**41**(2005), 175–221

**Smoothness of Solutions for Schr¨** **odinger** **Equations with Unbounded Potentials**

By

Shin-ichiDoi*∗*

*Dedicated to Professor Kunihiko Kajitani on his sixty-second birthday*

**Abstract**

We consider a Schr¨odinger equation with linearly bounded magnetic potentials
and a quadratically bounded electric potential when the coeﬃcients of the principal
part do not necessarily converge to constants near inﬁnity. Assuming that there exists
a suitable function *f(x) near inﬁnity which is convex with respect to the Hamilton*
vector ﬁeld generated by the (scalar) principal symbol, we show a microlocal smooth-
ing eﬀect, which says that the regularity of the solution increases for all time*t∈*(0, T]
at every point that is not trapped backward by the geodesic ﬂow if the initial data
decays in an incoming region in the phase space. Here*T* depends on the potentials;

we can choose*T*=*∞*if the magnetic potentials are sublinear and the electric poten-
tial is subquadratic. Our method regards the growing potentials as perturbations; so
it is applicable to matrix potentials as well.

**§****1.** **Introduction**

Let*H*(t) be a time dependent Schr¨odinger operator acting on**C*** ^{n}*-valued
functions:

*H*(t) =
*d*
*j,k=1*

(D*j**−a**j*(t, x))g* ^{jk}*(x)(D

*k*

*−a*

*k*(t, x)) +

*V*(t, x), (t, x)

*∈*

**R**

*×*

**R**

^{d}*.*

Communicated by T. Kawai. Received June 23, 2003. Revised November 6, 2003.

Partly supported by Grand-in-Aid for Young Scientists (B) 14740110, Japan Society of the Promotion of Science.

2000 Mathematics Subject Classiﬁcation(s): 35J10, 35B65.

*∗*Department of Mathematics, Graduate School of Science, Osaka University, 1-1
Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan.

e-mail: sdoi@math.sci.osaka-u.ac.jp

Here *D** _{j}* =

*−i∂*

*=*

_{j}*−i∂/∂x*

*;*

_{j}*M*

*(C) is the space of all*

_{n}*n×n*complex ma- trices;

*g*

*=*

^{jk}*g*

^{kj}*∈*

*C*

*(R*

^{∞}

^{d}*,*

**R), and (g**

*(x)) is positive deﬁnite for each*

^{jk}*x;*

*∂*_{x}^{α}*a**j**, ∂*^{α}_{x}*V* *∈* *C(R**t**×***R**^{d}_{x}*, M**n*(C)) for all *α* *∈* **N**^{d}_{0}, and *a**j*(t, x), V(t, x) are
Hermitian matrices for each (t, x).

Under suitable conditions, the Cauchy problem for the Schr¨odinger equa- tion

(∂* _{t}*+

*iH(·*))u= 0 in

*D*

*(R*

^{}*×*

**R**

^{d}*,*

**C**

*),*

^{n}*u(t*

_{0}) =

*u*

_{0}

*,*

is well-posed in the scale of spaces associated with the oscillator *H*_{osc} = 1*−*

∆ +*|x|*^{2}. Let *S(t, t*0) (t, t0 *∈* **R) denote the propagator, or the solution op-**
erator. This paper is concerned with the smoothing eﬀect of *S(t, t*0) and the
smoothness of its distribution kernel*K(t, t*0*, x, y) under general conditions on*
the coeﬃcients, when

(a)*c*_{1}*I*_{d}*≤*(g* ^{jk}*(x))

*≤c*

_{2}

*I*

*on*

_{d}**R**

*for some*

^{d}*c*

_{1}

*, c*

_{2}

*>*0,

(b) (g* ^{jk}*(x)) does not necessarily converge to a constant matrix, and
(c)

*|a*

*j*(t, x)

*|*=

*O(|x|*) and

*|V*(t, x)

*|*=

*O(|x|*

^{2}) as

*|x| → ∞*uniformly on every compact time interval.

*Remark.* If**R*** ^{d}* has a positive density

*v(x)dx,v∈C*

*(R*

^{∞}*), it is natural to consider the Schr¨odinger operator of the following form*

^{d}*H*˜(t) =*v(x)*^{−}^{1}
*d*
*j,k=1*

(D*j**−a**j*(t, x))v(x)g* ^{jk}*(x)(D

*k*

*−a*

*k*(t, x)) + ˜

*V*(t, x),

where ˜*V* is a Hermitian potential like*V*. Then*v(x)*^{1/2}*H*˜(t)*v(x)*^{−}^{1/2} =*H*(t)
with *V*(t, x) = ˜*V*(t, x) + (^{1}_{2}∆*g,v*log*v(x)−* ^{1}_{4}*g**x*(dlog*v, d*log*v))I**n*. Here for
*f* *∈* *C** ^{∞}*(R

*) we set ∆*

^{d}*g,v*

*f*(x) =

*v(x)*

^{−}^{1}

*d*

*j,k=1**∂**j*

*v(x)g** ^{jk}*(x)∂

*k*

*f*(x) and

*g*

*x*(df, df) =

*d*

*j,k=1**g** ^{jk}*(x)(∂

*j*

*f*(x))(∂

*k*

*f*(x)).

What are our diﬃculties? When (g* ^{jk}*(x)) = (δ

*), the previous works have regarded the potentials of the maximal order in (c) as part of the principal part and used the Hamilton ﬂow of this “principal symbol” to construct important operators such as the fundamental solution, a parametrix, and a conjugate operator; this construction has called for deriving detailed estimates of the Hamilton ﬂow, which has required stronger conditions on the derivatives of the potentials. When (g*

^{jk}*(x)) does not converge to a constant matrix as*

^{jk}*|x| →*

*∞*, the nontrapped bicharacteristic curve of the principal symbol *h*0(x, ξ) =
*d*

*j,k=1**g** ^{jk}*(x)ξ

*j*

*ξ*

*k*has no asymptotic velocity in general, because the short- range condition,

*|∇*

*x*

*g*

*(x)*

^{jk}*|*=

*O(|x|*

^{−}^{1}

^{−}*) as*

^{ε}*|x| → ∞*for some

*ε >*0, fails; so it seems hopeless to derive detailed estimates, or precise asymptotic behavior,

of the Hamilton ﬂow of the “principal symbol” when the maximally growing
potentials are present. When*n≥*2, the “principal symbol” is no more scalar,
and hence the Hamilton ﬂow cannot be deﬁned. These are typical diﬃculties.

Our remedy is simple: we should regard the potentials of order (c) as
*perturbations* and use only qualitative properties of the Hamilton ﬂow of the
principal symbol. To control the asymptotic behavior of the Hamilton ﬂow,
we assume that there exists a suitable strictly convex function *f*_{cv}*∈C** ^{∞}*(R

*) near inﬁnity with respect to the Hamilton vector ﬁeld*

^{d}*H*

*h*

_{0}=

*d*

*j=1*

*∂h*0

*∂ξ*_{j}

*∂*

*∂x*_{j}*−*

*∂h*_{0}

*∂x*_{j}

*∂*

*∂ξ*_{j}

. Then we can regard the potentials of order (c) as perturbations, not
for all*t∈***R**in general, but for all*t∈*[0, T]. Here*T >*0 is the largest number
satisfying

*T·* lim

*R**→∞* sup

*t**∈*[0,T],*|**x**|≥**R*

*d*
*j=1*

*|∇**x**a**j*(t, x)*|*+*T*^{2}*·* lim

*R**→∞* sup

*t**∈*[0,T],*|**x**|≥**R*

*|∇**x**V*(t, x)*|*

*|x|* *≤c*
for a constant*c*=*c(d, h*0*, f**cv*)*>*0 independent of the potentials. On this inter-
val, we use a kind of positive commutator method by constructing a conjugate
operator as a time dependent scalar pseudodiﬀerential operator whose symbol is
an explicit function of*h*_{0},*r*=*√*

*f** _{cv}*, and their Poisson bracket

*{h*

_{0}

*, r}*:=

*H*

_{h}_{0}

*r.*

Thus we need no detailed estimates of the Hamilton ﬂow of either the principal symbol or the “principal symbol” (the latter should have been scalar, because the Hamilton ﬂow of a matrix-valued function makes no sense in general). As a by-product, we can largely relax the conditions on the derivatives of the coeﬃcients and handle the matrix potentials as well.

Why can we regard the potentials as perturbations? We shall heuristically
explain this when*n*= 1 and (g* ^{jk}*(x)) =

*I*

*d*outside a compact set (then we can choose

*r(x) =*

*x*:=

1 +*|x|*^{2}). Let *h(t) be the Weyl symbol of* *H*(t) and
Φ*ts* the (2-parameter) Hamilton ﬂow of *h(t). Let* *K(t) be an invertible, time*
dependent pseudodiﬀerential operator with Weyl symbol *k(t, x, ξ) =* *e** ^{λ(t,x,ξ)}*
for a nonnegative symbol

*λ*(0

*≤t≤T). Under suitable conditions, we have*

*K(t)(∂**t*+*iH(t))K(t)*^{−}^{1}=*∂**t*+*iH(t) +P(t) +Q(t),*

where the Weyl symbol of *P*(t) is *−*(∂*t**λ(t) +H*_{h(t)}*λ(t)), and* *Q(t)** ^{∗}*+

*Q(t) is*bounded. Setting

*u(t) =S(t,*0)u

_{0}, we can show the estimate

*K(t)u(t)*^{2}+
*t*

0

(P(τ)K(τ)u(τ), K(τ)u(τ))dτ *≤CK(0)u*0^{2}*,* *t∈*[0, T],
for a constant*C >* 0 independent of *u*_{0} and *t* *∈*[0, T]. If*−*(∂_{t}*λ(t) +H*_{h}*λ(t))*
is bounded from below, then we can obtain an eﬀective microlocal estimate of

*u(·*) in the set*A** _{T}* =

*{*(t, x, ξ)

*∈*[0, T]

*×T*

^{∗}**R**

^{d}*\ {*0

*}*;

*λ(t, x, ξ)>*0

*}*. Therefore we require

*A*

*to be backward invariant under the (2-parameter) Hamilton ﬂow of*

_{T}*h(t): Φ*

*ts*(x, ξ)

*∈A*

*T*if (s, x, ξ)

*∈*

*A*

*T*and 0

*≤t*

*≤s*

*≤*

*T*. Sometimes we can replace

*h(t) by another “principal symbol” in requiring the last condition.*

This is the case where the potentials are bounded with additional conditions
on the derivatives. Then we can choose*A** _{T}* = [0, T]

*×S, where*

*S*= (x, ξ)*∈T*^{∗}**R**^{d}*\*0;*x* *> R*^{}*,* *x·ξ*
*x |ξ|* *<−δ*^{}

(R* ^{}* 1, 0

*< δ*

*1) is backward invariant under the Hamilton ﬂow of*

^{}*|ξ|*

^{2}. However, when the potentials are unbounded as in (c), we cannot control the order of

*x*on [0, T]

*×S.*So we require that

*x ≤*

*CT|ξ|*on

*A*

*T*for a constant

*C >*0 independent of

*T*. In fact, we can choose

*A** _{T}* = (t, x, ξ)

*∈*[0, T]

*×T*

^{∗}**R**

^{d}*\*0;

*R*

^{}*<x*

*<*5(2T

*−t)|ξ|,*

*x·ξ*

*x |ξ|*

*<−δ*

^{}
*.*
Then this set is backward invariant under the (2-parameter) Hamilton ﬂow of
*h(t) ifc(d, h*0*, f**cv*) is suﬃciently small. On this set, we can compare the order of
the potentials with that of the principal part, because*x ≤*10T*|ξ|*holds there.

Therefore we can regard the potentials as perturbations when *c(d, h*0*, f**cv*) is
suﬃciently small.

Let us write the operator*H*(t) in the following form:

*H*(t) =

^{d}

*j,k=1*

*D*_{j}*g** ^{jk}*(x)D

_{k}

*I*_{n}*−*
*d*
*j=1*

(a* ^{j}*(t, x)D

*+*

_{j}*D*

_{j}*a*

*(t, x)) +*

^{j}*b(t, x);*

*a** ^{j}*(t, x) =

*d*

*k=1*

*g** ^{jk}*(x)a

*k*(t, x),

*b(t, x) =V*(t, x) +

*d*

*j,k=1*

*a**j*(t, x)g* ^{jk}*(x)a

*k*(t, x).

Then the Weyl symbol*h(t) ofH*(t) is

*h(t, x, ξ) =h*0(x, ξ)I*n*+*h*1(t, x, ξ) +*h*2(t, x, ξ);

*h*0(x, ξ) =
*d*
*j,k=1*

*g** ^{jk}*(x)ξ

*j*

*ξ*

*k*

*,*

*h*1(t, x, ξ) =

*−*2

*d*

*j=1*

*a** ^{j}*(t, x)ξ

*j*

*,*

*h*2(t, x, ξ) =*h*2(t, x) =*b(t, x) +*1
4

*d*
*j,k=1*

*∂**j**∂**k**g** ^{jk}*(x)

*I*

*n*

*.*

We recall related results when the operator is scalar (n= 1).

(i) Assume*g** ^{jk}*(x) =

*δ*

*and that with some*

^{jk}*ε >*0

*|∂*_{x}^{α}*a**j*(t, x)*|*+*|∂*_{x}* ^{α}*(∂

*t*

*a*

*j*(t, x) +

*∂*

*j*

*V*(t, x))

*| ≤C*

*α*

*,*

*t∈*

**R, x**

*∈*

**R**

^{d}*,*

*|∂*_{x}* ^{α}*(∂

_{k}*a*

*(t, x)*

_{j}*−∂*

_{j}*a*

*(t, x))*

_{k}*| ≤C*

_{α}*(1 +*

^{}*|x|*)

^{−}^{1}

^{−}

^{ε}*,*

*t∈*

**R, x**

*∈*

**R**

^{d}*,*for all

*α∈*

**N**

^{d}_{0}with

*|α| ≥*1. Then

*K(t, s, x, y) isC*

*in*

^{∞}*x, y*when 0

*<|t−s| ≤T*for some

*T >*0 (see [6] when

*a*

*j*= 0 and [24, 25] in the general case). Remark that

*V*can be eliminated by the change of the unknown function:

*u(t, x)*

*→*

*v(t, x) =u(t, x) exp(i*

*t*

0*V*(τ, x)dτ).

(ii) Assume*g** ^{jk}*(x) =

*δ*

*,*

^{jk}*a*

*= 0, and*

_{j}*R*lim*→∞* sup

*t**∈***R,***|**x**|≤**R*

*|∂*_{x}^{α}*V*(t, x)*|*= 0 if*|α|*= 2;

*|∂*_{x}^{α}*V*(t, x)*| ≤C**α**,* *t∈***R, x***∈***R**^{d}*,* if*|α| ≥*3.

Then*K(t, s, x, y) isC** ^{∞}* in

*x, y*when

*t*=

*s*([26]). See also [13].

(iii) Assume *d* = 1, *g*^{11}(x) = 1, *a*_{1} = 0, and *V*(t, x) = *V*(x) *≥* *C(1 +*

*|x|*)^{2+ε}near inﬁnity for some*ε >*0 as well as other technical conditions. Then
*K(t, s, x, y) is nowhereC*^{1}([26]).

(iv) Assume*g** ^{jk}*(x) =

*δ*

*,*

^{jk}*a*

*j*= 0,

*V*(t, x) =

*|x|*

^{2}+

*W*(t, x) with

*W*(t, x) =

*o(|x|*

^{2}) as

*|x| → ∞*. Then

*K(t,*0, x, y) is

*C*

*in*

^{∞}*x*for every

*y*

*∈*

**R**

*and nonresonant*

^{d}*t /∈*(π/2)Z under general conditions on

*W*, and shows various phenomena such as recurrence and dispersion of singularities for resonant

*t*

*∈*(π/2)Zdepending on the growth order of

*W*(x) ([14, 17, 21, 27, 28]).

(v) Assume for some*ε >*0 and*δ >*0

*|∂*_{x}* ^{α}*(g

*(x)*

^{jk}*−δ*

*)*

^{jk}*| ≤C*

*α*(1 +

*|x|*)

^{−}^{1}

^{−}

^{ε}

^{−|}

^{α}

^{|}*,*

*x∈*

**R**

^{d}*,*

and*a** ^{j}*(t, x) =

*a*

*(x) =*

^{j}*O(|x|*

^{1}

^{−}*), b(t, x) =*

^{δ}*b(x) =O(|x|*

^{1}

^{−}*) as*

^{δ}*|x| → ∞*as well as similar conditions on the derivatives. Then the

*H*

*microlocal regularity of a solution for the Cauchy problem increases for all*

^{s}*t >*0 at a point in

*T*

^{∗}**R**

^{d}*\*0 if the point is not trapped backward by the Hamilton ﬂow of

*h*0and if the initial data decays along the backward bicharacteristics through that point ([1]).

See [3, 5] for the absence of smoothing eﬀects due to the trapping of the Hamilton ﬂow of the principal symbol. See also [3, 4, 5, 9, 10, 11, 12, 15, 16, 18, 19, 20, 22, 23] for related results in other frameworks.

Our goal is to handle the mixed case of (i) ,(ii), and (v) under relaxed con- ditions, which allow (a), (b), and (c). The case (iv) will be discussed elsewhere.

We explain the plan of this paper. Section 2 states the main results: the well-posedness of the Cauchy problem for the Schr¨odinger equation (Subsec- tion 2.1) and the smoothing eﬀect of the associated propagator (Subsection

2.2). Section 3 recalls the Weyl calculus of pseudodiﬀerential operators and
proves related lemmas. Section 4 proves two well-posedness theorems of the
Cauchy problem: one for the Schr¨odinger equation in Section 1 and the other
for a more general Schr¨odinger equation appearing in Section 7. Section 5
shows how the Schr¨odinger operator is transformed when conjugated by an in-
vertible pseudodiﬀerential operator. Section 6 proves ﬁrst, a smoothing eﬀect
of the Schr¨odinger propagator, local in time and global in an incoming region
in *T*^{∗}**R**^{d}*\*0, by using Section 5; second, a smoothing eﬀect at every point of
*T*^{∗}**R**^{d}*\*0 that is not trapped backward by the Hamilton ﬂow of the principal
symbol by using the result from Appendix A. Section 7 proves all assertions
in Section 2 except for Theorem 2.8. Section 8 discusses the smoothing eﬀect
of order half, or the so-called local smoothing eﬀect, from which Theorem 2.8
follows. Appendix A shows an energy estimate along the Hamilton ﬂow of the
principal symbol for a general dispersive equation.

Finally I would like to thank the referee for many useful comments.

*Notation.* **N**0=**N***∪ {*0*}*. *C** ^{k}*(U, V) is the set of all

*C*

*maps from*

^{k}*U*to

*V*(k

*∈*

**N**

_{0}

*∪ {∞}*), and

*C(U, V*) =

*C*

^{0}(U, V);

*V*is omitted if

*V*=

**C. For locally**convex spaces

*E*and

*F,*

*L(E, F*) is the set of all continuous linear operators from

*E*to

*F*, and

*L(E) =*

*L(E, E);*

*L(C*

*) is identiﬁed with*

^{n}*M*

*(C). The symbol (*

_{n}*·,·*) denotes the inner product of

*L*

^{2}(R

*) or*

^{d}*L*

^{2}(R

^{d}*,*

**C**

*) by abuse of notation, and*

^{n}*·*the norm. For

*v*

*∈*

**R**

*,*

^{n}*v*= (1 +

*|v|*

^{2})

^{1/2}. For a subset

*A*of

*T*

^{∗}**R**

*, set cone(A) =*

^{d}*{*(x, tξ); (x, ξ)

*∈A, t >*0

*}*.

**§****2.** **Main Results**

**§****2.1.** **Well-posedness of the Cauchy problem**

Throughout Section 2, we assume that the following conditions (H1)–(H4)
hold for some 0*< δ <*1.

(H1)*c*_{1}*I*_{d}*≤*(g* ^{jk}*(x))

*≤c*

_{2}

*I*

*on*

_{d}**R**

*for some*

^{d}*c*

_{1}

*, c*

_{2}

*>*0.

(H2) For every*α∈***N**^{d}_{0} with*|α| ≥*1, there is*C**α*(g)*>*0 such that

*|∂*_{x}^{α}*g** ^{jk}*(x)

*| ≤C*

*(g)*

_{α}*x*

^{−}^{1+δ(}

^{|}

^{α}

^{|−}^{1)}

*,*

*x∈*

**R**

^{d}*,*

*j, k*= 1, . . . , d.

(H3) For every compact set*I⊂***R**and*α∈***N**^{d}_{0}, there is*C** _{α}*(a, I)

*>*0 such that

*|a** ^{j}*(t, x)

*| ≤C*0(a, I)

*x,*

*t∈I, x∈*

**R**

^{d}*, j*= 1, . . . , d;

*|∂*_{x}^{α}*a** ^{j}*(t, x)

*| ≤C*

*(a, I)*

_{α}*x*

^{δ(}

^{|}

^{α}

^{|−}^{1)}

*,*

*t∈I, x∈*

**R**

^{d}*, j*= 1, . . . , d, if

*|α| ≥*1.

(H4) For every compact set*I⊂***R**and*α∈***N**^{d}_{0}, there is*C** _{α}*(b, I)

*>*0 such that

*|b(t, x)| ≤C*0(b, I)*x*^{2}*,* *t∈I, x∈***R*** ^{d}*;

*|∂*_{x}^{α}*b(t, x)| ≤C** _{α}*(b, I)

*x*

^{1+δ(}

^{|}

^{α}

^{|−}^{1)}

*,*

*t∈I, x∈*

**R**

^{d}*,*if

*|α| ≥*1.

The condition (H1) implies that the Hamilton vector ﬁeld *H**h*_{0} is com-
plete on *T*^{∗}**R*** ^{d}*, because

*h*0 is constant on each integral curve. Let Φ

*t*= exp(tH

*h*

_{0}) (t

*∈*

**R) be the Hamilton ﬂow of**

*H*

*h*

_{0}; in other words, Φ

*t*(y, η) = (x(t, y, η), ξ(t, y, η)) is the solution of the system of ordinary diﬀerential equa- tions

˙

*x**j*(t) =*∂**ξ*_{j}*h*0(x(t), ξ(t)), *x**j*(0) =*y**j**,*

*ξ*˙* _{j}*(t) =

*−∂*

_{x}

_{j}*h*

_{0}(x(t), ξ(t)),

*ξ*

*(0) =*

_{j}*η*

*(1*

_{j}*≤j≤d).*

Next we deﬁne Sobolev spaces*B** ^{s}*(R

*) (s*

^{d}*∈*

**R) (cf. [7]). Let**

*H*osc be the self-adjoint extension of the operator 1

*−*∆ +

*|x|*

^{2}with domain

*C*

_{0}

*(R*

^{∞}*). Then for every*

^{d}*s*

*∈*

**R,**

*H*osc

*is continuous on*

^{s/2}*S*(R

*) and extends to a continuous linear operator on*

^{d}*S*

*(R*

^{}*) (with the weak* topology), denoted also by*

^{d}*H*osc

*. We set*

^{s/2}*B** ^{s}*(R

*) =*

^{d}*{u∈ S*

*(R*

^{}*);*

^{d}*H*

_{osc}

^{s/2}*u∈L*

^{2}(R

*)*

^{d}*}.*These spaces are characterized as follows:

*B** ^{s}*(R

*) =*

^{d}*{u∈L*

^{2}(R

*);*

^{d}*x*

^{s}*u∈L*

^{2}(R

*),*

^{d}*D*

^{s}*u∈L*

^{2}(R

*)*

^{d}*}*(s

*≥*0);

*B** ^{s}*(R

*) =*

^{d}*B*

^{−}*(R*

^{s}*)*

^{d}*(s*

^{}*≤*0).

The vector-valued Sobolev spaces*B** ^{s}*(R

^{d}*,*

**C**

*) are similarly deﬁned.*

^{n}After preparing the Weyl calculus in Section 3, we shall prove in Lemma
4.1 that for every*s∈***R**there is*L(s)*1 such that the operator*E** _{s}*with Weyl
symbol

*e**s*(x, ξ) = (h0(x, ξ) +*|x|*^{2}+*L(s)*^{2})^{s/2}

is a homeomorphism from*B** ^{r+s}*(R

*) to*

^{d}*B*

*(R*

^{r}*) for all*

^{d}*r∈*

**R. We use**

*E*

*s*

*·*as a norm of

*B*

*(R*

^{s}*) (or*

^{d}*B*

*(R*

^{s}

^{d}*,*

**C**

*)), where*

^{n}*·*=

*·*

*L*

^{2}(R

*)(or*

^{d}*·*

*L*

^{2}(R

^{d}*,C*

*)).*

^{n}Now we state our two theorems on the well-posedness of the Cauchy prob- lem.

**Theorem 2.1.** *Let* *s∈***R,** *I*= [t1*, t*2] (t1*< t*2), *andt*0 *∈I. For every*
*u*0*∈ B** ^{s}*(R

^{d}*,*

**C**

*)*

^{n}*andf*

*∈L*

^{1}(I,

*B*

*(R*

^{s}

^{d}*,*

**C**

*)),*

^{n}*there existsu∈C(I,B*

*(R*

^{s}

^{d}*,*

**C**

*))*

^{n}*satisfying*

(2.1) (∂*t*+*iH(·*))u=*f* *inD** ^{}*((t1

*, t*2)

*×*

**R**

^{d}*,*

**C**

*),*

^{n}*u(t*0) =

*u*0

*,*

*which is unique in* *C(I,S** ^{}*(R

^{d}*,*

**C**

*)). Moreover,*

^{n}*the solution*

*u*

*satisﬁes the*

*following estimate*

(2.2) *e*^{−}^{γ}^{|}^{t}^{−}^{t}^{0}^{|}*E**s**u(t) ≤ E**s**u(t*0)+
*t*

*t*_{0}

*e*^{−}^{γ}^{|}^{τ}^{−}^{t}^{0}^{|}*E**s**f*(τ)*dτ*

*,* *t∈I.*

*Hereγ≥*0*depends ons∈***R***and on the constantsc*_{1}*, c*_{2}*, C** _{α}*(g), C

*(a, I),*

_{α}*and*

*C*

*α*(b, I)

*in*(H1)–(H4),

*but not onf*,

*u*0,

*oru. In particular,γ*= 0

*if*

*s*= 0.

**Theorem 2.2.** *Let* *S(t, t*_{0})*∈L(S** ^{}*(R

^{d}*,*

**C**

*)) (t, t*

^{n}_{0}

*∈*

**R)**

*be the operator*

*mapping*

*u*0

*∈ S*

*(R*

^{}

^{d}*,*

**C**

*)*

^{n}*tou(t)∈ S*

*(R*

^{}

^{d}*,*

**C**

*),*

^{n}*where*

*u∈C(R,S*

*(R*

^{}

^{d}*,*

**C**

*))*

^{n}*is the solution of the Cauchy problem*

(2.3) (∂*t*+*iH(·*))u= 0*inD** ^{}*(R

*×*

**R**

^{d}*,*

**C**

*),*

^{n}*u(t*0) =

*u*0

*.*

(1)*S*(t, t) = 1*andS(t, s)S(s, r) =S(t, r)on* *S** ^{}*(R

^{d}*,*

**C**

*) (t, s, r*

^{n}*∈*

**R).**

(2) *For every compact intervalI,{S(t, t*0)*|*_{B}* ^{s}*(R

^{d}*,C*

*);*

^{n}*t, t*0

*∈I}*

*is bounded*

*inL(B*

*(R*

^{s}

^{d}*,*

**C**

*)).*

^{n}(3)**R***×***R***× B** ^{s}*(R

^{d}*,*

**C**

*)(t, t*

^{n}_{0}

*, u*

_{0})

*→S(t, t*

_{0})u

_{0}

*∈ B*

*(R*

^{s}

^{d}*,*

**C**

*)*

^{n}*is contin-*

*uous.*

(4)*S*(t, t0)*|**L*^{2}(R^{d}*,C** ^{n}*)

*∈L(L*

^{2}(R

^{d}*,*

**C**

*))*

^{n}*is unitary.*

(5) *If* *H* = *H(t)* *is time independent,* *then* *H|**C*_{0}* ^{∞}*(R

^{d}*,C*

*)*

^{n}*is essentially*

*self-adjoint. If*

*H*

*denotes also its self-adjoint extension,*

*then*

*e*

^{−}

^{i(t}

^{−}

^{t}^{0}

^{)H}

*u*0 =

*S(t, t*

_{0})u

_{0}

*for everyt, t*

_{0}

*∈*

**R**

*andu*

_{0}

*∈L*

^{2}(R

^{d}*,*

**C**

*).*

^{n}**§****2.2.** **Smoothing eﬀects**

The asymptotic behavior of Φ*t* plays an important role in the smoothing
eﬀect of the propagator *S(t, s).* We introduce several subsets of *T*^{∗}**R**^{d}*\*0
consisting of the points which are trapped forward or backward by Φ* _{t}*:

*T*_{+}=*{X* *∈T*^{∗}**R**^{d}*\*0; lim

*t**→∞**|*Φ* _{t}*(X)

*|*=

*∞},*

*T*

*=*

_{−}*{X*

*∈T*

^{∗}**R**

^{d}*\*0; lim

*t**→−∞**|*Φ*t*(X)*| *=*∞}*;

*T**cpt,+*=*{X* *∈T*^{∗}**R**^{d}*\*0;*{*Φ*t*(X);*t≥*0*}*is relatively compact*},*
*T*_{cpt,}* _{−}*=

*{X*

*∈T*

^{∗}**R**

^{d}*\*0;

*{*Φ

*(X);*

_{t}*t≤*0

*}*is relatively compact

*}.*Put

*T*

*cpt*=

*T*

*cpt,+*

*∩T*

*cpt,*

*−*. To control the asymptotic behavior of Φ

*t*, we assume the following condition (H5) in addition to (H1)–(H4) stated at the beginning of this section.

(H5) (convexity near inﬁnity). There exists*f*_{cv}*∈* *C** ^{∞}*(R

*), lim*

^{d}

_{|}

_{x}

_{|→∞}*f*

*(x) =*

_{cv}*∞*,

*f*

_{cv}*≥*1, such that for every

*α∈*

**N**

^{d}_{0}with

*|α| ≥*2,

*∂*

^{α}*f*

_{cv}*∈L*

*(R*

^{∞}*) and that for some*

^{d}*σ >*0, R >0

*H*_{h}^{2}_{0}*f**cv**≥*2σ^{2}*h*0 on*{*(x, ξ)*∈T*^{∗}**R*** ^{d}*;

*r(x) :=*

*f**cv*(x)*≥R}.*

*Remark.* The function*f**cv* in (H5) satisﬁes*f**cv*(x)^{−}^{1}=*O(|x|*^{−}^{2}) as*|x| →*

*∞*. In fact, take*M >*0 such that*{x∈***R*** ^{d}*;

*|x| ≥M} ⊂ {x∈*

**R**

*;*

^{d}*f*

*cv*(x)

*≥R*

^{2}

*}*. For

*x*

*∈*

**R**

^{d}*,*

*|x|*

*> M*, take

*T >*0 and (y, η)

*∈*

*T*

^{∗}**R**

*such that*

^{d}*|y|*=

*M*,

*h*

_{0}(y, η) = 1,

*|x(t, y, η)|> M*(0

*< t < T*), and

*x(T, y, η) =x, where Φ*

*(y, η) = (x(t, y, η), ξ(t, y, η)). This is possible because Φ*

_{t}*is a complete geodesic ﬂow.*

_{t}Then*T* *≥c|x−y|*for some*c >*0 independent of*T, x, y*by (H1), and
*f**cv*(x) =*f**cv*(y) + (H*h*_{0}*f**cv*)(y, η)T+

1 0

(1*−θ)(H*_{h}^{2}_{0}*f**cv*)(Φ*θT*(y, η))dθ T^{2}

*≥f**cv*(y) + (H*h*_{0}*f**cv*)(y, η)T+*σ*^{2}*T*^{2}
by (H5). Therefore lim inf_{|}*x**|→∞**f**cv*(x)/*|x|*^{2}*≥c*^{2}*σ*^{2}.

*Remark.* If *|∇**x**g** ^{jk}*(x)

*|*=

*o(|x|*

^{−}^{1}) as

*|x| → ∞*, then (H5) holds with

*f*

*cv*(x) = 1 +

*|x|*

^{2}.

*Remark.* Let *a* *∈* *C** ^{∞}*([1,

*∞*)) such that

*C*

^{−}^{1}

*≤*

*a*

*≤*

*C*with

*C >*0 and

*∂*

^{k}*a(r) =*

*O(r*

^{−}^{1}) for all

*k*

*∈*

**N.**Assume lim sup

_{r}

_{→∞}*a*

*(r)r/a(r)*

^{}*<*

1. If (g* ^{jk}*(x)) =

*a(|x|*)

^{2}

*I*near inﬁnity, then (H5) is satisﬁed with

*f*

*cv*(x) = (

_{|}*x*

*|*

1 *a(r)*^{−}^{1}*dr)*^{2} near inﬁnity. In fact, using the coordinates *t* =*r*

1 *a(s)*^{−}^{1}*ds*
(r=*|x|*) and*ω*=*x/|x| ∈S*^{d}^{−}^{1}, we have*f** _{cv}*=

*t*

^{2}and

*h*

_{0}=

*τ*

^{2}+

*α(t)*

^{2}

*p, where*

*τ*is the dual variable of

*t,−p*is the principal symbol of the Laplacian on

*S*

^{d}

^{−}^{1}, and

*α(t) =a(r)/r. HenceH*

_{h}^{2}

0*t*^{2}= 8τ^{2}+ 8α(t)^{2}*pt/r·*(a(r)*−ra** ^{}*(r))

*≥ch*0 near inﬁnity for some

*c >*0.

For example, when*a(r) = 1 +c*sin(εlog*r) withc∈***R**and*ε >*0 satisfying
*c*^{2}(1 +*ε*^{2})*<*1, then (H5) holds.

The requirement that*∂*^{α}*f*_{cv}*∈L** ^{∞}*(R

*) for all*

^{d}*|α| ≥*3 is not essential in (H5), as the following lemma shows.

**Lemma 2.3.** *Let* *f* *∈* *C*^{2}(R* ^{d}*),

*f*

*≥*1, lim

_{|}

_{x}

_{|→∞}*f*(x) =

*∞*,

*such that*

*for everyα∈*

**N**

^{d}_{0}

*with|α|*= 2,

sup

*x**∈***R**^{d}

*|∂*^{α}*f*(x)*|<∞,* lim

*|**h**|→*+0 sup

*x**∈***R**^{d}

*|∂*^{α}*f*(x+*h)−∂*^{α}*f*(x)*|*= 0,

*and that for some* ˜*σ >*0,*R >*˜ 0,

*H*_{h}^{2}_{0}*f* *≥*2˜*σ*^{2}*h*0 *on{*(x, ξ)*∈T*^{∗}**R*** ^{d}*;

*f*(x)

*≥R*˜

^{2}

*}.*

*Then for every* 0 *< σ <σ*˜ *and* *R >R,*˜ *there exists* *f**cv* *∈* *C** ^{∞}*(R

*)*

^{d}*such that*(H5)

*holds with theseσ,*

*R,*

*andf*

*cv*

*.*

The condition (H5) ensures the existence of a positively (or negatively)
invariant set *S*_{+}(R^{}*, σ** ^{}*) (or

*S*

*(R*

_{−}

^{}*, σ*

*)) deﬁned below, which asymptotically includes every positive (or negative) orbit that is not relatively compact. The role of this set becomes clearer in Section 6. Let*

^{}*S*

^{∗}**R**

*=*

^{d}*{X∈T*

^{∗}**R**

*;*

^{d}*h*0(X) = 1

*}*. Remark that

*h*0

*◦*Φ

*t*=

*h*0.

**Proposition 2.4** [5, Theorem 3.2]. *ForR*^{}*≥R,*0*< σ*^{}*< σ, set*
*S*+(R^{}*, σ** ^{}*) =

*{X*= (x, ξ)

*∈S*

^{∗}**R**

*;*

^{d}*r(x)> R*

^{}*, H*

*h*

_{0}

*r(X)> σ*

^{}*},*

*S*

*(R*

_{−}

^{}*, σ*

*) =*

^{}*{X*= (x, ξ)

*∈S*

^{∗}**R**

*;*

^{d}*r(x)> R*

^{}*,*

*−H*

*h*

_{0}

*r(X)> σ*

^{}*},*

*whereR*

*andσ*

*are the constants in*(H5).

(1)_{+} Φ_{t}*S*_{+}(R^{}*, σ** ^{}*)

*⊂S*

_{+}(R

^{}*, σ*

*)*

^{}*if*

*t≥*0.

(1)* _{−}* Φ

_{t}*S*

*(R*

_{−}

^{}*, σ*

*)*

^{}*⊂S*

*(R*

_{−}

^{}*, σ*

*)*

^{}*ift≤*0.

(2)_{+} *For everyX*_{0}*∈S*^{∗}**R**^{d}*\T** _{cpt,+}*,

*there exists*

*T >*0

*such that*Φ

*(X*

_{t}_{0})

*∈*

*S*+(R

^{}*, σ*

*)*

^{}*ift≥T. In particular,*

*T*+ =

*T*

*cpt,+*

*.*

(2)_{−}*For everyX*0*∈S*^{∗}**R**^{d}*\T**cpt,**−*,*there exists* *T >*0*such that*Φ*t*(X0)*∈*
*S** _{−}*(R

^{}*, σ*

*)*

^{}*ift≤ −T. In particular,T*

*=*

_{−}*T*

*cpt,*

*−*

*.*

(3)*T**cpt**∩S*^{∗}**R**^{d}*is a compact subset of{*(x, ξ)*∈T*^{∗}**R**^{d}*\*0;*r(x)< R}.*
To state our main results, we need some notation. For a bounded interval
*I⊂***R, set**

*µ*_{1}(I, L) =
*d*
*j=1*

sup

*t**∈**I,**|**x**|≥**L*

*|∇**x**a** ^{j}*(t, x)

*|,*

*µ*

_{1}(I) = lim

*L**→∞**µ*_{1}(I, L);

*µ*2(I, L) = sup

*t**∈**I,**|**x**|≥**L*

*|∇**x**b(t, x)|*

*|x|* *,* *µ*2(I) = lim

*L**→∞**µ*2(I, L).

*Remark.* Set*µ*^{}_{2}(I, L) = sup_{t}_{∈}_{I,}_{|}_{x}_{|≥}_{L}^{|∇}^{x}^{h}_{|}_{x}^{2}^{(t,x)}_{|}* ^{|}*. Then lim

*L*

*→∞*

*µ*

^{}_{2}(I, L)

=*µ*2(I).

*Remark.* Set*µ*^{}_{1}(I, L) =*d*

*j=1*sup_{t}_{∈}_{I,}_{|}_{x}_{|≥}_{L}^{|}^{a}^{j}^{(t,x)}_{|}_{x}_{|}* ^{|}*and

*µ*

^{}_{1}(I) = lim

*L*

*→∞*

*µ*^{}_{1}(I, L). Then *µ*^{}_{1}(I) *≤* *µ*1(I), because the equation *a** ^{j}*(t, x) =

*a*

*(t, εx) + 1*

^{j}*ε* *∇**x**a** ^{j}*(t, θx)

*·x dθ*gives that

*µ*

^{}_{1}(I, L)

*≤*

*εµ*

^{}_{1}(I, εL) + (1

*−ε)µ*

_{1}(I, εL) for every 0

*< ε <*1 and

*L≥*1.

**Theorem 2.5.** *There existsc(d, h*_{0}*, r)>*0 *such that for every bounded*
*interval* *I* = [t_{1}*, t*_{2}] (t_{1} *< t*_{2}) *satisfying* *µ*_{1}(I)*|I|*+*µ*_{2}(I)*|I|*^{2} *≤* *c(d, h*_{0}*, r),* *the*
*assertion below holds:* *Ifa∈S*^{0}_{1,0}=*S(1,|dx|*^{2}+*ξ* ^{−}^{2}*|dξ|*^{2})*satisﬁes that*

supp*a∩T** _{−}*=

*∅*(resp. supp

*a∩T*

_{+}=

*∅*)

*and that*

*π(suppa)is relatively compact,then the mappings*

*x* ^{−}^{ρ}*B** ^{s}*(R

^{d}*,*

**C**

*)*

^{n}*u*0

*→ |t−t*1

*|*

^{ρ}*a*

^{w}*S*(t, t1)u0

*∈C(I*

*t*

*,B*

*(R*

^{s+ρ}

^{d}*,*

**C**

*)),*

^{n}*x*

^{−}

^{ρ}*B*

*(R*

^{s}

^{d}*,*

**C**

*)*

^{n}*u*0

*→ |t−t*_{1}*|*^{ρ}*a*^{w}*S*(t, t_{1})u_{0}*∈L*^{2}(I_{t}*,B** ^{s+ρ+1/2}*(R

^{d}*,*

**C**

*))*

^{n}*resp.x* ^{−}^{ρ}*B** ^{s}*(R

^{d}*,*

**C**

*)*

^{n}*u*0

*→ |t−t*2

*|*

^{ρ}*a*

^{w}*S*(t, t2)u0

*∈C(I*

*t*

*,B*

*(R*

^{s+ρ}

^{d}*,*

**C**

*)),*

^{n}*x*

^{−}

^{ρ}*B*

*(R*

^{s}

^{d}*,*

**C**

*)*

^{n}*u*0

*→ |t−t*2*|*^{ρ}*a*^{w}*S*(t, t2)u0*∈L*^{2}(I*t**,B** ^{s+ρ+1/2}*(R

^{d}*,*

**C**

*))*

^{n}*are continuous for alls∈***R***andρ∈*[0,*∞*). Here*π*:*T*^{∗}**R*** ^{d}*(x, ξ)

*→x∈*

**R**

^{d}*.*

*Remark.*Theorem 2.5 is a corollary of more general theorems (see Theorems 6.2 and 6.5). It suﬃces to assume that the initial data decays in an incoming region

*S*

*(R*

_{−}

^{}*, σ*

*) (resp. in an outgoing region*

^{}*S*+(R

^{}*, σ*

*)) in a sense.*

^{}**Corollary 2.6.** *Let* *c(d, h*_{0}*, r)* *>* 0 *be the constant in Theorem* 2.5.

*Then for every bounded interval* *I* = [t1*, t*2] (t1 *< t*2) *satisfying* *µ*1(I)*|I|*+
*µ*2(I)*|I|*^{2}*≤c(d, h*0*, r), the assertion below holds: For every* *u*0*∈ E** ^{}*(R

^{d}*,*

**C**

*)*

^{n}*W F*(S(t, t0)u0)*⊂T*_{−}*,* *t*1*≤t*0*< t≤t*2;
*W F*(S(t, t0)u0)*⊂T*+*,* *t*1*≤t < t*0*≤t*2*.*

**Corollary 2.7.** *Let* *c(d, h*0*, r)* *>* 0 *be the constant in Theorem* 2.5.

*Then for every bounded interval* *I* = [t1*, t*2] (t1 *< t*2) *satisfying* *µ*1(I)*|I|*+
*µ*2(I)*|I|*^{2}*≤c(d, h*0*, r),*

*W F*(K(t, t0))*⊂*(T_{−}*×T** _{−}*)

*∪*(0

*×T*

*)*

_{−}*∪*(T

_{−}*×*0),

*t*1

*≤t*0

*< t≤t*2;

*W F*(K(t, t

_{0}))

*⊂*(T

_{+}

*×T*

_{+})

*∪*(0

*×T*

_{+})

*∪*(T

_{+}

*×*0),

*t*

_{1}

*≤t < t*

_{0}

*≤t*

_{2}

*.*

*Here*0

*is the zero section ofT*

^{∗}**R**

^{d}*.*

**Theorem 2.8** (smoothing eﬀect of order half). *Let* *s∈***R***and*0*< ν*
1. Let*I*= [t1*, t*2] (t1*< t*2) *andt*0*∈I.*

(1) *If* *T** _{cpt}* =

*∅, then there exists*

*C >*0

*such that the following estimates*

*hold:*

*t*
*t*0

*x*^{−}^{(1+ν)/2}*E*_{s+1/2}*u(τ)*^{2}*dτ*

*≤CE**s**u(t*0)^{2}+*C*
*t*

*t*0

*E**s**f*(τ)*dτ*
2

*,*
*E**s**u(t)*^{2}+

*t*
*t*0

*x* ^{−}^{(1+ν)/2}*E*_{s+1/2}*u(τ)*^{2}*dτ*

*≤CE*_{s}*u(t*_{0})^{2}+*C*
*t*

*t*_{0}

*x* ^{(1+ν)/2}*E*_{s}_{−}_{1/2}*f*(τ)^{2}*dτ*
*for allt∈I* *andu∈C*^{1}(I,*S*(R^{d}*,***C*** ^{n}*))

*withf*(t) = (∂

*+*

_{t}*iH(t))u(t).*

(2)*For everya∈S(1,|dx|*^{2}+*|dξ|*^{2}*/X* ^{2})*satisfying*cone(supp*a)∩T** _{cpt}*=

*∅*,

*there existsC >*0

*such that the following estimate holds:*

*t*
*t*0

*x* ^{−}^{(1+ν)/2}*E*_{s+1/2}*a*^{w}*u(τ)*^{2}*dτ*

*≤CE**s**u(t*0)^{2}+*C*
*t*

*t*0

*E**s**f*(τ)*dτ*
2

*for allt∈I* *andu∈C*^{1}(I,*S*(R^{d}*,***C*** ^{n}*))

*withf*(t) = (∂

*t*+

*iH(t))u(t).*

*Remark.* In contrast to Theorem 2.5, Theorem 2.8 holds for every com-
pact interval*I* with no distinction between the forward, and backward, prop-
agators (especially, observe the condition cone(supp*a)∩T** _{cpt}* =

*∅*in (2)). See Section 8 for the comparison among various nontrapping conditions.

*Remark.* The smoothing eﬀect of order half fails at almost every point
in *T**cpt*. See [3, 5] for details in a little diﬀerent framework.

**§****3.** **Weyl Calculus**

In this section, we recall the Weyl calculus due to H¨ormander (see [8, Chapters 18.4-6] for details) and prove related lemmas.

For a Riemannian metric *g* on *V* = **R*** ^{N}* and a positive function

*m*

*∈*

*C(R*

*), the symbol space*

^{N}*S(m, g) is the set of alla∈C*

*(R*

^{∞}*) such that for every*

^{N}*k∈*

**N**0

*a**k,S(m,g)*=
*k*
*j=0*

sup

*|∂**v*1*· · ·∂**v**j**a(x)|*
*m(x)**j*

*i=1**g** _{x}*(v

*)*

_{i}^{1/2};

*x∈*

**R**

^{N}*,*0=

*v*

_{i}*∈*

**R**

^{N}*<∞,*

where *∂**v**f*(x) = (d/dt)*|**t=0**f*(x+*tv) and* *g**x*(v) = *g**x*(v, v). It is a Fr´echet
space with seminorms (*||·||**k,S(m,g)*)* _{k=0,1,...}*. A sequence (a

*)*

_{n}*in*

_{n=1,2,...}*S(m, g)*is said to converge to

*a*weakly in

*S(m, g), or simply*

*a*

*n*

*→*

*a*weakly in

*S(m, g), if (a** _{n}*) is bounded in

*S(m, g) and converges to*

*a*in

*C*

*(R*

^{∞}*) (or equivalently, in*

^{N}*D*

*(R*

^{}*)). Let*

^{N}*S(m, g;M*

*(C)) denote the*

_{n}*M*

*(C)-valued sym- bol space*

_{n}*S(m, g)⊗M*

*n*(C) =

*{*(a

*jk*)1

*≤*

*j,k*

*≤*

*n*;

*a*

*jk*

*∈*

*S(m, g)}*; the seminorms

*a*

*k,S(m,g;M*

*n*(C)) are deﬁned similarly to

*a*

*k,S(m,g)*except that

*|a(x)|*=

*a(x)*

*L(C*

*)in the former deﬁnition.*

^{n}From now on, we consider the case where *V* =**R**^{2d}*∼*=**R**^{d}*×*(R* ^{d}*)

*. Let*

^{}*σ*be the canonical 2-form on

**R**

^{2d}

*σ(X, Y*) =*ξ·y−η·x,*

where *X* = (x, ξ), Y = (y, η)*∈* **R**^{2d}. Let *g* be a Riemannian metric on **R**^{2d}.
The Riemannian metric*g** ^{σ}* on

**R**

^{2d}is deﬁned by

*g*_{X}* ^{σ}*(Y) = sup

*Y** ^{}*=0

*σ(Y, Y** ^{}*)

^{2}

*g*

*X*(Y

*)*

^{}*.*We consider three conditions on

*g.*

(G1) (slow variation). There are*c, C >*0 such that for every*X, Y, Z∈***R**^{2d}
*g**X*(Y)*≤c⇒C*^{−}^{1}*g**X*(Z)*≤g**X+Y*(Z)*≤Cg**X*(Z).

(G2) (σ temperance). There are *C, N >*0 such that for every *X, Y, Z* *∈*
**R**^{2d}

*g**Y*(Z)*≤Cg**X*(Z)(1 +*g*_{Y}* ^{σ}*(X

*−Y*))

^{N}*.*(G3) (uncertainty principle). For every

*X*

*∈*

**R**

^{2d}

*γ(X) =* sup

*Y**∈***R**^{2d}*,Y*=0

(g* _{X}*(Y)/g

^{σ}*(Y))*

_{X}^{1/2}

*≤*1.

*In the rest of this section, we ﬁx a Riemannian metricgsatisfying* (G1)–(G3).

A positive function*m*:**R**^{2d}*→*(0,*∞*) is said to be a *g*weight if it satisﬁes the
following conditions.

(M1) (gcontinuity). There are*c, C >*0 such that for every*X, Y* *∈***R**^{2d}
*g**X*(Y)*≤c⇒C*^{−}^{1}*≤m(X*+*Y*)/m(X)*≤C.*

(M2) (σ, g temperance). There are *C, N >* 0 such that for every*X, Y* *∈*
**R**^{2d}

*m(Y*)*≤Cm(X)(1 +g*^{σ}* _{Y}*(X

*−Y*))

^{N}*.*