Smoothing estimates for nondispersive equations
Mathematische Annalen
Michael Ruzhansky 0 1
Mitsuru Sugimoto 0 1
0 Graduate School of Mathematics, Nagoya University , Furocho, Chikusaku, Nagoya 4648602 , 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 nondispersive equations based on ideas of comparison principle and canonical transformation established in authors' previous paper (Ruzhansky and Sugimoto, Proc Lond Math Soc, 105:393423, 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 wellknown, while it is also known to fail for nondispersive 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
RPG201402. 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 nondispersive operators a(Dx ),
where ∇a(ξ ) may become zero on some set. Moreover, other types of such estimates,
and the case of timedependent 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 realvalued 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 BenArtzi and Devinatz [1], Constantin and Saut [7],
Sjölin [34] and Vega [38], the local, and then global smoothing estimates, together with
their application to nonlinear 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 nondispersive
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 nondispersive 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 L2norm 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 nondispersive 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 wavetrains (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
wavelengthdivisionmultiplexed 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 threedimensional 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
2dimensions. Strichartz estimates for the corresponding solutions have been analysed
by e.g. Ghidaglia and Saut [9] and by BenArtzi 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 BenArtzi 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 nondispersive cases. Since standard global
smoothing estimates are known to fail in nondispersive 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 nondispersive
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 nondispersive 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 nondispersive
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 nondispersive 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 nondispersive 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 wellknown model situations
(Theorem 3.3);
• around nondispersive points where the Hessian is nondegenerate, 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 timedecay 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 nondispersive situations. In Sect. 3, we
establish invariant estimates for several types of nondispersive equations. The case of
timedependent 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 realvalued. 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 nondispersive 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 BenArtzi 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 nonhomogenous 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 nonhomogeneous 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
L2boundedness 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 )) ≤ Cαξ m−1−α for all multiindices 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 )) ≤ Cαξ m−1−α for all multiindices 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 L2boundedness 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 multiindices 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 ) ≥ cξ 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 )1/2Dx −(m−1)/2 and ∇a(Dx )−1/2Dx (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 selfadjoint 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 onedimensional 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 nondispersive equations, namely of the smoothing
estimate obtained by Hoshiro [10]. If operator a(Dx ) has realvalued 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 −1/2Dx 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 −1/2Dx 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 nonhomogeneous 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 timeglobal 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 nondispersive equations
Canonical transformations and the comparison principle recalled for convenience in
Appendix A are still important tools when we discuss the smoothing estimates for
nondispersive equations
where realvalued 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 nondispersive 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 nondispersive 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
realvalued. 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 nonconstant 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 nondispersive points
Now we will present another approach to treat nondispersive 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 nondispersive 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 realvalued
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 nondispersive 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
realvalued 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) nonzero 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 realvalued. 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 ktimes, 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 nondegenerate 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
nondispersive equations:
Theorem 3.3 Let a ∈ C ∞(Rn\0) be realvalued 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 ) = ξξ233 , ξξ133 ,
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 nondispersive pseudodifferential 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 nondegenerate 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 nondegenerate, 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 , L2−1boundedness 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 nondegenerate:
a(ξ ) ∈ C ∞(Rn), ∇a(ξ ) ≥ C ξ m−1 (for large ξ ∈ Rn) for some C > 0,
∂α(a(ξ ) − am (ξ )) ≤ Cαξ m−1−α for all multiindices 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 realvalued and assume that it has finitely many
critical points, all of which are nondegenerate. 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 nondegenerate.
4 Equations with timedependent 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 timedependent 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 timeindependent 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 timedependent 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.
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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 cutoff 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 L2spaces. 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 realvalued 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 realvalued 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)/2eitDx m ϕ(x )
Dy (m−1)/2eit Dx Dym−1 ϕ(x , y)
for all x ∈ R, from which we straightforwardly obtain the following result:
x1 −s D1(m−1)/2eitD1m ϕ(x )
Suppose n ≥ 2, m > 0, and s > 1/2. Then we have
x1 −s Dn(m−1)/2eit D1Dnm−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) (nonelliptic 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 L2bounded for χ ∈ L∞,
we obtain the following result from Theorem 2.1 with a(ξ ) = ξ m :
Cfo∈roClla1r(yR5.)8b[e3r0e,aClovraollulaerdya7n.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 ) = ξ1m ,
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). Let f ∈ C 1(Rn) be
a realvalued function such that, for almost all ξ = (ξ2, . . . , ξn) ∈ Rn−1, f (ξ ) is
strictly monotone in ξ1 on supp χ . Let σ ∈ C 0(Rn) be such that for some A > 0 we
have
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