×

Generalized Carleson embeddings into weighted outer measure spaces. (English) Zbl 1475.42023

Summary: We prove generalized Carleson embeddings for the continuous wave packet transform from \(L^p(\mathbb{R}, w)\) into an outer \(L^p\) space over \(\mathbb{R} \times \mathbb{R} \times(0, \infty)\) for \(2 < p < \infty\) and weight \(w \in \mathbb{A}_{p / 2} \). This work is a weighted extension of the corresponding Lebesgue result in [Y. Do and C. Thiele, Bull. Am. Math. Soc., New Ser. 52, No. 2, 249–296 (2015; Zbl 1318.42016)] and generalizes a similar result in [Y. Do and M. Lacey, Stud. Math. 211, No. 2, 153–190 (2012; Zbl 1266.42031)]. The proof in this article relies on \(L^2\) restriction estimates for the wave packet transform which are geometric and may be of independent interest.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
28A12 Contents, measures, outer measures, capacities
42A20 Convergence and absolute convergence of Fourier and trigonometric series
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E15 Banach spaces of continuous, differentiable or analytic functions
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Benea, C.; Muscalu, C., Sparse domination via the helicoidal method (2017), arXiv preprint · Zbl 1472.42011
[2] Benea, C.; Muscalu, C., The helicoidal method, (Operator Theory: Themes and Variations. Operator Theory: Themes and Variations, Theta Ser. Adv. Math., vol. 20 (2018), Theta: Theta Bucharest), 45-96 · Zbl 07728577
[3] Carleson, L., On convergence and growth of partial sums of Fourier series, Acta Math., 116, 135-157 (1966) · Zbl 0144.06402
[4] Christ, M., Weak type \((1, 1)\) bounds for rough operators, Ann. Math. (2), 128, 1, 19-42 (1988) · Zbl 0666.47027
[5] Conde, J. M., A note on dyadic coverings and nondoubling Calderón-Zygmund theory, J. Math. Anal. Appl., 397, 2, 785-790 (2013) · Zbl 1254.42021
[6] Cruz-Uribe, D.; Martell, J. M., Limited range multilinear extrapolation with applications to the bilinear Hilbert transform, Math. Ann., 371, 1-2, 615-653 (2018) · Zbl 1425.42016
[7] Culiuc, A.; Di Plinio, F.; Ou, Y., Domination of multilinear singular integrals by positive sparse forms, J. Lond. Math. Soc. (2), 98, 2, 369-392 (2018) · Zbl 1402.42013
[8] Di Plinio, F.; Do, Y. Q.; Uraltsev, G. N., Positive sparse domination of variational Carleson operators, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), 18, 4, 1443-1458 (2018) · Zbl 1403.42010
[9] Di Plinio, F.; Ou, Y., A modulation invariant Carleson embedding theorem outside local \(L^2\), J. Anal. Math., 135, 2, 675-711 (2018) · Zbl 1440.42053
[10] Do, Y.; Lacey, M., Weighted bounds for variational Fourier series, Stud. Math., 211, 2, 153-190 (2012) · Zbl 1266.42031
[11] Do, Y.; Lacey, M., Weighted bounds for variational Walsh-Fourier series, J. Fourier Anal. Appl., 18, 6, 1318-1339 (2012) · Zbl 1271.42018
[12] Do, Y.; Oberlin, R.; Palsson, E. A., Variational bounds for a dyadic model of the bilinear Hilbert transform, Ill. J. Math., 57, 1, 105-119 (2013) · Zbl 1304.42033
[13] Do, Y.; Thiele, C., \( L^p\) theory for outer measures and two themes of Lennart Carleson united, Bull. Am. Math. Soc. (N.S.), 52, 2, 249-296 (2015) · Zbl 1318.42016
[14] Fefferman, C., Pointwise convergence of Fourier series, Ann. Math. (2), 98, 551-571 (1973) · Zbl 0268.42009
[15] Hunt, R. A., On the convergence of Fourier series, (Orthogonal Expansions and Their Continuous Analogues. Orthogonal Expansions and Their Continuous Analogues, Proc. Conf., Edwardsville, Ill., 1967 (1968), Southern Illinois Univ. Press: Southern Illinois Univ. Press Carbondale, Ill), 235-255 · Zbl 0159.35701
[16] Hytönen, T. P.; Lacey, M. T.; Pérez, C., Sharp weighted bounds for the q-variation of singular integrals, Bull. Lond. Math. Soc., 45, 3, 529-540 (2013) · Zbl 1271.42021
[17] Lacey, M.; Thiele, C., \( L^p\) estimates on the bilinear Hilbert transform for \(2 < p < \infty \), Ann. Math. (2), 146, 3, 693-724 (1997) · Zbl 0914.46034
[18] Lacey, M.; Thiele, C., On Calderón’s conjecture, Ann. Math. (2), 149, 2, 475-496 (1999) · Zbl 0934.42012
[19] Lacey, M.; Thiele, C., A proof of boundedness of the Carleson operator, Math. Res. Lett., 7, 4, 361-370 (2000) · Zbl 0966.42009
[20] Lerner, A. K., Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals, Adv. Math., 226, 5, 3912-3926 (2011) · Zbl 1226.42010
[21] Lerner, A. K., On sharp aperture-weighted estimates for square functions, J. Fourier Anal. Appl., 20, 4, 784-800 (2014) · Zbl 1310.42010
[22] X. Li, personal communication.
[23] Muscalu, C.; Tao, T.; Thiele, C., \( L^p\) estimates for the biest. I. The Walsh case, Math. Ann., 329, 3, 401-426 (2004) · Zbl 1073.42009
[24] Muscalu, C.; Tao, T.; Thiele, C., \( L^p\) estimates for the biest. II. The Fourier case, Math. Ann., 329, 3, 427-461 (2004) · Zbl 1073.42010
[25] Oberlin, R.; Seeger, A.; Tao, T.; Thiele, C.; Wright, J., A variation norm Carleson theorem, J. Eur. Math. Soc., 14, 2, 421-464 (2012) · Zbl 1246.42016
[26] Thiele, C., Wave Packet Analysis, CBMS Regional Conference Series in Mathematics, vol. 105 (2006), American Mathematical Society: American Mathematical Society Providence, RI, published for the Conference Board of the Mathematical Sciences, Washington, DC · Zbl 1101.42001
[27] Thiele, C.; Treil, S.; Volberg, A., Weighted martingale multipliers in the non-homogeneous setting and outer measure spaces, Adv. Math., 285, 1155-1188 (2015) · Zbl 1332.42005
[28] Uraltsev, G., Variational Carleson embeddings into the upper 3-space (2016), preprint
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.