Smoothing estimates for non-dispersive equations

Mathematische Annalen, Jun 2016

This paper describes an approach to global smoothing problems for non-dispersive equations based on ideas of comparison principle and canonical transformation established in authors’ previous paper (Ruzhansky and Sugimoto, Proc Lond Math Soc, 105:393–423, 2012), where dispersive equations were treated. For operators \(a(D_x)\) of order m satisfying the dispersiveness condition \(\nabla a(\xi )\ne 0\) for \(\xi \not =0\), the global smoothing estimate $$\begin{aligned} {\left\| {{\left\langle {x}\right\rangle }^{-s}|D_x|^{(m-1)/2}e^{ita(D_x)} \varphi (x)}\right\| }_{L^2{\left( {{\mathbb R}_t\times {\mathbb R}^n_x}\right) }} \le C{\left\| {\varphi }\right\| }_{L^2{\left( {{\mathbb R}^n_x}\right) }} \quad (s>1/2) \end{aligned}$$ is well-known, while it is also known to fail for non-dispersive operators. For the case when the dispersiveness breaks, we suggest the estimate in the form $$\begin{aligned} {\left\| {{\left\langle {x}\right\rangle }^{-s}|\nabla a(D_x)|^{1/2} e^{it a(D_x)}\varphi (x)}\right\| }_{L^2{\left( {{\mathbb R}_t\times {\mathbb R}^n_x}\right) }} \le C{\left\| {\varphi }\right\| }_{L^2{\left( {{\mathbb R}^n_x}\right) }} \quad (s>1/2) \end{aligned}$$ which is equivalent to the usual estimate in the dispersive case and is also invariant under canonical transformations for the operator \(a(D_x)\). We show that this estimate and its variants do continue to hold for a variety of non-dispersive operators \(a(D_x)\), where \(\nabla a(\xi )\) may become zero on some set. Moreover, other types of such estimates, and the case of time-dependent equations are also discussed.

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Smoothing estimates for non-dispersive equations

Mathematische Annalen Michael Ruzhansky 0 1 Mitsuru Sugimoto 0 1 0 Graduate School of Mathematics, Nagoya University , Furocho, Chikusa-ku, Nagoya 464-8602 , Japan 1 Department of Mathematics, Imperial College London, 180 Queen's Gate , London SW7 2AZ , UK This paper describes an approach to global smoothing problems for non-dispersive equations based on ideas of comparison principle and canonical transformation established in authors' previous paper (Ruzhansky and Sugimoto, Proc Lond Math Soc, 105:393-423, 2012), where dispersive equations were treated. For operators a( Dx ) of order m satisfying the dispersiveness condition ∇a(ξ ) = 0 for ξ = 0, the global smoothing estimate Mitsuru Sugimoto - x −s | Dx |(m−1)/2eita(Dx )ϕ(x ) L2(Rt ×Rnx ) ≤ C ϕ L2(Rnx ) is well-known, while it is also known to fail for non-dispersive operators. For the case when the dispersiveness breaks, we suggest the estimate in the form x −s |∇a( Dx )|1/2eita(Dx )ϕ(x ) L2(Rt ×Rnx ) ≤ C ϕ L2(Rnx ) M. Ruzhansky was supported in parts by the EPSRC Grant EP/K039407/1 and by the Leverhulme Grant RPG-2014-02. No new data was collected or generated during the course of the research. M. Sugimoto was supported in parts by the JSPS KAKENHI 26287022 and 26610021. B Michael Ruzhansky which is equivalent to the usual estimate in the dispersive case and is also invariant under canonical transformations for the operator a(Dx ). We show that this estimate and its variants do continue to hold for a variety of non-dispersive operators a(Dx ), where ∇a(ξ ) may become zero on some set. Moreover, other types of such estimates, and the case of time-dependent equations are also discussed. 1 Introduction Various kinds of smoothing estimates for the solutions u(t, x ) = eita(Dx )ϕ(x ) to equations of general form where a(Dx ) is the corresponding Fourier multiplier to a real-valued function a(ξ ), have been extensively studied under the ellipticity (a(ξ ) = 0) or the dispersiveness (∇a(ξ ) = 0) conditions, exclusive of the origin ξ = 0 (i.e. for ξ = 0) when a(ξ ) is a homogeneous function. Such conditions include Schrödinger equation as a special case (a(ξ ) = |ξ |2). Since Kato’s local gain of one derivative for the linearised KdV in [12], and the independent works by Ben-Artzi and Devinatz [1], Constantin and Saut [7], Sjölin [34] and Vega [38], the local, and then global smoothing estimates, together with their application to non-linear problem have been intensively investigated in a series of papers such as [2,4,5,10,11,13–21,27–30,33,35,36,39–41], to mention a few. Among them, a comprehensive analysis is presented in our previous paper [30] for global smoothing estimates by using two useful methods, that is, canonical transformations and the the comparison principle. Canonical transformations are a tool to transform one equation to another at the estimate level, and the comparison principle is a tool to relate differential estimates for solutions to different equations. These two methods work very effectively under the dispersiveness conditions to induce a number of new or refined global smoothing estimates, as well as many equivalences between them. Using these methods, the proofs of smoothing estimates are also considerably simplified. The objective of this paper is to continue the investigation by the same approach in the case when the dispersiveness breaks down. We will conjecture what we may call an ‘invariant estimate’ extending the smoothing estimates to the non-dispersive case. Such an estimate yields the known smoothing estimates in dispersive cases, it is invariant under canonical transforms of the problem, and we will show its validity for a number of non-dispersive evolution equations of several different types. The most typical example of a global smoothing estimate is of the form where m denotes the order of the operator a(Dx ), and this estimate, together with other similar kind of global smoothing estimates, has been already justified under appropriate dispersiveness assumptions (see Sect. 2.1). Throughout this paper we use the standard notation x = (1 + |x |2)1/2 and Dx = (1 − Note that the L2-norm of the solution is always the same as that of the Cauchy data ϕ for any fixed time t , but estimate (1.2) means that the extra gain of regularity of order (m − 1)/2 in the spacial variable x can be observed if we integrate the solution eita(Dx )ϕ(x ) to Eq. (1.1) in the time variable t . One interesting conclusion in [30] is that our method allowed us to carry out a global microlocal reduction of estimate (1.2) to the translation invariance of the Lebesgue measure. On the other hand, despite their natural appearance in many problems, quite limited results are available for non-dispersive equations while the dispersiveness condition was shown to be necessary for most common types of global smoothing estimates (see [11]). To give an example, coupled dispersive equations are of high importance in applications while only limited analysis is available. Let v(t, x ) and w(t, x ) solve the following coupled system of Schrödinger equations: This is the simplest example of Schrödinger equations coupled through linearised operators b(Dx ), c(Dx ). Such equations appear in many areas in physics. For example, this is a model of wave packets with two modes (in the presence of resonances), see Tan and Boyd [37]. In fibre optics they appear to describe certain types of a pair of coupled modulated wave-trains (see e.g. Manganaro and Parker [22]). They also describe the field of optical solitons in fibres (see Zen and Elim [42]) as well as Kerr dispersion and stimulated Raman scattering for ultrashort pulses transmitted through fibres. In these cases the linearised operators b and c would be of zero order. In models of optical pulse propagation of birefringent fibres and in wavelength-divisionmultiplexed systems they are of the first order (see Pelinovsky and Yang [25]). They may be of higher orders as well, for example in models of optical solitons with higher order effects (see Nakkeeran [24]). Suppose now that we are in the simplest situation when system (1.3) can be diagonalised. Its eigenvalues are and the system uncouples into scalar equations of type (1.1) with operators a(Dx ) = a±(Dx ). Since the structure of operators b(Dx ), c(Dx ) may be quite involved, this motivates the study of scalar equations (1.1) with operators a(Dx ) of rather general form. Not only the presence of lower order terms is important in time global problems, the principal part may be rather general since we may have ∇a± = 0 at some points. Let us also briefly mention another concrete example. Equations of the third order often appear in applications to water wave equations. For example, the Shrira equation [32] describing the propagation of a three-dimensional packet of weakly nonlinear internal gravity waves leads to third order polynomials in two dimensions. The same types of third order polynomials in two variables also appear in the Dysthe equation as well as in the Hogan equation, both describing the behaviour of deep water waves in 2-dimensions. Strichartz estimates for the corresponding solutions have been analysed by e.g. Ghidaglia and Saut [9] and by Ben-Artzi et al. [3] by reducing the equations to pointwise estimates for operators in normal forms. In general, by linear changes of variables, polynomials a(ξ1, ξ2) of order 3 are reduced to one of the following normal forms: modulo polynomials of order one. Strichartz estimates have been obtained for operators having their symbols in this list except for the cases a(ξ1, ξ2) = ξ13 and ξ1ξ22, see Ben-Artzi et al. [3]. Some of normal forms listed here satisfy dispersiveness assumptions and can be discussed by existing results listed in Sect. 2.1. Indeed, a(ξ1, ξ2) = ξ13 + ξ23 and ξ13 − ξ1ξ22 are homogeneous and satisfy the dispersiveness ∇a(ξ1, ξ2) = 0 except for the origin [which is assumption (H) in Sect. 2.1]. However the dispersiveness assumption is sensitive to the perturbation by polynomials of order one. For example, still satisfies the dispersiveness ∇a(ξ1, ξ2) = 0 everywhere [see assumption (L) in Sect. 2.1], but it breaks for Furthermore the other normal forms do not satisfy neither of these assumptions, with the corresponding equation losing dispersiveness at some points (see Examples 3.2 and 3.5). We now turn to describing an estimate which holds in such cases even when the dispersiveness fails at some points in the phase space. In this paper, based on the methods of comparison principle and canonical transforms, we develop several approaches to getting smoothing estimates in such non-dispersive cases. Since standard global smoothing estimates are known to fail in non-dispersive cases by [11], we will suggest an invariant form of global smoothing estimates instead, for the analysis, and call them invariant estimates, which we expect to continue to hold even in non-dispersive cases. As an example of it, estimate (1.2) may be rewritten in the form Indeed the normal form a(ξ1, ξ2) = ξ13 listed in (1.4) is known to satisfy this estimate (see estimate (5.6) in Corollary 5.7). Other types of invariant estimates are also suggested in Sect. 2.2. Such estimate has a number of advantages: • in the dispersive case it is equivalent to the usual estimate (1.2); • it does continue to hold for a variety of non-dispersive equations, where ∇a(ξ ) may become zero on some set and when (1.2) fails; • it does take into account possible zeros of the gradient ∇a(ξ ) in the non-dispersive case, which is also responsible for the interface between dispersive and nondispersive zone (e.g. how quickly the gradient vanishes); • it is invariant under canonical transformations of the equation. We will also try to justify invariant estimates for non-dispersive equations. The combination of the proposed two methods (canonical transformations and the comparison principles) has a good power again on the occasion of this analysis. Besides the simplification of the proofs of global smoothing estimates for standard dispersive equations, we have the following advantage in treating non-dispersive equations where the dispersiveness condition ∇a(ξ ) = 0 breaks: • in radially symmetric cases, we can use the comparison principle of radially symmetric type (Theorem 3.1); • in polynomial cases we can use the comparison principle of one dimensional type (Theorem 3.2); • in the homogeneous case with some information on the Hessian, we can use canonical transformation to reduce the general case to some well-known model situations (Theorem 3.3); • around non-dispersive points where the Hessian is non-degenerate, we can microlocalise and apply the canonical transformation based on the Morse lemma (Theorem 3.4); In particular, in the radially symmetric cases, we will see that estimate (1.5) is valid in a generic situation (Theorem 3.1). And as another remarkable result, it is also valid for any differential operators with real constant coefficients, including operators corresponding to normal forms listed in (1.4) with perturbation by polynomials of order one (Theorem 3.2). Some normal forms are also covered by Theorem 3.4. In addition, we will derive estimates for equations with time dependent coefficients. In general, the dispersive estimates for equations with time dependent coefficients may be a delicate problem, with decay rates heavily depending on the oscillation in coefficients (for a survey of different results for the wave equation with lower order terms see, e.g. Reissig [26]; for more general equations and systems and the geometric analysis of the time-decay rate of their solution see [31] or [8]). However, we will show in Sect. 4 that the smoothing estimates still remain valid if we introduce an appropriate factor into the estimate. Such estimates become a natural extension of the invariant estimates to the time dependent setting. We will explain the organisation of this paper. In Sect. 2, we list typical global smoothing estimates for dispersive equations, and then discuss their invariant form which we expect to remain true also in non-dispersive situations. In Sect. 3, we establish invariant estimates for several types of non-dispersive equations. The case of time-dependent coefficients will be treated in Sect. 4. In Appendix A, we review our fundamental tools, that is, the canonical transformation and the comparison principle, which is used for the analysis in Sect. 3. Finally we comment on the notation used in this paper. When we need to specify the entries of the vectors x , ξ ∈ Rn, we write x = (x1, x2, . . . , xn), ξ = (ξ1, ξ2, . . . , ξn) without any notification. As usual we will denote ∇ = (∂1, . . . , ∂n) where ∂ j = ∂x j , Dx = (D1, D2 . . . , Dn) where D j = −√−1 ∂ j ( j = 1, 2, . . . , n), and view operators a(Dx ) as Fourier multipliers. We denote the set of the positive real numbers (0, ∞) by R+. Constants denoted by letter C in estimates are always positive and may differ on different occasions, but will still be denoted by the same letter. 2 Invariant smoothing estimates for dispersive equations In this section we collect known smoothing estimates for dispersive equations under several different assumptions. Then we show that the invariant estimate (1.5) holds in these cases and is, in fact, equivalent to the known estimates. Thus, let us consider the solution u(t, x ) = eita(Dx )ϕ(x ) where we always assume that function a(ξ ) is real-valued. We sometimes decompose the initial data ϕ into the sum of the low frequency part ϕl and the high frequency part ϕh , where with sufficiently large R > 0. First we review a selection of known results on global smoothing estimates established in [30] when the dispersiveness assumption ∇a(ξ ) = 0 for ξ = 0 is satisfied, and then rewrite them in a form which is expected to hold even in non-dispersive situations. 2.1 Smoothing estimates for dispersive equations First we collect known results for dispersive equations. Let am (λξ ) = λm am (ξ ) for all λ > 0 and ξ = 0. Theorem 2.1 Assume (H). Then there exists a constant C > 0 such that the following estimates hold true. • Suppose n ≥ 1, m > 0, and s > 1/2. Then we have x −s |Dx |(m−1)/2eita(Dx )ϕ(x ) L2(Rt ×Rnx) ≤ C ϕ L2(Rnx ). • Suppose m > 0 and (m − n + 1)/2 < α < (m − 1)/2. Or, in the elliptic case a(ξ ) = 0 (ξ = 0), suppose m > 0 and (m − n)/2 < α < (m − 1)/2. Then we have L2(Rt ×Rnx) ≤ C ϕ L2(Rnx). • Suppose n − 1 > m > 1, but in the elliptic case a(ξ ) = 0 (ξ = 0) suppose n > m > 1. Then we have x −m/2 Dx (m−1)/2eita(Dx )ϕ(x ) L2(Rt ×Rnx ) ≤ C ϕ L2(Rnx ). In the Schrödinger equation case a(ξ ) = |ξ |2 and for n ≥ 3, estimate (2.1) was obtained by Ben-Artzi and Klainerman [2]. It follows also from a sharp local smoothing estimate by Kenig et al. [15, Theorem 4.1]), and also from the one by Chihara [5] who treated the case m > 1. For the range m > 0 and any n ≥ 1 it was obtained in [30, Theorem 5.1]. Compared to (2.1), the estimate (2.2) is scaling invariant with the homogeneous weights |x |−s instead of non-homogenous ones x −s . The estimate (2.2) was obtained in [30, Theorem 5.2], and it is a generalisation of the result by Kato and Yajima [14] ξ 2 with n ≥ 3 and 0 ≤ α < 1/2, or with n = 2 and who treated the case a(ξ ) = | | 0 < α < 1/2, and also of the one by Sugimoto [35] who treated elliptic a(ξ ) of order m = 2 with n ≥ 2 and 1 − n/2 < α < 1/2. The smoothing estimate (2.3) is of yet another type replacing |Dx |(m−1)/2 by its non-homogeneous version Dx (m−1)/2, obtained in [30, Corollary 5.3]. It is a direct consequence of (2.1) with s = m/2 and (2.2) with α = 0 (note also the L2-boundedness of Dx (m−1)/2(1 + |Dx |(m−1)/2)−1), and it also extends the result by Kato and Yajima [14] who treated the case a(ξ ) = |ξ |2 and n ≥ 3, the one by Walther [40] who treated the case a(ξ ) = |ξ |m , and the one by the authors [27] who treated the elliptic case with m = 2. We can also consider the case that a(ξ ) has lower order terms, and assume that a(ξ ) is dispersive in the following sense: a(ξ ) ∈ C ∞(Rn ), ∇a(ξ ) = 0 (ξ ∈ Rn), ∇am (ξ ) = 0 (ξ ∈ Rn \0), |∂α(a(ξ ) − am (ξ ))| ≤ Cα|ξ |m−1−|α| for all multi-indices α and all |ξ | ≥ 1. Condition (L) may be formulated equivalently in the following way a(ξ ) ∈ C ∞(Rn ), |∇a(ξ )| ≥ C ξ m−1 (ξ ∈ Rn ) for some C > 0, |∂α(a(ξ ) − am (ξ ))| ≤ Cα|ξ |m−1−|α| for all multi-indices α and all |ξ | ≥ 1. The last line of these assumptions simply amount to saying that the principal part am of a is positively homogeneous of order m for |ξ | ≥ 1. Then we have the following estimates: Theorem 2.2 Assume (L). Then there exists a constant C > 0 such that the following estimates hold true. • Suppose n ≥ 1, m > 0, and s > 1/2. Then we have • Suppose n ≥ 1, m ≥ 1 and s > 1/2. Then we have x −s Dx (m−1)/2eita(Dx )ϕ(x ) L2(Rt ×Rnx ) ≤ C ϕ L2(Rnx ). x −s |Dx |(m−1)/2eita(Dx )ϕ(x ) L2(Rt ×Rnx) ≤ C ϕ L2(Rnx ). The estimate (2.4) was established in [30, Theorem 5.4]. Consequently, (2.5) is a straightforward consequence of (2.4) and the L2-boundedness of the Fourier multiplier |Dx |(m−1)/2 Dx −(m−1)/2 with m ≥ 1. It is an analogue of (2.1) for operators a(Dx ) with lower order terms, and also a generalisation of [15, Theorem 4.1] who treated essentially polynomial symbols a(ξ ). For (2.5) in its full generality we refer to [30, Corollary 5.5]. Assumption (L) requires the condition ∇a(ξ ) = 0 (ξ ∈ Rn) for the full symbol, besides the same one ∇am (ξ ) = 0 (ξ = 0) for the principal term. To discuss what happens if we do not have the condition ∇a(ξ ) = 0, we can introduce an intermediate assumption between (H) and (L): a(ξ ) = am (ξ ) + r (ξ ), ∇am (ξ ) = 0 (ξ ∈ Rn \0), r (ξ ) ∈ C ∞(Rn) |∂αr (ξ )| ≤ C ξ m−1−|α| for all multi-indices α. Theorem 2.2 remain valid if we replace assumption (H) by (HL) and functions ϕ(x ) in the estimates by their (sufficiently large) high frequency parts ϕh (x ). However we cannot control the low frequency parts ϕl (x ), and so have only the time local estimates on the whole, which we now state for future use: Theorem 2.3 [30, Theorem 5.6] Assume (HL). Suppose n ≥ 1, m > 0, s > 1/2, and T > 0. Then we have x −s Dx (m−1)/2eia(Dx )ϕ(x ) dt ≤ C ϕ 2L2(Rn), where C > 0 is a constant depending on T > 0. 2.2 Invariant estimates Let us now suggest an invariant form of time global smoothing estimates which might remain valid also in some areas without dispersion ∇a(ξ ) = 0, where standard smoothing estimates are known to fail (see Hoshiro [11]). We can equivalently rewrite estimates above under the dispersiveness assumption (H) or (L) in the form w(x )ζ (|∇a(Dx )|)eita(Dx )ϕ(x ) L2(Rt ×Rnx) ≤ C ϕ L2(Rnx ) (s > 1/2), (2.6) where w is a weight function of the form w(x ) = |x |δ or x δ, and the smoothing is given by the function ζ on R+ of the form ζ (ρ) = ρη or 1 + ρ2 η/2, with some δ, η ∈ R. For example, we can rewrite estimate (2.1) of Theorem 2.1 as well as estimate (2.5) of Theorem 2.2 for the dispersive equations in the form x −s |∇a(Dx )|1/2eita(Dx )ϕ(x ) L2(Rt ×Rnx ) ≤ C ϕ L2(Rnx ). Similarly we can rewrite estimate (2.2) of Theorem 2.1 in the form L2(Rt ×Rnx ) ≤ C ϕ L2(Rnx ) (m = 1), and estimate (2.3) of Theorem 2.1 (with s = −m/2) as well as estimate (2.4) of Theorem 2.2 (with s > 1/2) in the form x −s ∇a(Dx ) 1/2eita(Dx )ϕ(x ) L2(Rt ×Rnx) ≤ C ϕ L2(Rnx). Indeed, under assumption (H) we clearly have |∇a(ξ )| ≥ c|ξ |m−1, so the equivalence between estimate (2.7) and estimate (2.1) in Theorem 2.1 follows from the fact that the Fourier multipliers |∇a(Dx )|1/2|Dx |−(m−1)/2 and |∇a(Dx )|−1/2|Dx |(m−1)/2 are bounded on L2(Rn). Under assumption (L) the same argument works for large |ξ |, while for small |ξ | both ξ (m−1)/2 and |∇a(ξ )|1/2 are bounded away from zero. Thus we have the equivalence between estimate (2.7) and estimate (2.5) in Theorem 2.2. The same is true for the other equivalences. As we will see later (Theorem 5.2), estimate (2.6), and hence estimates (2.7)–(2.9) are invariant under canonical transformations. On account of it, we will call estimate (2.6) an invariant estimate, and indeed we expect invariant estimates (2.7), (2.8), and (2.9) to hold without dispersiveness assumption (H) or (L), for s > 1/2, (m − n)/2 < α < (m − 1)/2, and s = −m/2 (n > m > 1), respectively in ordinary settings (elliptic case for example), where m > 0 is the order of a(Dx ). We will discuss and establish them in Sect. 3 in a variety of situations. Here is an intuitive understanding of the invariant estimate (2.7) with s > 1/2 by spectral argument. Let E (λ) be the spectral family of the self-adjoint realisation of a(Dx ) on L2(Rn), that is Then its spectral density d 1 A(λ) = dλ E (λ) = 2π i (R(λ + i 0) − R(λ − i 0)), which continues to have a meaning even if ∇a(ξ ) may vanishes (although this is just a formal observation). On the other hand, the right hand side of this identity with f = g has the uniform estimate in λ ∈ R by the one-dimensional Sobolev embedding (It can be justified at least when ∇a(ξ ) = 0. See [6, Lemma 1]). If once we justify them, we could have the estimate where H ∗ is the adjoint operator of H = x −s |∇a(Dx )|1/2. Estimate (2.10) can be regarded as the a(Dx )-smooth property which is equivalent to invariant estimate (2.7) (see [12, Theorem 5.1]). Or we may proceed as in [1] or [2] to obtain the same conclusion. In fact, we have L2(Rt ×Rnx ) = ≤ ( A(λ)ϕ, ϕ)1/2 A(λ)H ∗ψ (λ, ·), H ∗ψ (λ, ·) 1/2 by estimate (2.10). Hence by Schwartz inequality, Plancherel’s theorem, and the fact −∞∞ ( A(λ)ϕ, ϕ) dλ = ϕ 2L2(Rnx ), we have the estimate H eita(Dx )ϕ(x ), ψ (t, x ) L2(Rt ×Rnx ) ≤ C ϕ L2(Rnx ) ψ (t, x ) L2(Rt ×Rnx ) which is again equivalent to invariant estimate (2.7). Let us now show that the invariant estimate (2.7) with s > 1/2 is also a refinement of another known estimate for non-dispersive equations, namely of the smoothing estimate obtained by Hoshiro [10]. If operator a(Dx ) has real-valued symbol a = a(ξ ) ∈ C 1(Rn) which is positively homogeneous of order m ≥ 1 and no dispersiveness assumption is made, Hoshiro [10] showed the estimate But once we prove (2.7) with s > 1/2, we can have a better estimate with respect to the number of derivatives. In fact, using the Euler’s identity we see that this estimate trivially follows from x −s Dx −1/2|Dx |1/2|∇a(D)|1/2eita(Dx )ϕ(x ) ≤ C ϕ L2(Rnx) (s > 1/2), which in turn follows from (2.7) with s > 1/2 because the Fourier multiplier operator Dx −1/2|Dx |1/2 is L2(Rn)-bounded. In fact, estimate (2.11) holds only because of the homogeneity of a, since in this case by Euler’s identity zeros of a contain zeros of ∇a. In general, estimate (2.11) cuts off too much, and therefore does not reflect the nature of the problem for non-homogeneous symbols, as (2.7) still does. In terms of invariant estimates, we can also give another explanation to the reason why we do not have time-global estimate in Theorem 2.3. The problem is that the symbol of the smoothing operator Dx (m−1)/2 does not vanish where the symbol of ∇a(Dx ) vanishes, as should be anticipated by the invariant estimate (2.7). If zeros of ∇a(Dx ) are not taken into account, the weight should change to the one as in estimate (2.3). 3 Smoothing estimates for non-dispersive equations Canonical transformations and the comparison principle recalled for convenience in Appendix A are still important tools when we discuss the smoothing estimates for non-dispersive equations where real-valued functions a(ξ ) fail to satisfy dispersive assumption (H) or (L) in Sect. 2.1. However, the secondary comparison tools stated in Appendix A.3 work very effectively in such situations. In Corollary 5.8 for example, even if we lose the dispersiveness assumption at zeros of f , the estimate is still valid because σ must vanish at the same points with the order determined by condition |σ (ρ)| ≤ A| f (ρ)|1/2. The same is true in other comparison results in Corollary 5.9. In this section, we will treat the smoothing estimates of non-dispersive equations based on these observations. 3.1 Radially symmetric case The following result states that we still have estimate (2.7) of invariant form suggested in Sect. 2.2 even for non-dispersive equations in a general setting of the radially symmetric case. Remarkably enough, it is a straight forward consequence of the second comparison method of Corollary 5.8, and in this sense, it is just an equivalent expression of the translation invariance of the Lebesgue measure (see Appendix A.3): Theorem 3.1 Suppose n ≥ 1 and s > 1/2. Let a(ξ ) = f (|ξ |), where f ∈ C 1(R+) is real-valued. Assume that f has only finitely many zeros. Then we have x −s |∇a(Dx )|1/2eita(Dx )ϕ(x ) L2(Rt ×Rnx ) ≤ C ϕ L2(Rnx ). Proof Noticing |∇a(ξ )| = | f (|ξ |)|, use Corollary 5.8 for σ (ρ) = | f (ρ)|1/2 in each interval where f is strictly monotone. Example 3.1 As a consequence of Theorem 3.1, we have the estimate of invariant form (2.7) if a(ξ ) is a real polynomial of |ξ |. This fact is not a consequence of Theorems 2.1 or 2.2. For example, let with f (ρ) being a non-constant real polynomial on R. The principal part am (ξ ) of a(ξ ) is a power of |ξ |2 multiplied by a constant, hence it satisfies ∇am (ξ ) = 0 (ξ = 0). If f (ρ) is a homogeneous polynomial, then a(ξ ) satisfies assumption (H) and we have estimate (2.7) by Theorem 2.1. In the case when f (ρ) is not homogeneous, trivially a(ξ ) does not satisfy (H). Furthermore a(ξ ) does not always satisfy assumption (L) either since vanishes on the set |ξ |2 = c such that f (c) = 0 or f (c) = 0 as well as at the origin ξ = 0. Hence Theorem 2.2 does not assure the estimate (2.7) for a(ξ ) = (|ξ |2 − 1)2 (we take f (ρ) = ρ − 1), for example, but even in this case, we have the invariant smoothing estimate (3.1) by Theorem 3.1. 3.2 Polynomial case Another remarkable fact is that we can obtain invariant estimate (2.7) for all differential equations with real constant coefficients if we use second comparison method of Corollary 5.9 (hence again it is just an equivalent expression of the translation invariance of the Lebesgue measure): x −s |∇a(Dx )|1/2eita(Dx )ϕ(x ) L2(Rt ×R2x ) ≤ C ϕ L2(R2x ). Proof We can assume that the polynomial a is not a constant since otherwise the estimate is trivial. Thus, let m ≥ 1 be the degree of polynomial a(ξ ), and we write it in the form a(ξ ) = p0ξ1m + p1(ξ )ξ1m−1 + · · · + pm−1(ξ )ξ1 + pm (ξ ), where pk (ξ ) is a real polynomial in ξ = (ξ2, . . . , ξn) of degree k (k = 0, 1, . . . , m). The polynomial equation ∂1a(ξ ) = mp0ξ1m−1 + (m − 1) p1(ξ )ξ1m−2 + · · · + pm−1(ξ ) = 0 in ξ1 has at most m −1 real roots. For k = 0, 1, . . . , m −1, let Uk be the set of ξ ∈ Rn−1 for which it has k distinct real simple roots λk,1(ξ ) < λk,2(ξ ) · · · < λk,k (ξ ), and let k,l := (ξ1, ξ ) ∈ Rn : ξ ∈ Uk , λk,l (ξ ) < ξ1 < λk,l+1(ξ ) for l = 0, 1, . . . , k, regarding λk,0 and λk,k+1 as −∞ and ∞ respectively. Then we have the decomposition and a(ξ ) is strictly monotone in ξ1 on each k,l . Hence by taking χ in Corollary 5.9 to be the characteristic functions χk,l of the set k,l , we have the estimate Taking the sum in k, l and noting the fact that the Lebesgue measure of the complement of the set is zero, we have x1 −s |∂1a(Dx )|1/2eita(Dx )ϕ(x ) x j −s ∂ j a(Dx ) 1/2eita(Dx )ϕ(x ) and combining them all we have x −s |∂1a(Dx )|1/2 + · · · + |∂na(Dx )|1/2 eita(Dx )ϕ(x ) η(ξ ) = |∇a(ξ )|1/2 |∂1a(ξ )|1/2 + · · · + |∂na(ξ )|1/2 −1, Example 3.2 Some of normal forms listed listed in (1.4) in Introduction satisfy dispersiveness assumptions. Indeed, a(ξ1, ξ2) = ξ13 + ξ23 and ξ13 − ξ1ξ22 are homogeneous and satisfy the assumption (H). The other normal forms however satisfy neither (H) nor (L). For example, a(ξ1, ξ2) = ξ13 and ξ1ξ22 are homogeneous but ∇a(ξ1, ξ2) = 0 ξw1hξe22n+ξξ112=are0naontdhoξ2mo=ge0nereosupseacntidvesalyti.sOfyn∇thae(ξo1t,hξe2r)h=an0daat (thξ1e,oξr2i)gi=n. Sξe13e+Exξa22mapnlde 3.5 for others. But even for them we still have invariant smoothing estimate (2.7) with s > 1/2 by Theorem 3.2. 3.3 Hessian at non-dispersive points Now we will present another approach to treat non-dispersive equations. Recall that, in [30, Section 5], the method of canonical transformation effectively works to reduce smoothing estimates for dispersive equations listed in Sect. 2.1 to some model estimates. For example, as mentioned in the beginning of Appendix A.3, estimate (2.1) in Theorem 2.1 is reduced to model estimates (5.6) and (5.7) in Corollary 5.7. We explain here that this strategy works for non-dispersive cases as well. We will however look at the rank of the Hessian ∇2a(ξ ), instead of the principal type assumption ∇a(ξ ) = 0. Assume now that a = a(ξ ) ∈ C ∞(Rn\0) is real-valued and positively homogeneous of order two. It can be noted that from Euler’s identity we obtain since ∇a(ξ ) is homogeneous of order one (here ξ is viewed as a row). Then the condition rank ∇2a(ξ ) = n implies ∇a(ξ ) = 0 (ξ = 0), and as we have already explained, we have invariant estimates (2.7), (2.8) and (2.9) by Theorem 2.1 in this favourable case. We will show that in the non-dispersive situation the rank of ∇2a(ξ ) still has a responsibility for smoothing properties. We assume rank ∇2a(ξ ) ≥ k whenever ∇a(ξ ) = 0 (ξ = 0) with some 1 ≤ k ≤ n − 1. We note that condition (3.4) is invariant under the canonical transformation in the following sense: Lemma 3.1 Let a = σ ◦ ψ , with ψ : U → Rn satisfying det Dψ (ξ ) = 0 on an open set U ⊂ Rn. Then, for each ξ ∈ U , ∇a(ξ ) = 0 if and only if ∇σ (ψ (ξ )) = 0. Furthermore the ranks of ∇2a(ξ ) and ∇2σ (ψ (ξ )) are equal on whenever ξ ∈ U and ∇a(ξ ) = 0. Proof Differentiation gives and we have the first assertion. Another differentiation gives To fix the notation, we assume ∇a(en) = 0 and rank ∇2a(en) = k (1 ≤ k ≤ n − 1), where en = (0, . . . , 0, 1). Then we have a(en) = 0 by Euler’s identity 2a(ξ ) = ξ · ∇a(ξ ). We claim that there exists a conic neighbourhood ⊂ Rn\0 of en and a homogeneous C ∞-diffeomorphism ψ : → (satisfying ψ (λξ ) = λψ (ξ ) for all λ > 0 and ξ ∈ ) as appeared in Appendix A.1 such that we have the form a(ξ ) = (σ ◦ ψ )(ξ ), σ (η) = c1η1 + · · · + ck ηk2 + r (ηk+1, . . . , ηn ), 2 where η = (η1, . . . , ηn) and c j = ±1 ( j = 1, 2, . . . , k). We remark that r must be real-valued and positively homogeneous of order two. We will prove the existence of such ψ that will satisfy (3.6). By (3.3), (3.5), and the symmetricity, all the entries of the matrix ∇2a(en ) are zero except for the (perhaps) non-zero upper left (n − 1) × (n − 1) corner matrix. Moreover, by a linear transformation involving only the first (n − 1) variables of ξ = (ξ1, . . . , ξn−1, ξn ), we may assume ∂2a/∂ξ12(en) = 0. We remark that (3.5) still holds under this transformation. Then, by the Malgrange preparation theorem, we can write a(ξ ) = ±c(ξ )2(ξ12 + a1(ξ )ξ1 + a2(ξ )), ξ = (ξ2, . . . , ξn ). locally in a neighbourhood of en, where c(ξ ) > 0 is some strictly positive function, while function a1 and a2 are smooth and real-valued. Restricting this expression to the hyperplane ξn = 1, and using the homogeneity we can extend the expression (3.7) to a conic neighbourhood of en, so that functions c(ξ ), a1(ξ ) and a2(ξ ) are positively homogeneous of orders zero, one, and two, respectively. Let us define ψ0(ξ ) = c(ξ )ξ and where we write η = (η1, η ), η = (η2, . . . , ηn). Furthermore, let us define ψ1(ξ ) = (ξ1 + 21 a1(ξ ), ξ ), so that τ (ξ ) = (σ ◦ ψ1)(ξ ) with σ (η) = η12 + r (η ), where r (η ) = a2(η ) − 41 a1(η )2 is positively homogeneous of degree two. Then we have a = σ ◦ ψ , where ψ = ψ1 ◦ ψ0, and thus we have the expression (3.6) with k = 1. We note that, by the construction, we have where c(en ) > 0 and en = (0, . . . , 0, 1) ∈ Rn−1. Then we can see that the function r (η ) of (n − 1)-variables is defined on a conic neighbourhood of en in Rn−1. On account of this fact and Lemma 3.1, we can apply the same argument above to r (η ), and repeating the process k-times, we have the expression (3.6). To complete the argument, we check that det Dψ0(ξ ) = c(ξ )n, which clearly implies det Dψ (ξ ) = c(ξ )n, and assures that it does not vanish on a sufficiently narrow . We observe first that where In is the identity n by n matrix. We note that if we consider the matrix A = (αi β j )i, j=1 = t α · β, where α = (α1, . . . , αn), β = (β1, . . . , βn), then A has rank one, so its eigenvalues are n − 1 zeros and some λ. But Tr A is also the sum of the eigenvalues, hence λ = Tr A. Now, let α = ξ , β = ∇c(ξ ), and A = t α · β. Since c(ξ ) is homogeneous of order zero, by Euler’s identity we have hence all eigenvalues of A are zero. It follows now that there is a non-degenerate matrix S such that S−1 AS is strictly upper triangular. But then det Dψ0(ξ ) = det(c(ξ )In + S−1 AS), where matrix c(ξ )In + S−1 AS is upper triangular with n copies of c(ξ ) at the diagonal. Hence det Dψ0(ξ ) = c(ξ )n. On account of the above observations, we have the following result which states that invariant estimates (2.7), (2.8) and (2.9) with m = 2 still hold for a class of non-dispersive equations: Theorem 3.3 Let a ∈ C ∞(Rn\0) be real-valued and satisfy a(λξ ) = λ2a(ξ ) for all λ > 0 and ξ = 0. Assume that rank ∇2a(ξ ) ≥ n − 1 whenever ∇a(ξ ) = 0 and ξ = 0. • Suppose n ≥ 2 and s > 1/2. Then we have x −s |∇a(Dx )|1/2eita(Dx )ϕ(x ) L2(Rt ×Rnx ) ≤ C ϕ L2(Rnx ). • Suppose (4 − n)/2 < α < 1/2, or (3 − n)/2 < α < 1/2 in the elliptic case a(ξ ) = 0 (ξ = 0). Then we have L2(Rt ×Rnx) ≤ C ϕ L2(Rnx ). • Suppose n > 4, or n > 3 in the elliptic case a(ξ ) = 0 (ξ = 0). Then we have x −1 ∇a(Dx ) 1/2eita(Dx )ϕ(x ) L2(Rt ×Rnx) ≤ C ϕ L2(Rnx ). Proof By microlocalisation and an appropriate rotation, we may assume supp ϕ ⊂ , where ⊂ Rn\0 is a sufficiently narrow conic neighbourhood of the direction en = (0, . . . , 0, 1). Since everything is alright in the dispersive case ∇a(en ) = 0 by Theorem 2.1, we may assume ∇a(en ) = 0. We may also assume n ≥ 2 since ∇a(en ) = 0 implies ∇a(ξ ) = 0 for all ξ = 0 in the case n = 1. Then we have rank ∇2a(en) = n by the relation (3.3), hence rank ∇2a(en) = n − 1 by the assumption rank ∇2a(ξ ) ≥ n − 1. In the setting (3.5) and (3.6) above, we have by Lemma 3.1, where r (η) = r (ηk+1, . . . , ηn). Since k = n − 1 in our case, we can see that r is a function of one variable and r vanishes identically by (3.8) and the homogeneity of r . Then r is a polynomial of order one, but is also positively a(ξ ) = (σ ◦ ψ )(ξ ), σ (η) = c1η12 + · · · + cn−1ηn2−1. Now, we have the estimates L2(Rt ×Rnx ) ≤ C ϕ L2(Rnx ), |x |α−1|∇σ (Dx )|αeitσ (Dx )ϕ(x ) L2(Rt ×Rnx ) ≤ C ϕ L2(Rnx ), L2(Rt ×Rnx ) ≤ C ϕ L2(Rnx ), if we use the trivial inequalities x −s ≤ x −s , |x |α−1 ≤ |x |α−1 and x −1 ≤ x −1, Theorem 2.1 with respect to x , and the Plancherel theorem in xn , where x = (x , xn ) and x = (x1, . . . , xn−1). On account of Theorem 5.2 and the L2−s , L˙ 2α−1, L2−1boundedness of the operators Iψ,γ and Iψ−,1γ for (1/2 <)s < n/2, −n/2 < α − 1(< −1/2) (see Theorem 5.4), respectively, we have the conclusion. Example 3.3 The function satisfies the condition in Theorem 3.3, where b(ξ ) is a positively homogeneous function of order one such that ∇b(ξ ) = 0 (ξ = 0). Indeed, if b(ξ ) is elliptic, then ∇a(ξ ) = 2b(ξ )∇b(ξ ) = 0 (ξ = 0). If b(ξ0) = 0 at a point ξ0 = 0, then ∇a(ξ0) = 0 and further differentiation immediately yields ∇2a(ξ0) = 2t ∇b(ξ0)∇b(ξ0), and clearly we have rank ∇2a(ξ0) ≥ 1. Especially in the case n = 2, a(ξ ) meets the condition in Theorem 3.3. As an example, we consider Setting b(ξ ) = ξ1ξ2/|ξ |, we clearly have a(ξ ) = b(ξ )2 and ∇b(ξ ) = |ξξ2|33 , |ξξ1|33 , hence ∇b(ξ ) = 0 (ξ = 0). Although ∇a(ξ ) = 0 on the lines ξ1 = 0 and ξ2 = 0, we have invariant estimates (2.7), (2.8) and (2.9) in virtue of Theorem 3.3. This is an illustration of a smoothing estimate for the Cauchy problem for an equation like i ∂t u + D12 D22u = 0, which can be reduced to the second order non-dispersive pseudo-differential equation with symbol a(ξ ) above. Similarly, we have these estimates for more general case since we obtain rank ∇2a(ξ ) ≥ n − 1 from the observation above. 3.4 Isolated critical points Next we consider more general operators a(ξ ) of order m which may have some lower order terms. Then even the most favourable case det ∇2a(ξ ) = 0 does not imply the dispersive assumption ∇a(ξ ) = 0. The method of canonical transformation, however, can also allow us to treat this problem by obtaining localised estimates near points ξ where ∇a(ξ ) = 0. Assume that ξ0 is a non-degenerate critical point of a(ξ ), that is, that we have ∇a(ξ0) = 0 and det ∇2a(ξ0) = 0. Let us microlocalise around ξ0, so that we only look at what happens around ξ0. In this case, the order of the symbol a(ξ ) does not play any role and we do not distinguish between the main part and lower order terms. Let denote a sufficiently small open bounded neighbourhood of ξ0 so that ξ0 is the only critical point of a(ξ ) in . Since ∇2a(ξ0) is symmetric and non-degenerate, we may assume by a linear transformation. By Morse lemma for a(ξ ), there exists a diffeomorphism ψ : → ⊂ Rn with an open bounded neighbourhood of the origin such that a(ξ ) = (σ ◦ ψ )(ξ ), σ (η) = c1η1 + · · · + cnηn2, 2 where η = (η1, . . . , ηn) and c j = ±1 ( j = 1, 2, . . . , n). From Theorem 2.1 applied to operator σ (Dx ), we obtain the estimates x −s |∇σ (Dx )|1/2eitσ (Dx )ϕ(x ) L2(Rt ×Rnx) ≤ C ϕ L2(Rnx ) (s > 1/2), x −1 ∇σ (Dx ) 1/2eitσ (Dx )ϕ(x ) L2(Rt ×Rnx) ≤ C ϕ L2(Rnx ) (n > 2). Hence by Theorem 5.2, together with the L2−s , L2−1-boundedness of the operators Iψ,γ and Iψ−,1γ (which is assured by Theorem 5.3), we have these estimates with σ (Dx ) replaced by a(Dx ) assuming supp ϕ ⊂ . On the other hand, we have the same estimates for general ϕ by Theorem 2.2 if we assume condition (L). The above argument, however, assures that the following weak assumption is also sufficient if a(ξ ) has finitely many critical points and they are non-degenerate: a(ξ ) ∈ C ∞(Rn), |∇a(ξ )| ≥ C ξ m−1 (for large ξ ∈ Rn) for some C > 0, |∂α(a(ξ ) − am (ξ ))| ≤ Cα|ξ |m−1−|α| for all multi-indices α and all |ξ | 1. (L ) Thus, summarising the above argument, we have established the following result of invariant estimates (2.7) and (2.9): Theorem 3.4 Let a ∈ C ∞(Rn) be real-valued and assume that it has finitely many critical points, all of which are non-degenerate. Assume also (L ). • Suppose n ≥ 1, m ≥ 1, and s > 1/2. Then we have x −s |∇a(Dx )|1/2eita(Dx )ϕ(x ) L2(Rt ×Rnx ) ≤ C ϕ L2(Rnx ). • Suppose n > 2 and m ≥ 1. Then we have x −1 ∇a(Dx ) 1/2eita(Dx )ϕ(x ) L2(Rt ×Rnx) ≤ C ϕ L2(Rnx ). Example 3.4 It is easy to see that satisfies the assumption of Theorem 3.4. We remark that Morii [23] also established the first estimate in Theorem 3.4 under a more restrictive condition but which also allows this example. Example 3.5 Some normal forms listed in (1.4) are also covered by Theorem 3.4 although they are covered by Theorem 3.2. Indeed have their critical points at (ξ1, ξ2) = (0, 0), (−1/3, −1/3) and (ξ1, ξ2) = (0, 0), (1/3, ±1/√3), (−2/3, 0) respectively, and all of them are non-degenerate. 4 Equations with time-dependent coefficients If the symbol b(t, ξ ) is independent of t , invariants estimates (2.7), (2.8) and (2.9) say that ∇ξ b(t, Dx ) is responsible for the smoothing property. The natural question here is what quantity replaces it if b(t, ξ ) depends on t . We can give an answer to this question if b(t, ξ ) is of the product type where we only assume that c(t ) > 0 is a continuous function. In the case of dispersive and Strichartz estimates for higher order (in ∂t ) equations the situation may be very delicate and in general depends on the rates of oscillations of c(t ) (see e.g. Reissig [26] for the case of the time-dependent wave equation, or [8,31] for more general equations). For smoothing estimates, we will be able to state a rather general result in Theorem 4.1 below. The final formulae show that a natural extension of the invariant estimates of the previous section still remain valid in this case. In this special case, the Eq. (4.1) can be transformed to the equation with time-independent coefficients. In fact, by the assumption for c(t ), the function C (t ) = is strictly monotone and the inverse C −1(t ) exists. Then the function hence v(t, x ) solves the equation v(t, x ) := u(C −1(t ), x ) if u(t, x ) is a solution to Eq. (4.1). By this argument, invariant estimates for v(t, x ) = eita(Dx )ϕ(x ) should imply also estimates for the solution to Eq. (4.1). For example, if we notice the relations u(t, x ) = v(C (t ), x ) = ei 0t b(s,Dx ) ds ϕ(x ) v(·, x ) L2 = |c(·)|1/2u(·, x ) we obtain the estimate x −s |∇ξ b(t , Dx )|1/2ei 0t b(s,Dx ) ds ϕ(x ) L2(Rt ×Rnx ) ≤ C ϕ L2(Rnx ) from the invariant estimate (2.7). Estimate (4.2) is a natural extension of the invariant estimate (2.7) to the case of time-dependent coefficients, which says that ∇ξ b(t , Dx ) is still responsible for the smoothing property. From this point of view, we may call it an invariant estimate too. We can also note that estimate (4.2) may be also obtained directly, by formulating an obvious extension of the comparison principles to the time dependent setting. We also have similar estimates from the invariant estimates (2.8) and (2.9). The same method of the proof yields the following: We note that it is possible that α = −∞ and that β = +∞, in which case by continuity of c at such points we simply mean that the limits of c(t ) exist as t → α+ and as t → β−. To give an example of an estimate from Theorem 4.1, let us look at the case of the first statement of Theorem 2.1. In that theorem, we suppose that a(ξ ) satisfies assumption (H), and we assume n ≥ 1, m > 0, and s > 1/2. Theorem 2.1 assures that in this case we have the smoothing estimate (2.1), which is x −s | Dx |(m−1)/2eita(Dx )ϕ(x ) L2(Rt ×Rnx ) ≤ C ϕ L2(Rnx ). Theorem 4.1 states that solution u(t , x ) of Eq. (4.1) satisfies this estimate provided we replace L 2(Rt , Rnx ) by L 2([α, β], Rnx ), and insert |c(t )|1/2 in the left hand side norm. This means that u satisfies x −s |c(t )|1/2| Dx |(m−1)/2u(t , x ) The same is true with statements of any of Theorems 2.1, 2.2, 2.3, 3.1, 3.3 or 3.4. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Appendix A: Canonical transformation and comparison principle For convenience of the reader in this appendix we briefly recall two powerful tools introduced in [30] for getting smoothing estimates, that is, the canonical transformation and the comparison principle, which enable us to induce global smoothing estimates for dispersive equations rather easily, and formulate several corollaries of these methods to be used in the analysis of this paper. In particular, Theorem 5.2 explains the invariance of (1.5) and similar estimates under canonical transforms. Also, Corollary 5.9 is instrumental in treating equations with polynomial symbols (i.e. differential evolution equations) in Sect. 3.2. We remark that all known smoothing estimates from Sect. 2.1 were proved in [30] by using these two methods. A.1 Canonical transformation The first tool is the canonical transformation which transforms the equation with the operator a(Dx ) and the Cauchy data ϕ(x ) to that with σ (Dx ) and g(x ) at the estimate level, where a(Dx ) and σ (Dx ) are related with each other as a(ξ ) = (σ ◦ ψ )(ξ ). Let , ⊂ Rn be open sets and ψ : → be a C ∞-diffeomorphism (we do not assume them to be cones since we do not require homogeneity of phases). We always assume that for some C > 0. Let γ ∈ C ∞( ) and γ = γ ◦ ψ −1 ∈ C ∞( ) be cut-off functions which satisfy supp γ ⊂ , supp γ ⊂ . Then we set In the case that , ⊂ Rn\0 are open cones, we may consider the homogeneous ψ and γ which satisfy supp γ ∩ Sn−1 ⊂ ∩ Sn−1 and supp γ ∩ Sn−1 ⊂ ∩ Sn−1, where Sn−1 = {ξ ∈ Rn : |ξ | = 1}. Then we have the expressions for compositions Iψ,γ · σ (Dx ) = (σ ◦ ψ )(Dx ) · Iψ,γ , Iψ−,1γ · (σ ◦ ψ )(Dx ) = σ (Dx ) · Iψ−,1γ We also introduce the weighted L2-spaces. For a weight function w(x ), let L2w(Rn; w) be the set of measurable functions f : Rn → C such that the norm f L2(Rn;w) = is finite. Then we have the following fundamental theorem: Theorem 5.1 [30, Theorem 4.1] Assume that the operator Iψ,γ defined by (5.2) is L2(Rn; w)-bounded. Suppose that we have the estimate L2(Rt ×Rnx ) ≤ C g L2(Rnx ) is bounded. Then we have L2(Rt ×Rnx ) ≤ C ϕ L2(Rnx ) for all ϕ such that supp ϕ ⊂ supp γ , where a(ξ ) = (σ ◦ ψ )(ξ ). We remark that invariant estimate (2.6) introduced in Sect. 2.2, hence estimates (2.7)–(2.9) are invariant under canonical transformations by Theorem 5.1. More precisely, we have the following theorem: Theorem 5.2 Let ζ be a function on R+ of the form ζ (ρ) = ρη or 1 + ρ2 η/2 with some η ∈ R. Assume that the operators Iψ,γ and Iψ−,1γ defined by (5.2) are L2(Rn; w)bounded. Then the following two estimates w(x )ζ (|∇a(Dx )|)eita(Dx )ϕ(x ) L2(Rt ×Rnx ) ≤ C ϕ L2(Rnx ) (supp ϕ ⊂ supp γ ), w(x )ζ (|∇σ (Dx )|)eitσ (Dx )ϕ(x ) L2(Rt ×Rnx ) ≤ C ϕ L2(Rnx ) (supp ϕ ⊂ supp γ ) are equivalent to each other, where a = σ ◦ ψ ∈ C 1 on supp γ . Proof Note that ∇a(ξ ) = ∇σ (ψ (ξ ))Dψ (ξ ) and C |∇a(ξ )| ≤ |∇σ (ψ (ξ ))| ≤ C |∇a(ξ )| on supp γ with some C, C > 0, which is assured by the assumption (5.1). Then the result is obtained from Theorem 5.1. As for the L2(Rn; w)-boundedness of the operator Iψ,γ , we have criteria for some special weight functions. For κ ∈ R, let Lκ2 (Rn), L˙ κ2 (Rn) be the set of measurable functions f such that the norm is finite, respectively. Then we have the following: Theorem 5.3 [30, Theorem 4.2] Suppose κ ∈ R. Assume that all the derivatives of entries of the n × n matrix ∂ψ and those of γ are bounded. Then the operators Iψ,γ and Iψ−,1γ defined by (5.2) are Lκ2 (Rn)-bounded. Theorem 5.4 [30, Theorem 4.3] Let , ⊂ Rn\0 be open cones. Suppose |κ| < n/2. Assume ψ (λξ ) = λψ (ξ ), γ (λξ ) = γ (ξ ) for all λ > 0 and ξ ∈ . Then the operators Iψ,γ and Iψ−,1γ defined by (5.2) are Lκ2 (Rn)-bounded and L˙ κ2 (Rn)-bounded. A.2 Comparison principle The second tool is the comparison principle which relates the smoothing estimate for the solution u(t, x ) = eit f (Dx )ϕ(x ) with the operator f (Dx ) of the smoothing σ (Dx ) to that for v(t, x ) = eitg(Dx )ϕ(x ) with g(Dx ) of τ (Dx ): Theorem 5.5 [30, Theorem 2.5] Let f, g ∈ C 1(R+) be real-valued and strictly monotone on the support of a measurable function χ on R+. Let σ, τ ∈ C 0(R+) be such that, for some A > 0, we have for all ρ ∈ supp χ satisfying f (ρ) = 0 and g (ρ) = 0. Then we have χ (|Dx |)σ (|Dx |)eit f (|Dx |)ϕ(x ) L2(Rt ) ≤ A χ (|Dx |)τ (|Dx |)eitg(|Dx |)ϕ(x ) L2(Rt ) for all x ∈ Rn . Theorem 5.6 [30, Corollary 2.2] Let f, g ∈ C 1(Rn) be real-valued functions such that, for almost all ξ = (ξ2, . . . , ξn) ∈ Rn−1, f (ξ ) and g(ξ ) are strictly monotone in ξ1 on the support of a measurable function χ on Rn . Let σ, τ ∈ C 0(Rn) be such that, for some A > 0, we have for all ξ ∈ supp χ satisfying ∂1 f (ξ ) = 0 and ∂1g(ξ ) = 0. Then we have ≤ A χ (Dx )τ (Dx )eitg(Dx )ϕ(x1, x ) L2(Rt ×Rnx−1) L2(Rt ×Rnx−1) for all x1, x1 ∈ R, where x = (x2, . . . , xn ) ∈ Rn−1. for every x ∈ R. Here we have used the notation (x , y) = (x1, x2), and (Dx , Dy ) = (D1, D2). On the other hand, in the case n = 1, we have easily L2(Rt ) = ϕ L2(Rx ) for all x ∈ R, which is a straightforward consequence of the fact eit Dx ϕ(x ) = ϕ(x +t ) and the translation invariance of the Lebesgue measure. By using equality (5.5), we can estimate the right hand sides of equalities (5.3) and (5.4) with l = 1, and as a result, we have the following low dimensional pointwise estimates |Dx |(m−1)/2eit|Dx |m ϕ(x ) |Dy |(m−1)/2eit Dx |Dy|m−1 ϕ(x , y) for all x ∈ R, from which we straightforwardly obtain the following result: x1 −s |D1|(m−1)/2eit|D1|m ϕ(x ) Suppose n ≥ 2, m > 0, and s > 1/2. Then we have x1 −s |Dn|(m−1)/2eit D1|Dn|m−1 ϕ(x ) We remark that estimate (5.6) has been already the invariant estimate (2.7) for the normal form a(ξ ) = ξ1m (if we replace the weight x1 −s by a smaller one x −s ). A.3 Secondary comparison We remark that Corollary 5.7 is just a consequence of trivial equality (5.5), and the proof of Theorem 2.1 was carried out in [30] by reducing estimate (2.1) to estimate (5.6) (elliptic case) or estimate (5.7) (non-elliptic case) in Corollary 5.7 via canonical transformations discussed in Appendix A.1. Let us further compare estimates in Theorem 2.1 and Corollary 5.7 by using the comparison principle again to obtain secondary comparison results. In this sense, the results stated below are obtained from just the translation invariance of the Lebesgue measure via a combination use of the comparison principle and the canonical transformation. Now, in notation of Theorem 5.5, setting τ (ρ) = ρ(m−1)/2 and g(ρ) = ρm , we have |τ (ρ)|/|g (ρ)|1/2 = m−1/2. Hence, noticing that χ (Dx ) is L2-bounded for χ ∈ L∞, we obtain the following result from Theorem 2.1 with a(ξ ) = |ξ |m : Cfo∈roClla1r(yR5.)8b[e3r0e,aCl-ovraollulaerdya7n.d3]sStruicptplyosme onno≥to1n,eson>su1p/2p.χL.eLteχt σ∈ ∈L∞C(0R(R++).)Lbeet + such that for some A > 0 we have Similarly, in notation of Theorem 5.6, setting τ (ξ ) = |ξ1|(m−1)/2 and g(ξ ) = |ξ1|m , we have |τ (ξ )|/|∂g/∂ξ1(ξ )|1/2 = m−1/2. Then we obtain the following result from estimate (5.6) of Corollary 5.7: Corollary 5.9 Suppose n ≥ 1 and s > 1/2. Let χ ∈ L∞(Rn). 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Michael Ruzhansky, Mitsuru Sugimoto. Smoothing estimates for non-dispersive equations, Mathematische Annalen, 2016, 241-269, DOI: 10.1007/s00208-015-1281-1