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In this paper we study singular oscillatory integrals with a nonlinear phase function. We prove estimates of L2→L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2 \rightarrow L^2$$\end{document} and Lp→Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p\rightarrow L^p$$\end{document} type.


Introduction
Let K denote a singular kernel in R n . Singular integral operators T , defined by T f (x) = R n K (x − y) f (y)dy, x ∈ R n , f ∈ C ∞ 0 (R n ), have been studied for a very long time. Since approximately 1970 there has also been a lot of interest in oscillatory integral operators. The following theorem describes a typical result. Theorem 1.1 (see Stein [6], p. 377) Let ψ 1 ∈ C ∞ 0 (R n × R n ) and λ > 0 and let be real-valued and smooth. Set Communicated by Krzysztof Stempak.
The above choice of phase function is partially motivated by an application to an inhomogeneous Helmholtz equation where we give estimates for solutions. In this case we take γ = 1 (see [1], p. 544). It is also possible to use T λ to give L p -estimates for convolution operators. This will be studied in a forthcoming paper.
We shall point out a relation between the operators T λ and S λ . We choose γ > 1 and take K (z) = |z| −(n−m−1) , z ∈ R n \ {0}, and let T λ be defined as above. Then that is we obtain an operator of type S λ . The reason for introducing the homogeneous function ω in the above definition of T λ for 0 < γ ≤ 1 is that we want certain determinant conditions to be satisfied. This is discussed in [1, p. 539], and in this paper after the proof of Lemma 2.2.
We shall also make some remarks on an operator which is somewhat similar to S λ . Set where a > 0, a = 1, and α < n. Then L belongs to the space S (R n ) of tempered distributions and we set We say that the operator T is bounded on In Sjölin [5] the following theorem is proved.
We finally remark that Theorem 1.1 is due to Hörmander. In Sect. 2 we shall give the proofs of Theorems 1.3 and 1.4. In Sect. 3 we shall discuss the sharpness of the results in these theorems.

Proofs of Theorems 1.3 and 1.4
We shall apply the following theorem.
be real-valued, and assume that the determinant for f ∈ L 1 (R) and (x, y) ∈ R 2 . It follows that if q > 4 and 3/q + 1/r = 1.
We shall need an estimate of the norm of U N as an operator from L p (R) to L p (R 2 ). We denote this norm by ||U N || p . An application of Theorem 2.1 will give the inequalities in the following lemma.

Lemma 2.2 Let U N be defined as in Theorem 2.1. Then one has
Here ε is an arbitrary positive number and C depends on ϕ and p, and in the case 2 < p ≤ 4, also on ε.
Proof Assume that suppψ 1 and μ(t) = 1 for t ∈ B 1 . Now take q > 4 and assume that 3/q + 1/r = 1. It follows that 1 < r < 4 and using Hölder's inequality twice and Theorem 2.1 we obtain Hence for every ε > 0, where the constant depends on ε. Then we shall obtain an L 2 -estimate for the operator U N . From the condition on J in Theorem 2.1 it follows that there exists a number δ 0 > 0 such that and each μ j has support in a small cube. Here Q is a cube in R 3 with center at the origin and suppψ 1 ⊂ Q. It follows that Invoking Theorem 1.1 we get for every y. Integrating in y and summing over j we then obtain Interpolating between the inequalities (2.1) and (2.2) one has for every ε > 0. We then assume q > 4. Choosing μ as above we have U N ( f ) = U N (μf ) and it follows that where we have used Hölder's inequality. It remains to study the case 1 < p < 2.
Proof of Theorem 1.3. We shall estimate the norm of T λ where Stein [6,p. 393]). Since suppψ 0 is bounded it follows that there exists an integer k 0 such that We shall assume that k 0 = 0. The proof in the general case is the same as for We and invoking Hölder's inequality we obtain since the number of terms in the above sum is bounded. Setting y = 2 −k z we get We also have Now let χ ∈ C ∞ 0 (R) be so that χ = 1 on suppχ and supp χ ⊂ [−1, 1]. We then have and g(y ) = f (2 −k j + 2 −k y )χ (y ).
It is clear that ψ 1 has a support which is uniformly bounded in j and k, and the derivatives of ψ 1 can be bounded uniformly in j and k. Here we use the fact that k ≥ 0.
Invoking (2.6) we conclude that We set d = (|ξ − y | 2 + ξ 2 2 ) 1/2 . It follows from the definitions of ψ and ω that 1/2 ≤ d ≤ 2 and |ξ 2 | ≥ c > 0 on suppψ 1 . Hence the determinant J for the phase function satisfies |J | ≥ c > 0 on suppψ 1 , as we remarked after the proof of Lemma 2.2. We can therefore apply Lemma 2.2 and one obtains We have g = g j,k and and we obtain the inequality Making a trivial estimate we also have It is clear that B ≤ Cλ −(1/ p+m)/γ and in the case 1/ p + m < γβ(p) we get We remark that in the case p = 2 only the case 1/ p + m > γβ(p) can occur. The proof of Theorem 1.3 is complete.
It is clear that and for γ > 0, γ = 1, it follows that Proof of Theorem 1.4. We shall use the method in the proof of Theorem 1.3 and omit some details. We assume that where suppψ ⊂ {x ∈ R n−1 , 1/2 ≤ |x| ≤ 2}. One obtains We also have and χ ∈ C ∞ 0 (R n−1 ) is like χ in the proof of Theorem 1.3. The Schwarz inequality gives the estimate and arguing as in the proof of Theorem 1.3 we get 1 (y,ξ ) ψ 1 (y, ξ)g(y)dy

It follows that
Invoking the determinant condition (2.8) and Theorem 1.1 we conclude that where α = (n − 1)/2. Arguing as in the proof of Theorem 1.3 we then obtain and Theorem 1.4 follows easily from this inequality.
Setting f = χ E and taking x ∈ F we obtain Setting ρ = 1/γ we have Now taking c 0 small we obtain On the other hand The same proof works also in the case K (z) = |z| m−n+1 ω(z). In Theorems 1.2 and 1.3 we proved estimates of the type ||T λ || p ≤ Cλ −(1/ p+m)/γ and the inequality (3.1) shows that these estimates are sharp. In Theorem 1.4 we proved the estimate We shall now prove that also this estimate is sharp. We shall use the same method as in the above counter-example.
We take x 0 and y 0 in R n−1 with |x 0 − y 0 | = 100c 0 λ −ρ and set E = B(y 0 ; c 0 λ −ρ ) and F = B(x 0 ; c 0 λ −ρ ). Here E and F are balls in R n−1 . Setting f = χ E and arguing as above one obtains It follows that We conclude that ||S λ || 2 ≥ cλ −m/γ and it follows that (3.2) is sharp.
In Theorems 1.2 and 1.3 we have where x = (x , x n ) and ϕ(x, y ) = (|x − y | 2 + x 2 n ) γ /2 . We let a denote the point (0, 1) = (0, 0, . . . , 0, 1) in R n . We assume that ψ 0 (x, y ) = 1 in a neighbourhood of (a, 0) and let f = χ B where B = B(0; c 0 λ −1 ) is a ball in R n−1 . For x in a neighbourhood of a one obtains It follows from the mean value theorem that and choosing c 0 small we obtain where c 1 is small. It follows that there is no cancellation in the above integral and we get in a neighbourhood of a. Hence We have || f || 2 = c 4 λ −(n−1)/2 and we obtain and thus the estimates ||T λ || 2 ≤ Cλ −(n−1)/2 in Theorems 1.2 and 1.3 are sharp. We shall then construct a similar counter-example for the operator S λ in Theorem 1.4. Here we have where ϕ(x, y) = |x − y| γ . Take a = (0, 0, . . . , 0, 1) and assume that ψ 0 (x, y) = 1 in a neighbourhood of (a, 0). Also let f = χ B where B is as in the previous counterexample. The same argument as above then gives the estimate ||S λ || 2 ≥ cλ −(n−1)/2 and it follows that the estimate ||S λ || 2 ≤ Cλ −(n−1)/2 in Theorem 1.4 is sharp.
We shall then again consider the operator T λ in Theorem 1.3. Here we have n = 2 and the above counter-example also gives It follows that the estimate where ϕ(x, y, t) = (x − t) 2 + y 2 γ /2 and K (z) = |z| m−1 ω(z). Setting we get where (, ) 2 and (, ) 1 denote the inner products in L 2 (R 2 ) and L 2 (R). It follows that where 1/ p + 1/r = 1. We shall use this inequality for 4 ≤ p < ∞.
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