Oscillatory Singular Integral Operators with Hölder Class Kernels

Abstract

We establish the boundedness on $L^p({\mathbb {R}}^n)$ of oscillatory singular integral operators whose kernels are the products of an oscillatory factor with bilinear phase and a Calderón–Zygmund kernel K(x, y) satisfying a Hölder condition. Our results also hold on weighted $L^p$ spaces with $A_p$ weights.

1 Introduction

In [6], Phong and Stein studied oscillatory singular integrals with bilinear phases as a hybrid between the Fourier transforms and singular integral operators.

Let $n \in {\mathbb {N}},$$B = ( b_{j k} )$ be an $n \times n$ matrix with real entries. For $x = (x_1, \ldots , x_n), y = (y_1, \ldots , y_n) \in {\mathbb {R}}^n,$ let

$$\begin{aligned} B(x,y) = xBy^T = \sum _{j=1}^n \sum _{k=1}^n b_{j k}x_j y_k. \end{aligned}$$

For a singular kernel K(x, y),  the oscillatory singular integral operator $T_B,$ acting initially on test functions, is given by

$$\begin{aligned} T_B f(x) = \text{ p.v. }\int _{{\mathbb {R}}^n} e^{i B(x,y)} K(x,y) f(y) dy. \end{aligned}$$

(1)

Let’s begin by recalling the following:

Theorem 1.1

Let $A > 0$ and $\Delta = \{(x, x): x \in {\mathbb {R}}^n\}.$ Suppose that

• (i) For all $(x, y) \in ({\mathbb {R}}^n\times {\mathbb {R}}^n)\backslash \Delta ,$

  $$\begin{aligned} |K(x,y)| \le \frac{A}{|x - y|^n}; \end{aligned}$$

  (2)

• (ii) $K(x, y) \in C^1(({\mathbb {R}}^n \times {\mathbb {R}}^n)\backslash \Delta ),$ and for $(x, y) \in ({\mathbb {R}}^n\times {\mathbb {R}}^n)\backslash \Delta$

  $$\begin{aligned} |\nabla _x K(x,y)| + |\nabla _y K(x,y)| \le \frac{A}{|x - y|^{n+1}}; \end{aligned}$$

  (3)
• (iii)
  $$\begin{aligned} \Vert T_o\Vert _{L^2({\mathbb {R}}^n) \rightarrow L^2({\mathbb {R}}^n)} \le A, \end{aligned}$$

  (4)

  where

  $$\begin{aligned} T_o f(x) = \text{ p.v. } \int _{{\mathbb {R}}^n} K(x, y) f(y) dy. \end{aligned}$$

  (5)

  Then, for $1< p < \infty ,$ there exists a positive $C_p$ which may depend on p,  n and A,  but is independent of the matrix B,  such that

  $$\begin{aligned} \Vert T_B f \Vert _{L^p({\mathbb {R}}^n)} \le C_p \Vert f \Vert _{L^p({\mathbb {R}}^n)} \end{aligned}$$

  (6)

  for all $f \in L^p({\mathbb {R}}^n).$

The above result first appeared in [6, p. 130] for smooth convolutional kernels K,  while the fact it also holds for $C^1$ nonconvolutional kernels was pointed out in [7, p. 192].

Since the publication of [6], many papers have been written regarding oscillatory integrals with singular kernels, successfully extending the results of Phong and Stein to more general phase functions (see, for example, [1, 3, 5, 7, 8]). The focus of the current paper is to consider the $L^p$ boundedness of $T_B$ without assuming that K(x, y) is $C^1$ away from the diagonal [or that K(x, y) has a special form such as $K(x,y) = |x-y|^{-n}\Omega ((x-y)/|x-y|)$]. The $C^1$ condition on K(x, y) has been generally viewed as a key assumption due to the historically important role played by van der Corput type arguments. In our main result presented below, the $C^1$ assumption will be replaced by a well-known weaker condition of Hölder type on K(x, y).

Theorem 1.2

Let $A, \delta > 0.$ Suppose that

• (i) For all $(x, y) \in ({\mathbb {R}}^n\times {\mathbb {R}}^n)\backslash \Delta ,$

  $$\begin{aligned} |K(x,y)| \le \frac{A}{|x - y|^n}; \end{aligned}$$

  (7)
• (ii)
  $$\begin{aligned} | K(x,y) - K(x^\prime , y)| \le \frac{A|x - x^\prime |^\delta }{(|x - y| + |x^\prime - y|)^{n+\delta }} \end{aligned}$$

  (8)

  whenever $|x - x^\prime | < (1/2)\max \{|x-y|, |x^\prime - y|\},$ and

  $$\begin{aligned} | K(x,y) - K(x, y^\prime )| \le \frac{A|y - y^\prime |^\delta }{(|x - y| + |x - y^\prime |)^{n+\delta }} \end{aligned}$$

  (9)

  whenever $|y - y^\prime | < (1/2)\max \{|x-y|, |x - y^\prime |\};$

• (iii)
  $$\begin{aligned} \Vert T_o\Vert _{L^2({\mathbb {R}}^n) \rightarrow L^2({\mathbb {R}}^n)} \le A. \end{aligned}$$

  (10)

  Then, for $1< p < \infty ,$ there exists a positive $C_p$ which may depend on p,  n,  $\delta$ and A,  but is independent of the matrix B,  such that

  $$\begin{aligned} \Vert T_B f \Vert _{L^p({\mathbb {R}}^n)} \le C_p \Vert f \Vert _{L^p({\mathbb {R}}^n)} \end{aligned}$$

  (11)

  for all $f \in L^p({\mathbb {R}}^n).$

An extension of the above result to the weighted $L^p$ spaces with $A_p$ weights will be given in Sect. 3.

2 Proof of Theorem 1.2

For $B = (b_{jk})_{n\times n},$ let

$$\begin{aligned} b = \max \{ |b_{jk}|: 1 \le j, k \le n\}. \end{aligned}$$

(12)

If $b = 0,$ then $T_B = T_o.$ It is well-known that, under the conditions (7)–(10), $T_o$ is bounded on $L^p({\mathbb {R}}^n)$ for $1< p < \infty .$ Thus, from this point on, we may assume that $b > 0.$

For $m \in {\mathbb {N}},$$u \in {\mathbb {R}}^m$ and $r > 0,$ let $D_m(u, r) = \{v \in {\mathbb {R}}^{m}: |v - u| < r\}.$ Let $\phi$ be a real-valued $C^\infty$ function on $(0, \infty )$ such that $0 \le \phi \le 1,$

$$\begin{aligned} \text{ supp }(\phi ) \subset \bigg (\frac{1}{2}, 4\bigg ), \end{aligned}$$

(13)

and

$$\begin{aligned} \sum _{\nu \in {\mathbb {Z}}} \phi (2^{-\nu } t) = 1 \end{aligned}$$

(14)

for all $t > 0.$

For $\nu \ge 0,$ define the operator $S_\nu$ by

$$\begin{aligned} S_\nu f(x) = \int _{{\mathbb {R}}^n} e^{iB(x,y)} K(x, y) \phi (2^{-\nu }\sqrt{b}|x - y|) f(y) dy, \end{aligned}$$

(15)

and let

$$\begin{aligned} K_\nu (x, y) = \bigg (\frac{2^\nu }{\sqrt{b}}\bigg )^n K\bigg (\frac{2^\nu x}{\sqrt{b}}, \frac{2^\nu y}{\sqrt{b}}\bigg ). \end{aligned}$$

(16)

It is easy to see that (7)–(10) remain valid with the same constants A and $\delta$ if K(x, y) is substituted by $K_\nu (x, y).$ Clearly one may also assume that $\delta < 1.$

For $f \in L^2({\mathbb {R}}^n),$

$$\begin{aligned} \Vert S_\nu f\Vert ^2_{L^2({\mathbb {R}}^n)}= & {} \int _{{\mathbb {R}}^n} S_\nu f(z) \overline{ S_\nu f(z)} dz\nonumber \\= & {} \int _{{\mathbb {R}}^n}\int _{{\mathbb {R}}^n} L_\nu (x, y) f_\nu (x) \overline{f_\nu (y)} dx dy, \end{aligned}$$

(17)

where

$$\begin{aligned} f_\nu (x)= & {} \bigg (\frac{2^\nu }{\sqrt{b}}\bigg )^{n/2}f\bigg (\frac{2^\nu x}{\sqrt{b}}\bigg ), \end{aligned}$$

(18)

$$\begin{aligned} L_\nu (x, y)= & {} \int _{{\mathbb {R}}^n} e^{i (b^{-1} 2^{2\nu }) B(z, x - y)} K_\nu (z, x) \overline{K_\nu (z, y)} \phi (|z - x|) \phi (|z - y|) dz . \end{aligned}$$

(19)

Without loss of generality, we may assume that $b = \pm b_{1 k_0}$ holds for some $k_0 \in \{1, 2, \ldots , n\}.$ For $x \in {\mathbb {R}}^n,$ let ${\tilde{x}} = (x_2, \ldots , x_n),$$\displaystyle {P (x) = \sum \nolimits _{k=1}^n b_{1 k} x_k},$ and $G_\nu (x, y, z) =K_\nu (z, x) \overline{K_\nu (z, y)} \phi (|z - x|) \phi (|z - y|).$ Then,

$|L_\nu (x, y)| \le \chi _{D_n(0, 8)}(x - y)$

$$\begin{aligned} \times \int _{D_{n-1}({\tilde{x}}, 4) \cap D_{n-1}({\tilde{y}}, 4) }\bigg |\int _{{\mathbb {R}}} e^{i (b^{-1} 2^{2\nu })z_1 P(x - y)} G_\nu (x, y,z) dz_1\bigg | d{\tilde{z}} . \end{aligned}$$

(20)

Let $s \in {\mathbb {R}}.$ For $z = (z_1, {\tilde{z}}) \in {\mathbb {R}}^n,$ let $z^\prime = (z_1 + s, {\tilde{z}}).$ We will need the following inequality:

$$\begin{aligned} |G_\nu (x, y, z) - G_\nu (x, y, z^\prime )| \le C |s|^\delta , \end{aligned}$$

(21)

where C is independent of $s, \nu , x, y$ and z. We will first verify that

$$\begin{aligned} |K_\nu (z, x)\phi (|z-x|) - K_\nu (z^\prime ,x)\phi (|z^\prime - x|)| \le C |s|^\delta \end{aligned}$$

(22)

uniformly in $s, \nu , x$ and z.

When $|s| \ge 1/4,$ (22) follows trivially from (7) and (13). Thus, we may now assume that $|s| < 1/4.$

The first case to be examined is when $\phi (|z-x|)$ and $\phi (|z^\prime -x|)$ are both nonzero. Then, we have $|z-x| \ge 1/2,$$|z^\prime - x| \ge 1/2,$$|z - z^\prime | = |s| < (1/2) |z-x|,$ and

$$\begin{aligned} |K_\nu (z^\prime ,x)| \le C. \end{aligned}$$

Therefore, it follows from (8) that

$$\begin{aligned}&|K_\nu (z, x)\phi (|z-x|) - K_\nu (z^\prime ,x)\phi (|z^\prime - x|)| \\&\quad \le |K_\nu (z,x) - K_\nu (z^\prime ,x)||\phi (|z-x|)| + |K_\nu (z^\prime , x)| |\phi (|z-x|) - \phi (|z^\prime - x|)| \\&\quad \le C(|z - z^\prime |^\delta + \Vert \phi ^\prime \Vert _\infty |z-z^\prime |) \le C|s|^\delta . \end{aligned}$$

Next, if $\phi (|z-x|) \ne 0$ and $\phi (|z^\prime -x|) = 0,$ then $|z-x| \ge 1/2$ and

$$\begin{aligned} |K_\nu (z, x)\phi (|z-x|) - K_\nu (z^\prime ,x)\phi (|z^\prime - x|)|= & {} |K_\nu (z, x)| |\phi (|z-x|) - \phi (|z^\prime -x|)| \\\le & {} C \Vert \phi ^\prime \Vert _\infty |z-z^\prime | \le C|s|^\delta . \end{aligned}$$

Finally, the case of $\phi (|z-x|) = 0$ and $\phi (|z^\prime -x|) \ne 0$ can be treated in the same manner as above, which completes the proof of (22).

From (22), one gets

$$\begin{aligned} |\overline{K_\nu (z, y)}\phi (|z-y|) - \overline{K_\nu (z^\prime , y)}\phi (|z^\prime - y|)| \le C |s|^\delta \end{aligned}$$

(23)

uniformly in $s, \nu , y$ and z.

By (7), (13), (22) and (23), we have

$$\begin{aligned}&|G_\nu (x, y, z) - G_\nu (x, y, z^\prime )| \\&\quad \le |K_\nu (z, x)\phi (|z-x|) - K_\nu (z^\prime ,x)\phi (|z^\prime - x|)| |\overline{K_\nu (z, y)}\phi (|z-y|)| \\&\qquad +\; |\overline{K_\nu (z, y)}\phi (|z-y|) - \overline{K_\nu (z^\prime , y)}\phi (|z^\prime - y|)| |K_\nu (z^\prime ,x)\phi (|z^\prime - x|)| \\&\quad \le C|s|^\delta (|\overline{K_\nu (z, y)}\phi (|z-y|)| + |K_\nu (z^\prime ,x)\phi (|z^\prime - x|)|) \le C|s|^\delta , \end{aligned}$$

which proves (21).

By letting $s = \pi b [2^{2\nu } P(x - y)]^{-1}$ and using (22), we have

$$\begin{aligned}&\bigg |\int _{{\mathbb {R}}} e^{i (b^{-1} 2^{2\nu })z_1 P(x - y)} G_\nu (x, y,z) dz_1\bigg | \nonumber \\&\quad = (1/2) \bigg |\int _{{\mathbb {R}}} e^{i (b^{-1} 2^{2\nu })z_1 P(x - y)} (G_\nu (x, y, z) - G_\nu (x, y, z^\prime )) dz_1\bigg | \nonumber \\&\quad \le C |s|^\delta |D_1(x_1, 4) \cup D_1(x_1 - s, 4)| \nonumber \\&\quad \le C 2^{-2\nu \delta } b^\delta \bigg |\sum _{k=1}^n b_{1 k}(x_k - y_k)\bigg |^{-\delta }. \end{aligned}$$

(24)

It follows from (19), (20), (24) and the proposition on p. 182 of [7] that

$$\begin{aligned}&\sup _{x \in {\mathbb {R}}^n} \int _{{\mathbb {R}}^n} |L_\nu (x,y)|dy = \sup _{y \in {\mathbb {R}}^n} \int _{{\mathbb {R}}^n} |L_\nu (x,y)|dx \nonumber \\&\quad \le C2^{-2\nu \delta } b^\delta \int _{D_n(0, 8)} \bigg |\sum _{k=1}^n b_{1 k}x_k\bigg |^{-\delta } dx \nonumber \\&\quad \le C2^{-2\nu \delta } \bigg (\frac{b}{\sum _{k=1}^n |b_{1 k}|}\bigg )^\delta \le C2^{-2\nu \delta }. \end{aligned}$$

(25)

Thus, by (17), (18) and (25),

$$\begin{aligned} \Vert S_\nu f\Vert _{L^2({\mathbb {R}}^n)} \le (C2^{-2\nu \delta } \Vert f_\nu \Vert _{L^2({\mathbb {R}}^n)}^2)^{1/2} = C 2^{-\nu \delta } \Vert f\Vert _{L^2({\mathbb {R}}^n)}. \end{aligned}$$

By (7) and (13), we have

$$\begin{aligned} \Vert S_\nu f\Vert _{L^1({\mathbb {R}}^n)} \le C \Vert f\Vert _{L^1({\mathbb {R}}^n)}. \end{aligned}$$

By first interpolating between $L^1$ and $L^2$ and then using a duality argument, we obtain that, for $1< p < \infty$ and $\nu \in {\mathbb {N}},$

$$\begin{aligned} \Vert S_\nu f\Vert _{L^p({\mathbb {R}}^n)} \le C 2^{-\nu \delta _p}\Vert f\Vert _{L^p({\mathbb {R}}^n)}, \end{aligned}$$

(26)

where $\delta _p$ is the positive number given by

$$\begin{aligned} \delta _p = \left\{ \begin{array}{ll} \frac{2\delta (p-1)}{p} &{}\quad \text{ if } 1< p \le 2, \\ &{} \\ \frac{2\delta }{p} &{}\quad \text{ if } 2< p < \infty . \end{array} \right. \end{aligned}$$

Let

$$\begin{aligned} \psi (t) = \sum _{\nu = -\infty }^0 \phi (2^{-\nu } t) , \end{aligned}$$

and

$$\begin{aligned} {\tilde{S}} f(x) = \int _{{\mathbb {R}}^n} e^{i B(x,y)} K(x,y) \psi (\sqrt{b}|x - y|) f(y) dy. \end{aligned}$$

(27)

Since

$$\begin{aligned} \text{ supp }(\psi ) \subseteq [0, 4], \end{aligned}$$

the localization technique described on pp. 118–119 of [6] can be used to get

$$\begin{aligned} \Vert {\tilde{S}} f\Vert _{L^p({\mathbb {R}}^n)} \le C_p \Vert f\Vert _{L^p({\mathbb {R}}^n)} \end{aligned}$$

(28)

for $1< p < \infty .$ Since the argument, after proper scaling, uses the size condition (2) but not the smoothness condition (3), we will omit the details of the proof of (28).

It follows from (14), (26) and (28) that

$$\begin{aligned} \Vert T_B f \Vert _{L^p({\mathbb {R}}^n)}\le & {} \Vert {\tilde{S}} f \Vert _{L^p({\mathbb {R}}^n)} + \sum _{\nu = 1}^\infty \Vert S_\nu f \Vert _{L^p({\mathbb {R}}^n)} \\\le & {} C_p \bigg (1 + \sum _{\nu = 1}^\infty 2^{-\nu \delta _p}\bigg ) \Vert f\Vert _{L^p({\mathbb {R}}^n)} = C_p \Vert f\Vert _{L^p({\mathbb {R}}^n)} . \end{aligned}$$

Theorem 1.2 is proved.

3 Weighted $L^p$ Spaces

Theorem 1.2 can be extended to the weighted $L^p$ spaces with the Muckenhoupt $A_p$ weights. First, let us recall the definition of $A_p$ weights for $1< p < \infty .$ Let $w(\cdot )$ be a nonnegative, locally integrable function on ${\mathbb {R}}^n.$

Definition 3.1

For $1< p < \infty ,$w is said to be in the Muckenhoupt weight class $A_p({\mathbb {R}}^n)$ if there exists a constant $C > 0$ such that

$$\begin{aligned} \bigg (\frac{1}{|Q|} \int _Q w(y) dy\bigg ) \bigg (\frac{1}{|Q|} \int _Q w(y)^{-1/(p-1)} dy\bigg ) ^{p-1} \le C \end{aligned}$$

(29)

holds for all cubes Q in ${\mathbb {R}}^n.$ The smallest such constant C in (29) is the corresponding $A_p$ constant of w.

We recall that $A_{p_1}({\mathbb {R}}^n) \subset A_{p_2}({\mathbb {R}}^n)$ when $p_1 < p_2$ and

Lemma 3.1

[2] Let $p \in (1, \infty )$ and $w \in A_p({\mathbb {R}}^n).$ Then there exists a $\theta \in (0,1)$ such that $w^{1+\theta } \in A_p({\mathbb {R}}^n).$ Both $\theta$ and the $A_p$ constant of $w^{1+\theta }$ depend on n, p and the $A_p$ constant of w only.

Let $\displaystyle { L^p_w({\mathbb {R}}^n) = \bigg \{f: \int _{{\mathbb {R}}^n}|f(x)|^p w(x) dx < \infty \bigg \}}$ and

$$\begin{aligned} \Vert f\Vert _{L_w^p({\mathbb {R}}^n)} = \bigg (\int _{{\mathbb {R}}^n}|f(x)|^p w(x) dx\bigg )^{1/p}.\end{aligned}$$

Then, we have the following:

Theorem 3.1

Let the operator $T_B$ be given as in Theorem 1.2, $p \in (1, \infty )$ and $w \in A_p({\mathbb {R}}^n).$ Then there exists a positive $C_{p,w}$ which may depend on p,  n,  $\delta ,$A and the $A_p$ constant of w,  but is independent of the matrix B,  such that

$$\begin{aligned} \Vert T_B f \Vert _{L_w^p({\mathbb {R}}^n)} \le C_{p,w} \Vert f \Vert _{L_w^p({\mathbb {R}}^n)} \end{aligned}$$

(30)

for all $f \in L_w^p({\mathbb {R}}^n).$

We will end the paper with a brief description of the proof of Theorem 3.1.

First, a weighted version of (28) follows from the $L^p_w$ boundedness of $T_o$ (see [4, p. 712]) and an application of the localization technique mentioned earlier. Essentially all one needs now is to find a weighted analogue of (26), which can be done as follows.

By Lemma 3.1, there exists a $\theta > 0$ such that $w^{1+\theta } \in A_p({\mathbb {R}}^n).$ By (7), (13) and (15), we have

$$\begin{aligned} |S_\nu f (x)| \le C M_{HL} f(x), \end{aligned}$$

(31)

where $M_{HL}$ denotes the Hardy–Littlewood maximal operator. Thus,

$$\begin{aligned} \Vert S_\nu f \Vert _{L_{w^{1+\theta }}^p({\mathbb {R}}^n)} \le C_{p,w} \Vert f \Vert _{L_{w^{1+\theta }}^p({\mathbb {R}}^n)}. \end{aligned}$$

(32)

By using (26), (32) and an interpolation with change of measures (see [9]), one obtains that

$$\begin{aligned} \Vert S_\nu f\Vert _{L_w^p({\mathbb {R}}^n)} \le C_{p,w} 2^{-\nu \delta _p\theta /(1+\theta )} \Vert f\Vert _{L_w^p({\mathbb {R}}^n)}\end{aligned}$$

for all $\nu \in {\mathbb {N}}.$ The rest of the details are omitted.