, \[ \varphi (x,t):=w(x)t^p\ \textrm {for\ all}\ t\in [0,\infty )\ \textrm {and}\ x\in {\mathbb R}^n\] with $p\in (n/(n+\alpha ), 1]$ and $w\in A_{p(1+\alpha /n)}(\mathbb R^n)$, also essentially improve the known results.References
- Néstor Aguilera and Carlos Segovia, Weighted norm inequalities relating the $g^*_{\lambda }$ and the area functions, Studia Math. 61 (1977), no. 3, 293–303. MR 492276, DOI 10.4064/sm-61-3-293-303
- Z. Birnbaum and W. Orlicz, Über die verallgemeinerung des begriffes der zueinander konjugierten potenzen, Studia Math. 3 (1931), 1–67.
- Aline Bonami, Justin Feuto, and Sandrine Grellier, Endpoint for the DIV-CURL lemma in Hardy spaces, Publ. Mat. 54 (2010), no. 2, 341–358. MR 2675927, DOI 10.5565/PUBLMAT_{5}4210_{0}3
- Aline Bonami and Sandrine Grellier, Hankel operators and weak factorization for Hardy-Orlicz spaces, Colloq. Math. 118 (2010), no. 1, 107–132. MR 2600520, DOI 10.4064/cm118-1-5
- Aline Bonami, Sandrine Grellier, and Luong Dang Ky, Paraproducts and products of functions in $BMO(\Bbb R^n)$ and $\scr H^1(\Bbb R^n)$ through wavelets, J. Math. Pures Appl. (9) 97 (2012), no. 3, 230–241 (English, with English and French summaries). MR 2887623, DOI 10.1016/j.matpur.2011.06.002
- Aline Bonami, Tadeusz Iwaniec, Peter Jones, and Michel Zinsmeister, On the product of functions in BMO and $H^1$, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 5, 1405–1439 (English, with English and French summaries). MR 2364134, DOI 10.5802/aif.2299
- S. Campanato, Proprietà di una famiglia di spazi funzionali, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 18 (1964), 137–160 (Italian). MR 167862
- Lennart Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958), 921–930. MR 117349, DOI 10.2307/2372840
- Lennart Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547–559. MR 141789, DOI 10.2307/1970375
- Galia Dafni and Jie Xiao, Some new tent spaces and duality theorems for fractional Carleson measures and $Q_\alpha (\Bbb R^n)$, J. Funct. Anal. 208 (2004), no. 2, 377–422. MR 2035030, DOI 10.1016/S0022-1236(03)00181-2
- Matts Essén, Svante Janson, Lizhong Peng, and Jie Xiao, $Q$ spaces of several real variables, Indiana Univ. Math. J. 49 (2000), no. 2, 575–615. MR 1793683, DOI 10.1512/iumj.2000.49.1732
- C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
- G. B. Folland and Elias M. Stein, Hardy spaces on homogeneous groups, Mathematical Notes, vol. 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. MR 657581
- José García-Cuerva, Weighted $H^{p}$ spaces, Dissertationes Math. (Rozprawy Mat.) 162 (1979), 63. MR 549091
- José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
- John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
- Shaoxiong Hou, Dachun Yang, and Sibei Yang, Lusin area function and molecular characterizations of Musielak-Orlicz Hardy spaces and their applications, Commun. Contemp. Math. 15 (2013), no. 6, 1350029, 37. MR 3139410, DOI 10.1142/S0219199713500296
- Jizheng Huang and Yu Liu, Some characterizations of weighted Hardy spaces, J. Math. Anal. Appl. 363 (2010), no. 1, 121–127. MR 2559046, DOI 10.1016/j.jmaa.2009.07.054
- Svante Janson, Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation, Duke Math. J. 47 (1980), no. 4, 959–982. MR 596123
- R. Johnson and C. J. Neugebauer, Homeomorphisms preserving $A_p$, Rev. Mat. Iberoamericana 3 (1987), no. 2, 249–273. MR 990859, DOI 10.4171/RMI/50
- Luong Dang Ky, New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators, Integral Equations Operator Theory 78 (2014), no. 1, 115–150. MR 3147406, DOI 10.1007/s00020-013-2111-z
- Luong Dang Ky, Bilinear decompositions and commutators of singular integral operators, Trans. Amer. Math. Soc. 365 (2013), no. 6, 2931–2958. MR 3034454, DOI 10.1090/S0002-9947-2012-05727-8
- Andrei K. Lerner, Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals, Adv. Math. 226 (2011), no. 5, 3912–3926. MR 2770437, DOI 10.1016/j.aim.2010.11.009
- Andrei K. Lerner, On sharp aperture-weighted estimates for square functions, J. Fourier Anal. Appl. 20 (2014), no. 4, 784–800. MR 3232586, DOI 10.1007/s00041-014-9333-6
- Yiyu Liang, Jizheng Huang, and Dachun Yang, New real-variable characterizations of Musielak-Orlicz Hardy spaces, J. Math. Anal. Appl. 395 (2012), no. 1, 413–428. MR 2943633, DOI 10.1016/j.jmaa.2012.05.049
- Yiyu Liang and Dachun Yang, Musielak-Orlicz Campanato spaces and applications, J. Math. Anal. Appl. 406 (2013), no. 1, 307–322. MR 3062424, DOI 10.1016/j.jmaa.2013.04.069
- Julian Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983. MR 724434, DOI 10.1007/BFb0072210
- Eiichi Nakai and Yoshihiro Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal. 262 (2012), no. 9, 3665–3748. MR 2899976, DOI 10.1016/j.jfa.2012.01.004
- W. Orlicz, Über eine gewisse Klasse von Räumen vom Typus B, Bull. Int. Acad. Pol. Ser. A 8 (1932), 207–220.
- Jan-Olov Strömberg and Alberto Torchinsky, Weighted Hardy spaces, Lecture Notes in Mathematics, vol. 1381, Springer-Verlag, Berlin, 1989. MR 1011673, DOI 10.1007/BFb0091154
- Mitchell H. Taibleson and Guido Weiss, The molecular characterization of certain Hardy spaces, Representation theorems for Hardy spaces, Astérisque, vol. 77, Soc. Math. France, Paris, 1980, pp. 67–149. MR 604370
- Beatriz E. Viviani, An atomic decomposition of the predual of $\textrm {BMO}(\rho )$, Rev. Mat. Iberoamericana 3 (1987), no. 3-4, 401–425. MR 996824, DOI 10.4171/RMI/56
- Hua Wang and Heping Liu, Weak type estimates of intrinsic square functions on the weighted Hardy spaces, Arch. Math. (Basel) 97 (2011), no. 1, 49–59. MR 2820587, DOI 10.1007/s00013-011-0264-z
- Hua Wang and Heping Liu, The intrinsic square function characterizations of weighted Hardy spaces, Illinois J. Math. 56 (2012), no. 2, 367–381. MR 3161329
- Michael Wilson, The intrinsic square function, Rev. Mat. Iberoam. 23 (2007), no. 3, 771–791. MR 2414491, DOI 10.4171/RMI/512
- Michael Wilson, Weighted Littlewood-Paley theory and exponential-square integrability, Lecture Notes in Mathematics, vol. 1924, Springer, Berlin, 2008. MR 2359017
- M. Wilson, How fast and in what sense(s) does the Calderón reproducing formula converge?, J. Fourier Anal. Appl. 16 (2010), no. 5, 768–785. MR 2673708, DOI 10.1007/s00041-009-9109-6
- Michael Wilson, Convergence and stability of the Calderón reproducing formula in $H^1$ and $BMO$, J. Fourier Anal. Appl. 17 (2011), no. 5, 801–820. MR 2838108, DOI 10.1007/s00041-010-9165-y
- DaChun Yang and SiBei Yang, Local Hardy spaces of Musielak-Orlicz type and their applications, Sci. China Math. 55 (2012), no. 8, 1677–1720. MR 2955251, DOI 10.1007/s11425-012-4377-z
Additional Information
- Yiyu Liang
- Affiliation: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
- MR Author ID: 946733
- Email: yyliang@mail.bnu.edu.cn
- Dachun Yang
- Affiliation: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
- MR Author ID: 317762
- Email: dcyang@bnu.edu.cn
- Received by editor(s): February 12, 2013
- Published electronically: October 10, 2014
- Additional Notes: The second (corresponding) author was supported by the National Natural Science Foundation of China (Grant No. 11171027) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20120003110003).
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 3225-3256
- MSC (2010): Primary 42B25; Secondary 42B30, 42B35, 46E30
- DOI: https://doi.org/10.1090/S0002-9947-2014-06180-1
- MathSciNet review: 3314807