Journal of Fourier Analysis and Applications

, Volume 23, Issue 6, pp 1408–1425

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

• Per Sjölin
Open Access
Article

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.

Keywords

Singular integral Oscillatory integral Nonlinear phase function

42B20

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(xR) 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)
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.

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