Abstract
Let \(T_{a,\varphi }\) be a Fourier integral operator with symbol a and phase φ. In this paper, under the conditions \(a(x,\xi )\in L^{\infty }S^{n(\rho -1)/2}_{\rho }(\omega )\) and \(\varphi \in L^{\infty }\varPhi ^{2}\), the authors show that \(T_{a,\varphi }\) is bounded from \(L^{2}(\mathbb{R}^{n})\) to \(L^{2}(\mathbb{R}^{n})\).
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1 Introduction and main results
Fourier integral operator on \(\mathbb{R}^{n}\) has been studied extensively and is related to many areas in analysis and PDEs. In [1], Sogge considered the Cauchy problem of the hyperbolic equations via the \(L^{p}\)-estimates theory of the Fourier integral operators (also see, for the local smoothing estimates of wave equations, e.g., [2, 3] and the references therein for some recent developments). For the Fourier integral operators with smooth amplitude, the \(L^{2}\)-regularity theory is comparably more progress. In [4] and [5], Eskin and Hörmander found the local and global \(L^{2}\)-regularity theory for Fourier integral operators, respectively. There are also some results for the \(L^{p}\) boundedness of Fourier integral operators with classical symbol and phase (see Littman [6], Miyachi [7], Peral [8], and Beals [9]).
Let f̂ be the Fourier transform of f. A Fourier integral operator T is a linear operator of the form
with symbol \(a(x,\xi )\) and phase \(\varphi (x,\xi )\), respectively. In particular, for \(\varphi (x,\xi )=\langle x,\xi \rangle \), the operator \(T_{a}\) is a so-called pseudo-differential operator. In [10], Hörmander showed that \(T_{a}\) is bounded in \(L^{2}(\mathbb{R}^{n})\), when \(a\in S^{m}_{\rho ,\delta }\), \(\delta <1\) and \(m\leq n(\rho -\delta )/2\). For \(a\in S^{0}_{1,1}\), Ching [11] proved that \(T_{a}\) is not bounded in \(L^{2}(\mathbb{R}^{n})\). Moveover, for \(a\in S^{m}_{\rho ,1}\), Rodino [12] showed that \(T_{a}\) is bounded in \(L^{2}(\mathbb{R}^{n})\) if and only if \(m< n(\rho -1)/2\). However, the operator \(T_{a}\) is not always \(L^{2}\)-bounded for \(a\in S^{n(\rho -1)/2}_{\rho ,1}\); see, for example, [10–12]. The necessary and sufficient conditions of \(L^{2}\)-boundedness of \(T_{a}\) were obtained by Higuchi [13] as \(m= n(\rho -1)/2\). It is natural to ask if the corresponding results hold for the Fourier integral operators. Recently, Kenig, David, Salvador, and Wolfgang [14–16] have studied the Fourier integral operators with rough symbol and rough phases, both of which behave in the spatial variable x like an \(L^{\infty }\)-function. More precisely, the symbol belongs to the class \(L^{\infty }S^{m}_{\varrho }\) whose constituent element a obeys
Under this condition, for \(m=\min \{0,\frac{n}{2}(\rho -\delta )\}\), \(0\leq \rho \leq 1\), \(0\leq \delta <1\), and \(a\in S^{m}_{\rho ,\delta }\), Wolfgang [14] proved the global continuity on \(L^{p}\)-space with \(p\in [1,\infty ]\) of Fourier integral operators. A natural question is \(L^{2}\)-boundedness of Fourier integral operators for \(\delta =1\) and \(m= n(\rho -1)/2\). In this paper, we answer the question and prove the results for the Fourier integral operators.
Our main result could be stated as follows.
Theorem 1.1
Let\(T_{a,\varphi }\)be a Fourier integral operator given by (1.1) with symbol\(a(x,\xi )\in L^{\infty }S^{n(\rho -1)/2}_{\rho }(\omega )\)and phase function\(\varphi \in L^{\infty }\varPhi ^{2}\)satisfying the Lipschitz rough non-degeneracy condition. Then, for\(0\leq \rho \leq 1\), there exists a positive constantCsuch that
Here, the symbol class \(L^{\infty }S^{n(\rho -1)/2}_{\rho }(\omega )\) is defined by Definition 2.2, the phase class \(L^{\infty }\varPhi ^{2}\) is given by Definition 2.5, and the Lipschitz rough non-degeneracy condition is defined by Definition 2.6.
Remark 1.1
Here we remark that, for \(a\in S^{n(\rho -1)/2}_{\rho ,1}\), Higuchi and Nagase [13] pointed out that the boundedness of the pseudo-differential operator \(T_{a}\) from \(L^{2}(\mathbb{R}^{n})\) to \(L^{2}(\mathbb{R}^{n})\) is not always true. As the main result in this paper, we give an answer for this problem for the Fourier integral operator \(T_{a,\varphi }\). The main idea of our approach is treating the symbol class \(L^{\infty }S^{m}_{\rho }(\omega )\), where \(m=n(\rho -1)/2\). In particular, our results of \(L^{2}(\mathbb{R}^{n})\)-boundedness for \(T_{a,\varphi }\) are also the best as far as we know. We also remark that our methods are different from the previous methods; see, for example, [13].
Finally, we make some conventions on notation. Throughout this article, we denote by C a positive constant which is independent of the main parameters, but it may vary from line to line. We sometimes write \(A\lesssim B\) as shorthand for \(A\leq CB\). Let \(\mathbb{R}^{n}\) be an n-dimensional Euclidean space, \(x=(x_{1},\ldots ,x_{n})\) be a point in \(\mathbb{R}^{n}\), \(\mathbb{R}^{n}_{*}=\mathbb{R}^{n}\setminus \{0\}\), \(\mathbb{N}=\{1, 2,\ldots \}\), \(\mathbb{Z}_{+}=\mathbb{N}\cup \{0\}\), and \(\mathbb{Z}_{+}^{n}=(\mathbb{Z}_{+})^{n}\). For any multi-index \(\alpha = (\alpha _{1},\ldots ,\alpha _{n})\) and \(\beta = (\beta _{1},\ldots ,\beta _{n})\in \mathbb{Z}_{+}^{n}\), we let
and \(\nabla _{\xi }=(\partial _{\xi _{1}},\ldots ,\partial _{ \xi _{n}})\). Also, in the sequel we use the notation
2 Definitions, notations, and preliminaries
The following definition is just [17].
Definition 2.1
Let \(m\in \mathbb{R}\) and \(0\leq \delta \), \(\rho \leq 1\). For any two multi-indices α and β, we assume that the function \(a(x,\xi )\) satisfies the following condition:
where \(C_{\alpha \beta }\) is a positive constant only dependent on α and β. Let the smooth amplitude \(S^{m}_{\rho ,\delta }\) be the set of all smooth functions \(a(x,\xi )\) satisfying condition as in (2.1). Then the pseudo-differential operator \(T_{a}\) with the symbol \(a(x,\xi )\in S^{m}_{\rho ,\delta }\) is given formally by
The following definition for the class \(L^{\infty }S^{m}_{\rho }(\omega )\) plays an important role in our setting.
Definition 2.2
Let m be a real number. A function \(a(x,\xi )\), which is smooth in the frequency variable ξ and bounded measurable in the spatial variable x, belongs to the symbol class \(L^{\infty }S^{m}_{\rho }(\omega )\) if, for all multi-indices α, it satisfies
where \(\omega (t)\) satisfies
and \(\omega (t)\) is a nonnegative and decreasing function on \([1,\infty )\).
Remark 2.1
If \(\omega (t)\) satisfies (2.1), then \(\sum_{j=0}^{\infty }\omega ^{2}(2^{j})<\infty \).
David and Wolfgang [14] gave the class \(\varPhi ^{k}\) as follows.
Definition 2.3
([14], \(\varPhi ^{k}\))
A real-valued function \(\varphi (x,\xi )\) belongs to the class \(\varPhi ^{k}\) if \(\varphi (x,\xi )\in C^{\infty }(\mathbb{R}^{n}\times \mathbb{R}^{n}_{*})\) is positively homogeneous of degree 1 in the frequency variable ξ and satisfies the following condition: for any pair of multi-indices α and β, satisfying \(|\alpha |+|\beta |\geq k\), there exists a positive constant \(C_{\alpha ,\beta }\) such that
In connection to the problem of local boundedness of Fourier integral operators, one considers phase functions \(\varphi (x,\xi )\) that are positively homogeneous of degree 1 in the frequency variable ξ for which
The latter is referred to as the non-degeneracy condition. However, for the purpose of proving global regularity results, we require a stronger condition than the non-degeneracy condition above.
Definition 2.4
([14], The strong non-degeneracy condition)
A real-valued function \(\varphi (x,\xi )\in C^{2}(\mathbb{R}^{n}\times \mathbb{R}^{n}_{*})\) satisfies strong non-degeneracy condition if there exists a positive constant c such that
for all \((x,\xi )\in \mathbb{R}^{n}\times \mathbb{R}^{n}_{*}\).
Remark 2.2
The phases in class \(\varPhi ^{2}\) satisfying the strong non-degeneracy condition arise naturally in the study of the equations of hyperbolic type, namely
belongs to the class \(\varPhi ^{2}\) and satisfies the strong non-degeneracy condition.
In [14], they introduced the nonsmooth version of the class \(\varPhi ^{k}\) which will be used in our setting.
Definition 2.5
([14], \(L^{\infty }\varPhi ^{k}\))
A real-valued function \(\varphi (x,\xi )\) belongs to the phase class \(L^{\infty }\varPhi ^{k}\) if it is positively homogeneous of degree 1 and smooth on \(\mathbb{R}^{n}_{*}\) in the frequency variable ξ, bounded measurable in the spatial variable x, and if for all multi-indices \(|\alpha |\geq k\) it satisfies
Motivated by [14], we also need a Lipschitz rough non-degeneracy condition as follows.
Definition 2.6
(The Lipschitz rough non-degeneracy condition)
A real-valued function satisfies Lipschitz rough non-degeneracy condition if it is \(C^{\infty }\) on \(\mathbb{R}^{n}_{*}\) in the frequency variable ξ, bounded measurable in the spatial variable x, and there exist positive constants \(C_{1}\) and \(C_{2}\) such that, for all \(x,y\in \mathbb{R}^{n}\) and \(\xi \in \mathbb{R}^{n}_{*}\),
3 Proof of the main result
In this section, we shall prove the main result, i.e., Theorem 1.1.
First we need a dyadic partition of unity. Let A be the annulus \(A=\{\xi \in \mathbb{R}^{n};\frac{1}{2}\leq |\xi |\leq 2\}\) and
where \(\chi _{0}(\xi )\in C^{\infty }_{0}(B(0,2))\) and \(\chi _{j}(\xi )=\chi (2^{-j}\xi )\) when \(j\geq 1\) with \(\chi (\xi )\in C^{\infty }_{0}(A)\). Now we decompose the operator \(T_{a,\varphi }\) as follows:
The first term in (3.1) is bounded on \(L^{2}(\mathbb{R}^{n})\) from Theorem 1.1.8 in [14]. After a change of variables, we have
The kernel of the operator \(T_{j}(D)\) is given by
Let
Then
and it satisfies
We can confine ourselves to dealing with the high frequency component \(T_{j}\) of \(T_{a,\varphi }\). Here we shall use a \(S_{j}=T_{j}T_{j}^{*}\) argument, and therefore,
The kernel of the operator \(S_{j}=T_{j}T_{j}^{*}\) reads
Let \(b_{j}(x,y,\xi )= \chi _{j}^{2}(2^{j\varrho }\xi )a(x,2^{j\varrho } \xi )\overline{a(y,2^{j\varrho }\xi )}\). Then
We claim that
In fact,
Next we consider the following differential operators for \(j\in \mathbb{N}\):
where \(\varPhi (x,y,\xi )=\varphi (x,\xi )-\varphi (y,\xi )\). So \(L_{j}^{N}(x,y,D)e^{i2^{j\varrho }\varPhi }=e^{i2^{j\varrho }\varPhi }\) and
From this and (3.4), it follows that
Moreover, by (3.5), we see that
which implies that
Thus
where \(\varPhi _{\mu _{k}}=\partial _{\xi _{\mu _{k}}}\varPhi \). Because of the following equation
by Definition 2.6, we see that
From this, together with (3.3) and (3.6), we further obtain
Integration by parts yields
Thus,
which implies that
This further gives
By Young’s inequality, we obtain
Therefore, we have
Namely,
Next we need a Littlewood–Paley decomposition. Let \(\psi _{0}:\mathbb{R}^{n}\rightarrow \mathbb{R}\) be a smooth radial function which is equal to one on the unit ball centric at the origin and supported on its concentric double. Set \(\psi (\xi )=\psi _{0}(\xi )-\psi _{0}(2\xi )\) and \(\psi _{k}(\xi )=\psi (2^{-k}\xi )\). Then
and \(\operatorname{supp}\psi _{k}(\xi )\subset \{\xi :2^{k-1}\leq |\xi |\leq 2^{k+1} \}\) for \(k\geq 1\). And we further have
Then
For simplicity of notation, we write
where
From this, (3.7), Cauchy–Schwartz’s inequality, and Remark 2.1, it follows that
This finishes the proof of Theorem 1.1.
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Acknowledgements
The authors would like to thank the referees for their important comments and remarks.
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The research of the first author is supported by the Natural Science Foundation of Xinjiang Urgur Autonomous Region (2019D01C049, 62008031, 042312023) and the National Natural Science Foundation of China (11561065). The research of the second author is supported by the National Natural Science Foundation of China (11131005). The research of the third author is supported by the National Natural Science Foundation of China (11826202).
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Yang, J., Chen, W. & Zhou, J. On \(L^{2}\)-boundedness of Fourier integral operators. J Inequal Appl 2020, 173 (2020). https://doi.org/10.1186/s13660-020-02439-0
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DOI: https://doi.org/10.1186/s13660-020-02439-0