## Rocky Mountain Journal of Mathematics

### New real-variable characterizations of anisotropic weak Hardy spaces of Musielak-Orlicz type

#### Abstract

A real $n\times n$ matrix $A$ is called an expansive dilation if all of its eigenvalues $\lambda$ satisfy $|\lambda |\!\!>\!\!1$. Let $\varphi : \mathbb {R}^n\times [0,\infty )\to [0,\infty )$ be a Musielak-Orlicz function. The aim of this article is to find an appropriate general space which includes the weak Hardy space of Fefferman and Soria, the weighted weak Hardy space of Quek and Yang}, the anisotropic weak Hardy space of Ding and Lan, the Musielak-Orlicz Hardy space of Ky and the anisotropic Hardy space of Musielak-Orlicz type of Li, Yang and Yuan. For this reason, we introduce the anisotropic weak Hardy space of Musielak-Orlicz type $H^{\varphi , \infty }_{m,A}({\mathbb {R}}^n)$ with $m\in \mathbb {N}$ and obtain some new real-variable characterizations of $H^{\varphi , \infty }_{m,A}({\mathbb {R}}^n)$ in terms of the radial, the non-tangential and the tangential maximal functions via a new monotone convergence theorem adapted to the weak anisotropic Musielak-Orlicz space $L^{\varphi , \infty }({\mathbb {R}}^n)$. These maximal function characterizations generalize the known results on the anisotropic weak Hardy space $H^{p, \infty }_A({\mathbb {R}}^n)$ with $p\in (0, 1]$ and are new even for their weighted variants or weak Orlicz-Hardy variants. As an application, the authors show the boundedness of a class of multilinear operators formed by the anisotropic Calderon-Zygmund operators from product weighted Lebesgue space to $H^{\varphi , \infty }_{m,A}({\mathbb {R}}^n)$ with $\varphi (x,t):=t^p\omega (x)$ and $\omega \in \mathbb {A}_1(A)$, which is a weighted and non-isotropic extension of Grafakos.

#### Article information

Source
Rocky Mountain J. Math., Volume 48, Number 2 (2018), 607-637.

Dates
First available in Project Euclid: 4 June 2018

https://projecteuclid.org/euclid.rmjm/1528077635

Digital Object Identifier
doi:10.1216/RMJ-2018-48-2-607

Mathematical Reviews number (MathSciNet)
MR3809157

Zentralblatt MATH identifier
06883483

#### Citation

Qi, Chunyan; Zhang, Hui; Li, Baode. New real-variable characterizations of anisotropic weak Hardy spaces of Musielak-Orlicz type. Rocky Mountain J. Math. 48 (2018), no. 2, 607--637. doi:10.1216/RMJ-2018-48-2-607. https://projecteuclid.org/euclid.rmjm/1528077635

#### References

• J. Álvarez, $H^p$ and weak $H^p$ continuity of Calderón-Zygmund type operators, in Fourier analysis, Lect. Notes Pure Appl. Math. 157, Dekker, New York, 1994.
• T. Arai, Convex risk measures on Orlicz spaces: inf-convolution and shortfall, Math. Finan. Econ. 3 (2010), 73–88.
• Z. Birnbaum and W. Orlicz, Über die verallgemeinerung des begriffes der zueinander konjugierten potenzen, Stud. Math. 3 (1931), 1–67.
• A. Bonami, J. Feuto and S. Grellier, Endpoint for the DIV-CURL lemma in Hardy spaces, Publ. Mat. 54 (2010), 341–358.
• A. Bonami, S. Grellier and L. D. Ky, Paraproducts and products of functions in $\bmo(\rn)$ and $H^1(\rn)$ through wavelets, J. Math. Pure Appl. 97 (2012), 230–241.
• A. Bonami, T. Iwaniec, P. Jones and M. Zinsmeister, On the product of functions in $\bmo$ and $H^1$, Ann. Inst. Fourier (Grenoble) 57 (2007), 1405–1439.
• M. Bownik, Anisotropic Hardy spaces and wavelets, Mem. Amer. Math. Soc. 164 (2003).
• M. Bownik and K.-P. Ho, Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces, Trans. Amer. Math. Soc. 358 (2006), 1469–1510.
• M. Bownik, B. Li, D. Yang and Y. Zhou, Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators, Indiana Univ. Math. J. 57 (2008), 3065–3100.
• ––––, Weighted anisotropic product Hardy spaces and boundedness of sublinear operators, Math. Nachr. 283 (2010), 392–442.
• S.S. Byun, F. Yao and S. Zhou, Gradient estimates in Orlicz space for nonlinear elliptic equations, J. Funct. Anal. 255 (2008), 1851–1873.
• R.R. Coifman and L. Grafakos, Hardy space estimates for multilinear operators, I, Rev. Math. Iber. 8 (1992), 45–67.
• R.R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lect. Notes Math. 242, Springer-Verlag, Berlin, 1971.
• L. Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math. 129 (2005), 657–700.
• Y. Ding and S. Lan, Anisotropic weak Hardy spaces and interpolation theorems, Sci. China Math. 51 (2008), 1690–1704.
• ––––, Anisotropic Hardy space estimates for multilinear operators, Adv. Math. 38 (2009), 168–184.
• ––––, Hardy spaces estimates for a class of multilinear homogeneous operators, Sci. China Math. 42 (1999), 1270–1278.
• ––––, Hardy spaces estimates for multilinear operators with homogeneous kernals, Nagoya Math. J. 170 (2003), 117–133.
• C. Fefferman, N.M. Riviere and Y. Sagher, Interpolation between $H^p$ spaces: The real method, Trans. Amer. Math. Soc. 191 (1974), 75–81.
• C. Fefferman and E.M. Stein, $H^p$ spaces of several variables, Acta Math. 129 (1972), 137–193.
• R. Fefferman and F. Soria, The space weak $H^1$, Stud. Math. 85 (1987), 1–16.
• J. García-Cuerva and J.M. Martell, Wavelet characterization of weighted spaces, J. Geom. Anal. 11 (2001), 241–264.
• L. Grafakos, Hardy space estimates for multilinear operators, II, Rev. Math. Iber. 8 (1992), 69–72.
• R. Johnson and C.J. Neugebauer, Homeomorphisms preserving $A_p$, Rev. Mat. Iber. 3 (1987), 249–273.
• L.D. Ky, Bilinear decompositions and commutators of singular integral operators, Trans. Amer. Math. Soc. 365 (2013), 2931–2958.
• ––––, Bilinear decompositions for the product space $H^1_L\times BMO_L$, Math. Nachr. 287 (2014), 1288–1297.
• ––––, New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators, Int. Eq. Oper. Th. 78 (2014), 115–150.
• B. Li, D. Yang and W. Yuan, Anisotropic Hardy spaces of Musielak-Orlicz type with applications to boundedness of sublinear operators, The Scientific World Journal 2014, article ID 306214, 2014.
• Y. Liang, J. Huang and D. Yang, New real-variable characterizations of Musielak-Orlicz Hardy spaces, J. Math. Anal. Appl. 395 (2012), 413–428.
• Y. Liang, D. Yang and R. Jiang, Weak Musielak-Orlicz Hardy spaces and applications, Math. Nachr. 289 (2016), 634–677.
• H. Liu, The weak $H^p$ spaces on homogeneous groups, Lect. Notes Math. 1984, Springer-Verlag, Berlin, 1991.
• J. Liu, D. Yang and W. Yuan, Anisotropic Hardy-Lorentz spaces and their applications, Sci. China Math. 58 (2015), 1–52.
• N. Liu and Y. Ye, Weak Orlicz space and its convergence theorems, Acta Math. Sci. (China) 30 (2010), 1492–1500.
• A. Miyachi, Hardy spaces estimates for the product of singular integrals, Canad. J. Math. 52 (2000), 281–311.
• J. Musielak, Orlicz spaces and modular spaces, Lect. Notes Math. 1034, Springer-Verlag, Berlin, 1983.
• J. Orihuela and M. Ruiz Galán, Lebesgue property for convex risk measures on Orlicz spaces, Math. Finan. Econ. 6 (2012), 1–35.
• W. Orlicz, Über eine gewisse Klasse von Räumen vom Typus B, Bull. Int. Acad. Pol. 8 (1932), 207–220.
• T. Quek and D. Yang, Calderón-Zygmund-type operators on weighted weak Hardy spaces over $\rn$, Acta Math. Sinica 16 (2000), 141–160.
• M. Rao and Z. Ren, Theory of Orlicz spaces, Dekker, New York, 1991.
• E.M. Stein, Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, Princeton, NJ, 1993.
• J.O. Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces, Indiana Univ. Math. J. 28 (1979), 511–544.