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October  2021, 20(10): 3481-3497. doi: 10.3934/cpaa.2021115

## A new Carleson measure adapted to multi-level ellipsoid covers

 College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China

* Corresponding author

Received  December 2020 Revised  May 2021 Published  October 2021 Early access  June 2021

Fund Project: The project is supported by the Xinjiang Training of Innovative Personnel Natural Science Foundation of China grant 2020D01C048 and the National Natural Science Foundation of China grant 11861062

We develop highly anisotropic Carleson measure over multi-level ellipsoid covers $\Theta$ of $\mathbb{R}^n$ that are highly anisotropic in the sense that the ellipsoids can change rapidly from level to level and from point to point. Then we show that the Carleson measure $\mu$ is sufficient for which the integral defines a bounded operator from $H^p(\Theta)$ to $L^p(\mathbb{R}^{n+1}, \, \mu),\ 0 Citation: Ankang Yu, Yajuan Yang, Baode Li. A new Carleson measure adapted to multi-level ellipsoid covers. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3481-3497. doi: 10.3934/cpaa.2021115 ##### References:  [1] M. Bownik, Anisotropic Hardy spaces and wavelets, Mem. Amer. Math. Soc., 164 (2003), 1-122. doi: 10.1090/memo/0781. [2] M. Bownik, B. Li and J. Li, Variable anisotropic singular integral operators, arXiv: 2004.09707v2. [3] L. Carleson, Interpolation by bounded analytic functions and the corona problem, Ann. Math., 76 (1962), 547-559. doi: 10.2307/1970375. [4] A. P. Calderón and A. Torchinsky, Parabolic maximal functions associated with a distribution, Adv. Math., 16 (1975), 1-64. doi: 10.1016/0001-8708(75)90099-7. [5] W. Dahmen, S. Dekel and P. Petrushev, Two-level-split decomposition of anisotropic Besov spaces, Constr. Approx., 31 (2010), 149-194. doi: 10.1007/s00365-009-9058-y. [6] S. Dekel, Y. Han and P. Petrushev, Anisotropic meshless frames on${{{{{\mathbb R}}}^n}}$, J. Fourier Anal. Appl., 15 (2009), 634-662. doi: 10.1007/s00041-009-9070-4. [7] S. Dekel, P. Petrushev and T. Weissblat, Hardy spaces on${{{{{\mathbb R}}}^n}}$with pointwise variable anisotropy, J. Fourier Anal. Appl., 17 (2011), 1066-1107. doi: 10.1007/s00041-011-9176-3. [8] J. Duoandikoetxea, Fourier Analysis, Grad. Stud. Math, Providence, 2001. doi: 10.1090/gsm/029. [9] L. Grafakos, Modern Fourier Analysis, Springer-Verlag, New York, 2009. doi: 10.1007/978-0-387-09434-2. [10] S. Gadbois and T. Sledd, Careson measures on spaces of homogeneous type, Trans. Amer. Math Soc., 341 (1994), 841-862. doi: 10.1090/S0002-9947-1994-1149122-2. [11] E. Harboure, O. Salinas and B. Viviani, A look at$\text BMO_{\varphi}(\omega)$through Carleson measures, J. Fourier Anal. Appl., 13 (2007), 267-284. doi: 10.1007/s00041-005-5044-3. [12] S. Hou, D. Yang and S. Yang, Lusin area function and molecular characterizations of Musielak-Orlicz Hardy spaces and their applications, Commun. Contemp. Math., 15 (2013), 37 pp. doi: 10.1142/S0219199713500296. [13] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, N. J., 1993. show all references ##### References:  [1] M. Bownik, Anisotropic Hardy spaces and wavelets, Mem. Amer. Math. Soc., 164 (2003), 1-122. doi: 10.1090/memo/0781. [2] M. Bownik, B. Li and J. Li, Variable anisotropic singular integral operators, arXiv: 2004.09707v2. [3] L. Carleson, Interpolation by bounded analytic functions and the corona problem, Ann. Math., 76 (1962), 547-559. doi: 10.2307/1970375. [4] A. P. Calderón and A. Torchinsky, Parabolic maximal functions associated with a distribution, Adv. Math., 16 (1975), 1-64. doi: 10.1016/0001-8708(75)90099-7. [5] W. Dahmen, S. Dekel and P. Petrushev, Two-level-split decomposition of anisotropic Besov spaces, Constr. Approx., 31 (2010), 149-194. doi: 10.1007/s00365-009-9058-y. [6] S. Dekel, Y. Han and P. Petrushev, Anisotropic meshless frames on${{{{{\mathbb R}}}^n}}$, J. Fourier Anal. Appl., 15 (2009), 634-662. doi: 10.1007/s00041-009-9070-4. [7] S. Dekel, P. Petrushev and T. Weissblat, Hardy spaces on${{{{{\mathbb R}}}^n}}$with pointwise variable anisotropy, J. Fourier Anal. Appl., 17 (2011), 1066-1107. doi: 10.1007/s00041-011-9176-3. [8] J. Duoandikoetxea, Fourier Analysis, Grad. Stud. Math, Providence, 2001. doi: 10.1090/gsm/029. [9] L. Grafakos, Modern Fourier Analysis, Springer-Verlag, New York, 2009. doi: 10.1007/978-0-387-09434-2. [10] S. Gadbois and T. Sledd, Careson measures on spaces of homogeneous type, Trans. Amer. Math Soc., 341 (1994), 841-862. doi: 10.1090/S0002-9947-1994-1149122-2. [11] E. Harboure, O. Salinas and B. Viviani, A look at$\text BMO_{\varphi}(\omega)\$ through Carleson measures, J. Fourier Anal. Appl., 13 (2007), 267-284.  doi: 10.1007/s00041-005-5044-3. [12] S. Hou, D. Yang and S. Yang, Lusin area function and molecular characterizations of Musielak-Orlicz Hardy spaces and their applications, Commun. Contemp. Math., 15 (2013), 37 pp. doi: 10.1142/S0219199713500296. [13] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, N. J., 1993.
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