$L^p$-Estimates for Singular Oscillatory Integral Operators

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.

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

$$\begin{aligned} \mathcal {U}_\lambda f(x) = \int \limits _{{\mathbb R}^n} e^{i\lambda \Phi (x,\xi )} \psi _1 (x,\xi ) f(x) dx, \ \xi \in {\mathbb R}^n, \end{aligned}$$

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

$$\begin{aligned} || \mathcal {U}_\lambda f||_{L^2 ({\mathbb R}^n)} \le C \lambda ^{-n/2} ||f ||_{L^2 ({\mathbb R}^n)}. \end{aligned}$$

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

$$\begin{aligned} T_\lambda f(x) = \int \limits _{{\mathbb R}^{n-1}} e^{i\lambda |x-(y',0)|^\gamma } \psi _0 (x,y') K \big (x-(y',0) \big ) f(y') dy' \end{aligned}$$

for $x\in {\mathbb R}^n$, $\gamma >0$, and $\lambda \ge 2$. Here for $\gamma >1$ we set

$$\begin{aligned} K(z) = |z|^{-(n-m-1)}, \ z\in {\mathbb R}^n \setminus \{0\}, \end{aligned}$$

and for $0<\gamma \le 1$ we set

$$\begin{aligned} K(z) = |z|^{-(n-m-1)} \omega (z), \ z\in {\mathbb R}^n \setminus \{0\}, \end{aligned}$$

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

$$\begin{aligned} || T_\lambda ||_2 \le {\left\{ \begin{array}{ll} C \lambda ^{-(m+1/2)/\gamma }, &{} m<\gamma \alpha - 1/2 , \\ C \lambda ^{-\alpha } \log \lambda , &{} m=\gamma \alpha - 1/2 , \\ C \lambda ^{-\alpha }, &{} m>\gamma \alpha - 1/2 . \end{array}\right. } \end{aligned}$$

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

$$\begin{aligned} || T_\lambda ||_p \le {\left\{ \begin{array}{ll} C \lambda ^{-(1/p+m)/\gamma }, &{} 1/p+m<\gamma /2 , \\ C_\varepsilon \lambda ^{\varepsilon - 1/2} , &{} 1/p+m \ge \gamma /2 , \end{array}\right. } \end{aligned}$$

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

$$\begin{aligned} || T_\lambda ||_p \le {\left\{ \begin{array}{ll} C \lambda ^{-(1/p+m)/\gamma }, &{} 1/p+m<\gamma \beta (p) , \\ C \lambda ^{- \beta (p) } \log \lambda , &{} 1/p+m = \gamma \beta (p), \\ C \lambda ^{-\beta (p)} , &{}1/p+m > \gamma \beta (p) . \end{array}\right. } \end{aligned}$$

We shall also study the sharpness of the estimates in Theorem 1.3. We shall then estimate the operator $S_\lambda$ given by

$$\begin{aligned} S_\lambda f(x) = \int \limits _{{\mathbb R}^{n-1}} e^{i\lambda |x-y|^\gamma } \psi _0(x,y) K(x-y) f(y) dy, \ x\in {\mathbb R}^{n-1}, \end{aligned}$$

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

$$\begin{aligned} || S_\lambda ||_2 \le {\left\{ \begin{array}{ll} C \lambda ^{- m/ \gamma }, &{} m<\gamma \alpha , \\ C \lambda ^{- \alpha } \log \lambda , &{} m = \gamma \alpha , \\ C \lambda ^{-\alpha } , &{} m > \gamma \alpha , \end{array}\right. } \end{aligned}$$

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

$$\begin{aligned} T_\lambda f(x', 0) = \int \limits _{{\mathbb R}^{n-1}} e^{ i\lambda |x'-y'|^\gamma } \psi _0 (x',0,y') K(x' - y', 0) f(y') dy', \end{aligned}$$

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

$$\begin{aligned} L(x) = \frac{e^{i |x|^a}}{|x|^\alpha }, \ \ x\in {\mathbb R}^{n} \setminus \{0\}, \end{aligned}$$

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

$$\begin{aligned} T f = L\star f, \ \ f\in C_0^\infty ({\mathbb R}^n). \end{aligned}$$

We say that the operator T is bounded on $L^p({\mathbb R}^n)$ if

$$\begin{aligned} || T f ||_p \le C_p || f||_p, \ \ f\in C_0^\infty ({\mathbb R}^n). \end{aligned}$$

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

$$\begin{aligned} \mathcal {J} =\left| \begin{array}{cc} \varphi _{xt} &{} \varphi _{yt} \\ \varphi _{xtt} &{} \varphi _{ytt} \end{array} \right| \ne 0 \end{aligned}$$

on $\mathrm {supp} \psi _1$. Here $\varphi = \varphi (x,y,t)$ and $\varphi _{xt} = \frac{\partial ^2 \varphi }{\partial x \partial t}$ etc. Set

$$\begin{aligned} \mathcal {U}_N f (x,y) =\int \limits _{\mathbb R}e^{iN \varphi (x,y,t)} \psi _1(x,y,t) f(t) dt, \ N\ge 1, \end{aligned}$$

for $f\in L^1({\mathbb R})$ and $(x,y)\in {\mathbb R}^2$. It follows that

$$\begin{aligned} || \mathcal {U}_N f||_{L^q({\mathbb R}^2)} \le C N^{-2/q} (q/(q-4))^{1/4} ||f||_{L^r({\mathbb R})} \end{aligned}$$

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

$$\begin{aligned} ||\mathcal {U}_N ||_p \le C N^{-\beta (p)}, \ 1<p<\infty , \end{aligned}$$

where

$$\begin{aligned} \beta (p)= {\left\{ \begin{array}{ll} 1-1/p, &{} 1<p\le 2 , \\ 1/2 - \varepsilon , &{} 2<p\le 4 , \\ 2/p, &{} 4<p<\infty . \end{array}\right. } \end{aligned}$$

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

$$\begin{aligned} || \mathcal {U}_N f ||_4 \le C || \mathcal {U}_N f ||_q = C || \mathcal {U}_N(\mu f) ||_q&\le \\ C N^{-2/q} || \mu f||_r&\le C N^{-2/q} || \mu f||_4 \le C N^{-2/q} ||f ||_4. \end{aligned}$$

Hence

$$\begin{aligned} ||\mathcal {U}_N f ||_4 \le C N^{\varepsilon - 1/2} ||f ||_4 \end{aligned}$$

(2.1)

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

$$\begin{aligned} \delta _0 \le |\mathcal {J} | \le C_0(| \varphi _{xt} | +|\varphi _{yt}| ) \end{aligned}$$

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

$$\begin{aligned} \psi _1 = \sum \limits _{2}^M \psi _1 \mu _j = \sum \limits _{2}^M \psi _j, \end{aligned}$$

where $\psi _j = \psi _1 \mu _j$. Setting

$$\begin{aligned} \mathcal {U}_N^{(j)} f (x,y) = \int \limits _{\mathbb R}e^{i N \varphi (x,y,t)} \psi _j (x,y,t) f(t) dt \end{aligned}$$

we have

$$\begin{aligned} \mathcal {U}_N = \sum \limits _{j=2}^M \mathcal {U}_N^{(j)} \end{aligned}$$

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

$$\begin{aligned} \left( \int |\mathcal {U}_N^{(j)} f(x,y) |^2 dx \right) ^{1/2} \le C N^{-1/2} \left( \int |f(t)|^2 dt \right) ^{1/2} \end{aligned}$$

for every y. Integrating in y and summing over j we then obtain

$$\begin{aligned} || U_N f ||_{L^2({\mathbb R}^2)} \le C N^{-1/2} || f||_{L^2 ({\mathbb R})}. \end{aligned}$$

(2.2)

Interpolating between the inequalities (2.1) and (2.2) one has

$$\begin{aligned} || \mathcal {U}_N f ||_{L^p ({\mathbb R}^2)} \le C N^{\varepsilon - 1/2} ||f||_{L^p({\mathbb R})}, \ 2<p\le 4 \end{aligned}$$

(2.3)

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

$$\begin{aligned} || \mathcal {U}_n f ||_q \le C N^{-2/q} || \mu f||_r \le C N^{-2/q} ||\mu f ||_q \le C N^{-2/q} ||f||_q, \end{aligned}$$

(2.4)

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

$$\begin{aligned} || \mathcal {U}_n f ||_p \le C N^{-(1-1/p)} ||f||_p, \ 1<p<2, \end{aligned}$$

(2.5)

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

$$\begin{aligned} \mathcal {J}= \gamma ^2 (\gamma -2) y \big ( (\gamma -1) (x-t)^2 - y^2 \big ) \end{aligned}$$

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

$$\begin{aligned} T_\lambda f(x) = \int \limits _{\mathbb R}e^{i \lambda | x-(y',0) |^\gamma } \psi _0 (x,y') K \big (x-(y',0)\big ) f(y') dy', \end{aligned}$$

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

$$\begin{aligned} T_{\lambda ,k} f(x) = \int \limits _{\mathbb R}e^{i \lambda | x-(y',0) |^\gamma } \psi _0 (x,y') 2^{k(1-m)} \psi \big (2^k(x-(y',0))\big ) \omega (x-(y',0)) f(y') dy', \end{aligned}$$

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

$$\begin{aligned} |T_{\lambda ,k} f(x) |^p \le C \sum \limits _j |T_{\lambda ,k} f_j (x) |^p, \end{aligned}$$

since the number of terms in the above sum is bounded.

Setting $y' = 2^{-k} z'$ we get

$$\begin{aligned}&T_{\lambda ,k } f_j(x)\\&\quad = \int \limits _{\mathbb R}e^{i\lambda |x-(y',0)|^\gamma } 2^{k(1-m)} \psi _0 (x,y') \psi \big ( 2^k(x-(y',0)) \big ) \omega \big (x-(y',0)\big ) f_j(y') dy'\\&\quad =2^{-mk} \int \limits _{\mathbb R}e^{i\lambda |x-2^{-k}(z',0)|^\gamma } \psi _0 (x,2^{-k}z') \psi \big (2^k x-(z',0)\big ) \omega \big (x-2^{-k}(z',0) \big ) f_j(2^{-k}z') dz' \\&\quad =2^{-mk} \int \limits _{\mathbb R}e^{i\lambda 2^{-k \gamma } |2^k x-(z',0)|^\gamma } \psi _0 (x,2^{-k}z') \psi \big (2^kx-(z',0) \big ) \omega \big (2^k x- (z',0)\big ) f(2^{-k}z') \chi (z' - j) dz' \\&\quad = [\text {with } y'=z' - j ] 2^{-mk} \int \limits _{\mathbb R}e^{i\lambda 2^{-k \gamma }| 2^k x - (y'+j,0) |^\gamma } \psi _0 (x, 2^{-k} (y'+j) ) \psi ( 2^k x - (y'+j,0) ) \\&\qquad \times \omega \big (2^k x - (y'+j,0) \big ) f( 2^{-k} (y' +j) ) \chi (y' ) dy' = 2^{-mk} \int \limits _{{\mathbb R}} e^{i\lambda 2^{-k \gamma }| 2^k ( x - (2^{-k}j,0) ) - (y',0) |^\gamma } \\&\qquad \times \psi _0( x, 2^{-k} j + 2^{-k} y' ) \psi \big (2^{k} (x-(2^{-k}j,0)) - (y',0) \big ) \omega \big ( 2^k (x- (2^{-k}j,0)) - (y',0) \big ) \\&\qquad \times f(2^{-k} j + 2^{-k} y') \chi (y') dy'. \end{aligned}$$

We also have

$$\begin{aligned}&\int \limits _{{\mathbb R}^2} | T_{\lambda ,k} f_j(x) |^p dx = [\text {with } x=u+(2^{-k}j,0) ] \nonumber \\&\int \limits _{{\mathbb R}^2} \left| T_{\lambda ,k} f_j \big (u + (2^{-k}j, 0) \big ) \right| ^p du = [\text {with } \xi = 2^k u] \nonumber \\&2^{-2k} \int \limits _{{\mathbb R}^2} \left| T_{\lambda ,k} f_j \big (2^{-k} \xi +(2^{-k}j, 0) \big ) \right| ^p d\xi . \end{aligned}$$

(2.6)

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

$$\begin{aligned} T_{\lambda ,k} f_j \big (2^{-k} \xi +(2^{-k}j, 0) \big )= & {} 2^{-mk} \int \limits _{{\mathbb R}} e^{i\lambda 2^{-k \gamma } |\xi - (y',0)|^\gamma }\psi _0 ( 2^{-k} \xi \\&+(2^{-k}j,0), 2^{-k} j+ 2^{-k}y') \psi \big (\xi - (y',0)\big ) \\&\times \omega \big (\xi - (y',0) \big ) f(2^{-k}j + 2^{-k} y') \chi (y') \widetilde{\chi }(y') dy' \\= & {} 2^{-mk } \int \limits _{{\mathbb R}} e^{i\lambda 2^{-k \gamma } \Phi (y',\xi ) } \psi _1(y', \xi ) g(y') dy'\\= & {} 2^{- mk} \mathcal {U}_{\lambda 2^{-k\gamma }} g(\xi ), \end{aligned}$$

where

$$\begin{aligned} \Phi (y',\xi ) = |\xi - (y',0)|^\gamma = (|\xi ' - y'|^2 + \xi _2^2)^{\gamma /2}, \end{aligned}$$

$$\begin{aligned} \psi _1(y',\xi ) = \psi \big ( \xi - (y',0) \big ) \omega \big (\xi - (y',0)\big ) \psi _0 (2^{-k}\xi + (2^{-k}j,0), 2^{-k}j+2^{-k}y' ) \widetilde{\chi }(y'), \end{aligned}$$

and

$$\begin{aligned} g(y') = f( 2^{-k} j +2^{-k} y' ) \chi (y'). \end{aligned}$$

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

$$\begin{aligned} \left( \int \limits _{{\mathbb R}^2} |T_{\lambda , k} f_j (x) |^p dx \right) ^{1/p} = 2^{-2k/p} 2^{-mk} \left( \int \limits _{{\mathbb R}^2} | \mathcal {U}_{\lambda 2^{-k \gamma }} g(\xi ) |^p d\xi \right) ^{1/p}. \end{aligned}$$

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

$$\begin{aligned} \left( \int \limits _{{\mathbb R}^2} | \mathcal {U}_{\lambda 2^{-k \gamma }} g(\xi ) |^p d\xi \right) ^{1/p} \le C (\lambda 2^{-k \gamma })^{-\beta (p)} ||g||_{L^p({\mathbb R})}. \end{aligned}$$

We have $g=g_{j,k}$ and

$$\begin{aligned} \int \limits _{{\mathbb R}} |g_{j,k}|^p dy' \le \int \limits _{-1}^1 |f (2^{-k}j + 2^{-k} y') |^p dy' = 2^k \int \limits _{|z'|\le 2^{-k}} |f(2^{-k}j+z')|^p dz' \end{aligned}$$

and it follows that

$$\begin{aligned} \sum \limits _{j=-\infty }^\infty \int \limits _{{\mathbb R}} |g_{j,k}|^p dy' \le C 2^k || f||_p^p. \end{aligned}$$

Hence

$$\begin{aligned}&\int \limits _{{\mathbb R}^2} |T_{\lambda ,k} f|^p dx \le C \sum \limits _j \int \limits _{{\mathbb R}^2} |T_{\lambda ,k} f_j|^p dx \le C 2^{-2k} 2^{-mkp} (\lambda 2^{-k \gamma })^{-\beta (p) p} \\&\quad \sum \limits _j \int \limits _{\mathbb R}|g_{j,k}|^p dy' \le C 2^{-k} 2^{-mk p} (\lambda 2^{-k \gamma })^{-p \beta (p)} ||f||_p^p \end{aligned}$$

and we obtain the inequality

$$\begin{aligned} || T_{\lambda , k}||_p \le C 2^{-k/p} 2^{-mk} (\lambda 2^{-k \gamma })^{-\beta (p)}. \end{aligned}$$

Making a trivial estimate we also have

$$\begin{aligned} ||T_{\lambda ,k} ||_p \le C 2^{-k/p} 2^{- m k}. \end{aligned}$$

Invoking the inequality $|| T_\lambda ||_p \le \sum \limits _0^\infty || T_{\lambda , k} ||_p$ we obtain

$$\begin{aligned} || T_\lambda ||_p \le C \lambda ^{-\beta (p)} \sum \limits _{2^k \le \lambda ^{1/\gamma }} 2^{ k ( -1/p -m+\gamma \beta (p) ) } + C \sum \limits _{2^k \ge \lambda ^{1/\gamma }} 2^{-k (1/p+m)} =A+B. \end{aligned}$$

It is clear that $B\le C \lambda ^{-(1/p+m)/\gamma }$ and in the case $1/p + m<\gamma \beta (p)$ we get

$$\begin{aligned} A \le C \lambda ^{-\beta (p)} \lambda ^{(-1/p-m +\gamma \beta (p))/\gamma } = C\lambda ^{-(1/p +m)/\gamma } \end{aligned}$$

and

$$\begin{aligned} || T_\lambda ||_p \le C \lambda ^{-(1/p+m)/\gamma }. \end{aligned}$$

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

$$\begin{aligned} P(x', \xi ', \xi _n ) =\mathrm {det} \left( \frac{\partial ^2 \Phi }{\partial x_i \partial \xi _j} \right) _{i,j=1}^{n-1} \end{aligned}$$

for $1/2 \le d \le 2$. In [1] it is proved that

$$\begin{aligned} P(x', \xi ', \xi _n) = (-\gamma d^{\gamma -2})^{n-1} \frac{(\gamma -1) |\xi ' -x'|^2 + \xi _n^2 }{d^2}. \end{aligned}$$

(2.7)

Now let $\Phi _1 (x', \xi ') = |\xi ' - x'|^\gamma = d_1^\gamma$ where $d_1 = |\xi ' - x'|$. We shall need the determinant

$$\begin{aligned} P_1 (x', \xi ') =\mathrm {det} \left( \frac{\partial ^2 \Phi _1}{\partial x_i \partial \xi _j} \right) _{i,j=1}^{n-1}. \end{aligned}$$

It is clear that

$$\begin{aligned} P_1(x', \xi ') =P(x', \xi ', 0) = (-\gamma d_1^{\gamma - 2 })^{n-1}(\gamma -1) \end{aligned}$$

and for $\gamma >0$, $\gamma \ne 1$, it follows that

$$\begin{aligned} |P_1(x', \xi ')|\ge c>0 \text { for } 1/2\le d_1 \le 2. \end{aligned}$$

(2.8)

Proof of Theorem 1.4

We shall use the method in the proof of Theorem 1.3 and omit some details. We assume that

$$\begin{aligned} K(z) = \sum \limits _{k=0}^\infty 2^{k(n-1-m)} \psi (2^k z), \end{aligned}$$

where $\mathrm {supp} \psi \subset \{ x\in {\mathbb R}^{n-1}, \ 1/2\le |x| \le 2 \}$. One obtains

$$\begin{aligned} S_\lambda f = \sum \limits _{k=0}^\infty S_{\lambda , k } f \end{aligned}$$

where

$$\begin{aligned} S_{\lambda , k} f(x) = \int \limits _{{\mathbb R}^{n-1}} e^{i\lambda |x-y|^\gamma } \psi _0 (x,y) 2^{k(n-1-m)} \psi \big ( 2^k(x-y) \big ) f(y) dy. \end{aligned}$$

We also have

$$\begin{aligned} f=\sum \limits _{j\in {\mathbb Z}^{n-1}} f_j, \end{aligned}$$

where

$$\begin{aligned} f_j(t) = f(t) \chi \big (2^k(t-2^{-k}j)\big ), \ j\in {\mathbb Z}^{n-1}, \ t\in {\mathbb R}^{n-1}, \end{aligned}$$

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

$$\begin{aligned} | S_{\lambda ,k} f(x) |^2 \le C \sum \limits _j |S_{\lambda , k} f_j(x)|^2 \end{aligned}$$

and arguing as in the proof of Theorem 1.3 we get

$$\begin{aligned}&S_{\lambda , k } f_j(x) = 2^{-mk} \int \limits _{{\mathbb R}^{n-1}} e^{i\lambda 2^{-k \gamma } |2^k (x-2^{-k}j)-y|^\gamma } \psi _0(x, 2^{-k}j +2^{-k}y)\\&\quad \psi (2^k(x-2^{-k}j)-y) \times f(2^{-k}j+2^{-k }y) \chi (y)dy \end{aligned}$$

and

$$\begin{aligned} \int \limits _{{\mathbb R}^{n-1}} |S_{\lambda , k } f_j(x)|^2 dx = 2^{-k(n-1)} \int \limits _{{\mathbb R}^{n-1}} |S_{\lambda , k} f_j (2^{-k}\xi + 2^{-k} j)|^2 d\xi . \end{aligned}$$

It follows that

$$\begin{aligned} S_{\lambda , k} f_j (2^{-k}\xi + 2^{-k} j)&= 2^{-m k} \int \limits _{{\mathbb R}^{n-1}} e^{i \lambda 2^{-k \gamma } |\xi - y|^\gamma } \psi _0( 2^{-k}\xi + 2^{-k} j, 2^{-k} j+2^{-k}y ) \\&\ \quad \times \psi (\xi - y) f(2^{-k}j + 2^{-k} y) \chi (y) \widetilde{\chi }(y) dy\\&= 2^{- m k} \mathcal {U}_{\lambda 2^{-k \gamma }} g(\xi ) \\&=2^{-m k}\int \limits _{{\mathbb R}^{n-1}} e^{i\lambda 2^{-k\gamma } \Phi _1(y,\xi )} \psi _1(y,\xi ) g(y) dy&\end{aligned}$$

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

$$\begin{aligned} || \mathcal {U}_{\lambda 2^{-k \gamma } } g||_{L^2({\mathbb R}^{n-1})} \le C (\lambda 2^{-k \gamma })^{-\alpha } ||g||_{L^2({\mathbb R}^{n-1})} \end{aligned}$$

where $\alpha =(n-1)/2$. Arguing as in the proof of Theorem 1.3 we then obtain

$$\begin{aligned} || S_{\lambda , k}||_2 \le C 2^{-mk} \lambda ^{-\alpha } 2^{k \gamma \alpha } \end{aligned}$$

and $||S_{\lambda , k} ||_2 \le C 2^{- mk}$.

Hence

$$\begin{aligned} || S_\lambda ||_2 \le C \lambda ^{-\alpha } \sum \limits _{2^k \le \lambda ^{1/\gamma }} 2^{(\gamma \alpha - m)k} + \sum \limits _{2^k \ge \lambda ^{1/\gamma }} 2^{-mk} \end{aligned}$$

and Theorem 1.4 follows easily from this inequality. $\square$

3 Counter-examples

Assume $\gamma >0$, $1<p<\infty$, and

$$\begin{aligned} T_\lambda f(x) = \int \limits _{{\mathbb R}^{n-1}} e^{i\lambda |x-(y',0)|^\gamma } \psi _0(x,y') K \big ( x-(y',0) \big ) f(y') dy', \end{aligned}$$

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

$$\begin{aligned} T_\lambda f(x)= & {} \int \limits _E K \big ( x-(y',0) \big )dy' + \int \limits _E ( e^{i\lambda |x-(y',0)|^\gamma }-1 ) K \big (x-(y',0) \big ) dy' \\= & {} P(x) +R(x). \end{aligned}$$

Setting $\rho =1/\gamma$ we have

$$\begin{aligned} | e^{i\lambda |x-(y',0)|^\gamma }-1 | \le \lambda |x-(y',0)|^\gamma \le C c_0 \lambda \lambda ^{-\rho \gamma } =C c_0, \ y'\in E, \end{aligned}$$

and

$$\begin{aligned} |R(x)| \le C c_0 \int \limits _E K \big ( x-(y',0) \big ) dy'. \end{aligned}$$

Now taking $c_0$ small we obtain

$$\begin{aligned} | T_\lambda f(x) | \ge c \int \limits _E K \big (x-(y',0)\big ) dy' \ge c \int \limits _E \lambda ^{-\rho (m-n+1)} dy' = C \lambda ^{-\rho m} \end{aligned}$$

and

$$\begin{aligned} \int \limits _F |T_\lambda f(x) |^p dx \ge c \lambda ^{-\rho m } (\lambda ^{-\rho n})^{1/p} = c \lambda ^{-m/\gamma } \lambda ^{-n/ \gamma p }. \end{aligned}$$

On the other hand

$$\begin{aligned} ||f||_p = \left( \int \limits _E dy' \right) ^{1/p} = C \lambda ^{-\rho (n-1)/p} = C \lambda ^{-(n-1)/\gamma p} \end{aligned}$$

and we have

$$\begin{aligned} ||T_\lambda ||_p \ge c \frac{\lambda ^{-m/\gamma } \lambda ^{-n/\gamma p} }{\lambda ^{-(n-1)/\gamma p}}= c \lambda ^{-m/\gamma } \lambda ^{-1/\gamma p} = c\lambda ^{-(1/p+m)/\gamma }. \end{aligned}$$

(3.1)

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

$$\begin{aligned} ||T_\lambda ||_p \le C \lambda ^{-(1/p +m)/\gamma } \end{aligned}$$

and the inequality (3.1) shows that these estimates are sharp.

In Theorem 1.4 we proved the estimate

$$\begin{aligned} ||S_\lambda ||_{2} \le C \lambda ^{-m/\gamma }. \end{aligned}$$

(3.2)

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

$$\begin{aligned} |S_\lambda f(x) | \ge c \lambda ^{-\rho m} \text { for } x\in F. \end{aligned}$$

It follows that

$$\begin{aligned} ||S_\lambda f ||_2 \ge c \lambda ^{-m/\gamma } \lambda ^{-(n-1)/2\gamma } \end{aligned}$$

and

$$\begin{aligned} ||f||_2 = C \lambda ^{-(n-1)/2\gamma }. \end{aligned}$$

We conclude that

$$\begin{aligned} ||S_\lambda ||_2 \ge c \lambda ^{-m/\gamma } \end{aligned}$$

and it follows that (3.2) is sharp.

In Theorems 1.2 and 1.3 we have

$$\begin{aligned} T_\lambda f(x) = \int \limits _{{\mathbb R}^{n-1}} e^{i\lambda \varphi (x,y')} \psi _0(x,y') K \big (x-(y',0) \big ) f(y') dy' \end{aligned}$$

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

$$\begin{aligned} T_\lambda f(x) = \int \limits _B e^{i \lambda \varphi (x,y')} K \big ( x-(y',0) \big ) dy'. \end{aligned}$$

It follows from the mean value theorem that

$$\begin{aligned} |\varphi (x,y') - \varphi (x,0)|\le C c_0 \lambda ^{-1} \text { for } y'\in B \end{aligned}$$

and choosing $c_0$ small we obtain

$$\begin{aligned} |\lambda \varphi (x,y') - \lambda \varphi (x,0) |\le c_1 \text { for } y'\in B, \end{aligned}$$

where $c_1$ is small. It follows that there is no cancellation in the above integral and we get

$$\begin{aligned} |T_\lambda f(x) | \ge c_2 \lambda ^{-(n-1)} \end{aligned}$$

in a neighbourhood of a. Hence

$$\begin{aligned} ||T_\lambda f||_2 \ge c_3 \lambda ^{-(n-1)}. \end{aligned}$$

We have $||f||_2 = c_4 \lambda ^{-(n-1)/2}$ and we obtain

$$\begin{aligned} \frac{||T_\lambda ||_2 }{|| f||_2} \ge \frac{c_3 \lambda ^{-(n-1)}}{c_4 \lambda ^{-(n-1)/2}} = c_5 \lambda ^{-(n-1)/2}. \end{aligned}$$

Hence

$$\begin{aligned} ||T_\lambda ||_2 \ge c_5 \lambda ^{-(n-1)/2} \end{aligned}$$

(3.3)

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

$$\begin{aligned} S_\lambda f(x) = \int \limits _{{\mathbb R}^{n-1}} e^{i\lambda \varphi (x,y)} \psi _0(x,y ) K(x-y) f(y) dy, \ x\in {\mathbb R}^{n-1}, \end{aligned}$$

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

$$\begin{aligned} || T_\lambda ||_p \ge \frac{||T_\lambda f||_p }{|| f||_p} \ge c \frac{\lambda ^{-1}}{\lambda ^{-1/p}} = c\lambda ^{-(1-1/p)} \end{aligned}$$

for $1\le p <2$. It follows that the estimate

$$\begin{aligned} ||T_\lambda ||_p \le C \lambda ^{-\beta (p)} \end{aligned}$$

for $1<p<2$ in Theorem 1.3 is sharp (since $\beta (p) = 1-1/p$).

In Theorem 1.3 we have

$$\begin{aligned} T_\lambda f(x,y) = \int \limits _{\mathbb R}e^{i\lambda \varphi (x,y,t)}\psi _0(x,y,t) K(x-t, y) f(t) dt, \ (x,y)\in {\mathbb R}^2, \end{aligned}$$

where $\varphi (x,y,t) = \big ( (x-t)^2 + y^2 \big )^{\gamma /2}$ and $K(z) = |z|^{m-1} \omega (z)$.

Setting

$$\begin{aligned} T_\lambda ^* g(t) = \int \limits _{{\mathbb R}^2} e^{-i \lambda \varphi (x,y,t) } \overline{\psi _0(x,y,t)} K(x-t,y) g(x,y) dx dy, \ t\in {\mathbb R}, \end{aligned}$$

we get

$$\begin{aligned} ( T_\lambda f,g )_2 =(f,T_\lambda ^* g)_1, \ f\in C_0^\infty ({\mathbb R}), \ g\in C_0^\infty ({\mathbb R}^2), \end{aligned}$$

where $(,)_2$ and $(,)_1$ denote the inner products in $L^2({\mathbb R}^2)$ and $L^2({\mathbb R})$. It follows that

$$\begin{aligned} ||T_\lambda ||_p = || T_\lambda ||_{L^p({\mathbb R}) \rightarrow L^p({\mathbb R}^2) } \ge || T_\lambda ^*||_{L^r({\mathbb R}^2) \rightarrow L^r({\mathbb R})} \end{aligned}$$

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

$$\begin{aligned} ||g||_r \le \left( \iint \limits _B dx dy \right) ^{1/r} = c \lambda ^{-2/r} \end{aligned}$$

and choosing $\psi _0$ such that $\psi _0 (x,y,t) = 1$ in a neighbourhood of (0, 1, 0) we get

$$\begin{aligned} |T_\lambda ^* g(t) | \ge c \lambda ^{-2} \end{aligned}$$

in a neighbourhood of 0. Hence

$$\begin{aligned} || T_\lambda ^* g||_r \ge c \lambda ^{-2} \end{aligned}$$

and

$$\begin{aligned} || T_\lambda ^* ||_r \ge \frac{||T_\lambda ^* g ||_r }{ ||g ||_r }\ge c \frac{\lambda ^{-2}}{\lambda ^{-2/r}} = c \lambda ^{-2(1-1/r)}. \end{aligned}$$

Since $1-1/r = 1/p$ we conclude that

$$\begin{aligned} ||T_\lambda ||_p \ge c \lambda ^{-2/p}, \ 4\le p <\infty \end{aligned}$$

and it follows that the estimate

$$\begin{aligned} || T_\lambda ||_p \le C \lambda ^{-\beta (p)} , \ 4<p<\infty , \end{aligned}$$

in Theorem 1.3 is sharp (since $\beta (p) = 2/p$).

In Theorem 1.3 we also have an estimate of the type

$$\begin{aligned} || T_\lambda ||_p \le C \lambda ^{-1/2 + \varepsilon } \end{aligned}$$

for $2<p<4$. We shall finally discuss the sharpness of this estimate in the case $\gamma =1$. We shall study the statement

$$\begin{aligned} || T_\lambda ||_p \le C \lambda ^{-1/2 - \delta } \text { for some } p \text { with } 2<p<4 \text { and some } \delta >0. \end{aligned}$$

(3.4)

Omitting details we shall describe how (3.4) leads to a contradiction.

Following Stein [6], p. 393, we have

$$\begin{aligned} \frac{1}{|x|^{3/2}} = u(x) + \sum \limits _{k=1}^\infty 2^{-3k/2} \psi \left( \frac{x}{2^k} \right) , \ x\in {\mathbb R}^2 \setminus \{0\}, \end{aligned}$$

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

$$\begin{aligned} K_0(x) = \frac{e^{i|x|}}{|x|^{3/2}} = e^{i|x| } u(x) + \sum \limits _{k=1}^\infty 2^{-3k/2} e^{i|x|} \psi (x/2^k), \ x\in {\mathbb R}^2\setminus \{0\}, \end{aligned}$$

and $S_0 f =K_0 \star f$. We define the operator $V_k$ by setting

$$\begin{aligned} V_k f = 2^{-3k/2} 2^{2k} ( e^{i 2^k |x| } \psi ) \star f&= \\ 2^{k/2} ( e^{i 2^k |x| } \psi ) \star f&= \lambda ^{1/2} ( e^{i\lambda |x| } \psi )\star f, \end{aligned}$$

where $\lambda = 2^k$. Using (3.4) we can prove that

$$\begin{aligned} || V_k ||_p = || V_k||_{L^p({\mathbb R}^2 ) \rightarrow L^p ({\mathbb R}^2)} \le C \lambda ^{-\delta } = C 2^{-k \delta }, \end{aligned}$$

and the inequality

$$\begin{aligned} \sum \limits _{k=1}^\infty || V_k ||_p <\infty \end{aligned}$$

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.