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 and 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.

[2]

R. J. Bagby, An extended inequality for the maximal function, Proc. Amer. Math. Soc., 48 (1975), 419-422.  doi: 10.2307/2040276.

[3]

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.

[4]

C. Benea and C. Muscalu, Multiple vector-valued, mixed norm estimates for Littlewood–Paley square functions, preprint, arXiv: 1808.03248.

[5]

A. Benedek and R. Panzone, The space $L^p$, with mixed norm, Duke Math. J., 28 (1961), 301-324. 

[6]

M. Bownik, Anisotropic Hardy Spaces and Wavelets, Mem. Amer. Math. Soc., 164 (781) (2003), vi+122. doi: 10.1090/memo/0781.

[7]

M. BownikB. LiD. Yang and Y. Zhou, Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators, Indiana Univ. Math. J., 57 (2008), 3065-3100.  doi: 10.1512/iumj.2008.57.3414.

[8]

M. BownikB. LiD. Yang and Y. Zhou, Weighted anisotropic product Hardy spaces and boundedness of sublinear operators, Math. Nachr., 283 (2010), 392-442.  doi: 10.1002/mana.200910078.

[9]

M. Bownik and L. A. D. Wang, A PDE characterization of anisotropic Hardy spaces, preprint.

[10]

A. P. Calderón and A. Torchinsky, Parabolic maximal functions associated with a distribution, Adv. Math., 16 (1975), 1-64.  doi: 10.1016/0001-8708(75)90099-7.

[11]

S. Campanato, Proprietà di una famiglia di spazi funzionali, Ann. Scuola Norm. Super. Pisa (3), 18 (1964), 137–160.

[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.

[13]

T. Chen and W. Sun, Iterated and mixed weak norms with applications to geometric inequalities, J. Geom. Anal., arXiv: 1712.01064. doi: 10.1007/s12220-019-00243-x.

[14]

M. Christ, A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math., 60/61 (1990), 601-628.  doi: 10.4064/cm-60-61-2-601-628.

[15]

G. CleanthousA. G. Georgiadis and M. Nielsen, Anisotropic mixed-norm Hardy spaces, J. Geom. Anal., 27 (2017), 2758-2787.  doi: 10.1007/s12220-017-9781-8.

[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.

[17]

H. Dong and D. Kim, On $L^p$-estimates for elliptic and parabolic equations with $A_p$ weights, Trans. Amer. Math. Soc., 370 (2018), 5081-5130.  doi: 10.1090/tran/7161.

[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.

[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.

[20]

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

[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.

[22]

A. G. GeorgiadisJ. Johnsen and M. Nielsen, Wavelet transforms for homogeneous mixed-norm Triebel–Lizorkin spaces, Monatsh. Math., 183 (2017), 587-624.  doi: 10.1007/s00605-017-1036-z.

[23]

Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94.  doi: 10.1016/0022-1236(91)90136-S.

[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.

[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.

[26]

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

[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.

[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.

[29]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[43]

J. Liu, D. Yang and W. Yuan, Littlewood–Paley characterizations of weighted anisotropic Triebel–Lizorkin spaces via averages on balls Ⅰ & Ⅱ, Z. Anal. Anwend., 38 (2019), 397–418 & 39 (2020), 1–26. doi: 10.4171/ZAA/1643.

[44]

P. I. Lizorkin, Multipliers of Fourier integrals and estimates of convolutions in spaces with mixed norm. Applications, Izv. Akad. Nauk SSSR Ser. Mat., 34 (1970), 218–247.

[45]

S. MedaP. Sjögren and M. Vallarino, On the $H^1$-$L^1$ boundedness of operators, Proc. Amer. Math. Soc., 136 (2008), 2921-2931.  doi: 10.1090/S0002-9939-08-09365-9.

[46]

S. Müller, Hardy space methods for nonlinear partial differential equations, Tatra Mt. Math. Publ., 4 (1994), 159-168. 

[47]

T. Nogayama, Mixed Morrey spaces, Positivity, 23 (2019), 961-1000.  doi: 10.1007/s11117-019-00646-8.

[48]

W. Rudin, Functional Analysis, 2$^{nd}$ edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.

[49]

Y. Sawano, Sharp estimates of the modified Hardy–Littlewood maximal operator on the nonhomogeneous space via covering lemmas, Hokkaido Math. J., 34 (2005), 435-458.  doi: 10.14492/hokmj/1285766231.

[50]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, Vol. 43, Monographs in Harmonic Analysis III, Princeton University Press, Princeton, NJ, 1993.

[51]

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