1.1 Some relevant results regarding boundedness of nondegenerate oscillatory integral operators

We start by giving an overview of the previously known regularity results for oscillatory integral operators, which is to a large extent biased by their relevance to our current paper.

For simplicity, we confine ourselves to oscillatory integral operators of the form

(2)

with amplitude $a(x,\xi )$ and phase function $\varphi (x,\xi )$ . Here, following L. Hörmander [Reference Hörmander18] one can assume that the $a(x,\xi )$ belongs to the class $S^{m}_{\rho , \delta }(\mathbb {R}^n),$ which means that the amplitude is in $\mathcal {C}^{\infty }(\mathbb {R}^n \times \mathbb {R}^n)$ and satisfies

(3) $$ \begin{align} \left|\partial_{\xi}^{\alpha}\partial_{x}^{\beta} a(x,\xi)\right | \leq C_{\alpha , \beta} (1+|\xi|)^{m-\rho|\alpha|+\delta|\beta|}. \end{align} $$

It was shown by Hörmander [Reference Hörmander18] and G. I. Eskin [Reference Èskin10] that if $a(x,\xi )\in S^{0}_{1,0}(\mathbb {R}^n)$ is smooth and compactly supported in x and if $\varphi \in \mathcal {C}^{\infty }(\mathbb {R}^n \times \mathbb {R}^n \setminus \{0\})$ is positively homogeneous of degree 1 in $\xi $ and the mixed Hessian matrix of $\varphi (x,\xi )$ has a nonzero determinant on the support of $a(x,\xi )$ (the nondegeneracy condition), then the operator $T_a^\varphi $ is $L^2$ -bounded. These type of operators are examples of the so-called Fourier integral operators, which were officially introduced in [Reference Hörmander18]. The global extension of the $L^2$ -boundedness result of Eskin and Hörmander to all possible amplitudes in the class $S^{m}_{\rho , \delta }(\mathbb {R}^n)$ was done by D. Dos Santos Ferreira and W. Staubach [Reference Dos Santos Ferreira and Staubach9]. In that paper, it was shown that, if the phase function $\varphi $ is positively homogeneous of degree 1 in $\xi $ , the determinant of the mixed Hessian of $\varphi $ is globally bounded from below by a nonzero constant (the strong nondegeneracy condition), and $\varphi $ satisfies the bound

$$ \begin{align*} \sup_{(x,\,\xi) \in \mathbb{R}^n \times\mathbb{R}^n \setminus\{0\}} |\xi| ^{-1+\vert \alpha\vert}\left| \partial_{\xi}^{\alpha}\partial_{x}^{\beta}\varphi(x,\xi)\right| \leq C_{\alpha , \beta},\qquad |\alpha+\beta|\geq 2, \end{align*} $$

then the Fourier integral operator $T^\varphi _a$ is globally $L^2$ -bounded, provided that $\rho , \delta \in [0,1]$ , $\delta \neq 1$ 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, and therefore completes the study of Hörmander-class Fourier integral operators with nondegenerate phase functions.

Regarding operators $T_a^{\varphi }$ where the phase function $\varphi $ is inhomogeneous, it was shown by D. Fujiwara [Reference Fujiwara13] that if $\varphi \in \mathcal {C}^{\infty }(\mathbb {R}^n \times \mathbb {R}^n)$ satisfies the condition

(4) $$ \begin{align} \left |\partial^\alpha_\xi \partial^\beta_x \varphi (x,\xi)\right | \leq c_{\alpha,\beta} , \quad |\alpha+\beta|\geq 2, \end{align} $$

aforementioned satisfies the strong nondegeneracy condition, and if the amplitude belongs to the class $S^{0}_{0,0}(\mathbb {R}^{n})$ , then $T_a^\varphi $ is globally $L^2$ -bounded. In [Reference Cordero, Nicola and Rodino6], E. Cordero, F. Nicola and L. Rodino gave an elegant proof of this $L^2$ -boundednes result, which completely avoids many of the technicalities (e.g., the use of the Cotlar–Stein lemma) involved in the previous proofs and instead relies on techniques from the theory of modulation spaces. Therefore, one could assert that the $L^2$ -regularity of operators $T_a^\varphi $ with smooth amplitudes and smooth nondegenerate phase functions has been brought to completion. However, the extent of the impact of [Reference Cordero, Nicola and Rodino6] was not confined to the aforementioned $L^2$ -result, and indeed the investigations of Cordero–Nicola–Rodino also paved the way and inspired much activity in the field and, not least, some of the results of this paper.

Turning to the problem of $L^p$ -regularity for $p\neq 2$ , J. Peral [Reference Peral25] and A. Miyachi [Reference Miyachi22] studied the problem of $L^p$ -boundedness of Fourier multipliers of the form $m(\xi )= e^{i\varphi (\xi )} \, \sigma (\xi )$ , $\sigma (\xi )\in S^m_{1,0}(\mathbb {R}^n)$ and $\varphi $ positively homogeneous of degree one. It was realised that for a $\sigma \in S^m_{1,0}(\mathbb {R}^{n})$ the $L^p$ -boundedness can (in general) not hold if $m>-(n-1) |1/p-1/2|.$ M. Beals [Reference Beals2] extended their results to operators of the form (2) when the phase function is analytic in $\xi \in \mathbb {R}^n \setminus \{0\}$ , nondegenerate and positively homogeneous of degree one in $\xi $ .

For oscillatory Fourier multipliers of the form $m(\xi )=e^{i|\xi |^{k}} \psi (\xi )|\xi |^{m},$ $\psi $ smooth and vanishing near the origin, $0<k<1$ and $m<0$ , C. Fefferman and E.M. Stein proved the $L^p$ boundedness for $1<p<\infty $ in their seminal paper [Reference Fefferman and Stein12] on $H^p$ -spaces. The $L^p$ -boundedness of multipliers $m(\xi )$ in the case of $k>1$ was established by H. Ishii [Reference Ishii19]. All these results were further extended in an influential paper by Miyachi [Reference Israelsson, Mattsson and Staubach23], which has had a significant impact on the development of the regularity theory of oscillatory integral operators in Banach and quasi-Banach spaces. Miyachi showed that for $k=1$ the Fourier multiplier defined by $m(\xi )$ is $H^p$ -bounded for $0<p<\infty $ if and only if $m\leq -(n-1)|1/p-1/2|$ , and for $k>0$ (but $k\neq 1$ ) $T_a^\varphi $ is $H^p$ -bounded if and only if $m\leq -kn |1/p-1/2|.$ Moreover, Miyachi proves $L^\infty (\mathbb {R}^n) \to \mathrm {BMO}(\mathbb {R}^n)$ estimates as well as boundedness results in Lipschitz (or Hölder) spaces for the aforementioned Fourier multipliers.

In the range $1<p<\infty $ , Peral’s and Miyachi’s results for the ordinary wave operator were generalised by A. Seeger, C. Sogge and E. Stein [Reference Seeger, Sogge and Stein29] to Fourier integral operators with amplitudes $a(x, \xi )\in S^m_{1,0}(\mathbb {R}^{n})$ , $m\leq -(n-1)|1/p-1/2|$ , with compact spatial support and homogeneous of degree one (nondegenerate) phase functions $\varphi (x,\xi )$ . Using a novel method which was also partly inspired by Fefferman’s paper [Reference Fefferman11], Seeger–Sogge–Stein thus proved the optimal local $L^p$ -boundedness of Fourier integral operators. Extensions to global estimates in more general function spaces were carried out by A. Israelsson, S. Rodríguez-López and W. Staubach in [Reference Israelsson, Rodríguez-López and Staubach20]. Thus, the investigations mentioned above complete the picture regarding the regularity of Fourier integral operators of the form (2) with nondegenerate homogeneous of degree one phase functions and amplitudes in $S^m_{1,0}(\mathbb {R}^{n})$ .

For the so-called Schrödinger integral operators which are operators of the form (2) with nondegenerate phase functions $\varphi (x,\xi )$ satisfying equation (4), the question of boundedness has so far only been treated in the context of modulation spaces. Indeed Cordero, Nicola and Rodino [Reference Cordero, Nicola and Rodino6] showed that Schrödinger integral operators with amplitudes in the class $S^{0}_{0,0}(\mathbb {R}^n)$ are bounded from the modulation space $M_p$ to itself for $1\leq p< \infty .$ Regarding calculi for Schrödinger integral operators, Asada and Fujiwara [Reference Asada and Fujiwara1] studied the action of pseudodifferential operators on their class of oscillatory integrals and showed that their class is closed under composition with pseudodifferential operators of order zero. In 2013, E. Cordero, K-H. Gröchenig, F. Nicola and L. Rodino showed in [Reference Cordero, Gröchenig, Nicola and Rodino5] that the class of operators that we here refer to as Schrödinger integral operators (of order zero) is actually closed under composition.

1.2 Some relevant results in the theory of linear partial differential equations

In the paper [Reference Seeger, Sogge and Stein29], the authors prove optimal local regularity results for the solutions of strictly hyperbolic variable coefficients in $L^p$ -based Sobolev spaces for fixed time and their results were extended in [Reference Israelsson, Rodríguez-López and Staubach20] to global estimates in Besov–Lipschitz and Triebel–Lizorkin spaces.

The earlier investigations of B. Helffer and D. Robert [Reference Helffer and Robert16] and those of Helffer [Reference Helffer15] in connection to the study of propagators for Schrödinger equations (for example, the harmonic oscillator) have demonstrated the importance of Schrödinger integral operators; see, for example, the paper of Cordero–Nicola–Rodino [Reference Cordero, Nicola and Rodino7]. Furthermore, in the remarkable paper [Reference D’Ancona and Nicola8], P. D’Ancona and F. Nicola established sharp $L^p$ -frequency-truncated estimates (for certain $p'$ s) for the Schrödinger group $e^{itH}$ , where H is a nonnegative self-adjoint operator in $L^2(\mathbb {R}^n)$ whose heat operator $e^{-tH}$ satisfies a suitable off-diagonal algebraic decay estimate.

For dispersive equations of the form (1) with initial data $f(x)$ , the solutions are given by oscillatory integrals of the form

(5)

for which one can apply the results of [Reference Fefferman and Stein12] and [Reference Israelsson, Mattsson and Staubach23] to obtain certain regularity results in Hardy and Hölder spaces.

1.3 Synopsis of the results of this paper

Given the discussion above, some natural open problems are:

• • The extension of the results of Seeger–Sogge–Stein to global results that also accommodate the case of Klein–Gordon-type equations and also to the setting of both Banach and quasi-Banach spaces.

• • Development of the $L^p$ -regularity theory for oscillatory integral operators with phase functions in $\mathcal {C}^{\infty }(\mathbb {R}^n \times \mathbb {R}^n)$ that are associated to variable-coefficient Schrödinger equations.

• • The extension of the results of Miyachi to the variable-coefficient setting in the case of $k\neq 1$ . In other words, investigation of the regularity properties of oscillatory integral operators, with phase functions in $\mathcal {C}^{\infty }(\mathbb {R}^n \times \mathbb {R}^n \setminus \{0\})$ that fall beyond the scope of the theory of Fourier integral operators.

• • Estimates for variable coefficient Schrödinger as well as other dispersive equations in Triebel–Lizorkin and Besov–Lipschitz spaces.

In this paper, we have made progress in solving these problems and summarise our results as follows:

1. i) Established an optimal $L^p$ -regularity theory (and indeed even $L^p-L^q$ , $1< p\leq q<\infty $ ), for Schrödinger integral operators. We also go beyond these classes of operators and investigate the regularity of general oscillatory integral operators (see Definition 2.15) in classical function spaces. Furthermore, our results extend the range of validity of the estimates obtained by D’Ancona and Nicola in [Reference D’Ancona and Nicola8].

2. ii) Investigated the boundedness problem in both Banach and quasi-Banach spaces. Our regularity results are valid in Besov–Lipschitz and Triebel–Lizorkin spaces with both Banach and quasi-Banach scales.

3. iii) The abolition of the usual homogeneity assumption in the phase functions and improvement on the order of decay of the amplitudes. We show that it is enough for the amplitudes to merely belong to the class $S^{m}_{0,0}(\mathbb {R}^{n}),$ as opposed to the usual class $S^m_{1,0}(\mathbb {R}^{n})$ (which is used in all previously obtained $L^p-$ regularity results).

4. vi) Sharp boundedness results were obtained, namely we show that our results are optimal for the specific order of decay m that we choose. Our results are not only sharp regarding the order of the amplitudes but also optimal regarding their type, which is measured by the lowercase indices of the classes of amplitudes. Indeed, for those oscillatory integral operators that are not Fourier integral operators, choosing an amplitude in the better class $S^m_{1,0}(\mathbb {R}^{n})$ would not yield any improvement in the order of decay m (which is required for the regularity in various functions spaces).

5. v) Thorough analysis of the singular low-frequency part of globally defined oscillatory integral operators. The phase functions of oscillatory integrals such as those given by equation (5) have a singularity at origin when k is not an even positive integer. In previous investigations in the literature (i.e., Fefferman–Stein [Reference Fefferman and Stein12], Ishii [Reference Ishii19], Miyachi [Reference Israelsson, Mattsson and Staubach23], Peral [Reference Peral25] and Seeger–Sogge–Stein [Reference Seeger, Sogge and Stein29]), this singularity is always cut out in order to confine the analysis to the high-frequency part of the operator. However, in PDE theory, it is important to investigate the regularity of the whole operator (i.e., both the low- and high-frequency portions). In fact, the low-frequency part has a decisive impact on the range of the validity of global estimates in the quasi-Banach setting, as we have clearly demonstrated in the paper.

6. vi) Steps towards construction of a calculus for oscillatory integral operators. We prove a basic composition theorem (similar to that of Hörmander’s in the Fourier integral operator setting) for the composition of a pseudodifferential operator and a general oscillatory integral operator. The advantages of our ‘calculus’ as compared to Hörmander’s composition formula for Fourier multipliers and Fourier integral operators are (a) our composition result is global, (b) the composition works for operators with inhomogeneous phase functions and (c) our composition is parameter dependent and the dependency on the parameter is carefully tracked. We show the usefulness of this composition theorem in our demonstration of the regularity results in Besov–Lipschitz and Triebel–Lizorkin spaces.

7. vii) Sharp estimates for linear dispersive PDEs in Banach and quasi-Banach spaces. The oscillatory integrals that are studied in this paper can be used to construct solutions to various evolutionary PDEs and our oscillatory integral estimates provide new sharp estimates for the solutions to initial value problems in Besov–Lipshitz and Triebel–Lizorkin spaces.

The proofs of the results are divided into essentially two categories. Those for oscillatory integral operators whose phases have a singularity at the origin (the simplest examples are provided by equation (5)), and those oscillatory integrals whose phases are smooth everywhere (e.g., the case of Schrödinger integral operators). In the former case, we split the operator into, low-, middle-, and high-frequency pieces and deal with each piece separately. It turns out that the low-frequency portion requires an additional condition which is also reflected in the optimal range of the validity of the boundedness results. The high-frequency portion can be dealt with methods based on Littlwood–Paley theory. However, the differences here compared to that of the case of Fourier integral operators treated in [Reference Seeger, Sogge and Stein29] are that we deal with a larger class of amplitudes and also have to deal with inhomogeneous phase functions. This forces a different approach towards the proof of the boundedness of general oscillatory integral operators. Indeed, the usual approach of first showing the $H^1-L^1$ boundedness, which was carried out successfully for Fourier integral operators in [Reference Seeger, Sogge and Stein29], fails in our case. Moreover, the lack of a calculus (once again as opposed to the case of local Fourier integral operators in [Reference Seeger, Sogge and Stein29]) hampers the way of using duality arguments. To remedy this, we prove the boundedness of the operator and its adjoint in the quasi-Banach realm of the $h^p-L^p$ spaces ( $0<p<1$ and $h^p$ is the local Hardy space of D. Goldberg’s [Reference Goldberg14]). Thereafter, we lift these boundedness results to the $h^p-h^p$ boundedness and then use appropriate interpolation of Triebel–Lizorkin spaces to extend matters to all ranges of $p's$ . Thus, utilising the quasi-Banach Hardy spaces in this context is crucial for our goals.

Concerning the proof of the boundedness of Schrödinger integral operators under the mere assumptions of strong nondegeneracy of the phase function and condition (4), it is not enough to simply use the Littlewood–Paley theory. Here, we once again use quasi-Banach spaces and use yet another frequency localisation superimposed on the first Littlewood–Paley decomposition adapted specifically to Schrödinger integral operators. In this connection, we have a domain of influence for Littlewood–Paley pieces of the operator and one has to use an atomic decomposition of the Hardy spaces to be able to estimate the $L^p$ -norms of various pieces of the operator (and its adjoint) in the interior and the exterior of the aforementioned domain of influence. Putting the partial results together and summing up, we can show the global $h^p-L^p$ boundedness of the Schrödinger integral operators and use interpolation to obtain results for a wider range of $p's$ .

In obtaining regularity results in Besov–Lipschitz and Triebel–Lizorkin spaces, it behooves one to consider the action of Fourier multipliers on various oscillatory integral operators. To achieve that, one needs some sort of a calculus (or asymptotic expansion for the composition). The proof of this special form of calculus, which is also extended to the composition of a pseudodifferential operator and an oscillatory integral operator, is a rather technical task that has been carried out by a suitable splitting of the operators into various pieces and delicate oscillatory integral estimates.

The paper is organised as follows: In Section 2, we recall some of the basic definitions and facts from the theory of function spaces. We also define our classes of phase functions, amplitudes and the corresponding operators which will be treated in this paper. In Section 3, we state the main results of the paper and outline the proofs of the theorems. This includes both local and global regularity results in Besov–Lipschitz and Triebel–Lizorkin spaces, $L^p-L^q$ estimates and also our parameter-dependent composition theorem. Furthermore, in the same section we also provide some examples regarding applications of our results within harmonic analysis and partial differential equations. For example, we show the regularity of operators with phase functions of the form $x\cdot \xi + t(x)|\xi |^k$ with $0<k\leq 1$ and $x\cdot \xi + t(x)\left \langle \xi \right \rangle $ , with $t(x)$ being smooth and bounded together with all of its derivatives. The former is significant in the study of water-wave equation ( $k=1/2$ ), and the latter example is of significance in the study of Klein–Gordon equations. Since our regularity results are also valid for phase functions of the form $x\cdot \xi +|\xi |^k$ with $0<k<\infty $ , this enables us to prove sharp basic estimates (in both Banach and quasi-Banach scales) for the solutions of a large class of dispersive PDEs. Thereafter, we turn to variable-coefficient Schrödinger equations and show sharp estimates in classical function spaces for the solutions of Schrödinger equations with quadratic potentials (including the case of the harmonic oscillator).

Section 4 is devoted to the basic kernel estimates for oscillatory integral operators. In Section 5, we discuss the $L^2$ -regularity of the operators, and in Section 6, we deal with the boundedness of the low-frequency portion of the operators in Besov–Lipschitz and Triebel–Lizorkin spaces. Since we will sometimes divide the operators in question into low-frequency, middle-frequency and high-frequency portions, in Section 7 we treat the boundedness of the middle-frequency portion of the operators. In Section 8, we prove a local $h^p-L^p$ result for the oscillatory integral operators and Section 9 is devoted to the study of the $h^p-L^p$ boundedness of the high-frequency portion of the operators. The same problem for the Schrödinger integral operators is treated in Section 10. In Section 11, we prove a parameter-dependent composition formula and an expansion for the action of a pseudodifferential operator on an oscillatory integral operator. Section 12 and Section 13 are devoted to regularity results in Besov–Lipschitz and Triebel–Lizorkin spaces, respectively. The sharpness of the results is discussed in Section 14.