Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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Lp-SOBOLEV REGULARITY FOR INTEGRAL OPERATORS OVER CERTAIN HYPERSURFACES

  • Received : 2013.06.26
  • Published : 2014.07.31
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Abstract

In this paper we establish sharp $L^p$-regularity estimates for averaging operators with convolution kernel associated to hypersurfaces in $\mathbb{R}^d(d{\geq}2)$ of the form $y{\mapsto}(y,{\gamma}(y))$ where $y{\in}\mathbb{R}^{d-1}$ and ${\gamma}(y)={\sum}^{d-1}_{i=1}{\pm}{\mid}y_i{\mid}^{m_i}$ with $2{\leq}m_1{\leq}{\cdots}{\leq}m_{d-1}$.

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References

  1. M. Christ, Failure of an endpoint estimate for integrals along curves, Fourier analysis and partial differential equations (Miraflores de la Sierra, 1992), 163-168, Stud. Adv. Math. CRC, Boca Raton, FL, 1995.
  2. E. Ferreyra, T. Godoy, and M. Urciuolo, Endpoint bounds for convolution operators with singular measures, Colloq. Math. 76 (1998), no. 1, 35-47. https://doi.org/10.4064/cm-76-1-35-47
  3. A. Iosevich, E. Sawyer, and A. Seeger, On averaging operators associated with convex hypersurfaces of finite type, J. Anal. Math. 79 (1999), 159-187. https://doi.org/10.1007/BF02788239
  4. A. Nagel, A. Seeger, and S. Wainger, Averages over convex hypersurfaces, Amer. J. Math. 115 (1993), no. 4, 903-927. https://doi.org/10.2307/2375017
  5. A. Seeger, Some inequalities for singular convolution operators in Lp-spaces, Trans. Amer. Math. Soc. 308 (1988), no. 1, 259-272.
  6. A. Seeger and T. Tao, Sharp Lorentz space estimates for rough operators, Math. Ann. 320 (2001), no. 2, 381-415. https://doi.org/10.1007/PL00004479