Abstract
We prove the global \(L^p\)-boundedness of Fourier integral operators that model the parametrices for hyperbolic partial differential equations, with amplitudes in classical Hörmander classes \(S^{m}_{\rho , \delta }(\mathbb {R}^n)\) for parameters \(0\le \rho \le 1\), \(0\le \delta <1\). We also consider the regularity of operators with amplitudes in the exotic class \(S^{m}_{0, \delta }(\mathbb {R}^n)\), \(0\le \delta < 1\) and the forbidden class \(S^{m}_{\rho , 1}(\mathbb {R}^n)\), \(0\le \rho \le 1.\) Furthermore we show that despite the failure of the \(L^2\)-boundedness of operators with amplitudes in the forbidden class \(S^{0}_{1, 1}(\mathbb {R}^n)\), the operators in question are bounded on Sobolev spaces \(H^s(\mathbb {R}^n)\) with \(s>0.\) This result extends those of Y. Meyer and E. M. Stein to the setting of Fourier integral operators.
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1 Introduction
In this paper we investigate the local and global regularity of Fourier integral operators (FIOs) of the form
with amplitudes in Hörmander classes \(S^m_{\rho , \delta }(\mathbb {R}^n)\) consisting of functions in \(\mathcal {C}^{\infty }(\mathbb {R}^n \times \mathbb {R}^n)\) satisfying
for \(0\le \rho \le 1\), \(0\le \delta \le 1\). More specifically we consider the boundedness in \(L^p(\mathbb {R}^n)\) and \(H^s(\mathbb {R}^n)\) (Sobolev spaces) for FIOs that model the parametrices of variable coefficient wave equations where the \(\mathrm {rank}\,(\partial ^2_{\xi \xi }\varphi (x,\xi ))=n-1\). The corresponding investigation for the other extreme case, i.e. \(\mathrm {rank}\,(\partial ^2_{\xi \xi }\varphi (x,\xi ))=0\) which is the pseudodifferential operator-case, was carried out by J. Alvarez, and J. Hounie in [1]. In that paper the authors consider the \(L^p\)-boundedness of pseudodifferential operators with symbols in \(S^m_{\rho , \delta }(\mathbb {R}^{n})\) where \(0<\rho \le 1\), \(0\le \delta < 1\). It was then shown that for these ranges of \(\rho \) and \(\delta \) and \(1<p<\infty \) one has the \(L^p\)-boundedness of pseudodifferential operators provided that
However, we shall also emphasize that here we are not putting any assumptions on \(\mathrm {rank}\,(\partial ^2_{\xi \xi }\varphi (x,\xi ))\) in our theorems. Therefore the results that we obtain are also valid for pseudodifferential operators, although they are not necessarily optimal for these kind of operators. For instance, we obtain \(L^p\)-boundedness results for pseudodifferential operators with exotic amplitudes i.e. those with amplitudes in \(S^m_{0,\delta }(\mathbb {R}^n)\), \(0\le \delta <1\) (which was missing in [1]). The case of \(\rho =\delta =0\) was treated (using methods that are different than ours) by R. Coifman and Y. Meyer in [5], see also [24].
The first main result of this paper regarding the \(L^p\)-boundedness of FIOs is
Theorem 1
Let \(n\ge 1\), \(0\le \rho \le 1\), \(0 \le \delta < 1\),
and \(a\in S^m_{\rho ,\delta }(\mathbb {R}^{n})\). Assume that \(\varphi (x,\xi )\in \mathcal {C}^{\infty }(\mathbb {R}^n \times \mathbb {R}^n {\setminus }\{0\})\) is a strongly non-degenerate phase function, positively homogeneous of degree one in the frequency variable \(\xi \) satisfying the following estimate
for any pair of multi-indices \(\alpha \) and \(\beta \), with \(|\alpha |+|\beta |\ge 2.\) Then the FIO \(T_a^\varphi \) is \(L^p\)-bounded for \(1<p<\infty .\)
Prior to this investigation, the only source for results regarding \(L^p\)-regularity of FIOs in \(S^m_{\rho , \delta }\)-classes were those by A. Seeger. C. Sogge and E. M. Stein [23], where the authors established the local \(L^p\)-boundedness for \(\rho \in [1/2,1]\) and \(\delta =1-\rho .\)
Regarding global \(L^p\)-boundedness, the results of M. Ruzhansky and M. Sugimoto [22] are global extensions of those of Seeger-Sogge-Stein, however they are confined to the amplitudes with \(\rho =1\) and \(\delta =0\).
If one goes outside the aforementioned Hörmander classes of operators, then global boundedness results have been proven in various settings for example in the papers by S. Coriasco and M. Ruzhansky [9, 10] and E. Cordero, F. Nicola and L. Rodino [7, 8]. Recently, A. Hassell, P. Portal and J. Rozendaal [13] obtained results regarding global boundedness of Fourier integral operators, that go beyond those in [22]. More precisely in [13] the authors also establish the regularity of FIOs with amplitudes that decay faster than those in \(S^m_{\rho , 1-\rho }(\mathbb {R}^n)\) (with \(\rho \in [1/2,1]\)), when differentiated in the radial direction in the frequency variables. In [11] D. Dos Santos Ferreira and W. Staubach considered amplitudes in very rough classes (that also contain all the Hörmander classes \(S^m_{\rho , \delta }(\mathbb {R}^n)\)), and proved global \(L^p\)-boundedness of corresponding FIOs. However, due to the roughness of the amplitudes, the order m (which depends on \(\rho \) and \(\delta \)) is not as good as the expected one for smooth amplitudes, and further work is needed to achieve the right order of decay required for the \(L^p\)-boundedness of, for example, FIOs that yield parametrices for variable coefficient wave equations.
As one of the justifications of this investigation, we would like to mention that the work of R. Melrose and M. Taylor [18], and also the study of FIOs on certain nilpotent Lie groups (other than the Heisenberg group) motivates the consideration of FIOs with amplitudes in Hörmander classes \(S^m_{1/3, 2/3}(\mathbb {R}^n)\), for which, so far, no \(L^p\)-boundedness results have been available. Regarding \(L^2\)-boundedness of operators with general Hörmander-class amplitudes, in [11] Dos Santos Ferreira and Staubach showed that \(T^\varphi _a\) is globally \(L^2\)-bounded, provided that \(\rho , \delta \in [0,1]\), \(\delta \ne 1\) and \(m=\min (0,n(\rho -\delta )/2)\), or \(\rho \in [0,1],\) \(\delta =1\) and \(m<n(\rho -1)/2\). This result is sharp. In this paper we also discuss the global \(L^p\)-boundedness of FIOs with forbidden amplitudes \(S^m_{\rho ,1}(\mathbb {R}^n)\), \(0\le \rho \le 1\). For instance, it was shown in [11, Theorem 2.17] that FIOs with strongly non-degenerate phase functions and amplitudes in \(S^m_{1,1}(\mathbb {R}^n)\) are \(L^p\)-bounded if and only if \(m<-(n-1)|1/p-1/2|,\) a result which parallels the well-known facts about pseudodifferential operators with forbidden symbols. However, the endpoint case of \(S^0_{1,1}(\mathbb {R}^{n})\), which is not covered by the results above, is of particular interest. Indeed as Y. Meyer [19] and E. Stein (unpublished) have shown, despite the lack of, say \(L^2\)-boundedness ([24, Prop. 2, p. 272]), the pseudodifferential operators with symbols in \(S^0_{1,1}(\mathbb {R}^{n})\) map Sobolev spaces \(H^s(\mathbb {R}^n)\) with \(s>0\) continuously to themselves. This remarkable fact has had a large impact on the applications of J. M. Bony’s paradifferential calculus [2] to a systematic study of various nonlinear partial differential equations, see [25] for a comprehensive presentation. Our second main result in this paper is
Theorem 2
Let \(a\in S^0_{1,1}(\mathbb {R}^{n})\) and \(\varphi \) satisfy the same conditions as in Theorem 1 above. Then for \(s>0\) the FIO \(T_a^\varphi \) is bounded from the Sobolev space \(H^{s}(\mathbb {R}^n)\) to \(H^{s}(\mathbb {R}^n)\).
The paper is organised as follows; in Sect. 2 we recall some definitions, facts and results from microlocal and harmonic analysis that will be used throughout the paper. In Sect. 3 we reduce the FIOs to a form that is amenable for Ruzhansky-Sugimoto’s globalisation technique, which will in turn be adapted to general classes of Hörmander-class amplitudes. In Sect. 4 we first prove a general composition formula for the left-action of a Fourier multiplier on an FIO with amplitude in general Hörmander classes. Our result extends the known results to the global setting and all values of \(\rho , \delta \) (although the case of \(\delta =1\) has to be excluded). Thereafter, in Sect. 5, we extend the method of Seeger-Sogge-Stein to the case of FIOs with general classical Hörmander-class amplitudes, and decompose the Fourier integral operators into certain pieces for which we establish the basic kernel estimates. In Sect. 6 we prove Theorem 1 above (for FIOs with amplitudes in the exotic and classical Hörmander classes) as a combination of the results of Propositions 6.2 and 6.4. Finally in Sect. 7 we prove the \(H^s\)-boundedness for FIOs with amplitudes in the forbidden class \(S^0_{1,1}(\mathbb {R}^n)\) for \(s>0\) and thereby extend the result of Meyer and Stein to the FIO-setting. This is done by using an auxiliary class of amplitudes \(C^{r}_{*}S^0_{1,1}(\mathbb {R}^{n})\) which contains the class \(S^0_{1,1}(\mathbb {R}^n)\), for all \(r>0\) and showing that for \(\max (0, r-1/2)<s<r\), FIOs with amplitudes in \(C^{r}_{*}S^0_{1,1}(\mathbb {R}^{n})\) map \(H^{s}(\mathbb {R}^n)\) to itself.
2 Preliminaries
As is common practice, we will denote positive constants in the inequalities by C, which can be determined by known parameters in a given situation but whose value is not crucial to the problem at hand. Such parameters in this paper would be, for example, m, p, s, n, and the constants connected to the seminorms of various amplitudes or phase functions. The value of C may differ from line to line, but in each instance could be estimated if necessary. We also write \(a\lesssim b\) as shorthand for \(a\le Cb\) and moreover will use the notation \(a\approx b\) if \(a\lesssim b\) and \(b\lesssim a\).
Definition 2.1
Let \(\psi _0 \in \mathcal C_c^\infty (\mathbb {R}^n)\) be equal to 1 on B(0, 1) and have its support in B(0, 2). Then let
where \(j\ge 1\) is an integer and \(\psi (\xi ) := \psi _1(\xi )\). Then \(\psi _j(\xi ) = \psi \left( 2^{-(j-1)}\xi \right) \) and one has the following Littlewood-Paley partition of unity
It is sometimes also useful to define a sequence of smooth and compactly supported functions \(\Psi _j\) with \(\Psi _j=1\) on the support of \(\psi _j\) and \(\Psi _j=0\) outside a slightly larger compact set. One could for instance set
with \(\psi _{-1}:=\psi _{0}\).
In what follows we define the Littlewood-Paley operators by
where denotes the normalised Lebesgue measure \({\, \mathrm {d}\xi }/{(2\pi )^n}\) and
is the Fourier transform of f. Using the Littlewood-Paley decomposition of Definition 2.1, we define the Sobolev space \(H^s(\mathbb {R}^{n})\) in a somewhat unusual way. One can however show that this is equivalent to the standard definition of \(H^s(\mathbb {R}^{n})\).
Definition 2.2
Let \(s \in {\mathbb R}\). The Sobolev space is defined by
where \(\mathscr {S}'(\mathbb {R}^n)\) denotes the space of tempered distributions.
Remark 2.3
Different choices of the sequence \(\{\psi _j\}_{j=0}^\infty \) in Definition 2.1 give equivalent norms of \(H^s(\mathbb {R}^n)\) in Definition 2.2, see e.g. [26]. We will use either \(\{\psi _j\}_{j=0}^\infty \) or \(\{\Psi _j\}_{j=0}^\infty \) to define the norm of \(H^s(\mathbb {R}^n)\).
Remark 2.4
By Fubini’s theorem, one can change the order of the norms in Definition 2.2, i.e.
Also, using fairly standard Littlewood-Paley theory one can show the following well-known result:
Lemma 2.5
Let \(\{f_j\}_{j=0}^\infty \subset {\mathscr {S}}'(\mathbb {R}^{n})\) be such that
Then, for \(s > 0\), one has
For a proof, see e.g. [25].
In proving the \(L^p\)-boundedness of FIOs (\(1<p<\infty \)), the standard procedure is to first show the boundedness of the operator (and its adjoint) from the Hardy space \(\mathscr {H}^1(\mathbb {R}^n)\) to \(L^1(\mathbb {R}^n)\) and thereafter interpolate the results with the \(L^2\)-boundedness. In proving the Hardy space boundedness, the main tool is to use the so-called Hardy space atoms.
Definition 2.6
Let \(p\in (0,1].\) A function \(\mathfrak {a}\) is called an \(\mathscr {H}^p\)-atom if for some \(x_0\in \mathbb {R}^n\) and \(r>0\) the following three conditions are satisfied:
-
(i)
\({{\,\mathrm{supp}\,}}\mathfrak {a}\subset B(x_{0}, r)\),
-
(ii)
\( |\mathfrak {a}(x)|\le |B(x_{0}, r)|^{-1/p},\)
-
(iii)
\( \int _{\mathbb {R}^n} x^{\alpha }\, \mathfrak {a}(x)\, \mathrm {d}x=0\) for all \(|\alpha |\le N\) for some \(N\ge n(1/p-1)\).
Then a distribution \(f\in \mathscr {H}^p (\mathbb {R}^n)\), has an atomic decomposition
where the \(\lambda _{j}\) are constants with
and the \(\mathfrak {a}_{j}\) are \(\mathscr {H}^p\)-atoms.
Remark 2.7
Different choices of N in (iii) above give equivalent definitions of the \(\mathscr {H}^p\)-norm.
Next we define the building blocks of the FIOs and the pseudodifferential operators. These are the amplitudes (symbols in the pseudodifferential setting) and the phase functions. The class of amplitudes considered in this paper were first introduced by L. Hörmander in [14].
Definition 2.8
Let \(m\in \mathbb {R}\) and \(\rho , \delta \in [0,1]\). An amplitude (symbol) \(a(x,\xi )\) in the class \(S^m_{\rho ,\delta }(\mathbb {R}^n)\) is a function \(a\in \mathcal {C}^\infty (\mathbb {R}^n\times \mathbb {R}^n)\) that verifies the estimate
for all multi-indices \(\alpha \) and \(\beta \) and \((x,\xi )\in \mathbb {R}^n\times \mathbb {R}^n\), where \(\langle \xi \rangle := (1+|\xi |^2)^{1/2}.\) We shall henceforth refer to m as the order of the amplitude. Following the folklore in harmonic and microlocal analysis, we shall refer to the class \(S_{0,\delta }^m(\mathbb {R}^{n})\) as the exotic class and to \(S_{\rho ,1}^m(\mathbb {R}^{n})\) as the forbidden class of amplitudes.
Towards the end of this paper, in connection with the amplitudes with low spatial regularity and also the forbidden amplitudes, we will use the Zygmund class \(C_{*}^{r} (\mathbb {R}^n)\) whose definition we now recall.
Definition 2.9
Let \(r \in \mathbb {R}\). The Zygmund class is defined by
If \(C^r(\mathbb {R}^n)\), \(r\in \mathbb {R}_+\), denotes the Hölder space, and \(\mathcal {C}^r(\mathbb {R}^n)\) denotes the space of continuous functions with continuous derivatives of orders up to and including r, then one also has that
In connection to the definition of the Zygmund class, there is another class of amplitudes which have low regularity in the x-variable, which were considered by G. Bourdaud in [3].
Definition 2.10
Let \(m\in \mathbb {R}\), \(0\le \delta \le 1\) and \(r>0\). An amplitude (symbol) \(a(x,\xi )\) is in the class \(C_{*}^{r} S_{1, \delta }^{m}(\mathbb {R}^n)\) if it is \(\mathcal {C}^\infty ( \mathbb {R}^n)\) in the \(\xi \) variable and verifies the estimates
and
for all multi-indices \(\alpha \) and \(\xi \in \mathbb {R}^n\). Here \(C_{*}^{r}(\mathbb {R}^n)\) is the Zygmund class of Definition 2.9.
It is important to note that \(S^m_{1,1} (\mathbb {R}^n)\subset C_{*}^{r} S_{1,1}^{m}(\mathbb {R}^n),\) for all \(r>0\), which follows from (3).
Given the symbol classes defined above, one associates to the symbol its Kohn-Nirenberg quantisation as follows:
Definition 2.11
Let a be a symbol. Define a pseudodifferential operator (\(\Psi \mathrm {DO}\) for short) as the operator
a priori defined on the Schwartz class \(\mathscr {S}(\mathbb {R}^n).\)
In order the define the Fourier integral operators that are studied in this paper, following [11], we also define the classes of phase functions.
Definition 2.12
A phase function \(\varphi (x,\xi )\) in the class \(\Phi ^k\) is a function \(\varphi (x,\xi )\in \mathcal {C}^{\infty }(\mathbb {R}^n \times \mathbb {R}^n {\setminus }\{0\})\), positively homogeneous of degree one in the frequency variable \(\xi \) satisfying the following estimate
for any pair of multi-indices \(\alpha \) and \(\beta \), satisfying \(|\alpha |+|\beta |\ge k.\) In this paper we will mainly use phases in class \(\Phi ^2\) and \(\Phi ^1\).
We will also need to consider phase functions that satisfy certain non-degeneracy conditions. These conditions have to be adapted to the case of local and global boundedness in an appropriate way. Following [23], in connection to the investigation of the local results, that is, under the assumption that the x-support of the amplitude \(a(x,\xi )\) lies within a fixed compact set \(\mathcal {K}\), the non-degeneracy condition is formulated as follows:
Definition 2.13
Let \(\mathcal {K}\) be a fixed compact subset of \(\mathbb {R}^n\). One says that the phase function \(\varphi (x,\xi ) \) satisfies the non-degeneracy condition if
Following the approaches in e.g. [11, 21, 22], for the global \(L^p\)-boundedness results that were established in those papers, we also define the following somewhat stronger notion of non-degeneracy:
Definition 2.14
One says that the phase function \(\varphi (x,\xi )\) satisfies the strong non-degeneracy condition (or \(\varphi \) is \(\mathrm {SND}\) for short) if
Having the definitions of the amplitudes and the phase functions at hand, one has
Definition 2.15
A Fourier integral operator (\(\mathrm {FIO}\) for short) \(T_a^\varphi \) with amplitude a and phase function \(\varphi \), is an operator defined (once again a-priori on \(\mathscr {S}(\mathbb {R}^n)\)) by
where \(\varphi (x,\xi )\in \mathcal {C}^{\infty }(\mathbb {R}^n \times \mathbb {R}^n{\setminus }\{0\})\) and is positively homogeneous of degree one in \(\xi \).
In this paper, the basic \(L^2\)-boundedness result which we shall utilise for the FIOs, is the following proposition which could be found in [11] as Theorems 2.2 and 2.7.
Proposition 2.16
Let \(\rho , \delta \in [0,1]\), \(\delta \ne 1\). Assume that \(a(x,\xi )\in S^{m}_{\rho ,\delta }(\mathbb {R}^n)\) and \(\varphi (x,\xi )\) is in the class \(\Phi ^2\) and is SND. Then the FIO \(T_a^\varphi \) is bounded on \(L^2(\mathbb {R}^n)\) if and only if \(m=-n \, \max (0,(\delta -\rho )/2)\). In case \(\rho \in [0,1],\) \(\delta =1\) then the \(L^2\)-boundedness is valid if and only if \(m<n(\rho -1)/2\).
A global result concerning the boundedness of FIOs with amplitudes of order zero, which will be used in the proof of Proposition 7.2 goes as follows:
Lemma 2.17
Let \(a (x,\xi )\in S_{1,0}^{0}(\mathbb {R}^n)\). Assume also that \(\varphi (x,\xi ) \in \Phi ^2,\) is \(\mathrm {SND}\). Then for \(s\in \mathbb {R}\), the FIO \(T_a^\varphi \) is bounded from the Sobolev space \(H^{s}(\mathbb {R}^n)\) to \(H^{s}(\mathbb {R}^n)\).
Proof
This follows immediately from [16, Theorem 5.7 part (ii)], by noting that the Besov-Lipschitz space \(B^s_{p,q}(\mathbb {R}^n)\) in that result reduces to the Sobolev space \(H^s(\mathbb {R}^n)\) when \(p=q=2\). \(\square \)
We also state the following version of the non-stationary phase lemma, whose proof can be found in [21, Lemma 3.2].
Lemma 2.18
Let \(\mathcal {K}\subset \mathbb {R}^n\) be a compact set and \(\Omega \supset \mathcal {K}\) an open set. Assume that \(\Phi \) is a real valued function in \(\mathcal C^{\infty }(\Omega )\) such that \(|\nabla \Phi |>0\) and
for all multi-indices \(\alpha \) with \(|\alpha |\ge 1\). Then, for any \(F\in \mathcal C^\infty _c (\mathcal {K})\), \(\lambda >0\), and any integer \(k\ge 0\),
Finally we recall a composition result, whose proof can be found in [20, Theorem 4.2], or in a more general setting in [4, Theorem 3.11], which will enable us to keep track of the parameter while a parameter-dependent \(\Psi \)DO acts from the left on a parameter-dependent FIO. This will be crucial in the proof of the boundedness of FIOs with forbidden amplitudes on Sobolev spaces.
Proposition 2.19
Let \(m \le 0\), \(\displaystyle 0<\varepsilon <1/2\) and \(\Omega := \mathbb {R}^n \times \{|\xi | > 1\}\). Suppose that \( a_t(x, \xi )\in S^m_{1,0} (\mathbb {R}^n)\) uniformly in \(t \in (0, 1]\) and it is supported in \(\Omega \), \(b(\xi )\in S^0_{1,0}(\mathbb {R}^n)\) and \(\varphi \in \mathcal C^\infty (\Omega )\) is such that
-
(i)
for constants \(C_1, C_2 > 0\), \(C_1|\xi | \le |\nabla _x \varphi (x, \xi )| \le C_2|\xi |\) for all \((x, \xi ) \in \Omega \), and
-
(ii)
for all \(|\alpha |, |\beta | \ge 1\), \(|\partial _x^\alpha \varphi (x, \xi )|\lesssim \langle \xi \rangle \) and \(|\partial _\xi ^\alpha \partial _x^\beta \varphi (x, \xi )| \lesssim |\xi |^{1-|\alpha |}\), for all \((x, \xi ) \in \Omega \).
Consider the parameter dependent Fourier integral operator \(T_{a_t}^\varphi \), given by (6) with amplitude \(a_t(x,\xi )\), and the parameter dependent Fourier multiplier
Then the composition \(b (tD)T_{a_t}^\varphi \) is also an FIO with phase \(\varphi \) and amplitude \(\sigma _t\) which is given by
Moreover, for each \(M \ge 1\), we can write \(\sigma _t\) as
for \(t \in (0, 1)\). Moreover, for all multi-indices \(\beta , \gamma \) one has
and
3 Reduction
We start by describing how the problem of \(L^p\)-boundedness of FIOs with SND phase functions that belong to the class \(\Phi ^2\), can be reduced to the case of operators that are well-suited for the Ruzhansky-Sugimoto’s globalisation procedure.
Thus let \(T_a^{\varphi }\) be an FIO given by (6), with \(\varphi \in \Phi ^2\) and SND, and an amplitude \(a(x,\xi )\in S^{m}_{\rho , \delta }(\mathbb {R}^n)\) with \(m\in \mathbb {R}\), \(\rho , \delta \in [0,1]\). We start by localising the amplitude in the \(\xi \) variable by introducing an open convex covering \(\{U_l\}_{l=1}^{M}\) of the unit sphere \(\mathbb {S}^{n-1}\), where M is finite (due to compactness of \(\mathbb {S}^{n-1}\)) but large enough, and the sets \(U_l\) have diameters of at most d. Let \(\Xi _{l}\) be a smooth partition of unity subordinate to the covering \(U_l\) and set
Define
and fix a point \(\zeta _l \in U_l.\) Then for any \(\xi \in U_l\), Taylor’s formula and Euler’s homogeneity formula yield
where \(\lambda \) is the remainder term and its dependence on \(\zeta _l\) has been suppressed due the fact that \(\zeta _l\) was fixed. Thus for \(\xi \in U_l\), using the fact that \(\partial _{\xi _k} \varphi (x, \xi )\) is homogeneous of degree zero in \(\xi \), we have
and hence the mean-value theorem and the definition of class \(\Phi ^2\) yield
for all \(|\beta |\ge 0\), and for \(|\alpha |\ge 2\). One also observes that due to the homogeneity of \(\lambda (x,\xi )\) in \(\xi \) and the mean-value theorem, one also has that \(|\nabla _x\lambda (x, \xi )|\lesssim |\xi |\). We shall now extend the function \(\lambda (x,\xi )\) to the whole of \(\mathbb {R}^{n}\times \mathbb {R}^{n}{\setminus } \{ 0 \}\), preserving its properties and we denote this extension by \(\lambda (x,\xi )\) again. Now this \(\lambda \) belongs to the class \(\Phi ^1\). Hence the Fourier integral operators \(T_l\) defined by
are the localised pieces of the original Fourier integral operator T and therefore
Now, let us investigate the \(L^p\)-boundedness of each piece \(T_l\). To this end we observe that due to the SND assumption on \(\varphi \), the map \(\mathbf{t}_{l}(x):= \nabla _\xi \varphi (x,\zeta _l)\) is a global diffeomorphism and composing \(T_l f(x)\) with the inverse of \(\mathbf{t}_{l}\) results in the FIO
Observe that all the derivatives of \(\mathbf{t}^{-1}_{l}\) are bounded and indeed the phase function \(\lambda (\mathbf{t}^{-1}_{l}(x),\xi )+x\cdot \xi \) is SND (note also that the diameters d can be picked as small as we like). Therefore the study of the global \(L^p\)-boundedness of \(T_l\) is reduced to the study of the global \(L^p\)-boundedness of FIOs of the form
where \(\sigma (x,\xi )\) belongs to the same amplitude-class as \(a(x,\xi )\) (i.e. \(S^m_{\rho , \delta }(\mathbb {R}^n)\)), \(\theta \in \Phi ^1\) and \(\theta (x,\xi )+ x\cdot \xi \) being SND.
In a similar way, one can show that, for an FIO of the form
with \(\varphi \in \Phi ^2\), matters can be reduced to FIOs of the form
where \(\sigma (y,\xi )\) belongs to the same class as \(a(y,\xi )\) and \(\theta (y,\xi )\in \Phi ^{1}.\)
4 Composition of Fourier multipliers and FIOs
We start with a composition theorem which allows us to left-compose a Fourier multiplier with an FIO. The difference between Theorem 4.1 below and Proposition 2.19 lies in the fact that, although the latter deals with the parameter dependent case, it only covers amplitudes in \(S^m_{1,0}(\mathbb {R}^n)\). Also the method of proof of Proposition 2.19 is quite different from that of the following theorem whose proof is not just a modification of the former. The difficulties arise exactly when \(\delta \ge \rho ,\) but they can be overcome. However, as we shall see, the forbidden case of \(\delta =1\) has to be excluded.
Note that the composition theorem below also allows us to compose our more general FIOs with Bessel potential operators (see i.e. Lemma 6.3) in order to obtain crucial \(\mathscr {H}^q-L^2\) estimates that are in turn used in the proofs of the \(L^p\)-boundedness results (Proposition 6.4).
Theorem 4.1
Let \(m, m'\in \mathbb {R}\), \(\rho \in [0,1], \delta \in [0,1)\) and \(\Omega := \mathbb {R}^n \times {\{|\xi | > 1\}}\). Suppose that \( a(x, \xi )\in S_{\rho ,\delta }^m(\mathbb {R}^{n}) \), and it is supported in \(\Omega \), \(\gamma (\xi )\in S^{m'}_{1,0}(\mathbb {R}^{n})\) and \(\varphi \in \mathcal C^\infty (\Omega )\) is such that
-
(i)
for constants \(C_1, C_2 > 0\), \(C_1|\xi | \le |\nabla _x \varphi (x, \xi )| \le C_2|\xi |\) for all \((x, \xi ) \in \Omega \), and
-
(ii)
for all \(|\alpha |, |\beta | \ge 1\), \(|\partial _x^\alpha \varphi (x, \xi )|\lesssim \langle \xi \rangle \) and \(|\partial _\xi ^\alpha \partial _x^\beta \varphi (x, \xi )| \lesssim 1\), for all \((x, \xi ) \in \Omega \).
Consider the Fourier multiplier and the Fourier integral operator
Then the composition operator \(T_{b}^\varphi :=\gamma (D)T_{a}^\varphi \) is also an FIO (with the same phase as \(T^\varphi _a\)), and with amplitude given by
Moreover \(b\in S^{m+m'}_{\rho , \delta }(\mathbb {R}^{n})\).
Remark 4.2
It is easy to show that if a phase function \(\varphi \in \Phi ^2\) is SND then it satisfies all the requirements of Theorem4.1.
Proof of Theorem 4.1
The expression in (7) can easily be derived through a simple calculation. Set
and
Then the following estimates are valid:
Indeed the first equality is trivial, and for the second one, setting
we note that for \(|\xi |<R\) and \(R\ge 1\), triangle inequality and condition (i) on the phase yield that
On the other hand, for \(|\xi |<R\) and \(R<1\), condition (i) on the phase implies
To show the third estimate in (8) we observe that condition (i) on the phase yields
Finally to show the fourth estimate in (8) we observe that the mean-value theorem and condition (ii) on the phase yield that
Using Faà di Bruno’s formulae and estimate (8) we can also show that
Introduce the differential operators
and integrating by parts we have
for large positive N and M. Note that
for all multi-indices \(\alpha ,\, \beta ,\, \gamma \). It follow from this and (8) that \(b(x,\xi )\) is bounded on \(\mathbb {R}_{x}^n \times B(0,R)\) for \(R>0\), and using (9) one can also show that, for any multi-indices \(\alpha ,\, \beta \), \(\partial ^\alpha _\xi \partial ^{\beta }_x b(x, \xi )\) is also bounded on \(\mathbb {R}_{x}^n \times B(0,R)\) and therefore \(b(x,\xi )\in \mathcal {C}^{\infty }_{b}(\mathbb {R}_{x}^n \times B(0,R))\) (the subscript b indicates the boundedness of all the derivatives). It also follows from condition (i) on the phase that
Now let \(\chi \in \mathcal {C}_c^{\infty }(\mathbb {R}^n)\) be such that \(0\le \chi (x)\le 1\) and \(\chi (x)=1\) when \(|x|\le 1/2\) and \(\chi (x)=0\) when \(|x|\ge 2/3.\) Set
Since \(|\eta |\le 2 C_1\langle \xi \rangle /3\) on the support of \(\chi _1\), (11) yields (on \({{\,\mathrm{supp}\,}}\chi _1\))
At this point, we observe that since
it follows that there exists \(R_1>0\) and \(C>0\) such that on \({{\,\mathrm{supp}\,}}\chi _1 \cap \{|\xi |\ge R_1\}\) one has
Setting
and integrating by parts yields
Since \(b(x,\xi )\in \mathcal {C}^{\infty }_{b}(\mathbb {R}_{x}^n \times B(0,R))\), estimates (9) and (12) yield that \(b_1(x,\xi )\in S^{-\infty }(\mathbb {R}^{n})\) (first we show using (10) that \(|b_2(x, \xi )|\lesssim \langle \xi \rangle ^{-N}\), for all \(N\ge 0\), then using (9) we boost up this result to all the derivatives of \(b(x, \xi ),\) for more details on these type of calculations, see e.g. the somewhat similar proof of Theorem 3.11 in [4]).
Now define \(\chi _2: =1-\chi _1\) and consider
To simplify the calculations from here we set
Noting that \(\nabla _x \varphi (x,\xi )=\nabla _x \Lambda (x,\xi )+\xi \), rewriting
making the change of variables
and then defining
we obtain
Moreover, \(b_2(x,\xi )\) in turn can be split into \(b_3(x,\xi )+ b_4(x,\xi )\) with
and
We observe that on the support of \(\chi _2(\zeta , \xi )\) one has that
and on the support of \(\chi _2 (I(x,y,\xi )+ \xi +\zeta , \xi )\),
Therefore integrating by parts we can show that
to conclude that \(b_3(x,\xi )\in S^{-\infty }(\mathbb {R}^{n}).\)
Defining
we see that
and \(b_4(x,\xi )\) can be written as
Using the definition of \(c_2(y,\xi ,\eta )\) we see that
on the support of \(c_2(y,\xi ,\eta )\).
In what follows we shall denote the derivative of \(c_2(y,\xi ,\eta )\) with respect to y by \(\partial _1 c_2\), the derivative of \(c_2(y,\xi ,\eta )\) w.r.t. \(\xi \) by \(\partial _2 c_2\), and the derivative of \(c_2(y,\xi ,\eta )\) w.r.t. \(\eta \) by \(\partial _3 c_2\). Now using (13) and Taylor’s formula we have that
For \(s\in [0,1]\), set
Let us now study the behaviour of the derivatives of
To this end observe that
Now for \(M>n/2\) we write
and integration by parts yields that
with
We also have that
Divide the domain of integration in \(\zeta \) in (14) into three pieces \(\mathbf {A}:=\{|\zeta |\le \langle \xi \rangle ^\delta /2\}\), \(\mathbf {B}:=\{\langle \xi \rangle ^\delta /2\le |\zeta |\le \langle \xi \rangle /2\}\) and \(\mathbf {C}:=\{|\zeta |\ge \langle \xi \rangle /2\}.\)
We observe that for \(s\in [0,1]\), \(|\zeta |\le \langle \xi \rangle / 2\) we get
since \(|\partial _{\xi _j} \langle \xi +ts\zeta \rangle |\le 1\). This implies that for \(s\in [0,1]\), \(|\zeta |\le \langle \xi \rangle /2\) we have
Moreover (16) and (17) also yield that in \(\mathbf {A}\cup \mathbf {B}\) we have
Hence
Next we observe that since
one has
Therefore
On the set \(\mathbf {C}\) we have \(\langle \xi \rangle \le 2|\zeta |\) and for all \(s\in [0,1]\) that
Hence the definition of \(R_s\) in (15) and (16), (18), yield that
Therefore integrating by parts and choosing N so large that
and
(observe once again that \(\delta <1\)), we obtain
This concludes the proof. \(\square \)
5 Decomposition of the FIOs
In connection to the study of the \(L^p\)-regularity of FIOs, based on an idea of C. Fefferman [12], Seeger, Sogge and Stein [23] introduced a second dyadic decomposition superimposed on a preliminary Littlewood-Paley decomposition.
Note that, since we are dealing with FIOs with amplitudes in general Hörmander classes, the constructions in [23] have to be generalised to this setting. Here we follow the rendition of the Seeger-Sogge-Stein decomposition, as was given in Dos Santos Ferreira-Staubach [11].
To start, one considers an FIO \(T^{\varphi }_{a}\) with amplitude \(a(x, \xi )\in S^{m}_{\rho , \delta }(\mathbb {R}^n)\), \(0<\rho \le 1\), \(0\le \delta <1\), \(m\in \mathbb {R}\) and the SND phase \(\varphi \in \Phi ^2\) and its Littlewood-Paley decomposition
where the kernel \(K_j\) of \(T_j\) is given by
Here each \(\psi _j\) is supported in a dyadic shell \(\left\{ 2^{j-1}\le \vert \xi \vert \le 2^{j+1}\right\} \) (as in Definition 2.1) so the \(\xi \)-support of the integrand in the second integral is
The shells \(A_j\) will in turn be decomposed into truncated cones using the following construction:
Definition 5.1
For each \(j\in \mathbb {N}\) and \(0<\rho \le 1\), we fix a collection of unit vectors \(\big \{\xi ^{\nu }_{j}\big \} \) that satisfy the following two conditions.
-
(i)
\( \big | \xi ^{\nu }_{j}-\xi ^{\nu '}_{j} \big |\ge 2^{-j\rho /2},\) if \(\nu \ne \nu '\).
-
(ii)
If \(\xi \in \mathbb {S}^{n-1}\), then there exists a \( \xi ^{\nu }_{j}\) so that \(\big \vert \xi -\xi ^{\nu }_{j} \big \vert <2^{-j\rho /2}\).
To do this we take a collection \(\{\xi _{j}^{\nu }\}\) which is maximal with respect to the first property of Definition5.1.
Let \(\Gamma ^{\nu }_{j}\) denote the cone in the \(\xi \)-space, with the apex at the origin, whose central direction is \(\xi ^{\nu }_{j}\), i.e.
One also defines
where
and \(\phi \) is a non-negative function in \( \mathcal C_c^\infty (\mathbb {R}^n)\) with \(\phi (u)=1\) for \(|u|\le 1\) and \(\phi (u)=0\) for \(\left| {u}\right| \ge 2\).
Now turning back to the shells (20), we decompose each \(A_j\) into truncated cones \(\Gamma ^{\nu }_{j}\cap A_j\) and observe that since the diameter of the set \(\Gamma ^{\nu }_{j}\cap \{\xi ;\, |\xi |= 1\}\) is of size\(2^{-j\rho /2}\), using elementary geometry, it is clear that the diameter of the set \(\Gamma ^{\nu }_{j} \cap \{\xi ;\, |\xi |= 2^{j(1-\rho )+1}\}\) is of size \(2^{j(1-\rho )}2^{-j \rho / 2} \). Therefore, for each truncated cone \(\Gamma ^{\nu }_{j}\cap A_j\) there are \(n-1\) directions with length (roughly) equal to \(2^{j(1-\rho )}2^{-j \rho / 2} \) and one direction with length roughly equal to \(2^{j(1-\rho )}\) (which is the thickness of \(A_j\)). Hence we infer that
Using this, it also follows that there are \(O\big (2^{j\rho (n-1)/2}\big )\) such truncated cones needed to cover one shell \(A_j\), and thus there are at most \(O\big (2^{j \rho (n-1)/2}\big )\) elements in the collection \(\{\xi _{j}^{\nu }\}\), with regard to \(\nu \)’s.
For the following lemma and throughout the rest of the paper, we choose the coordinate axes in \(\xi \)-space such that \(\xi _1\) is in the direction of \(\xi ^{\nu }_{j}\) and \(\xi ':=(\xi _2 , \dots , \xi _{n})\) is perpendicular to \(\xi ^{\nu }_{j}\).
Lemma 5.2
The functions \(\chi _j^\nu \) belong to \(\mathcal C^\infty (\mathbb {R}^n{\setminus } \{0\} )\) and are supported in the cones \(\Gamma _j^\nu \). They sum to 1 in \(\nu \):
and moreover they satisfy the estimates
for all multi-indices \(\alpha \) and
Proof
In proving (23) we note that the argument of \(\eta _j^\nu \) contains a factor of \(2^{j\rho /2}\) followed by a factor that is homogeneous of degree zero. Hence \(\alpha \) derivatives yield a factor of \(2^{j\rho |\alpha |/2}\) and a function that is homogeneous of degree \(-\left| {\alpha }\right| \). To prove (24) one observes that in the support of \(\chi ^\nu _j\) one can write
where \(\partial _{r}\) is the radial derivative and \(\partial _{r}^N \chi ^\nu _j=0\) since \(\chi ^\nu _j\) is homogeneous of degree zero. \(\square \)
We will split the phase \(\varphi (x,\xi )-y\cdot \xi \) into two different pieces, \((\nabla _\xi \varphi (x,\xi _j^\nu )-y)\cdot \xi \) (which is linear in \(\xi \)), and \(\varphi (x,\xi ) - \nabla _\xi \varphi (x,\xi _j^\nu ) \cdot \xi \). The following lemma yields an estimate for the nonlinear second piece.
Lemma 5.3
For \(j, \nu \ge 1,\) define
Then for \(0<\rho \le 1\) and for \(\xi \) in \(A_j \cap \Gamma ^{\nu }_{j}\) (see (20) and (21)), one has that
and for \(N\ge 1\)
Proof
The proof is based on simple Taylor expansions and homogeneity considerations, see [24, p. 407]. Also observe that \(\left| {\xi '}\right| \lesssim 2^{j(1-3\rho /2)}\). \(\square \)
Remark 5.4
Note that by the previous lemma, one has the estimate
for all \(\xi \in \Gamma ^{\nu }_j,\) due to the fact that \(\nabla _{\xi } h^{\nu }_j (x, \xi )\) is homogeneous of degree zero in \(\xi .\)
In [23] the authors define an “influence set” associated to the SND phase function \(\varphi \). We have to make a similar definition but it has to be fitted to the more general classes of amplitudes that we are considering here. To this end we have
Definition 5.5
Assume that \(\varphi \) is an SND phase function in the class \(\Phi ^2\), and let \(\bar{y}\in \mathbb {R}^n\) be the centre of a ball B with radius \(r< 1\), and \(\rho \in (0,1]\). Define for \(j, \, \nu \ge 1,\)
where \(\pi _j^\nu \) is the orthogonal projection in the direction \(\xi _j^\nu \) and c is a large constant depending on the size of the Hessian matrix of \(\varphi \) but independent of j, to be specified later. We also define \(R_j^\nu \) as the preimage of \(\tilde{R}_j^\nu \) under the mapping \(x\rightarrow \nabla _\xi \varphi (x,\xi _j^\nu )\), i.e.
Now set
and recall from the discussion following Definition5.1 that the number of \(\nu \)’s in the union above is \(O\big (2^{j\rho (n-1)/2}\big ).\)
In this connection we have the following estimates.
Lemma 5.6
Let \(\bar{y}\in \mathbb {R}^n\) be the centre of a ball B with radius \(r<1\)
and let \(B^*\) be defined as in (27). Then for \(\rho \in (0,1]\)
-
(i)
One has the following estimate for the measure of \(B^*\)
$$\begin{aligned}\begin{aligned}\left| {B^*}\right| \lesssim r^{\rho }.\end{aligned}\end{aligned}$$ -
(ii)
If \(x\in \mathbb {R}^n {\setminus } B^*\), \(y\in B(\bar{y}, r)\) and the integer \(k\ge 1\) is uniquely chosen in such a way that \(2^{-(k-1)}\le r< 2^{-k}\), then for c in (26) large enough, we have
$$\begin{aligned} 2^{j\rho }\left| {(\nabla _\xi \varphi (x,\xi _j^\nu )-y)_1}\right| + 2^{j\rho /2}|(\nabla _\xi \varphi (x,\xi _j^\nu )-y)'| \gtrsim 2^{(j-k)\rho /2}, \quad j \ge k. \end{aligned}$$(28)
Proof
(i) Since \(R_j^\nu \) is of size \(O(2^{- j\rho })\) in the \(\xi _j^\nu \)-direction and \(O(2^{- j\rho /2})\) in the other \(n-1\) directions, we have for \(0<\rho \le 1\)
(ii) Observe that (28) is equivalent to
Moreover, it is enough to show that
Since \(\xi ^{\nu }_j \in \mathbb {S}^{n-1},\) by Definition 5.1 part (ii), there exists a unit vector \(\xi ^{\nu '}_k\) such that
Furthermore, since \(\mathbb {R}^n {\setminus } B^*\subset \mathbb {R}^n {\setminus } R_k^{\nu '}\) then the assumption \(x\in \mathbb {R}^n {\setminus } B^*\) yields that \(x\in \mathbb {R}^n {\setminus } R_k^{\nu '}\) and therefore by definition (26) we have
which in turn implies that
for c sufficiently large. Moreover we also have that
Indeed, we observe that since \(\varphi \in \Phi ^2\) one has that \(|\partial ^2_{\xi \xi }\varphi (x,\xi )|\lesssim |\xi |^{-1},\) for all x and all \(\xi \ne 0.\) Therefore the mean-value theorem and (30) yield that for some \(t\in [0,1]\) and all \(k\ge 1\) one has
Note that (32) trivially implies
We also claim that (30) yields
To see this, using the homogeneity of \(\varphi \) (dictated by the \(\Phi ^2\)-condition) we have
where
Moreover, by (25) one has the estimate
for all \(\xi \in \Gamma ^{\nu '}_{k}=\{\xi ; \big |\frac{\xi }{|\xi |}- \xi ^{\nu '}_k\big |<2\cdot 2^{-k\rho /2}\}\) (which is a cone with vertex at the origin and central direction \(\xi ^{\nu '}_{k}\)). Recalling that \(\xi ^{\nu }_j\) is in \(\mathbb {S}^{n-1}\), (30) shows that \(\xi ^{\nu }_{j}\) also belongs to the cone \(\Gamma ^{\nu '}_{k}\), and so for all \(t\in [0,1]\), the expression \(\xi ^{\nu }_j+t(\xi ^{\nu '}_k-\xi ^{\nu }_j)\) which represents the line segment joining \(\xi ^{\nu }_j\) and \(\xi ^{\nu '}_k\) belongs to \(\Gamma ^{\nu '}_{k},\) due to the convexity of the cone. Therefore, (35) yields that for all \(t\in [0,1]\)
Now we also observe that \(h^{\nu '}_k(x,\xi ^{\nu '}_k)=0\). Hence, using the mean-value theorem and (30), one readily sees that \(h^{\nu '}_k(x,\xi ^\nu _j)= O(2^{-k\rho }).\) For the term \( (\nabla _\xi \varphi (x,\xi ^{\nu }_j) - \nabla _\xi \varphi (x,\xi ^{\nu '}_k))\cdot (\xi ^{\nu '}_k- \xi ^{\nu }_j)\) we just use the Cauchy-Schwarz inequality, (30) and (32), which concludes the proof of (34).
Now having (33) we claim that
This is because our initial convention that \(\xi ^{\nu }_j\) lies along the \(\xi _1\)-axis, the triangle inequality, the Cauchy-Schwarz inequality, (30), (32) and (34), yield that
Finally to show (29) (which as we mentioned above implies the desired estimate (28)), we use the triangle inequality, (31), (33) and (36) to obtain
where the constant A stems from estimate (36) and C from (33). Therefore picking c large enough we obtain (28). \(\square \)
6 \(L^p\)-results
In this section we prove our main \(L^p\)-boundedness results for FIOs with general Hörmander-class amplitudes. This generalizes the results of Seeger–Sogge–Stein in two ways. First of all this is a global regularity result and opposed to the local one in [23]. Second, we consider all possible values of \(\rho \)’s and \(\delta \)’s as opposed to just \(\rho \in [1/2, 1]\) and \(\delta =1-\rho .\)
Remark 6.1
We note that in what follows we can confine ourselves to the case of amplitudes \(a(x, \xi )\) that vanish in a neighbourhood of the \(\xi =0\). To see this we take \(\psi _0 \in \mathcal C_c^\infty (\mathbb {R}^n)\) be equal to 1 on B(0, 1) and have its support in B(0, 2), and split the amplitude into the pieces \(a(x, \xi )\psi _0(\xi )+a(x, \xi )(1-\psi _0(\xi ))\) and observe that the first term is in \(S^m_{1,0}\) for all m. Thus as was shown in [11, Theorem 1.18], the FIO with amplitude \(a(x, \xi )\psi _0(\xi )\) is an \(L^p\)-bounded operator for \(1\le p\le \infty \). This means that our analysis can be concentrated on the amplitude \(a(x, \xi )(1-\psi _0(\xi ))\) which belongs to \(S^{m}_{\rho , \delta }(\mathbb {R}^n)\) and vanishes near \(\xi =0.\)
6.1 Exotic amplitudes
We start by proving the \(L^p\)-boundedness of exotic FIOs with amplitudes in \(a\in S^m_{\rho ,\delta }(\mathbb {R}^{n})\) with \(\rho =0\), \(0 \le \delta < 1\).
Proposition 6.2
Let \(n\ge 1\), \(0\le \delta <1\), \(a\in S^m_{0,\delta }(\mathbb {R}^{n})\), and assume that \(\varphi \in \Phi ^2\) is SND. Then for
the FIO \(T_a^\varphi \) is \(L^p\)-bounded for \(1<p<\infty .\)
Proof
Using the discussion in Sect. 3, we shall from now on assume that \(T_a^{\varphi }\) is of the form
where \(\theta \in \Phi ^1\) and \(\theta (x,\xi )+x\cdot \xi \) is SND.
Since the result has already been proven for the case when \(p=2\) (see Proposition 2.16) it only remains to show that T and its adjoint map \(\mathscr {H}^p(\mathbb {R}^{n})\) to \(L^p(\mathbb {R}^{n})\) continuously, for some \(p<1\), and thereafter interpolate these with the \(L^2\)-boundedness (see [6] for interpolation of Hardy spaces).
Due to the atomic decomposition in Definition 2.6 of an element of \(\mathscr {H}^p(\mathbb {R}^{n})\), we would need to show that
is uniformly bounded for every \(\mathscr {H}^p\)-atom \(\mathfrak {a}\), where the atom is supported in the ball \(B:=B(\bar{y} ,r)\). To prove the assertion in the case \(2<p<\infty \) we also need the uniform boundedness of (38) for the adjoint operator \(T^*\). However since the proof is almost identical to the case of T, we confine ourselves to this case.
We split \(\mathbb {R}^{n}\) into 2B and \(\mathbb {R}^{n}{\setminus } 2B\) and start with the case of 2B (note that 2B is ball \(B(\bar{y}, 2r)\). By Hölder’s inequality and the \(L^2\)-boundedness of T, we have
We proceed to the boundedness of \(\Vert T \mathfrak {a}\Vert _{L^{p}( \mathbb {R}^{n}{\setminus } 2B)}\). Now we consider what we call a "generic Littlewood-Paley piece" of the operator T given by (37). The generic Littlewood-Paley piece is the FIO with the amplitude \(a_j\), where \(a_j(x,\xi ) := a(x,\xi )\,\psi _j(\xi )\), and \(j\ge 1\) due to Remark 6.1, and the phase function \(x\cdot \xi + \theta (x, \xi )\). We denote this operator by \(S_j\) and note that the integral kernel of \(S_j\) is given by
We claim that
for all multi-indices \(\alpha \) and \(\beta \). Since differentiating (40) \(\beta \) times in y will only introduce factors of the size \(2^{j|\beta |},\) it is enough to establish (41) for \(\beta =0\). Now the global \(L^2\)-boundedness (41) of the kernel can be formulated as the \(L^2\)-boundedness of a kernel of the form
To this end, take \({\Psi }_j\) as in Definition 2.1, integrate by parts and rewrite
where \(S_{j}^{\alpha _1, \alpha _2, \lambda _1, \dots \lambda _r}\) is an FIO with the phase function \(\theta (x,\xi )+x\cdot \xi \) and amplitude \(b_j^{\alpha _1,\alpha _2, \lambda _1, \dots , \lambda _r}(x,\xi )\) given by
Moreover \(|\lambda _j| \ge 1\) and \(\tau _{-y}\) is a translation by \(-y\).
We observe that \(b_j^{\alpha _1,\alpha _2, \lambda _1, \dots , \lambda _r}(x,\xi ) \in S^{-n\delta /2}_{0,\delta }(\mathbb {R}^{n})\) uniformly in j, since \(a \in S^{m}_{0,\delta }(\mathbb {R}^{n})\) and \(\theta \in \Phi ^1\).
Therefore by Proposition 2.16, \(S_{j}^{\alpha _1, \alpha _2, \lambda _1, \dots \lambda _r}\) is an \(L^2\)-bounded FIO, so
which proves (41).
Now, the estimate in (41) yields that for any integer M, if one sums over \(\left| {\alpha }\right| \le M\),
We now observe that for \(t\in [0,1],\) \(x\in \mathbb {R}^{n}{\setminus } 2B\) and \(y\in B\), one has
Next we introduce
where \(M > n/q\) and \(1/q=1/p-1/2\). The Hölder and the Minkowski inequalities together with (42) and (43) (with \(t=1\)) yield
since \( m=-n(1/p-1/2)-n\delta /2\).
On the other hand, taking \(N:=[n(1/p-1)]\) (note that \(N > n/p-n-1\)), a Taylor expansion of the kernel at the point \(y=\overline{y}\) yields that
and due to vanishing moments of the atom in Definition 2.6, (iii), we may express the operator as
Noting that \(\left| { (y-\bar{y})^\beta }\right| \lesssim r^{N+1}\) and applying the same procedure as above together with estimates (41) and (43), we obtain
Now we split the proof in two different cases, namely when the radius r of the support of the atom \(\mathfrak {a}\) is less than or greater or equal to one.
For \(r\ge 1\), (44) yields that
Assume now that \(r < 1\). Choose \(\ell \in \mathbb {Z}_+\) such that \(2^{-\ell -1} \le r < 2^{-\ell }\). Using the facts that \(2^{-\ell }\approx r\), \(N+1+n-n/p>0\), \(n-n/p<0\), together with (44) and (45) we conclude that
Putting this together with (39), yields the uniform boundedness of (38).
The proof of the adjoint case is identical, except for the fact that (41) becomes
and when applying \(\beta \) derivatives in the y-variable the y-dependence in both arguments has to be taken into consideration. \(\square \)
It is also evident that Proposition 6.2 yields the \(L^p\)-boundedness of pseudodifferential operators with exotic symbols and thereby completes the investigation in [1].
6.2 Classical amplitudes
We proceed by proving a global \(L^p\)-boundedness result for classical FIOs with amplitudes in \(a\in S^m_{\rho ,\delta }(\mathbb {R}^{n})\) with \(0<\rho \le 1\), \(0 \le \delta < 1\).
Before doing that, we need the following lemma which provides \(\mathscr {H}^q-L^2\) estimates for FIOs with amplitudes in general Hörmander classes.
Lemma 6.3
Let \(m_0\le 0\), \(n\ge 1,\) \(\rho \in [0,1]\), \(\delta \in [0,1)\) and
Suppose that \(a\in S^m_{\rho ,\delta }(\mathbb {R}^{n})\) and that \(a(x,\xi )\) vanishes in a neighborhood of \(\xi =0\). Also, let \(\varphi \) be an SND phase function in the class \(\Phi ^2\). Then \(T_a^\varphi \), defined in (6), satisfies
Also for the adjoint operator one has
Proof
Since the operator \(T_a^\varphi (1-\Delta )^{-m_0/2}\) is an FIO with the phase \(\varphi \) and an amplitude in \(S^{-n \max (0,(\delta -\rho )/2)}_{\rho ,\delta }(\mathbb {R}^{n})\) it is \(L^2\)-bounded by Proposition 2.16. This \(L^2\)-boundedness together with the estimates for the Bessel potential operators reformulated in terms of embedding of Triebel-Lizorkin spaces (see [27, Corollary 2.7]) yield
with \(1/q-1/2= -m_0/n,\) which proves (46). Here observe that the choice of the range of \(m_0\) implies that \(0< q\le 2\).
Next we prove (47). By Theorem 4.1 the composition \( (1-\Delta )^{-m_0 /2} T_a^\varphi \) is an FIO with the phase \(\varphi \) and an amplitude in \(S^{-n \max (0,(\delta -\rho )/2)}_{\rho ,\delta }(\mathbb {R}^{n})\), and therefore \(L^2\)-bounded. Finally, observing that
one can proceed as above. \(\square \)
Now we are ready to state and prove our main \(L^p\)-estimate for FIOs with general classical Hörmander-type amplitudes.
Proposition 6.4
Let \(n\ge 1\), \(a\in S^m_{\rho ,\delta }(\mathbb {R}^{n})\), \(\varphi \) be an SND phase function in the class \(\Phi ^2\) and let \(T_a^\varphi \) be given as in Definition 2.15. For \(0<\rho \le 1\), \(0 \le \delta < 1\) and
the FIO \(T_a^\varphi \) is \(L^p\)-bounded for \(1<p<\infty .\)
Remark 6.5
In Proposition 6.4 it is not possible to consider the case \(\rho =0\) for several reasons. First, the definitions of the rectangles in (26) turn out to be inadequate. Second, the choice of M in the proof of (53) below would not be possible. Third, the choice of \(L> n/\rho \) in (68) would be problematic.
Proof of Proposition 6.4
For \(n=1\), it is well known that FIOs are special cases of pseudodifferential operators and hence the result follows from the corresponding theory for those operators (see e.g. [24]). Therefore, from now on we concentrate on the case \(n\ge 2.\) We will initially assume that \(a(x,\xi )\) is supported in a fixed compact set in the x-variable. This will however be removed later on in the proof. Since the result has already been proven for the case when \(p=2\) in Proposition 2.16, the only thing that is left to prove is that \(T_a^\varphi \) and its adjoint map \(\mathscr {H}^1(\mathbb {R}^{n})\) to \(L^1(\mathbb {R}^{n})\) continuously, when \(a(x,\xi )\in S^m_{\rho , \delta }(\mathbb {R}^n)\) with
Due to the atomic decomposition in Definition 2.6 of a member of \(\mathscr {H}^1(\mathbb {R}^{n})\) we need to show that
is uniformly bounded for every \(\mathscr {H}^1\)-atom \(\mathfrak {a}\), where the atom is supported in the ball \(B(x_0 ,r)\).
Step 1 - Estimates of \(\mathbf {\Vert T_a^\varphi \mathfrak {a}\Vert _{L^1(\mathbb {R}^n)}}\) when \(\mathbf {r< 1}\)
Recalling the set \(B^*\) in (27), we split (48) into two pieces, namely
Using the first part of Lemma 5.6, the Cauchy-Schwarz inequality, and Lemma 6.3 we can deduce that \({\text {I}}\) is uniformly bounded. Indeed take \(\mathfrak b = |B|^{1- 1/q}\mathfrak {a}\) with \(q={2n}/{(n-2m')}\). Note that since \(\Vert \mathfrak b\Vert _{\mathscr {H}^{q}(\mathbb {R}^{n})}= 1\), we have
provided that
Now we have to deal with the last and most complicated part of the proof, that is the boundedness of \({\text {II}}\). To do this, we use a partition of unity and decompose \(T_a^\varphi \) as in (19). Recall that for \(j\ge 1\) the kernel \(K_j\) of \(T_j\) is given by
By Remark 6.1 it is enough to consider \(j\ge 1\). We would first like to prove that
which immediately yields
Now we make a second dyadic decomposition of \(K_j\) using Lemma 5.2, where each piece has the form,
where \(\chi _j^\nu (\xi )\) are as in Lemma 5.2. To justify (50), set
and observe that
with the vector \(b_j^\nu (x,\xi )\) given by
Note that on the \(\xi -\)support of each component of \(b_j^\nu (x,\xi )\) one has that \(|\xi |\sim 2^{j(1-\rho )}.\) Therefore to estimate the size of the derivatives of components of \(b_j^\nu (x,\xi )\), we use this information and Lemmas 5.2 and 5.3. The estimates have been summarised in Table 1.
Hence if we define the differential operator
it is clear that for \(N\ge 1\)
Finally using the information given in Table 1, we deduce that
Now using integration by parts
where \(\Gamma ^{\nu }_j \cap A_j\) is the support of \(b_j^\nu (x,\xi )\) and \(A_j\) and \(\Gamma ^{\nu }_j\) were defined in (20) and (21) respectively. Let \(\widehat{g}_j^\nu \) be a smooth cut-off function that is constantly equal to one on the \(\xi \)-support of \(b_j^\nu \) and vanishes outside a compact set which is slightly larger the aforementioned \(\xi \)-support. Now set \(\mathbf{t}(x):= \nabla _\xi \varphi (x,\xi _j^\nu )\) and define
Then because of (52), the choice of \(m'\) in (49), and that \(\mathbf{t}\) is a diffeomorphism, \(S_{j,y}^{\nu ,N}\) is a \(\Psi \)DO of order \(- n\max (0,(\delta -\rho )/2)\) and hence \(L^2\)-bounded, by Proposition 2.16, uniformly in y and j. Observe that \(\nabla _yK_j^\nu (x,y)\) can be rewritten as
Now using the compact x-support, Cauchy-Schwarz inequality and that \(\mathbf{t}(x)\) is a diffeomorphism, we have
where recalling (22) we have that \(\left| {\Gamma _j^\nu \cap A_j}\right| \sim 2^{j(n+\rho (1-3n)/2)}\). Therefore, summing in \(\nu \) and observing that since there are roughly \(2^{j\rho (n-1)/2}\) terms involved, we obtain
which is (50).
Our next goal is to show that
To this end, define
with
We observe that the same reasoning as in the proof of (52) reveals that for \(N\ge 0\)
which in turn implies that \( \widetilde{S}_{j,y}^{\nu ,N}\) is a \(\Psi \)DO of order \(- n\max (0,(\delta -\rho )/2)\) and hence once again \(L^2\)-bounded.
A similar calculation as in the case of \(\nabla _y K_j(x,y)\) and estimate (28) yield that
Hence,
Finally, given an atom \(\mathfrak {a}\) with support in the ball \(B:= B(x_0,r)\) with \(r<1\) we write
where as before, r is the radius of the support of the atom \(\mathfrak {a}\). Now observe that property (iii) of Definition 2.6 implies that
so this together with Minkowski’s inequality, (51) and (53) yield that
The corresponding proof of the \(\mathscr {H}^{1}-L^1\) boundedness of the adjoint \((T_a^\varphi )^*\) is similar to the one above with few modifications. First, regarding the \(L^2\)-boundedness in Lemma 6.3, estimate (46) in that Lemma has to be replaced by (47). Second, the x and y dependencies of the kernel are reversed. This means the following replacements:
Otherwise the proof remains the same.
Step 2 - Estimates of \(\mathbf {\Vert T_a^\varphi \mathfrak {a}\Vert _{L^1(\mathbb {R}^n)}}\) when \(\mathbf {r\ge 1}\)
Now we turn our attention to atoms with supports in balls of radii \(r\ge 1\). In this case, using the compact support of the amplitude a and the \(L^2\)-boundedness of \(T^{\varphi }_a\) (Proposition 2.16) we have
To prove the boundedness of the adjoint \((T_a^\varphi )^*\), we split the \(L^1\)-norm into two pieces, namely
where \(B'\) is the ball centered at the origin with radius 2K and
We treat the first term of (55) as in (54). For the second term we observe that the kernel of \((T_a^\varphi )^*\) satisfies
for \(|x|>2K\). This follows from the fact that, on the support of \(a(y,\xi ),\) the modulus of the gradient of the phase of the oscillatory integral above satisfies
Now if
is a Littlewood-Paley partition of unity with \({{\,\mathrm{supp}\,}}\psi \) inside a fixed annulus (see Definition 2.1), then using Remark 6.1 (i.e. omitting the term where \(j=0\)) we have
with \(\lambda := 2^j |x|\),
and
with compact support in y and annulus-support in \(\xi \). Now since for all multi-indices \(\alpha \), \(|\partial ^{\alpha }_\xi b(y,\xi )|\lesssim 2^{jm}\) and since for \((y,\xi ) \in {{\,\mathrm{supp}\,}}b(y,\xi )\), (57) yields that \(|\nabla _\xi \Phi (x,y,\xi )|= \frac{1}{|x|} |x-\nabla _\xi \varphi (y,\xi )|\gtrsim 1\), the non-stationary phase estimate of Lemma 2.18 could be used to deduce that
for any \(N>0\). Thus using this in (58) and summing in j, (56) follows. Hence
Step 3 - Globalisation of Steps 1 & 2
At this point we once again use the conditions on the phase function to reduce our analysis to the case of operators \(T_{a}^{\varphi }\) with a phase function \(\varphi (x, \xi )= \theta (x,\xi )+x\cdot \xi \), \(\theta \in \Phi ^1\) and \(\theta (x,\xi )+x\cdot \xi \) is SND (see Sect. 3).
As far as the amplitude of \(T_{a}^{\varphi }\) is concerned, we assume that \(a(x, \xi )\) is as in the previous steps of the proof, albeit without the assumption of the compact support in x.
Our actual goal here is to globalise the results that we have obtained so far for both \(T_a^\varphi \) and \((T_a^\varphi )^* \) at the same time. Whenever we write T we refer to both \(T_a^\varphi \) and \((T_a^\varphi )^* \).
Now with the reduction mentioned above we proceed to describe the globalisation procedure. In [22], Ruzhansky and Sugimoto developed a new technique to transfer local boundedness of Fourier integral operators, which was proven by Seeger, Sogge and Stein [23], to a global result, where the amplitudes of the corresponding operators do not have compact spatial supports.
Let us consider the FIO given by
where \(\sigma (x, y,\xi )\in \mathcal {C}^{\infty }(\mathbb {R}^{n}\times \mathbb {R}^{n}\times \mathbb {R}^{n})\) satisfies the estimate
with \(m\le 0\), \(0<\rho \le 1\) , \(0\le \delta \le 1\) for all multi-indices \(\alpha \), \(\beta \) and \(\gamma \) and \((x,y,\xi )\in \mathbb {R}^n\times \mathbb {R}^n \times \mathbb {R}^{n}\) (thus note that the case of \(\rho =0\) is excluded due to a technical reason in the proof of Lemma 6.7 below). Here the phase function \(\vartheta (x,y,\xi )\) is either \(\theta (x,\xi )\) or \(-\theta (y,\xi )\), with \(\theta \in \Phi ^1\) and is assumed to be smooth on the support of the amplitude \(\sigma (x, y, \xi ).\)
Then one defines the function
and
We also define
and
and set
and
Here we observe that \(N_L <\infty \) by the \(\Phi ^1\)-condition on \(\theta \) above. Given these definitions one has the following lemma.
Lemma 6.6
Let \(r\ge 1\) and \(L\ge 1\). Then we have \(\mathbb {R}^n {\setminus } \widetilde{\Delta }_{2r} \subset \{z:\, |z|<(2+N_L )r \}\). Furthermore for \(r>0\), \(x\in \widetilde{\Delta }_{2r}\) and \(|y|\le r\) we have
and therefore \((x,y,x-y)\in \Delta _r\).
Proof
For \(z\in \mathbb {R}^n{\setminus }\widetilde{\Delta }_{2r}\), we have \(\widetilde{H}(z) <2r\). Hence, there exist \(x_0,y_0,\xi _0\in \mathbb {R}^n\) such that
Since, \(r\ge 1\), this yields that
The claim that \((x,y,x-y)\in \Delta _r\) follows from (63) and the definition of \( \Delta _r\). Therefore it only remains to prove (63). Now, if \(|y|\le r\) and \(x\in \widetilde{\Delta }_{2r}\) then since \( \widetilde{H}(x)\ge 2r\), we have that
From this, (63) follows at once. \(\square \)
In order to prove the global boundedness, the following result is of particular importance.
Lemma 6.7
Let \(\sigma (x,y,\xi )\) satisfy (60) with some \(\rho \in (0, 1]\). The kernel associated to the operator T in (59) that is given by
is smooth on \(\cup _{r>0}\Delta _r\). Moreover, for all \(L> n/\rho \) and \(r\ge 1\) it satisfies
where \(C( L, M_L ,N_{L+1})\) is a positive constant depending only on L, \(M_L\) and \(N_{L+1}\). For \( L> n\) and \(r\ge 1\), the function \(\widetilde{H}(z)\) satisfies the bound
Proof
If one introduces the differential operator
with the transpose \(D^*\), then integrating by parts L times yields
Now (65) follows from the relation
which is valid for \((x,y,z)\in \Delta _r\) and \(\xi \in \mathbb {R}^n\). Moreover
for any \(\xi \not =0\), which yields that
Hence for \(|z|\ge 2N_{L+1}\) one has
and therefore
Using this we get
which proves (66) . \(\square \)
Now at this point we have all the tools that would help us to achieve our globalisation, and this amounts to prove that
when there is no requirement on the support of the amplitude. To this end, first we observe that a global norm estimate for \(T\mathfrak {a}\) with \(\mathfrak {a}\) supported in a ball with an arbitrary centre, would follow from a norm-estimate that is uniform in s for \(\tau _s^* T\tau _s \mathfrak {a}\), with an atom \(\mathfrak {a}\) whose support is inside a ball centred at the origin. Note that here \(\tau _s\) is the operator of translation by \(s\in \mathbb {R}^n\). This is because by translation invariance of the \(L^1\)-norm one has that
Thus our goal is to establish that
where the estimate is uniform in s and \(\mathfrak {a}\) has its support in a ball centred at the origin.
Now let \(r\ge 1\), \( L>n/\rho \) and \(s\in \mathbb {R}^n\) and suppose \(\mathfrak {a}\) is an \(\mathscr {H}^1\)-atom supported in a ball B, centred at the origin, with radius r. We use the notions that were introduced in connection to the globalisation procedure in Section 3 and split the \(L^1\)-norm of \(\tau _s^* T\tau _s \mathfrak {a}\) into the following two pieces:
where \(\widetilde{\Delta }_{r}\) is defined in (62). First let us show that
By Lemma 6.6, for \(x\in \widetilde{\Delta }_{2r}\) and \(|y|\le r\), we have
and \((x,y,x-y)\in \Delta _r\). Now since for the kernel of the operator T given by (64) one has that
(67) and Lemma 6.7 yield for any atom \(\mathfrak {a}\) supported in B(0, r) that
since \( \Vert \mathfrak {a}\Vert _{L^1(\mathbb {R}^n)} \le 1.\) Therefore, if \(r\ge 1\), choosing \( L> n/\rho \), Lemma 6.7 and the monotonicity of \(\Delta _r\) yield
Observe that the phase function and the amplitude of \(\tau _s^* T\tau _s\) are of the form \( \theta (x+ s, \xi )+(x-y)\cdot \xi \) and \(\sigma (x+s, \xi )\) respectively when \(T= T_a^\varphi \) (a similar property is also true for \((T_a^\varphi )^*\)). Therefore the conjugation of T by \(\tau _s\) renders the constants \(M_L\) and \(N_{L+1}\) unchanged and therefore the estimate above also yields the very same one for \(\tau _s^* T\tau _s\). This means that
On the other hand for \(\left\| \tau _s^* T\tau _s \mathfrak {a}\right\| _{L^1 (\mathbb {R}^n {\setminus } \widetilde{\Delta }_{2r})}\), Lemma 6.6, Hölder’s inequality and the properties of the atom \(\mathfrak {a}\) yield that
Now if the atom is supported in a ball of radius \(r\le 1\) then clearly \({{\,\mathrm{supp}\,}}\mathfrak {a}\subset B(0,1).\) Now write \(\mathbb {R}^n = \widetilde{\Delta }_2 \cup (\mathbb {R}^n {\setminus } \widetilde{\Delta }_2)\) and observe that we can now use Lemma 6.7 with \(r=1\) to conclude that
which in turn yields that
Finally, in view of the first part of Lemma 6.6 we see that \(\mathbb {R}^n {\setminus } \widetilde{\Delta }_2 \subset B(0, 2+N_{L})\) which together with the local boundedness result that we established in Steps 1 and 2 implies
Now that we have boundedness from \(\mathscr {H}^1(\mathbb {R}^{n})\) to \(L^1(\mathbb {R}^{n})\) for both \(T_a^\varphi \) itself and its adjoint, as well as \(L^2\)-boundedness, we can use a standard Riesz-Thorin interpolation argument to conclude that \(T_a^\varphi \) is bounded from \(L^p(\mathbb {R}^n)\) to itself. \(\square \)
6.3 Forbidden amplitudes
The case of operators with amplitudes in \(S^m_{\rho , 1}(\mathbb {R}^{n})\) with \(0\le \rho \le 1\) is rather special since the involved FIOs are, in general, not \(L^2\)-bounded. However, Proposition 2.16 yields that if \(m< -n(1-\rho )/2\) then the associated FIO is indeed \(L^2\)-bounded, and this result is sharp. Here, only for the sake of completeness of exposition we state the result proven in [11] regarding the \(L^p\)-boundedness of FIOs with forbidden amplitudes.
Proposition 6.8
Let \(a\in S^m_{\rho ,1}(\mathbb {R}^{n})\), \(\varphi \) be an SND phase function in the class \(\Phi ^2\) and let \(T_a^\varphi \) be given as in Definition 2.15. For \(0 \le \rho \le 1\) and
the FIO \(T_a^\varphi \) is \(L^p\)-bounded for \(1\le p\le \infty .\)
Proof
See [17, Propositions 2.3 and 2.5] for the case \(n=1\), which is essentially the pseudodifferential case, and [11, Theorem 2.17] for \(n\ge 2\). \(\square \)
7 Sobolev space boundedness of FIOs with \(S^{0}_{1,1}\)-amplitudes
It turns out that just as in the case of pseudodifferential operators, the FIOs with forbidden amplitudes, say in \(S^0_{1,1}(\mathbb {R}^{n})\), despite failing to be \(L^2\)-bounded are bounded on \(H^s(\mathbb {R}^{n})\) with \(s>0\). As it was mentioned in the introduction, the proof of the Sobolev-boundedness in the pseudodifferential case goes back to E. Stein and independently to Y. Meyer. Other proofs were given by Bourdaud [3] and Hörmander [15]. Following Bourdaud, we establish the Sobolev boundedness of FIOs with amplitudes in the class \(S^{0}_{1,1}(\mathbb {R}^n).\)
Proposition 7.1
Let \(r>0\), \(a\in C_{*}^{r} S^0_{1,1}(\mathbb {R}^{n})\) and \(\varphi \) be an \(\mathrm {SND}\) phase function in the class \(\Phi ^2\). Then for \(\max (0,r-1/2)<s< r\), the FIO \(T_a^\varphi \) is bounded from the Sobolev space \(H^{s}(\mathbb {R}^n)\) to \(H^{s}(\mathbb {R}^n)\).
Proof
We divide the proof into five steps.
Step 1 - Reduction of the FIOs with amplitudes in \(\mathbf {C^{r}_{*}S^0_{1,1}(\mathbb {R}^{n})}\) class
Following [3], for an amplitude in \(C^{r}_{*}S^0_{1,1}(\mathbb {R}^{n})\) one has the decomposition
where \(\psi _{k}\) was introduced in Definition 2.1 and \(M_{k}(x)\) satisfies
The \(C_{*}^{r}\)-norm is given in Definition 2.9.
We treat the case \(k=0\) (the low frequency portion of the FIO) separately, so for now assume that \(k\ge 1\).
Using the Littlewood-Paley partition of unity \(I=\sum _{j=0}^\infty \psi _{j} (D)\) and setting
(the amplitude of the FIO \(T_{1}^\varphi \) is identically equal to one) we have that for \(k\ge 1\) and \(j\ge 0\)
Denoting the inverse Fourier transform of a function \(\psi \) by \(\psi ^{\vee }\), the first estimate in (71) can be shown using the bound \(\left\| M_{k}\right\| _{C_{*}^{r}(\mathbb {R}^{n})} \lesssim 2^{k r}\), the properties of the Zygmund class given in (3) and that
as follows
for \(j>0\). For \(j=0\) this is a consequence of the \(L^\infty \)-boundedness of \(\psi _0(D)\) and the first estimate in (70).
The second estimate in (71) is of course a direct consequence of the \(L^2\)-boundedness of FIOs with amplitudes in \(S^{0}_{1,0}(\mathbb {R}^n).\)
Using the above notation we can now decompose \(T^{\varphi }_{a}\) as
At this point, taking into account the properties of the SND phase function \(\varphi \in \Phi ^2\) (i.e. \(|\nabla _x \varphi (x, \xi )|\approx |\xi |\)), Proposition 2.19 and choosing a smooth annulus-supported \(\tilde{\psi }\) with \(\psi \cdot (\tilde{\psi }\circ \nabla _x\varphi (x,\cdot ))= \psi \), we have for any integer \(N_1>0\) and any \(0<\varepsilon <1/2\)
with
and
where the estimates above are uniform in k.
Thus
Step 2 - Analysis of \(\mathbf {\sum _{k=1}^\infty M_k(x)\, F^{1}_k (x)}\)
Now to analyse \(F^{1}_k\) we write
and split the sum in j into the following pieces
Firstly, we establish the \(H^s\)-boundedness of \(\mathbf {A}\). To this end we have
The Fourier transform of \(b_k\) is given by
By recalling the defining property (2) of the Littlewood-Paley decomposition and the telescoping property of their partial sums, we see that the Fourier transform of \(b_k\) satisfies
Note using the information in Definition 2.1 regarding the support of \(\psi _0\) we see that \(|\eta -\xi |\le 2^{k-1}\), and due to the fact that \(\tilde{\psi }\) is supported in an annulus, we have \(|\xi |\sim 2^k\). Therefore the support of \(\widehat{b_k}\) is bounded above by \(|\eta |\le |\eta -\xi |+|\xi |\lesssim 2^k(2^{-1}+1)\), and from below by \(|\eta | \ge |\xi |-|\eta -\xi |\gtrsim 2^k(1-2^{-1})\). From this it follows that the spectrum of \(b_k\) is contained in an annulus \(|\eta |\approx 2^k\). Hence by Lemma 2.5 it follows that for \(s>0\),
where we have used Remark 2.4 and that
Now for term \(\mathbf {B}\) in (73), applying Lemma 2.5 to
we obtain (using Fubini’s theorem for sums, (71) and Young’s inequality for discrete convolutions)
because of Remarks 2.3, 2.4 and \(r>s\).
Step 3 - Analysis of \(\mathbf {\sum _{k=1}^\infty M_k(x)\, F^{2}_k (x)}\)
To establish the Sobolev boundedness for the term \(\sum _{k=1}^\infty M_k(x)\, F^{2}_k (x)\), we need to analyse the action of the Littlewood-Paley operator \(\psi _j(D)\) on this term in order to use Definition 2.2 together with Remark 2.4. Here we also assume that \(r\in (s, s+\varepsilon )\), where \(\varepsilon \in (0,1/2)\) is given in representation of \(F_k (x)\) in (72).
Then, for an integer \(N_2>0\) and \(0<\varepsilon '<1/2\), write
with
and
where both estimates above are uniform in j and k. Moreover, since in the decomposition (69) of \(a(x,\xi )\), we are at present considering the parts supported outside a neighbourhood of the origin in the \(\xi \)-variable, i.e. those for which \(k\ge 1\), we also have that \(r_{j,k}(x,\xi )\) vanishes in a neighbourhood of \(\xi =0\).
For term \(\mathrm {I}\), and in view of the support properties of \(\sigma _{\alpha ,\beta ,k,j}\), we claim that (uniformly in j and k)
where \(\Psi _j\) is a Littlewood-Paley-type frequency localisation that is equal to one on the support of \(\sigma _{\alpha ,\beta ,k,j}.\) Therefore \(T^\varphi _{\sigma _{\alpha ,\beta ,k,j}} f= T^\varphi _{\sigma _{\alpha ,\beta ,k,j}} \Psi _j(D) f,\) and it is enough to show that
uniformly in k and j. To see this, we proceed by studying the boundedness of \(S_j:=T^\varphi _{\sigma _{\alpha ,\beta ,k,j}}(T^\varphi _{\sigma _{\alpha ,\beta ,k,j}})^{*}\). A simple calculation shows that
with
Now, since \(\varphi \) is homogeneous of degree one in the \(\xi \) variable, \(K_j (x,y)\) can be written as
with
and
Observe that the \(\xi \)-support of \(b_{j }(x,y,2^{j }\xi )\) lies in the compact set \(\mathcal {K}:=\left\{ C_1\le \left| {\xi }\right| \le C_2\right\} \). From the SND condition (5) it also follows that
Assume that \(N_3>n\) is an integer, fix \(x\ne y\) and set \(\phi (\xi ):=\Phi (x,y, \xi )\), \(\vartheta :=\left| {\nabla _\xi \phi }\right| ^2\). By the mean value theorem, (4) and (77), for any multi-index \(\alpha \) with \(\left| {\alpha }\right| \ge 1\) and any \(\xi \in \mathcal {K}\),
On the other hand, since
it follows that, for any \(\left| {\alpha }\right| \ge 0\), \(\left| {\partial ^\alpha _{\xi } \vartheta }\right| \lesssim \vartheta \). We estimate the kernel \(K_j\) in two different ways. For the first estimate, (77) and Lemma 2.18 with \(F(\xi ):=b_{j } (x,y,2^{j }\xi ),\) yield
where the fact that the \(\xi \)-support of \(b_j \) lies in a ball of radius \(\approx 2^j \) and that for \(|\alpha |\ge 0\)
have been used. By (79) we also obtain
and when combining estimates (78) and (80) one has
Thus, using (81) and Minkowski’s inequality we have
The Cauchy-Schwarz inequality yields
Therefore
and (76) is proven.
Observe that (70), Cauchy-Schwarz inequality, Fubini’s theorem, (75), and the definition of the Sobolev norm yield
For term \(\mathrm {II}\), we decompose \(T^{\varphi }_{r_{j,k}}\) into Littlewood-Paley pieces as follows:
where the \(\psi _\ell \)’s are defined in Definition 2.1. By a proof identical to the one of (75), we see that
Thus
Note that the estimate (83) is uniform in j. Then we claim that for \(s>0\) one has
Indeed the Cauchy-Schwarz inequality yields
The last step follows from the definition of \(H^s(\mathbb {R}^n)\), see Remark 2.3. Thus, for \(N_2>\frac{s}{\varepsilon '}\),
For term \(\mathrm {III}\) of (74), the second estimate in (70) and relations (3) yield that
for \(j>0\), where \(\mathcal {M}\) is the Hardy-Littlewood maximal function. For \(j=0\) this follows from the \(L^\infty \)-boundedness of \(\psi _0(D)\) and the first estimate of (70). Therefore using the \(L^2\)-boundedness of Hardy-Littlewood maximal function, Proposition 2.16 (on \(T^{\varphi }_{\sigma _{\alpha ,k}}\)) and the commutator estimate above, we obtain
Hence, if \(r\in (s, s+\varepsilon )\), by Cauchy-Schwarz’s inequality we obtain
Thus putting this, (82) and (84) together, we obtain
We note that since we assumed that \(s>0\) and \(r\in (s, s+\varepsilon ),\) and since \(\varepsilon \in (0, 1/2)\), these force s to belong to the interval \(\max (0,r-1/2)<s< r\), which is the claimed range of s in the proposition.
Step 4 - Analysis of \(\mathbf {\sum _{k=1}^\infty M_k(x)\, F^{3}_k (x)}\)
Finally we turn to the Sobolev boundedness of the term
Once again, using the definition of \(F^3_k\) above, we have
The commutator term can be treated as term \(\mathrm {III}\) in Step 3, since we can choose arbitrarily fast decay in k. After taking the sum in k of the first term, using (70), then taking the \(L^2\)-norm, multiplying with \(4^{js}\) and then taking the \(\ell ^2\)-norm in j one has the estimate
where we have also used Lemma 2.17.
Step 5 - Analysis of the case \(\varvec{k=0}\)
We have, using the second estimate in (70), a similar argument as in (85), and the Sobolev-boundedness result in Lemma 2.17
\(\square \)
As a corollary we obtain Theorem 2 in the introduction of the paper, namely
Corollary 7.2
Let \(a\in S^0_{1,1}(\mathbb {R}^{n})\) and \(\varphi \) be an \(\mathrm {SND}\) phase function in the class \(\Phi ^2\). Then for \(s>0\) the FIO \(T_a^\varphi \) is bounded from the Sobolev space \(H^{s}(\mathbb {R}^n)\) to \(H^{s}(\mathbb {R}^n)\).
Proof
Note that by Proposition 7.1 we have the \(H^{s}\)-boundedness of \(T^{\varphi }_{a}\) with \(a(x,\xi )\in C_{*}^{r} S_{1,1}^{0}(\mathbb {R}^{n})\), for \(\max (0,r-1/2)<s< r\). Moreover it is well-known that \( S_{1,1}^{0}(\mathbb {R}^{n}) \subset C_{*}^{r} S_{1,1}^{0}(\mathbb {R}^{n})\) for all \(r>0\), see [3, 19]. Therefore the arbitrariness of r would then yield the desired result for amplitudes in \(S_{1,1}^{0}(\mathbb {R}^{n})\). \(\square \)
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The authors are indebted to the referees for their meticulous reading of the manuscript, and for suggestions that improved the overall presentation of the paper. In particular, we are grateful for comments concerning one of our initial results, which prompted us to improve it further and transform it to a much better one.
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Castro, A.J., Israelsson, A. & Staubach, W. Regularity of Fourier integral operators with amplitudes in general Hörmander classes. Anal.Math.Phys. 11, 121 (2021). https://doi.org/10.1007/s13324-021-00552-x
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DOI: https://doi.org/10.1007/s13324-021-00552-x