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1993 | 106 | 1 | 1-44
Tytuł artykułu

Nonconvolution transforms with oscillating kernels that map $Ḃ_{1}^{0,1}$ into itself

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EN
We consider operators of the form $(Ωf)(y) = ʃ_{-∞}^∞ Ω(y,u)f(u)du$ with Ω(y,u) = K(y,u)h(y-u), where K is a Calderón-Zygmund kernel and $h ∈ L^∞$ (see (0.1) and (0.2)). We give necessary and sufficient conditions for such operators to map the Besov space $Ḃ^{0,1}_1$ (= B) into itself. In particular, all operators with $h(y) = e^{i|y|^a}$, a > 0, a ≠ 1, map B into itself.
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  • Mathematics Department, Auburn University, Auburn, Alabama 36849-5201, U.S.A.
Bibliografia
  • [1] S. Chanillo and M. Christ, Weak (1,1) bounds for oscillating singular integrals, Duke Math. J. 55 (1987), 141-155.
  • [2] S. Chanillo and M. Christ, Weak $L^1$ and Mischa Cotlar, preprint, 1985.
  • [3] S. Chanillo, D. S. Kurtz and G. Sampson, Weighted weak (1,1) estimates and weighted $L^p$ estimates for oscillating kernels, Trans. Amer. Math. Soc. 295 (1986), 127-145.
  • [4] R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645.
  • [5] G. David and J.-L. Journé, A boundedness criterion for generalized Calderón-Zygmund operators, Ann. of Math. 120 (1984), 371-397.
  • [6] G. S. De Souza, Spaces formed by special atoms, Ph.D. dissertation, S.U.N.Y. at Albany, 1980.
  • [7] G. S. De Souza and A. Gulisashvili, Special atom decompositions of Besov spaces on fractal sets, preprint, 1990.
  • [8] G. S. De Souza and G. Sampson, A real characterization of the pre-dual of Bloch functions, J. London Math. Soc. (2) 27 (1983), 267-276.
  • [9] 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., 1991.
  • [10] Y. -S. Han and S. Hofmann, T1 theorems for Besov and Triebel-Lizorkin spaces, Trans. Amer. Math. Soc., to appear.
  • [11] Y.-S. Han, B. Jawerth, M. Taibleson and G. Weiss, Littlewood-Paley theory and ε-families of operators, Colloq. Math. 60/61 (1) (1990), 321-359.
  • [12] 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.
  • [13] M. Meyer, Continuité Besov de certains opérateurs intégraux singuliers, Ark. Mat. 27 (1989), 305-318.
  • [14] Y. Meyer, La minimalité de l'espace de Besov $Ḃ_1^{0,1}$ et la continuité des opérateurs définis par des intégrales singulières, Monografías de Matemáticas 4, Univ. Autónoma de Madrid, 1986.
  • [15] M. Reyes, An analytic study of the functions defined on the self-similar fractals, preprint, 1990.
  • [16] F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals, 1. Oscillatory integrals, J. Funct. Anal. 73 (1987), 179-194.
  • [17] G. Sampson, Oscillating kernels that map $H^1$ into $L^1$, Ark. Mat. 18 (1980), 125-144.
  • [18] G. Sampson, On classes of real analytic families of singular integrals on $H^1$ and $L^p$, J. London Math. Soc. (2) 23 (1981), 433-441.
  • [19] G. Sampson, $H^p$ and $L^2$ weighted estimates for convolutions with singular and oscillating kernels, ibid. 43 (1991), 465-484.
  • [20] G. Sampson, Operators mapping $Ḃ_1^0,1$ into $Ḃ_1^0,1$ and operators mapping $L^2$ into $L^2$, preprint, 1991.
  • [21] P. Sjölin, Convolution with oscillating kernels, Indiana Univ. Math. J. 30 (1981), 47-56.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv106i1p1bwm
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