Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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CONVOLUTION OPERATORS WITH THE AFFINE ARCLENGTH MEASURE ON PLANE CURVES

  • Published : 1999.01.01
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Abstract

Let ${\gamma}$ : Ilongrightarrow R2 be a sufficiently smooth curve and $\sigma$${\gamma}$ be the affine arclength measure supported on ${\gamma}$. In this paper, we study the Lp - improving properties of the convolution operators T$\sigma$${\gamma}$ associated with $\sigma$${\gamma}$ for various curves ${\gamma}$. Optimal results are obtained for all finite type plane curves and homogeneous curves (possibly blowing up at the origin). As an attempt to extend this result to infinitely flat curves we give and example of a family of flat curves whose affine arclength measure has same Lp-improvement property. All of these results will be based on uniform estimates of damping oscillatory integrals.

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References

  1. Endpoint bounds for singular fractional integral operators M. Christ
  2. Math. Proc. Camb. Phil. Soc. v.108 Degenerate curves and harmonic analysis S. W. Drury
  3. Math. Proc. Camb. Phil. Soc. v.110 Convolution estimates related to surfaces of half the ambient dimension S. W. Drury;K. Guo
  4. Generalized functions;Properties and operations v.Ⅰ I. M. Gelfand;G. E. Shilov
  5. Differential geometry H. W. Guggenheimer
  6. Proc. Sympos. Pura Math. ⅩⅩⅢ;Amer. Math. Soc. $L^p-L^q$-estimates for singular integral operators arising from hyperbolic equations W. Littman
  7. J. Funct. Anal. v.86 Harmonic analysis on nilpotent groups and singular integrals, Ⅲ, Fractional integration along manifolds F. Ricci;E. M. Stein
  8. Harmonic Analysis: real-variable methods, orthogonality, and oscillatory integrals E. M. Stein
  9. Introduction to Fourier analysis in Euclidean spaces E. M. Stein;G. Weiss