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On weighted weak type norm inequalities for one-sided oscillatory singular integrals. (English) Zbl 1257.42022

The authors consider one-sided weighted classes of Muckenhoupt type and study the weighted weak (1,1) norm inequalities for certain one-sided oscillatory singular integrals with smooth kernel.

One-sided oscillatory singular integral operators T + and T - are defined by

T + f(x)=lim ε0+ x+ε e iP(x,y) K(x-y)f(y)dy,T - f(x)=lim ε0+ - x-ε e iP(x,y) K(x-y)f(y)dy,

where P(x,y) is a real polynomial defined on ×, and K is a one-sided Calderón-Zygmund kernel. The following theorem is the main result of the paper.

Let ω ¯A 1 + , the class of one-sided A 1 weights. Then there exists a constant C depending on the degree of P and A 1 + (ω ¯) such that

sup λ>0 λω ¯({x:|T + f(x)|>λ})Cf L 1 (ω ¯)

for all Schwartz functions f.

MSC:
42B20Singular and oscillatory integrals, several variables
42B25Maximal functions, Littlewood-Paley theory