Singular integrals with highly oscillating kernels on product spaces

Elena Prestini

Colloquium Mathematicae (2000)

  • Volume: 86, Issue: 1, page 9-13
  • ISSN: 0010-1354

Abstract

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We prove the L 2 ( 2 ) boundedness of the oscillatory singular integrals P 0 f ( x , y ) = ∫ D x e i ( M 2 ( x ) y ' + M 1 ( x ) x ' ) ο v e r x ' y ' f ( x - x ' , y - y ' ) d x ' d y ' for arbitrary real-valued L ∞ functions M 1 ( x ) , M 2 ( x ) and for rather general domains D x ⊆ 2 whose dependence upon x satisfies no regularity assumptions.

How to cite

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Prestini, Elena. "Singular integrals with highly oscillating kernels on product spaces." Colloquium Mathematicae 86.1 (2000): 9-13. <http://eudml.org/doc/210844>.

@article{Prestini2000,
abstract = {We prove the $L^\{2\}(^\{2\})$ boundedness of the oscillatory singular integrals $P_\{0\} f(x,y)=\int _\{D_\{x\}\} \{\{e^\{i(M_2(x)y^\{\prime \} + M_1(x)x^\{\prime \})\}\}οver\{x^\{\prime \}y^\{\prime \}\}\} f(x-x^\{\prime \},y-y^\{\prime \})dx^\{\prime \}dy^\{\prime \}$ for arbitrary real-valued $L^\{∞\}$ functions $M_\{1\}(x), M_\{2\}(x)$ and for rather general domains $D_\{x\} ⊆ ^\{2\}$ whose dependence upon x satisfies no regularity assumptions.},
author = {Prestini, Elena},
journal = {Colloquium Mathematicae},
keywords = {Fourier series; oscillatory singular integrals; Hardy-Littlewood maximal function; Carleson maximal operator; boundedness},
language = {eng},
number = {1},
pages = {9-13},
title = {Singular integrals with highly oscillating kernels on product spaces},
url = {http://eudml.org/doc/210844},
volume = {86},
year = {2000},
}

TY - JOUR
AU - Prestini, Elena
TI - Singular integrals with highly oscillating kernels on product spaces
JO - Colloquium Mathematicae
PY - 2000
VL - 86
IS - 1
SP - 9
EP - 13
AB - We prove the $L^{2}(^{2})$ boundedness of the oscillatory singular integrals $P_{0} f(x,y)=\int _{D_{x}} {{e^{i(M_2(x)y^{\prime } + M_1(x)x^{\prime })}}οver{x^{\prime }y^{\prime }}} f(x-x^{\prime },y-y^{\prime })dx^{\prime }dy^{\prime }$ for arbitrary real-valued $L^{∞}$ functions $M_{1}(x), M_{2}(x)$ and for rather general domains $D_{x} ⊆ ^{2}$ whose dependence upon x satisfies no regularity assumptions.
LA - eng
KW - Fourier series; oscillatory singular integrals; Hardy-Littlewood maximal function; Carleson maximal operator; boundedness
UR - http://eudml.org/doc/210844
ER -

References

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  1. [1] L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1968), 135-157. Zbl0144.06402
  2. [2] L. K. Chen, Singular integrals with highly oscillating kernels on product domains, Colloq. Math. 64 (1993), 293-302. Zbl0816.42008
  3. [3] C. Fefferman, Pointwise convergence of Fourier series, Ann. of Math. 98 (1973), 551-572. 
  4. [4] R. A. Hunt, On the convergence of Fourier series, in: Orthogonal Expansions and their Continuous Analogues (Edwardsville, IL, 1967), Southern Illinois Univ. Press, Carbondale, IL, 1968, 235-255. 
  5. [5] R. A. Hunt and W. S. Young, A weighted norm inequality for Fourier series, Bull. Amer. Math. Soc. 80 (1974), 274-277. Zbl0283.42004
  6. [6] E. Prestini, Variants of the maximal double Hilbert transform, Trans. Amer. Math. Soc. 290 (1985), 761-771. Zbl0524.44002
  7. [7] E. Prestini, Singular integrals on product spaces with variable coefficients, Ark. Mat. 25 (1987), 276-287. Zbl0639.42014
  8. [8] E. Prestini, L 2 boundedness of highly oscillatory integrals on product domains, Proc. Amer. Math. Soc. 104 (1988), 493-497. 
  9. [9] E. Prestini, A contribution to the study of the partial sums operator S N N 2 for double Fourier series, Ann. Mat. Pura Appl. 134 (1983), 287-300. Zbl0535.42012
  10. [10] E. Prestini, Uniform estimates for families of singular integrals and double Fourier series, Austral. J. Math. 41 (1986), 1-12. Zbl0623.42008
  11. [11] E. Prestini, Singular integrals on product spaces related to Carleson operator, preprint. Zbl1103.42010
  12. [12] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1970. Zbl0207.13501

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