Oscillatory kernels in certain Hardy-type spaces

Lung-Kee Chen; Dashan Fan

Studia Mathematica (1994)

  • Volume: 111, Issue: 2, page 195-206
  • ISSN: 0039-3223

Abstract

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We consider a convolution operator Tf = p.v. Ω ⁎ f with Ω ( x ) = K ( x ) e i h ( x ) , where K(x) is an (n,β) kernel near the origin and an (α,β), α ≥ n, kernel away from the origin; h(x) is a real-valued C ∞ function on ℝ n ∖ 0 . We give a criterion for such an operator to be bounded from the space H 0 p ( ℝ n ) into itself.

How to cite

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Chen, Lung-Kee, and Fan, Dashan. "Oscillatory kernels in certain Hardy-type spaces." Studia Mathematica 111.2 (1994): 195-206. <http://eudml.org/doc/216128>.

@article{Chen1994,
abstract = {We consider a convolution operator Tf = p.v. Ω ⁎ f with $Ω(x) = K(x)e^\{ih(x)\}$, where K(x) is an (n,β) kernel near the origin and an (α,β), α ≥ n, kernel away from the origin; h(x) is a real-valued $C^∞$ function on $ℝ^n ∖ \{0\}$. We give a criterion for such an operator to be bounded from the space $H^\{p\}_\{0\}(ℝ^n)$ into itself.},
author = {Chen, Lung-Kee, Fan, Dashan},
journal = {Studia Mathematica},
keywords = {Hardy-type spaces; oscillating integrals},
language = {eng},
number = {2},
pages = {195-206},
title = {Oscillatory kernels in certain Hardy-type spaces},
url = {http://eudml.org/doc/216128},
volume = {111},
year = {1994},
}

TY - JOUR
AU - Chen, Lung-Kee
AU - Fan, Dashan
TI - Oscillatory kernels in certain Hardy-type spaces
JO - Studia Mathematica
PY - 1994
VL - 111
IS - 2
SP - 195
EP - 206
AB - We consider a convolution operator Tf = p.v. Ω ⁎ f with $Ω(x) = K(x)e^{ih(x)}$, where K(x) is an (n,β) kernel near the origin and an (α,β), α ≥ n, kernel away from the origin; h(x) is a real-valued $C^∞$ function on $ℝ^n ∖ {0}$. We give a criterion for such an operator to be bounded from the space $H^{p}_{0}(ℝ^n)$ into itself.
LA - eng
KW - Hardy-type spaces; oscillating integrals
UR - http://eudml.org/doc/216128
ER -

References

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  1. [1] S. Chanillo, D. Kurtz and G. Sampson, Weighted weak (1,1) and weighted L p estimates for oscillating kernels, Trans. Amer. Math. Soc. 295 (1986), 127-145. Zbl0594.42007
  2. [2] R. Coifman, A real variable characterization of H p , Studia Math. 51 (1974), 269-274. Zbl0289.46037
  3. [3] D. Fan, An oscillating integral on the Besov space B 0 1 , 0 , J. Math. Anal. Appl., to appear. 
  4. [4] D. Fan and Y. Pan, Boundedness of certain oscillatory singular integrals, Studia Math., to appear. Zbl0886.42008
  5. [5] C. Fefferman and E. M. Stein, H p spaces of several variables, Acta Math. 129 (1972), 137-193. Zbl0257.46078
  6. [6] M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley theory and the study of function spaces, CBMS Regional Conf. Ser. in Math. 79, Amer. Math. Soc., 1992. Zbl0757.42006
  7. [7] Y. Han, Certain Hardy-type spaces, Ph.D. thesis, Washington University, St Louis, 1984. 
  8. [8] Y. Han, A class of Hardy-type spaces, Chinese Quart. J. Math. 1 (2) (1986), 42-64. 
  9. [9] W. B. Jurkat and G. Sampson, The complete solution to the ( L p , L q ) mapping problem for a class of oscillating kernels, Indiana Univ. Math. J. 30 (1981), 403-413. Zbl0507.47013
  10. [10] R. H. Latter, A characterization of H p ( ℝ n ) in terms of atoms, Studia Math. 62 (1978), 93-101. Zbl0398.42017
  11. [11] P. Sjölin, Convolution with oscillating kernels on H p spaces, J. London Math. Soc. 23 (1981), 442-454. Zbl0426.46034

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