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.
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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
For a singular kernel K(x, y), the oscillatory singular integral operator \(T_B,\) acting initially on test functions, is given by
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
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 \( 0 \le \phi \le 1,\)
and
for all \(t > 0.\)
For \(\nu \ge 0,\) define the operator \(S_\nu \) by
and let
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 \(f \in L^2({\mathbb {R}}^n),\)
where
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,
\(|L_\nu (x, y)| \le \chi _{D_n(0, 8)}(x - y)\)
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:
where C is independent of \(s, \nu , x, y\) and z. We will first verify that
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
Therefore, it follows from (8) that
Next, if \(\phi (|z-x|) \ne 0\) and \(\phi (|z^\prime -x|) = 0,\) then \(|z-x| \ge 1/2\) and
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
uniformly in \(s, \nu , y\) and z.
By (7), (13), (22) and (23), we have
which proves (21).
By letting \(s = \pi b [2^{2\nu } P(x - y)]^{-1}\) and using (22), we have
It follows from (19), (20), (24) and the proposition on p. 182 of [7] that
By first interpolating between \(L^1\) and \(L^2\) and then using a duality argument, we obtain that, for \( 1< p < \infty \) and \(\nu \in {\mathbb {N}},\)
where \(\delta _p\) is the positive number given by
Let
and
Since
the localization technique described on pp. 118–119 of [6] can be used to get
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
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
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
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
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
where \(M_{HL}\) denotes the Hardy–Littlewood maximal operator. Thus,
By using (26), (32) and an interpolation with change of measures (see [9]), one obtains that
for all \(\nu \in {\mathbb {N}}.\) The rest of the details are omitted.
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We thank the referees for their helpful comments.
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Communicated by Arieh Iserles.
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Al-Qassem, H., Cheng, L. & Pan, Y. Oscillatory Singular Integral Operators with Hölder Class Kernels. J Fourier Anal Appl 25, 2141–2149 (2019). https://doi.org/10.1007/s00041-018-09660-y
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DOI: https://doi.org/10.1007/s00041-018-09660-y