Journal of Geometric Analysis

, Volume 21, Issue 1, pp 96–117

On Radial and Conical Fourier Multipliers

Article

Abstract

We investigate connections between radial Fourier multipliers on ℝd and certain conical Fourier multipliers on ℝd+1. As an application we obtain a new weak type endpoint bound for the Bochner–Riesz multipliers associated with the light cone in ℝd+1, where d≥4, and results on characterizations of LpLp inequalities for convolutions with radial kernels.

Keywords

Radial Fourier multipliers Cone multiplier Weak type estimates 

Mathematics Subject Classification (2000)

42B15 

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References

  1. 1.
    Chang, S.Y.A., Fefferman, R.: A continuous version of duality of H 1 and BMO on the bidisc. Ann. Math. 112, 179–201 (1980) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Christ, M.: Weak type (1,1) bounds for rough operators. Ann. Math. (2) 128(1), 19–42 (1988) CrossRefMathSciNetGoogle Scholar
  3. 3.
    Fefferman, C., Stein, E.M.: Some maximal inequalities. Am. J. Math. 93, 107–115 (1971) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Garrigós, G., Seeger, A.: Characterizations of Hankel multipliers. Math. Ann. 342(1), 31–68 (2008) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Garrigós, G., Seeger, A.: Plate decompositions for cone multipliers. Proc. Edinb. Math. Soc. 52, 1–21 (2009) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Garrigós, G., Seeger, A., Schlag, W.: Improvements in Wolff inequality for decompositions of cone multipliers Google Scholar
  7. 7.
    Heo, Y.: Improved bounds for high dimensional cone multipliers. Indiana Univ. Math. J. 58(3), 1187–1202 (2009) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Heo, Y., Hong, S., Yang, C.: An endpoint estimate for the cone multiplier. Proc. Am. Math. Soc. 138, 1333–1347 (2010) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Heo, Y., Nazarov, F., Seeger, A.: Radial Fourier multipliers in high dimensions. Acta Math. (to appear) Google Scholar
  10. 10.
    Hong, S.: Weak type estimates for cone multipliers on H p spaces, p<1. Proc. Amer. Math. Soc. 128(12), 3529–3539 (2000) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hong, S.: Some weak type estimates for cone multipliers. Il. J. Math. 44(3), 496–515 (2000) MATHGoogle Scholar
  12. 12.
    Jodeit, M.: A note on Fourier multipliers. Proc. Am. Math. Soc. 27, 423–424 (1971) MATHMathSciNetGoogle Scholar
  13. 13.
    Łaba, I., Wolff, T.: A local smoothing estimate in higher dimensions. J. Anal. Math. 88, 149–171 (2002) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Peetre, J.: On spaces of Triebel–Lizorkin type. Ark. Mat. 13, 123–130 (1975) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Sagher, Y.: On analytic families of operators. Isr. J. Math. 7, 350–356 (1969) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Seeger, A.: Remarks on singular convolution operators. Stud. Math. 97, 91–114 (1990) MATHMathSciNetGoogle Scholar
  17. 17.
    Tao, T.: The weak-type endpoint Bochner–Riesz conjecture and related topics. Indiana Univ. Math. J. 47, 1097–1124 (1998) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Triebel, H.: Theory of Function Spaces. Monographs in Mathematics, vol. 78. Birkhäuser, Basel (1983) CrossRefGoogle Scholar
  19. 19.
    Wolff, T.: Local smoothing type estimates on L p for large p. Geom. Funct. Anal. 10(5), 1237–1288 (2000) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA

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