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On the Fourier transform of measures carried by submanifolds of finite type. (English) Zbl 0897.42006

Let \(M\) be a compact finite-type submanifold of \({\mathbb{R}}^n\) of dimension \(m\), with surface measure \(\sigma\). The atomic Hardy space associated to \(M\) is denoted by \(H^1(M)\). The main theorem of this elegant paper is that, if \(\phi \in H^1(M)\), then \(\int_0^{\infty} | (\psi \sigma)\widehat{\phantom{x}} (t\xi)| t^{-1} dt \leq C \| \psi \| _{H^1}\). The proof involves observing that it suffices to take \(\psi\) with very small support, and then applying a change of variables to change the integral involved in computing \((\psi \sigma)\widehat{\phantom{x}}\) to an integral over a small subset of \({\mathbb{R}}^m\), where the simple exponential \(e^{it \xi \cdot x}\) of the Fourier transform is replaced by an exponential with a nonlinear phase. Oscillatory integral techniques are then employed.
Reviewer: M.Cowling (Sydney)

MSC:

42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42B30 \(H^p\)-spaces
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[1] Blank, B., Nontangential maximal functions over compact Riemannian manifolds, Proc. Amer. Math. Soc, 103, 999-1002 (1988) · Zbl 0665.58046 · doi:10.2307/2046892
[2] Bruna, J.; Nagel, A.; Wainger, S., Convex hypersurfaces and Fourier transforms, Ann. of Math. (2), 127, 333-365 (1988) · Zbl 0666.42010 · doi:10.2307/2007057
[3] Coifman, R.; Weiss, G., Extension of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83, 569-645 (1977) · Zbl 0358.30023 · doi:10.1090/S0002-9904-1977-14325-5
[4] Colzani, L., Hardy spaces on spheres, Ph.D Thesis (1982), St. Louis: Washington University, St. Louis
[5] Colzani, L., Lipschitz spaces on compact rank one symmetric spaces, Harmonic Analysis (Cortona, 1982), 139-160 (1983), Berlin-New York: Springer, Berlin-New York · Zbl 0518.43007
[6] Colzani, L.; Travaglini, G., Hardy-Lorentz spaces and expansions in eigenfunctions of the Laplace-Beltrami operator on compact manifolds, Coll. Math, LVIII, 305-315 (1990) · Zbl 0707.46013
[7] Cowling, M.; Disney, S.; Mauceri, G.; Müller, D., Damping oscillatory integrals, Invent. Math, 101, 237-260 (1990) · Zbl 0733.42011 · doi:10.1007/BF01231503
[8] Duoandikoetxea, J.; de Francia, J. L. Rubio, Maximal and singular integral operators via Fourier transform estimates, Invent. Math, 84, 541-561 (1986) · Zbl 0568.42012 · doi:10.1007/BF01388746
[9] Fan, D., Restriction theorems related to atoms, Illinois J. Math, 40, 13-20 (1996) · Zbl 0883.42011
[10] Herz, C., Fourier transforms related to convex sets, Ann. of Math. (2), 75, 81-92 (1962) · Zbl 0111.34803 · doi:10.2307/1970421
[11] Hlawka, E., über Integrale auf Konvexen Körper I, Monatsh. Math, 54, 1-36 (1950) · Zbl 0036.30902 · doi:10.1007/BF01304101
[12] Hörmander, L., The Analysis of Linear Partial Differential Operators, I (1983), Berlin: Springer-Verlag, Berlin · Zbl 0521.35001
[13] Liftman, W., Fourier transforms of surface-carried measures and differentiability of surface averages, Bull. Amer. Math. Soc, 69, 766-770 (1963) · Zbl 0143.34701
[14] Oberlin, D., Oscillatory integrals with polynomial phase, Math. Scand, 69, 45-56 (1991) · Zbl 0761.42007
[15] Pan, Y., Boundedness of oscillatory singular integrals on Hardy spaces: II, Indiana Univ. Math. J, 41, 279-293 (1992) · Zbl 0779.42008 · doi:10.1512/iumj.1992.41.41016
[16] Phong, D. H.; Stein, E. M., Oscillatory integrals with polynomial phases, Invent. Math, 110, 39-62 (1992) · Zbl 0829.42014 · doi:10.1007/BF01231323
[17] Randol, B., On the Fourier transform of the indicator function of a planar set, Trans. Amer. Math. Soc, 139, 271-278 (1969) · Zbl 0183.26904 · doi:10.2307/1995319
[18] Randol, B., On the asymptotic behavior of the Fourier transform of the indicator function of a convex set, Trans. Amer. Math. Soc, 139, 279-285 (1969) · Zbl 0183.26905 · doi:10.2307/1995320
[19] Ricci, F.; Stein, E. M., Harmonic analysis on nilpotent groups and singular integrals I: Oscillatory integrals, J. Funct. Anal, 73, 179-194 (1987) · Zbl 0622.42010 · doi:10.1016/0022-1236(87)90064-4
[20] Sogge, C.; Stein, E. M., Averages of functions over hypersurfaces in R^n, Invent. Math, 82, 543-556 (1985) · Zbl 0626.42009 · doi:10.1007/BF01388869
[21] Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals (1993), Princeton, NJ: Princeton University Press, Princeton, NJ · Zbl 0821.42001
[22] Svensson, I., Estimates for the Fourier transform of the characteristic function of a convex set, Ark. Mat, 9, 11-22 (1971) · Zbl 0221.52001 · doi:10.1007/BF02383634
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