Abstract
In this paper we study singular oscillatory integrals with a nonlinear phase function. We prove estimates of \(L^2 \rightarrow L^2\) and \(L^p\rightarrow L^p\) type.
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1 Introduction
Let K denote a singular kernel in \({\mathbb R}^n\). Singular integral operators T, defined by \( T f(x) = \int \limits _{{\mathbb R}^n} K(x-y) f(y) dy\), \(x\in {\mathbb R}^n\), \(f\in C_0^\infty ({\mathbb R}^n)\), 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 \(\psi _1\in C_0^\infty ({\mathbb R}^n \times {\mathbb R}^n)\) and \(\lambda >0\) and let \(\Phi \) be real-valued and smooth. Set
and assume that \(\mathrm {det}\left( \frac{\partial ^2 \Phi }{\partial x_i \partial \xi _j} \right) \ne 0\) on \(\mathrm {supp} \psi _1\). Then one has
We shall here consider singular oscillatory integral operators, that is operators defined by integrals containing both a singular kernel and an oscillating factor. Operators of this type have been much studied in the theory of convergence of Fourier series and also in for instance Phong and Stein [4]. We shall continue this study.
Let \(\psi _0 \in C_0^\infty ({\mathbb R}^n \times {\mathbb R}^{n-1})\) and \(n\ge 2\). For \(f\in L^2({\mathbb R}^{n-1})\) set
for \(x\in {\mathbb R}^n\), \(\gamma >0\), and \(\lambda \ge 2\). Here for \(\gamma >1\) we set
and for \(0<\gamma \le 1\) we set
where \(\omega \in C^\infty ({\mathbb R}^n \setminus \{0\})\), \(\omega \) is homogeneous of degree 0, and \(\omega (z) = 0\) for all z with \(|z|=1\) and \(|z_n|\le \varepsilon _0\) for some given \(\varepsilon _0 >0\). We also assume that \(0<m<n-1\).
We shall study the norm of \(T_\lambda \) as an operator from \(L^p({\mathbb R}^{n-1})\) to \(L^p({\mathbb R}^n)\) and denote this norm by \(|| T_\lambda ||_p\). In Aleksanyan et al. [1] the following theorem was proved.
Theorem 1.2
Set \(\alpha =(n-1)/2 \) and assume \(\gamma \ge 1\). Then one has
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 \(\gamma =1\) (see [1], p. 544). It is also possible to use \(T_\lambda \) to give \(L^p\)-estimates for convolution operators. This will be studied in a forthcoming paper.
In [1] it is also proved that \(||T_\lambda ||_2 \ge c \lambda ^{-(m+1/2)/\gamma } \) for \(\gamma >1\), where c denotes a positive constant. We shall here prove that this also holds for \(\gamma =1\) and that \(|| T_\lambda ||_2 \ge c \lambda ^{-\alpha }\) for \(\gamma \ge 1\). It follows that the results in Theorem 1.2 are essentially sharp.
In this paper we shall first study the case \(n=2\) and \(1<p<\infty \). We have the following theorem.
Theorem 1.3
Assume \(n=2\) and \(0<\gamma \le 1\). Then \(||T_\lambda ||_2 \le C \lambda ^{-1/2}\), and for \(2<p\le 4\) one has
where \(\varepsilon \) denotes an arbitrary positive number. Also set \(\beta (p) = 1-1/p\) for \(1<p<2\), and \(\beta (p)= 2/p\) for \(4<p<\infty \). For \(1<p<2\) and \(4<p<\infty \) one has
We shall also study the sharpness of the estimates in Theorem 1.3. We shall then estimate the operator \(S_\lambda \) given by
where \(n\ge 2\), \(\psi _0 \in C_0^\infty ({\mathbb R}^{n-1} \times {\mathbb R}^{n-1})\), and \(K(z) = |z|^{-(n-m-1)}\), \(z\in {\mathbb R}^{n-1} \setminus \{0\}\). We let \(||S_\lambda ||_p\) denote the norm of \(S_\lambda \) as an operator from \(L^p({\mathbb R}^{n-1})\) to \(L^p({\mathbb R}^{n-1})\). We shall prove the following theorem.
Theorem 1.4
Assume \(n\ge 2\), \(0<m<n-1\), \(\gamma >0\), and \(\gamma \ne 1\). Then
where \(\alpha = (n-1)/2\). Here the constant C depends on n, m, and \(\gamma \).
We shall point out a relation between the operators \(T_\lambda \) and \(S_\lambda \). We choose \(\gamma >1\) and take \(K(z) = |z|^{-(n-m-1)}\), \(z \in {\mathbb R}^n\setminus \{0\}\), and let \(T_\lambda \) be defined as above. Then setting \(x=(x', x_n)\), where \(x'=(x_1,x_2,\ldots ,x_{n-1})\) we obtain
that is we obtain an operator of type \(S_\lambda \). The reason for introducing the homogeneous function \(\omega \) in the above definition of \(T_\lambda \) for \(0<\gamma \le 1\) 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 \(S_\lambda \). Set
where \(a>0\), \(a\ne 1\), and \(\alpha < n\). Then L belongs to the space \(\mathcal {S}' ({\mathbb R}^n)\) of tempered distributions and we set
We say that the operator T is bounded on \(L^p({\mathbb R}^n)\) if
In Sjölin [5] the following theorem is proved.
Theorem 1.5
If \(\alpha \ge n (1- a/2)\) set \(p_0 = n a /(na- n +\alpha ) \). Then T is bounded on \(L^p({\mathbb R}^n)\) if and only if \(p_0 \le p \le p_0'\). If \(\alpha < n(1-a/2)\) then T is not bounded on any \(L^p({\mathbb R}^n)\), \(1\le p \le \infty \).
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.
2 Proofs of Theorems 1.3 and 1.4
We shall apply the following theorem.
Theorem 2.1
(see Hörmander [3], p. 3) Let \(\psi _1\in C_0^\infty ({\mathbb R}^3)\), let \(\varphi \in C^\infty ({\mathbb R}^3)\) be real-valued, and assume that the determinant
on \(\mathrm {supp} \psi _1\). Here \(\varphi = \varphi (x,y,t)\) and \(\varphi _{xt} = \frac{\partial ^2 \varphi }{\partial x \partial t}\) etc. Set
for \(f\in L^1({\mathbb R})\) and \((x,y)\in {\mathbb R}^2\). It follows that
if \(q>4\) and \(3/q + 1/r=1\).
We shall need an estimate of the norm of \(\mathcal {U}_N\) as an operator from \(L^p({\mathbb R})\) to \(L^p({\mathbb R}^2)\). We denote this norm by \(||\mathcal {U}_N||_p\). An application of Theorem 2.1 will give the inequalities in the following lemma.
Lemma 2.2
Let \(\mathcal {U}_N\) be defined as in Theorem 2.1. Then one has
where
Here \(\varepsilon \) is an arbitrary positive number and C depends on \(\varphi \) and p, and in the case \(2<p\le 4\), also on \(\varepsilon \).
Proof
Assume that \(\mathrm {supp} \psi _1 \subset B_2 \times B_1\), where \(B_1\) is a ball in \({\mathbb R}\) and \(B_2\) a ball in \({\mathbb R}^2\). We then have \(\mathcal {U}_N f = \mathcal {U}_N (\mu f)\) if \(\mu \in C_0^\infty ({\mathbb R})\) and \(\mu (t) =1\) for \(t\in B_1\). 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 \(\varepsilon >0\), where the constant depends on \(\varepsilon \). Then we shall obtain an \(L^2\)-estimate for the operator \(\mathcal {U}_N\). From the condition on \(\mathcal {J}\) in Theorem 2.1 it follows that there exists a number \(\delta _0 >0\) such that
on \(\mathrm {supp} \psi _1\), where \(C_0\) depends on \(\varphi \).
Choose \(\mu _j \in C_0^\infty ({\mathbb R}^3)\), \(j=2,3,\ldots ,M\), such that \(\sum \limits _{2}^M \mu _j(x,y,t) = 1\) for \((x,y,t)\in Q\) and each \(\mu _j\) has support in a small cube. Here Q is a cube in \({\mathbb R}^3\) with center at the origin and \(\mathrm {supp}\psi _1 \subset Q \). It follows that
where \(\psi _j = \psi _1 \mu _j\). Setting
we have
and shall estimate each \(\mathcal {U}_N^{(j)}\).
If \((x_0, y_0, t_0) \in \mathrm {supp} \psi _j\) then \((x_0, y_0, t_0) \in \mathrm {supp} \psi _1\) and \(|\varphi _{xt} | \ge \delta /2\) or \(|\varphi _{yt}| \ge \delta /2\) at \((x_0, y_0, t_0)\), where \(\delta = \delta _0 / C_0\). Say that \(|\varphi _{xt}| \ge \delta /2\). Then \(|\varphi _{xt}| \ge \delta /4\) on \(\mathrm {supp} \psi _j\) since \(\mathrm {supp} \psi _j\) is contained in a small cube.
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 \(\varepsilon >0\).
We then assume \(q>4\). Choosing \(\mu \) as above we have \( \mathcal {U}_N (f) = \mathcal {U}_N (\mu f) \) and it follows that
where we have used Hölder’s inequality. It remains to study the case \(1<p<2\). Interpolating between (2.2) and the trivial estimate \( ||\mathcal {U}_N f ||_1 \le C || f||_1 \) one obtains
and Lemma 2.2 follows from (2.2), (2.3), (2.4), and (2.5). \(\square \)
Now let \(\varphi (x,y,t) = d^\gamma \), where \(d=((x-t)^2 + y^2)^{1/2}\) and \(0<\gamma \le 1\). A computation shows that
for \(d=1\). Since \(\mathcal {J}\) is a homogeneous function of degree \(2\gamma -5\) of \((x_0, y)\) where \(x_0 = x-t\), we conclude that if \(1/2 \le d \le 2\) and \(|y|\ge c>0\) on \(\mathrm {supp} \psi _1\), then \(|\mathcal {J}| \ge c_1 >0\) on \(\mathrm {supp} \psi _1\). Hence (2.2)–(2.5) hold in this case.
We remark that in the case \(\gamma =1\) \(\mathcal {J}\) was computed in Carleson and Sjölin [2], and that in the case \(\gamma =1\) (2.2) and (2.3) are proved in [2] in the case \(\psi _1(x,y,t) = \chi _1(t) \chi _2(x,y)\), where \(\chi _1\) is the characteristic function for the interval [0, 1] and \(\chi _2\) is the characteristic function for the square \([0,1]\times [2,3]\). We shall now prove Theorem 1.3.
Proof of Theorem 1.3
We shall estimate the norm of \(T_\lambda \) where
where \(x\in {\mathbb R}^2\). Here \(\lambda \ge 2\), \(0<\gamma \le 1\), and \(\psi _0 \in C_0^\infty ({\mathbb R}^2 \times {\mathbb R})\). Also \(K(z) = |z|^{m-1} \omega (z) \), \(z\in {\mathbb R}^2\setminus \{0\}\), where \(0<m<1\) and \(\omega \) is described in the introduction.
We first observe that there exists \(\psi \in C_0^\infty ({\mathbb R}^2)\), with support in \(\{ x\in {\mathbb R}^2: \ 1/2 \le |x| \le 2 \}\) such that \(K(z) = \sum \limits _{k=-\infty }^\infty 2^{k(1-m) } \psi (2^k z) \omega (z) \) (see Stein [6, p. 393]). Since \(\mathrm {supp} \psi _0\) is bounded it follows that there exists an integer \(k_0\) such that \(K(z) = \sum \limits _{k=k_0 }^\infty 2^{k(1-m) } \psi (2^k z) \omega (z) \) for all \(z=x-(y',0)\) with \((x,y')\in \mathrm {supp} \psi _0\). We shall assume that \(k_0 = 0\). The proof in the general case is the same as for \(k_0 = 0\). Also choose \(\chi \in C_0^\infty ({\mathbb R})\) such that \( \mathrm {supp} \chi \subset [-1/2 -1/10, 1/2+1/10] \) and \(\sum \limits _{j=-\infty }^\infty \chi (t-j) =1 \).
We have \(T_\lambda f = \sum \limits _{k=0}^\infty T_{\lambda ,k} f\) where
Also \(T_{\lambda ,k} f = \sum \limits _{j} T_{\lambda ,k} f_j \) where \(f_j(t) = f(t) \chi \big ( 2^k(t-2^{-k}j) \big )\). Assuming \(1<p<\infty \) 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 \(\widetilde{\chi }\in C_0^\infty ({\mathbb R})\) be so that \(\widetilde{\chi } =1\) on \(\mathrm {supp} \chi \) and \(\mathrm {supp} \widetilde{\chi } \subset [-1,1]\). We then have
where
and
Here \(\xi =(\xi _1, \xi _2)=(\xi ', \xi _2)\).
It is clear that \(\psi _1\) has a support which is uniformly bounded in j and k, and the derivatives of \(\psi _1\) can be bounded uniformly in j and k. Here we use the fact that \(k\ge 0\).
Invoking (2.6) we conclude that
We set \(d=(|\xi ' - y'|^2 + \xi _2^2)^{1/2}\). It follows from the definitions of \(\psi \) and \(\omega \) that \(1/2 \le d \le 2\) and \(|\xi _2|\ge c>0\) on \(\mathrm {supp} \psi _1 \). Hence the determinant \(\mathcal {J}\) for the phase function \(\Phi \) satisfies \(|\mathcal {J}|\ge c>0\) on \(\mathrm {supp} \psi _1\), 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 it follows that
Hence
and we obtain the inequality
Making a trivial estimate we also have
Invoking the inequality \(|| T_\lambda ||_p \le \sum \limits _0^\infty || T_{\lambda , k} ||_p \) we obtain
It is clear that \(B\le C \lambda ^{-(1/p+m)/\gamma }\) and in the case \(1/p + m<\gamma \beta (p)\) we get
and
In the case \(1/p + m = \gamma \beta (p)\) we get \(A\le C \lambda ^{-\beta (p) } \log \lambda \) and \( || T_\lambda ||_p \le C \lambda ^{-\beta (p)} \log \lambda \).
Finally, in the case \(1/p + m >\gamma \beta (p)\) we have \( A \le C \lambda ^{-\beta (p)} \) and \(|| T_\lambda ||_p \le C \lambda ^{-\beta (p)}.\)
We remark that in the case \(p=2\) only the case \(1/p +m >\gamma \beta (p)\) can occur. The proof of Theorem 1.3 is complete. \(\square \)
Before proving Theorem 1.4 we shall make a preliminary observation. Set \(\xi =(\xi ', \xi _n)\) where \(\xi ' = (\xi _1,\xi _2,\ldots ,\xi _{n-1})\) and \(n \ge 2\). Also set \(x'= (x_1,x_2,\ldots ,x_{n-1})\) and \(\Phi (x', \xi ) = d^\gamma \) where \(\gamma >0\) and \(d=( |\xi ' - x'|^2 + \xi _n^2 )^{1/2}\). In [1, Section 4.1], we studied the determinant
for \(1/2 \le d \le 2\). In [1] it is proved that
Now let \(\Phi _1 (x', \xi ') = |\xi ' - x'|^\gamma = d_1^\gamma \) where \(d_1 = |\xi ' - x'|\). We shall need the determinant
It is clear that
and for \(\gamma >0\), \(\gamma \ne 1\), 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 \(\mathrm {supp} \psi \subset \{ x\in {\mathbb R}^{n-1}, \ 1/2\le |x| \le 2 \}\). One obtains
where
We also have
where
and \(\chi \in C_0^\infty ({\mathbb R}^{n-1})\) is like \(\chi \) 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
and
It follows that
where \(\Phi _1(y,\xi ) = |\xi - y|^\gamma \), \(\psi _1(y,\xi ) = \psi ( \xi - y) \psi _0( 2^{-k}\xi + 2^{-k}j, 2^{-k} j+2^{-k }y ) \widetilde{\chi } (y)\), and \(g(y) = f(2^{-k} j +2^{-k} y) \chi (y)\).
Invoking the determinant condition (2.8) and Theorem 1.1 we conclude that
where \(\alpha =(n-1)/2\). Arguing as in the proof of Theorem 1.3 we then obtain
and \(||S_{\lambda , k} ||_2 \le C 2^{- mk}\).
Hence
and Theorem 1.4 follows easily from this inequality. \(\square \)
3 Counter-examples
Assume \(\gamma >0\), \(1<p<\infty \), and
where \(x \in {\mathbb R}^n\), \(n\ge 2\), and \(K(z) = |z|^{m-n+1}\) with \(0<m<n-1\). We shall estimate the norm \(|| T_\lambda ||_p = || T_\lambda ||_{L^p({\mathbb R}^{n-1}) \rightarrow L^p({\mathbb R}^n)} \) from below. We take \(y_0'\in {\mathbb R}^{n-1}\) and set \(E=B(y_0'; c_0 \lambda ^{-\rho })\) where B(x; R) denotes a ball with center x and radius R. Also let F denote a cube in \({\mathbb R}^n\) with center \((y_0', 100 c_0 \lambda ^{-\rho } )\) and side length \(c_0 \lambda ^{-\rho }\). We assume that \(\psi _0(x,y') = 1\) for \(x\in F\) and \(y'\in E\).
Setting \(f=\chi _E\) and taking \(x\in F\) we obtain
Setting \(\rho =1/\gamma \) we have
and
Now taking \(c_0\) small we obtain
and
On the other hand
and we have
The same proof works also in the case \(K(z) = |z|^{m-n+1} \omega (z)\).
In Theorems 1.2 and 1.3 we proved estimates of the type
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 \({\mathbb R}^{n-1}\) with \(|x_0 -y_0|=100 c_0 \lambda ^{-\rho }\) and set \(E= B(y_0; c_0 \lambda ^{-\rho })\) and \(F = B(x_0; c_0 \lambda ^{-\rho })\). Here E and F are balls in \({\mathbb R}^{n-1}\). Setting \(f=\chi _E\) and arguing as above one obtains
It follows that
and
We conclude that
and it follows that (3.2) is sharp.
In Theorems 1.2 and 1.3 we have
where \(x=(x',x_n)\) and \(\varphi (x,y') = ( |x'-y'|^2 + x_n^2 )^{\gamma /2} \).
We let a denote the point \((0,1) = (0,0,\ldots ,0,1)\) in \({\mathbb R}^n\). We assume that \(\psi _0(x,y') =1\) in a neighbourhood of (a, 0) and let \(f=\chi _B\) where \(B=B(0; c_0 \lambda ^{-1})\) is a ball in \({\mathbb R}^{n-1}\). 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 \(||f||_2 = c_4 \lambda ^{-(n-1)/2}\) and we obtain
Hence
and thus the estimates \( ||T_\lambda ||_2 \le C \lambda ^{-(n-1)/2}\) in Theorems 1.2 and 1.3 are sharp.
We shall then construct a similar counter-example for the operator \(S_\lambda \) in Theorem 1.4. Here we have
where \(\varphi (x,y) = |x-y|^\gamma \). Take \(a=(0,0,\ldots ,0,1)\) and assume that \(\psi _0(x,y)=1\) in a neighbourhood of (a, 0). Also let \(f=\chi _B\) where B is as in the previous counter-example. The same argument as above then gives the estimate \(||S_\lambda ||_2 \ge c \lambda ^{-(n-1)/2}\) and it follows that the estimate \(||S_\lambda ||_2 \le C \lambda ^{-(n-1)/2}\) in Theorem 1.4 is sharp.
We shall then again consider the operator \(T_\lambda \) in Theorem 1.3. Here we have \(n=2\) and the above counter-example also gives
for \(1\le p <2\). It follows that the estimate
for \(1<p<2\) in Theorem 1.3 is sharp (since \(\beta (p) = 1-1/p\)).
In Theorem 1.3 we have
where \(\varphi (x,y,t) = \big ( (x-t)^2 + y^2 \big )^{\gamma /2}\) and \(K(z) = |z|^{m-1} \omega (z)\).
Setting
we get
where \((,)_2\) and \((,)_1\) denote the inner products in \(L^2({\mathbb R}^2)\) and \(L^2({\mathbb R})\). It follows that
where \(1/p + 1/r =1\). We shall use this inequality for \(4\le p<\infty \).
Let B denote a disc in \({\mathbb R}^2\) with center (0, 1) and radius \(c_0 \lambda ^{-1}\). Take \(g\in C_0^\infty ({\mathbb R}^2)\) with support in B, \(0\le g \le 1\), and \(g=1\) in \(\frac{1}{2} B\). Then
and choosing \(\psi _0\) such that \(\psi _0 (x,y,t) = 1\) in a neighbourhood of (0, 1, 0) we get
in a neighbourhood of 0. Hence
and
Since \(1-1/r = 1/p\) we conclude that
and it follows that the estimate
in Theorem 1.3 is sharp (since \(\beta (p) = 2/p\)).
In Theorem 1.3 we also have an estimate of the type
for \(2<p<4\). We shall finally discuss the sharpness of this estimate in the case \(\gamma =1\). We shall study the statement
Omitting details we shall describe how (3.4) leads to a contradiction.
Following Stein [6], p. 393, we have
where \(u\in L^1({\mathbb R}^2)\), \(\psi \) is smooth, and \(\mathrm {supp} \psi \subset \{x\in {\mathbb R}^2; \ 1/2 \le |x|\le 2 \}\). We set
and \(S_0 f =K_0 \star f\). We define the operator \(V_k\) by setting
where \(\lambda = 2^k\). Using (3.4) we can prove that
and the inequality
implies that \(S_0\) is a bounded operator on \(L^p({\mathbb R}^2)\). It follows that the characteristic function of the unit disc is a Fourier multiplier for \(L^p({\mathbb R}^2)\). This contradicts Fefferman’s multiplier theorem.
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Communicated by Krzysztof Stempak.
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Sjölin, P. \(L^p\)-Estimates for Singular Oscillatory Integral Operators. J Fourier Anal Appl 23, 1408–1425 (2017). https://doi.org/10.1007/s00041-016-9507-5
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DOI: https://doi.org/10.1007/s00041-016-9507-5