Weighted Estimates for Oscillatory Singular Integrals

We establish uniformbounds for oscillatory singular integrals as well as oscillatory singular integral operators.We allow the singular kernel to be given by a function in the Hardy space H(S), while such results were known previously only for kernels in L log L(S), a proper subspace of H(S). One of our results established a Lp(w) → Lp(w) bound for certain weights. At the same time, it provides a solution to an open problem in Lu (2005).


Introduction
In this paper we establish uniform bounds for oscillatory singular integrals.We consider two types of oscillatory singular integrals, which will be described later.
Let  ≥ 2 and S −1 denote the unit sphere in R  equipped with the induced Lebesgue measure .For an integrable function Ω : we define where   = /|| for  ∈ R  \ {0}.For ,  ∈ N, let P (, ) = { : R  → R :  be a polynomial with deg () ≤ }. ( Type I.An oscillatory integral of type I is given by where  is given by (2) and  is a polynomial on R  .For a given Ω : S −1 → C and  ∈ N the main concern is to establish a bound for sup ∈P(,) (Ω, )     .
Type II.A type II oscillatory singular integral is actually an integral operator of the form where  is given by (2) and  is a real-valued polynomial on R  × R  .Ricci and Stein [3] showed that, if Ω ∈  1 (S −1 ),  Ω, is bounded on   (R  ).Subsequently Lu and Zhang [4] and Jiang and Lu [5] established the same bounds for ‖ Ω, ‖ , under the weaker conditions Ω ∈  1+ (S −1 ) and Ω ∈  log (S −1 ), respectively.We will now state our main results, beginning with oscillatory singular integrals of Type II.
A set  in R  is called a rectangle if there is an orthonormal basis { 1 , . . .,   } of R  (which may depend on ) such that In other words, what we call a rectangle in R  is simply any rotation of an arbitrary , and let  be a nonnegative, locally integrable function on R  .We say that  is in the weight class It is easy to see that   is a subcollection of the well-known weight class   of Muckenhoupt [6,7].Examples of weights in   include all weights of the form |()|  , where () is a polynomial in R  and −1 <  deg() <  − 1.
Theorem 2. Let (, ) be a real-valued polynomial on R  × R  .Suppose that  ∈   , Ω ∈  1 (S −1 ) and Ω satisfies (1).Then the operator  Ω, is bounded on   (R  , ) for 1 <  < ∞, with a bound on its norm which may depend on the degree of  but is otherwise independent of the coefficients of .
The space  1 (S −1 ) is the Hardy space on the unit sphere.Since  log (S −1 ) is a proper subspace of  1 (S −1 ), Theorem 2 represents an improvement over results mentioned earlier.By taking  = 1, it answers an open question in [8, page 52] in the affirmative.
Our second result has the same flavor as the first, but it concerns Type I oscillatory singular integrals instead.Theorem 3. Suppose that Ω ∈  1 (S −1 ) and Ω satisfies (1).
where   is a constant independent of  and Ω.
Our result in this regard is built on the work of Papadimitrakis and Parissis who gave the following bound in [2]: They also showed the logarithmic growth of the bound in  to be best possible.Our bound, while dependent on the dimension , provides an improvement over the factor (‖Ω‖  log  + 1).

Proofs of Theorems 2 and 3
We will begin by recalling the atomic decomposition for  1 (S −1 ).