Weighted weak type (1,1) estimates for oscillatory singular integrals

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Title

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Department of Mathematics, Faculty of Education, Kanazawa University, Kanazawa 920-1192, Japan

We consider the

A_{1}

-weights and prove the weighted weak type (1,1) inequalities for certain oscillatory singular integrals.

rough operators, oscillatory singular integrals

S. Chanillo, Weighted norm inequalities for strongly singular convolution operators, Trans. Amer. Math. Soc. 281 (1984), 77-107.
S. Chanillo and M. Christ, Weak (1,1) bounds for oscillatory singular integrals, Duke Math. J. 55 (1987), 141-155.
S. Chanillo, D. S. Kurtz and G. Sampson, Weighted weak (1,1) and weighted $L^{p}$ estimates for oscillating kernels, Trans. Amer. Math. Soc. 295 (1986), 127-145.
M. Christ, Hilbert transforms along curves, I: Nilpotent groups, Ann. of Math. 122 (1985), 575-596.
M. Christ, Weak type (1,1) bounds for rough operators, ibid. 128 (1988), 19-42.
M. Christ, Weak type endpoint bounds for Bochner-Riesz multipliers, Rev. Mat. Iberoamericana 3 (1987), 25-31.
M. Christ and J. L. Rubio de Francia, Weak type (1,1) bounds for rough operators, II, Invent. Math. 93 (1988), 225-237.
J.-L. Journé, Calderón-Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral of Calderón, Lecture Notes in Math. 994, Springer, 1983.
F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals, I, J. Funct. Anal. 73 (1987), 179-194.
S. Sato, Some weighted weak type estimates for rough operators, Math. Nachr. 187 (1997), 211-240.
A. Vargas, Weighted weak type (1,1) bounds for rough operators, J. London Math. Soc. (2) 54 (1996), 297-310