On weighted weak type norm inequalities for one-sided oscillatory singular integrals

Abstract

The purpose of this paper is to establish the weighted norm inequalities of one-sided oscillatory integral operators by the aid of interpolation of operators with change of measures.

Some one-sided estimates for oscillatory singular integrals

“…The proof of (b) is similar to that of Theorem 1.5 in [12]. This argument can now be applied again for the completeness of this paper.…”

“…Combining Lemma 2.4 with Lemma 2.5, we can prove Lemma 2.6 by a similar analysis as in [15], corresponding argument, see also [11] and [33].…”

“…Recently, we proved the weighted L p (1 < p < ∞) estimates for T + in [12] and showed that for any real polynomial P (x, y), T + is of type (L p (w), L p (w)) for w ∈ A + p . It is easy to see that when P (x, y) is trivial, for example, P (x, y) = 0, then T + is T + .…”

“…Inspired by theory of oscillatory singular operators and one-sided operators, the authors of this paper defined the one-sided oscillatory integral operator in [11] (see also [12]), which is just the object of this paper. We first recall its definitions as T + f (x) = lim e iP (x,y) K(x − y)f (y)dy, where P (x, y) is a real polynomial defined on R × R, and K is a one-sided Calderón-Zygmund kernel with support in R − = (−∞, 0) and R + = (0, +∞), respectively.…”

“…In [40], Sawyer studied the weighted theory of onesided maximal Hardy-Littlewood operators in depth for the first time. Since then, numerous papers have appeared, among which we choose to refer to [2], [3], [11], [24], [25] and [37] about one-sided operators, [1], [10], [30], [34], [35], and [38] about one-sided spaces, respectively. Interestingly, lots of results show that for a class of smaller operators (one-sided operators) and a class of wider weights (one-sided weights), many famous results in harmonic analysis still hold.…”