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Choquard equations via nonlinear rayleigh quotient for concave-convex nonlinearities
A new Carleson measure adapted to multi-level ellipsoid covers
College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China |
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
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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. |
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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.
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J. Duoandikoetxea, Fourier Analysis, Grad. Stud. Math, Providence, 2001.
doi: 10.1090/gsm/029. |
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L. Grafakos, Modern Fourier Analysis, Springer-Verlag, New York, 2009.
doi: 10.1007/978-0-387-09434-2. |
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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. |
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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. |
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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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