Fourier decay of fractal measures on surfaces of co-dimension two in \(\mathbb{R}^5\). (English) Zbl 1534.42019
Summary: In this paper, we study the Fourier decay of fractal measures on the quadratic surfaces of high co-dimensions. Unlike the case of co-dimension 1, quadratic surfaces of high co-dimensions possess some special scaling structures and degenerate characteristics. We will adopt the strategy from X. Du and R. Zhang [Ann. Math. (2) 189, No. 3, 837–861 (2019; Zbl 1433.42010)], combined with the broad-narrow analysis with different dimensions as divisions, to obtain a few lower bounds of Fourier decay of fractal measures on quadratic surfaces of co-dimension two in \(\mathbb{R}^5\).
MSC:
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 42B37 | Harmonic analysis and PDEs |
| 28A80 | Fractals |
| 28A75 | Length, area, volume, other geometric measure theory |
Citations:
Zbl 1433.42010References:
| [1] | Bak, J.-G.; Lee, J.; Lee, S., Bilinear restriction estimates for surfaces of codimension bigger than 1, Anal. PDE, 10, 8, 1961-1985, (2017) · Zbl 1376.42015 |
| [2] | Bak, J.-G.; Lee, S., Restriction of the Fourier transform to a quadratic surface in \(\mathbb{R}^n\), Math. Z., 247, 2, 409-422, (2004) · Zbl 1073.42006 |
| [3] | Barron, A.; Erdoǧan, M. B.; Harris, T. L.J., Fourier decay of fractal measures on hyperboloids, Trans. Am. Math. Soc., 374, 2, 1041-1075, (2021) · Zbl 1455.42023 |
| [4] | Bennett, J.; Carbery, A.; Christ, M.; Tao, T., The Brascamp-Lieb inequalities: finiteness, structure and extremals, Geom. Funct. Anal., 17, 5, 1343-1415, (2008) · Zbl 1132.26006 |
| [5] | Bennett, J.; Carbery, A.; Christ, M.; Tao, T., Finite bounds for Hölder-Brascamp-Lieb multilinear inequalities, Math. Res. Lett., 17, 4, 647-666, (2010) · Zbl 1247.26029 |
| [6] | Bennett, J.; Carbery, A.; Tao, T., On the multilinear restriction and Kakeya conjectures, Acta Math., 196, 2, 261-302, (2006) · Zbl 1203.42019 |
| [7] | Bourgain, J., Decoupling, exponential sums and the Riemann zeta function, J. Am. Math. Soc., 30, 1, 205-224, (2017) · Zbl 1352.11065 |
| [8] | Bourgain, J.; Demeter, C., The proof of the \(l^2\) decoupling conjecture, Ann. Math., 182, 1, 351-389, (2015) · Zbl 1322.42014 |
| [9] | Bourgain, J.; Demeter, C., Decouplings for surfaces in \(\mathbb{R}^4\), J. Funct. Anal., 270, 4, 1299-1318, (2016) · Zbl 1332.53007 |
| [10] | Bourgain, J.; Demeter, C., Decouplings for curves and hypersurfaces with nonzero Gaussian curvature, J. Anal. Math., 133, 279-311, (2017) · Zbl 1384.42016 |
| [11] | Bourgain, J.; Demeter, C.; Guth, L., Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three, Ann. Math., 184, 2, 633-682, (2016) · Zbl 1408.11083 |
| [12] | Bourgain, J.; Guth, L., Bounds on oscillatory integral operators based on multilinear estimates, Geom. Funct. Anal., 21, 6, 1239-1295, (2011) · Zbl 1237.42010 |
| [13] | Carbery, A.; Christ, M.; Wright, J., Multidimensional van der Corput and sublevel set estimates, J. Am. Math. Soc., 12, 4, 981-1015, (1999) · Zbl 0938.42008 |
| [14] | De Carli, L., Unique continuation for elliptic operators with non-multiple characteristics, Isr. J. Math., 118, 15-27, (2000) · Zbl 0962.47021 |
| [15] | Cho, C.; Ham, S.; Lee, S., Fractal Strichartz estimate for the wave equation, Nonlinear Anal., 150, 61-75, (2017) · Zbl 1355.35005 |
| [16] | Christ, M., Restriction of the Fourier transform to submanifolds of low codimension, (1982), University of Chicago, Ph.D. thesis |
| [17] | Christ, M., On the restriction of the Fourier transform to curves: endpoint results and the degenerate case, Trans. Am. Math. Soc., 287, 1, 223-238, (1985) · Zbl 0563.42010 |
| [18] | Demeter, C.; Guo, S.; Shi, F., Sharp decouplings for three dimensional manifolds in \(\mathbb{R}^5\), Rev. Mat. Iberoam., 35, 2, 423-460, (2019) · Zbl 1420.42008 |
| [19] | Du, X.; Guth, L.; Li, X., A sharp Schrödinger maximal estimate in \(\mathbb{R}^2\), Ann. Math., 186, 2, 607-640, (2017) · Zbl 1378.42011 |
| [20] | Du, X.; Guth, L.; Li, X.; Zhang, R., Pointwise convergence of Schrödinger solutions and multilinear refined Strichartz estimate, Forum Math. Sigma, 6, e14, 1-18, (2018) · Zbl 1395.42063 |
| [21] | Du, X.; Zhang, R., Sharp \(L^2\) estimate of Schrödinger maximal function in higher dimensions, Ann. Math., 189, 3, 837-861, (2019) · Zbl 1433.42010 |
| [22] | Eroǧan, M. B., A note on the Fourier transform of fractal measures, Math. Res. Lett., 11, 3, 299-313, (2004) · Zbl 1083.42006 |
| [23] | Falconer, K. J., On the Hausdorff dimensions of distance sets, Mathematika, 32, 2, 206-212, (1985) · Zbl 0605.28005 |
| [24] | Guo, S.; Oh, C., Fourier restriction estimates for surfaces of co-dimension two in \(\mathbb{R}^5\), J. Anal. Math., 148, 471-499, (2022) · Zbl 1505.42011 |
| [25] | Guo, S.; Oh, C.; Roos, J.; Yung, P.-L.; Zorin-Kranich, P., Decoupling for two quadratic forms in three variables: a complete characterization, Rev. Mat. Iberoam., 39, 1, 283-306, (2023) · Zbl 1514.42023 |
| [26] | Guo, S.; Zorin-Kranich, P., Decoupling for certain quadratic surfaces of low co-dimensions, J. Lond. Math. Soc., 102, 1, 319-344, (2020) · Zbl 1454.42015 |
| [27] | Guo, Shaoming; Oh, Changkeun; Zhang, Ruixiang; Zorin-Kranich, Pavel, Decoupling inequalities for quadratic forms, Duke Math. J., 172, 2, 387-445, (2023) · Zbl 1507.42016 |
| [28] | Guth, L., A short proof of the multilinear Kakeya inequality, Math. Proc. Camb. Philos. Soc., 158, 1, 147-153, (2015) · Zbl 1371.42007 |
| [29] | Guth, L.; Wang, H.; Zhang, R., A sharp square function estimate for the cone in \(\mathbb{R}^3\), Ann. Math., 192, 2, 551-581, (2020) · Zbl 1450.35156 |
| [30] | Harris, T. L.J., Improved decay of conical averages of the Fourier transform, Proc. Am. Math. Soc., 147, 11, 4781-4796, (2019) · Zbl 1423.42047 |
| [31] | Lee, J.; Lee, S., Restriction estimates to complex hypersurfaces, J. Math. Anal. Appl., 506, 2, Article 125702 pp., (2020) · Zbl 1475.42026 |
| [32] | Maldague, D., Regularized Brascamp-Lieb inequalities and an application, Q. J. Math., 73, 1, 311-331, (2022) · Zbl 1497.26033 |
| [33] | Mattila, P., Spherical averages of Fourier transforms of measures with finite energy; dimension of intersections and distance sets, Mathematika, 34, 2, 207-228, (1987) · Zbl 0645.28004 |
| [34] | Mockenhaupt, G., Bounds in Lebesgue spaces of oscillatory integral operators, (1996), University of Siegen, Habilitation Ph.D. thesis · Zbl 0916.42009 |
| [35] | Oh, C., Decouplings for three-dimensional surfaces in \(\mathbb{R}^6\), Math. Z., 290, 1-2, 389-419, (2018) · Zbl 1420.42007 |
| [36] | Ou, Y.; Wang, H.; Wilson, B.; Du, X.; Guth, L.; Zhang, R., Weighted restriction estimates and application to Falconer distance set problem, Am. J. Math., 143, 1, 175-211, (2021) · Zbl 1461.28003 |
| [37] | Sjölin, P., Estimate of averages of Fourier transforms of measures with finite energy, Ann. Acad. Sci. Fenn., Math., 22, 1, 227-236, (1997) · Zbl 0865.42007 |
| [38] | Wolff, T. H., Decay of circular means of Fourier transforms of measures, Int. Math. Res. Not., 1999, 10, 547-567, (1999) · Zbl 0930.42006 |
| [39] | Wongkew, R., Volumes of tubular neighbourhoods of real algebraic varieties, Pac. J. Math., 159, 177-184, (1993) · Zbl 0803.14029 |
| [40] | Wooley, T. D., The cubic case of the main conjecture in Vinogradov’s mean value theorem, Adv. Math., 294, 532-561, (2016) · Zbl 1365.11097 |
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