doi: 10.3934/cpaa.2021115
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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 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<p\leq 1 $. Finally, we give several equivalent Carleson measures adapted to multi-level ellipsoid covers and obtain a specific Carleson measure induced by the highly anisotropic BMO functions.

Citation: Ankang Yu, Yajuan Yang, Baode Li. A New carleson measure adapted to multi-level ellipsoid covers. Communications on Pure &amp; Applied Analysis, doi: 10.3934/cpaa.2021115
References:
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S. DekelP. 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.  Google Scholar

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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.  Google Scholar

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E. HarboureO. 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.  Google Scholar

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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.  Google Scholar

[13] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, N. J., 1993.   Google Scholar

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.  Google Scholar

[2]

M. Bownik, B. Li and J. Li, Variable anisotropic singular integral operators, arXiv: 2004.09707v2. Google Scholar

[3]

L. Carleson, Interpolation by bounded analytic functions and the corona problem, Ann. Math., 76 (1962), 547-559.  doi: 10.2307/1970375.  Google Scholar

[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.  Google Scholar

[5]

W. DahmenS. 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.  Google Scholar

[6]

S. DekelY. 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.  Google Scholar

[7]

S. DekelP. 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.  Google Scholar

[8]

J. Duoandikoetxea, Fourier Analysis, Grad. Stud. Math, Providence, 2001. doi: 10.1090/gsm/029.  Google Scholar

[9]

L. Grafakos, Modern Fourier Analysis, Springer-Verlag, New York, 2009. doi: 10.1007/978-0-387-09434-2.  Google Scholar

[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.  Google Scholar

[11]

E. HarboureO. 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.  Google Scholar

[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.  Google Scholar

[13] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, N. J., 1993.   Google Scholar
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