Journal of the Mathematical Society of Japan

$L^p$-bounds for Stein's square functions associated to operators and applications to spectral multipliers

Peng CHEN, Xuan Thinh DUONG, and Lixin YAN

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Abstract

Let $(X, d, \mu)$ be a metric measure space endowed with a metric $d$ and a nonnegative Borel doubling measure $\mu$. Let $L$ be a non-negative self-adjoint operator of order $m$ on $X$. Assume that $L$ generates a holomorphic semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ satisfy Gaussian upper bounds but without any assumptions on the regularity of space variables $x$ and $y$. Also assume that $L$ satisfies a Plancherel type estimate. Under these conditions, we show the $L^p$ bounds for Stein's square functions arising from Bochner-Riesz means associated to the operator $L$. We then use the $L^p$ estimates on Stein's square functions to obtain a Hörmander-type criterion for spectral multipliers of $L$. These results are applicable for large classes of operators including sub-Laplacians acting on Lie groups of polynomial growth and Schrödinger operators with rough potentials.

Article information

Source
J. Math. Soc. Japan, Volume 65, Number 2 (2013), 389-409.

Dates
First available in Project Euclid: 25 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1366896639

Digital Object Identifier
doi:10.2969/jmsj/06520389

Mathematical Reviews number (MathSciNet)
MR3055591

Zentralblatt MATH identifier
1277.42011

Subjects
Primary: 42B15: Multipliers
Secondary: 42B25: Maximal functions, Littlewood-Paley theory 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47)

Keywords
Stein's square function Bochner-Riesz means non-negative self-adjoint operator heat semigroup space of homogeneous type

Citation

CHEN, Peng; DUONG, Xuan Thinh; YAN, Lixin. $L^p$-bounds for Stein's square functions associated to operators and applications to spectral multipliers. J. Math. Soc. Japan 65 (2013), no. 2, 389--409. doi:10.2969/jmsj/06520389. https://projecteuclid.org/euclid.jmsj/1366896639


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References

  • G. Alexopoulos, Spectral multipliers on Lie groups of polynomial growth, Proc. Amer. Math. Soc., 120 (1994), 973–979.
  • P. Auscher, On Necessary and Sufficient Conditions for $L^p$-Estimates of Riesz Transforms Associated to Elliptic Operators on ${\mathbb R}^n$ and Related Estimates, Mem. Amer. Math. Soc., 186, Amer. Math. Soc., Providence, RI, 2007.
  • P. Auscher, T. Coulhon, X. T. Duong and S. Hofmann, Riesz transform on manifolds and heat kernel regularity, Ann. Sci. École Norm. Sup. (4), 37, (2004), 911–957.
  • F. Bernicot and J. Zhao, New abstract Hardy spaces, J. Funct. Anal., 255 (2008), 1761–1796.
  • A. Carbery, The boundedness of the maximal Bochner-Riesz operator on $L^4({\mathbb R}^2)$, Duke Math. J., 50 (1983), 409–416.
  • A. Carbery, G. Gasper and W. Trebels, Radial Fourier multipliers of $L^p({\mathbb R}^2)$, Proc. Nat. Acad. Sci. U.S.A., 81 (1984), 3254–3255.
  • M. Christ, On almost everywhere convergence of Bochner-Riesz means in higher dimensions, Proc. Amer. Math. Soc., 95 (1985), 16-20.
  • M. Christ, $L^p$ bounds for spectral multipliers on nilpotent groups, Trans. Amer. Math. Soc., 328 (1991), 73–81.
  • R. R. Coifman and G. Weiss, Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes, Lecture Notes in Math., 242, Springer-Verlag, Berlin, New York, 1971.
  • E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Math., 92, Cambridge University Press, 1989.
  • L. De Michele and G. Mauceri, $H^p$ multipliers on stratified groups, Ann. Mat. Pura Appl. (4), 148 (1987), 353–366.
  • H. Dappa, A Marcinkiewicz criterion for $L^p$-multipliers, Pacific J. Math., 111 (1984), 9–21.
  • X. T. Duong and A. McIntosh, Singular integral operators with non-smooth kernels on irregular domains, Rev. Mat. Iberoamericana, 15 (1999), 233–265.
  • X. T. Duong, E. M. Ouhabaz and A. Sikora, Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal., 196 (2002), 443–485.
  • X. T. Duong, A. Sikora and L. Yan, Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers, J. Funct. Anal., 260 (2011), 1106–1131.
  • G. B. Folland and E. M. Stein, Hardy spaces on Homogeneous Groups, Math. Notes, 28, Princeton University Press, 1982.
  • C. Guillarmou, A. Hassell and A. Sikora, Restriction and spectral multiplier theorems on asymptotically conic manifolds, Anal. PDE, to appear.
  • L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math., 104 (1960), 93–140.
  • A. Hulanicki and E. M. Stein, Marcinkiewicz multiplier theorem for stratified groups, unpublished manuscript.
  • S. Igari, A note on the Littlewood-Paley function $g^{\ast}(f)$, Tôhoku Math. J. (2), 18 (1966), 232–235.
  • S. Igari and S. Kuratsubo, A sufficient condition for $L^p$-multipliers, Pacific J. Math., 38 (1971), 85–88.
  • S. Lee, K. M. Rogers and A. Seeger, Improved bounds for Stein's square functions, preprint arXiv:1012.2159.
  • E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Math. Soc. Monogr., 31, Princeton University Press, 2005.
  • M. Reed and B. Simon, Methods of Modern Mathematical Physics. I, Academic Press, 1980.
  • D. W. Robinson, Elliptic Operators and Lie Groups, Oxford Math. Monogr., The Clarendon Press, Oxford University Press, New York, 1991.
  • A. Sikora, On the ${L}\sp 2\to {L}\sp \infty$ norms of spectral multipliers of “quasi-homogeneous” operators on homogeneous groups, Trans. Amer. Math. Soc., 351 (1999), 3743–3755.
  • B. Simon, Maximal and minimal Schrödinger forms, J. Operator Theory, 1 (1979), 37–47.
  • E. M. Stein, Localization and summability of multiple Fourier series, Acta Math., 100 (1958), 93–147.
  • E. M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser., 43, Princeton University Press, Princeton, NJ, 1993.
  • E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser., 32, Princeton University Press, Princeton, NJ, 1971.
  • T. Tao, A sharp bilinear restrictions estimate for paraboloids, Geom. Funct. Anal., 13 (2003), 1359–1384.
  • N. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, Cambridge Tracts in Math., 100, Cambridge University Press, Cambridge, 1992.
  • K. Yosida, Functional Analysis (Fifth edition), Grundlehren Math. Wiss., 123, Springer-Verlag, Berlin, 1978.
  • A. Zygmund, Trigonometric Series. 2nd ed., 2, Cambridge University Press, New York, 1959.