June  2020, 19(6): 3033-3082. doi: 10.3934/cpaa.2020132

Real-variable characterizations of new anisotropic mixed-norm Hardy spaces

1. 

Laboratory of Mathematics and Complex Systems, (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

2. 

School of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China

* Corresponding author

Received  April 2019 Revised  December 2019 Published  March 2020

Fund Project: This project is supported by the National Natural Science Foundation of China (Grant Nos. 11971058, 11761131002, 11671185 and 11871100). Jun Liu is also supported by the Scientific Research Foundation of China University of Mining and Technology (Grant No. 102519054)

Let $ \vec{p}\in(0, \infty)^n $ and $ A $ be a general expansive matrix on $ \mathbb{R}^n $. In this article, via the non-tangential grand maximal function, the authors first introduce the anisotropic mixed-norm Hardy spaces $ H_A^{\vec{p}}(\mathbb{R}^n) $ associated with $ A $ and then establish their radial or non-tangential maximal function characterizations. Moreover, the authors characterize $ H_A^{\vec{p}}(\mathbb{R}^n) $, respectively, by means of atoms, finite atoms, Lusin area functions, Littlewood–Paley $ g $-functions or $ g_{\lambda}^\ast $-functions via first establishing an anisotropic Fefferman–Stein vector-valued inequality on the mixed-norm Lebesgue space $ L^{\vec{p}}(\mathbb{R}^n) $. In addition, the authors also obtain the duality between $ H_A^{\vec{p}}(\mathbb{R}^n) $ and the anisotropic mixed-norm Campanato spaces. As applications, the authors establish a criterion on the boundedness of sublinear operators from $ H_A^{\vec{p}}(\mathbb{R}^n) $ into a quasi-Banach space. Applying this criterion, the authors then obtain the boundedness of anisotropic convolutional $ \delta $-type and non-convolutional $ \beta $-order Calderón–Zygmund operators from $ H_A^{\vec{p}}(\mathbb{R}^n) $ to itself [or to $ L^{\vec{p}}(\mathbb{R}^n) $]. As a corollary, the boundedness of anisotropic convolutional $ \delta $-type Calderón–Zygmund operators on the mixed-norm Lebesgue space $ L^{\vec{p}}(\mathbb{R}^n) $ with $ \vec{p}\in(1, \infty)^n $ is also presented.

Citation: Long Huang, Jun Liu, Dachun Yang, Wen Yuan. Real-variable characterizations of new anisotropic mixed-norm Hardy spaces. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3033-3082. doi: 10.3934/cpaa.2020132
References:
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show all references

References:
[1]

V. AlmeidaJ. J. Betancor and L. Rodríguez-Mesa, Anisotropic Hardy–Lorentz spaces with variable exponents, Canadian J. Math., 69 (2017), 1219-1273.  doi: 10.4153/CJM-2016-053-6.  Google Scholar

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C. Benea and C. Muscalu, Multiple vector-valued inequalities via the helicoidal method, Anal. Partial Differ. Equ., 9 (2016), 1931-1988.  doi: 10.2140/apde.2016.9.1931.  Google Scholar

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[12]

C. CabrelliU. Molter and J. Romero, Non-uniform painless decompositions for anisotropic Besov and Triebel–Lizorkin spaces, Adv. Math., 232 (2013), 98-120.  doi: 10.1016/j.aim.2012.09.026.  Google Scholar

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[16]

G. CleanthousA. G. Georgiadis and 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.  doi: 10.1016/j.acha.2017.10.001.  Google Scholar

[17]

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[18]

H. Dong, D. Kim and T. Phan, Boundary Lebesgue mixed-norm estimates for non-stationary Stokes systems with VMO coefficients, preprint, arXiv: 1910.00380. Google Scholar

[19]

H. Dong and N. V. Krylov, Fully nonlinear elliptic and parabolic equations in weighted and mixed-norm Sobolev spaces, Calc. Var. Partial Differ. Equ., 58 (2019), Art. 145, 1–32. doi: 10.1007/s00526-019-1591-3.  Google Scholar

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C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-193.  doi: 10.1007/BF02392215.  Google Scholar

[21]

D. Foschi and S. Klainerman, Bilinear space-time estimates for homogeneous wave equations, Ann. Sci. École Norm. Sup., 33 (2000), 211–274. doi: 10.1016/S0012-9593(00)00109-9.  Google Scholar

[22]

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[24]

L. GrafakosL. Liu and D. Yang, Maximal function characterizations of Hardy spaces on RD-spaces and their applications, Sci. China Ser. A, 51 (2008), 2253-2284.  doi: 10.1007/s11425-008-0057-4.  Google Scholar

[25]

J. HartR. H. Torres and X. Wu, Smoothing properties of bilinear operators and Leibniz-type rules in Lebesgue and mixed Lebesgue spaces, Trans. Amer. Math. Soc., 370 (2018), 8581-8612.  doi: 10.1090/tran/7312.  Google Scholar

[26]

L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math., 104 (1960), 93-140.  doi: 10.1007/BF02547187.  Google Scholar

[27]

L. HuangJ. LiuD. Yang and W. Yuan, Atomic and Littlewood–Paley characterizations of anisotropic mixed-norm Hardy spaces and their applications, J. Geom. Anal., 29 (2019), 1991-2067.  doi: 10.1007/s12220-018-0070-y.  Google Scholar

[28]

L. HuangJ. LiuD. Yang and W. Yuan, Dual spaces of anisotropic mixed-norm Hardy spaces, Proc. Amer. Math. Soc., 147 (2019), 1201-1215.  doi: 10.1090/proc/14348.  Google Scholar

[29]

L. Huang and D. Yang, On function spaces with mixed norms — a survey, J. Math. Study, arXiv: 1908.03291. Google Scholar

[30]

T. Jakab and M. Mitrea, Parabolic initial boundary value problems in nonsmooth cylinders with data in anisotropic Besov spaces, Math. Res. Lett., 13 (2006), 825-831.  doi: 10.4310/MRL.2006.v13.n5.a12.  Google Scholar

[31]

F. John and L. Nirenberg, On functions of bounded mean oscillation, Commun. Pure Appl. Math., 14 (1961), 415-426.  doi: 10.1002/cpa.3160140317.  Google Scholar

[32]

J. JohnsenS. Munch Hansen and W. Sickel, Anisotropic Lizorkin–Triebel spaces with mixed norms – traces on smooth boundaries, Math. Nachr., 288 (2015), 1327-1359.  doi: 10.1002/mana.201300313.  Google Scholar

[33]

C. E. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle, Commun. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.  Google Scholar

[34]

D. Kim, Elliptic and parabolic equations with measurable coefficients in $L_p$-spaces with mixed norms, Meth. Appl. Anal., 15 (2008), 437-468.  doi: 10.4310/MAA.2008.v15.n4.a3.  Google Scholar

[35]

N. V. Krylov, Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms, J. Funct. Anal., 250 (2007), 521-558.  doi: 10.1016/j.jfa.2007.04.003.  Google Scholar

[36]

L. D. Ky, New Hardy spaces of Musielak–Orlicz type and boundedness of sublinear operators, Integr. Equ. Oper. Theory, 78 (2014), 115-150.  doi: 10.1007/s00020-013-2111-z.  Google Scholar

[37]

B. LiM. Bownik and D. Yang, Littlewood–Paley characterization and duality of weighted anisotropic product Hardy spaces, J. Funct. Anal., 266 (2014), 2611-2661.  doi: 10.1016/j.jfa.2013.12.017.  Google Scholar

[38]

B. LiX. Fan and D. Yang, Littlewood–Paley characterizations of anisotropic Hardy spaces of Musielak–Orlicz type, Taiwan. J. Math., 19 (2015), 279-314.  doi: 10.11650/tjm.19.2015.4692.  Google Scholar

[39]

Y. LiangY. SawanoT. UllrichD. Yang and W. Yuan, New characterizations of Besov–Triebel–Lizorkin–Hausdorff spaces including coorbits and wavelets, J. Fourier Anal. Appl., 18 (2012), 1067-1111.  doi: 10.1007/s00041-012-9234-5.  Google Scholar

[40]

J. LiuF. WeiszD. Yang and W. Yuan, Variable anisotropic Hardy spaces and their applications, Taiwan. J. Math., 22 (2018), 1173-1216.  doi: 10.11650/tjm/171101.  Google Scholar

[41]

J. LiuD. Yang and W. Yuan, Anisotropic Hardy–Lorentz spaces and their applications, Sci. China Math., 59 (2016), 1669-1720.  doi: 10.1007/s11425-016-5157-y.  Google Scholar

[42]

J. LiuD. Yang and W. Yuan, Anisotropic variable Hardy–Lorentz spaces and their real interpolation, J. Math. Anal. Appl., 456 (2017), 356-393.  doi: 10.1016/j.jmaa.2017.07.003.  Google Scholar

[43]

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