$\dot{\nearrow}’$
エ
$7^{\iota}.\triangleright$t ノ
/-$. \bigwedge_{\prime},\overline{\mathrm{A}}$
Ja
$\zeta,$ $\cdot\hslash \mathrm{a}^{\gamma,z_{\backslash }}\{^{\mathrm{A}}‘/\nearrow-\eta’r\ovalbox{\tt\small REJECT}.\not\leq$,
東
K-
数埋刺字
(
$\rho f\supset$森正 紳伐
$\mathrm{r}S\mathrm{k}\mathrm{i}\mathrm{r}\gamma\approx$
.
$p\mathit{4}\mathit{0}_{t_{1}}^{\cdot}t\circ\dot{k}y$$0$
.
INTRODUCTION -DEFINITIONS
AND
TIIEOREMS-We
define
a class of wavelct
transforms
as a
continuous md
micro-local version of the Littlewood-Paley decompositions.
H\"orn)ander’s
wave front sets
$\iota 1\mathrm{S}$well
as Besov
$i\iota \mathrm{n}\mathrm{d}$
Triebel-Lizorkin
spaces
may be
$\mathrm{c}1_{1}L\backslash .\mathrm{r}\mathrm{a}\mathrm{C}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{d}$
in tcrms of our wavelet transforms. We remark
$\mathrm{t}\mathrm{h}‘\backslash \dagger$.
our
decompositions can be
regarded
$1\mathrm{i}\mathrm{n}\mathrm{e}\iota Tr$ly independent.
This
$\mathrm{P}^{\iota 1}1^{)\mathrm{e}\mathrm{r}}$consists of two parts.
The
former part is the comparison
between the wave front sets
defined
by
our
wavelct
transforms
$\dot{\mathrm{c}}\mathrm{u}\mathrm{u}\mathrm{d}$H\"ormander’s
wave front sets. The latter part is the characterization of
Besov,
Triebel-Lizorkin
spaces by
using
our wavelet
trmsforms.
First,
we define our
wavelet transforms
as
follows;
Definition
1.
Suppose that the function
$\psi(x)$
(called
wavelct) luas the
following properties;
$\psi(x)\in S(\mathbb{R}^{\iota}’),$
$’\hat{\psi}(\xi)\in C_{0}^{\infty}(\mathbb{R}n)\mathrm{c}\mathrm{t}\prime \mathrm{n}\mathrm{d}\hat{\psi}(\xi)\geqq 0$
.
Let
$\Omega=\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\cdot\hat{\psi}(\xi)$,
$(0, \cdot\cdot, 0,1)$
is the central
axis of
$\Omega$, and
$\gamma\xi$
is any rotation which sends
$\xi/|\xi|$
to
$(0, \cdot\cdot, 0,1)$
.
When
$n=1,$
$\Omega\subset(0, \infty)$
and when
$n\geqq 2,$
$\Omega$is
connected, does
not contain the
origin
$0$
and
$\psi(x)=\psi(?\cdot x)$
for
any
$r\in SO(n)$
s\v{c}rtisfying
$r(0, \cdot\cdot, 0,1)=(0, \cdot\cdot, 0,1)$
.
Then
our
wavelet
trans-form is
defined
$\mathrm{c}\lambda \mathrm{S}$follows;
for
$f(t)\in S’(\mathbb{R}^{\iota}’),$
$(.x, \xi)\in \mathbb{R}^{2n}$
,
$\nu V_{\psi}f(x, \xi)=\{$
$\int_{\mathbb{R}}f(i\int_{\mathrm{R}^{\nu:}}f()|\xi t)|\xi|^{1/2}\psi|n/\overline{2^{\frac{(\xi(t-x))dt}{\psi(|\xi|r\epsilon(t-X))}}.}’ dt$,
if
$n\geqq\underline{9}*$
if
$n=1$
,
Remark
1.
$\nu V_{\psi}f(x, \xi)$
is rewritten
as
follows;
$\int_{\mathrm{R}^{\mathfrak{n}}}\hat{f}(\tau)\cdot|\xi|^{-\frac{\mathfrak{n}}{2}\hat{\psi}}(\frac{r_{\zeta}}{|\xi|}\tau)\cdot edirx\mathcal{T}$
.
From
this, thc lneiuuing of
our wavelet transforms is clear.
$\acute{\mathrm{R}}$
emark
2. Our
wavelet
$\mathrm{t}\mathrm{r}_{C}\backslash \mathrm{n}\mathrm{s}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{n}1\dot{\mathrm{s}}$in
$\mathbb{R}^{n}$are the reduced versions
of those
defined
by
R.Murenzi
$(\mathrm{s}_{\mathrm{e}}\mathrm{e},[\zeta])$.
Our
purpose is to carry out the
analogy of the
$\mathrm{m}\mathrm{i}_{\mathrm{C}\mathrm{r}\mathrm{o}}1\mathrm{o}\mathrm{C}\mathrm{a}\underline{1}-i\iota \mathrm{n}\mathrm{a}\mathrm{i}\mathrm{y}\mathrm{s}\mathrm{i}\mathrm{s}$L.H\"ormander
succeeded
in
$\beta\downarrow$.
Remark
3.
The
domain of a
wavelet transformation is
usually
the
$L_{2^{-}}\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{c}(\mathrm{S}\mathrm{e}\mathrm{e},[t])$
,
but
can be
e.x
tended to
$S’,\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}$is, the dual space of
$S$
.
It
is easy to see that the image of
$S$
by this
$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{n}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}$is also
$S$
.
Now,
we
define
our
wave front set
$WF_{\psi}(f)(\subset \mathrm{R}_{x}^{n}\cross \mathrm{R}_{\xi}^{n})$
of
$f\in s’(\mathbb{R}^{1}’)$
as follows.
Definition 2.
$(x_{0}, \xi^{0})\not\in WF_{\psi}(f)$
is
defined
as follows:
there
exists a neighbourhood
$U(x_{0})$
of
$x_{0}$
and
a
conic neighbourhood
$\Gamma(\xi^{0})$
of
$\xi^{0}$such that
$|\nu V\psi f(x, \xi)|=O(|\xi|^{-N})$
as
$|\xi|$
tends
to
$\infty$for
any
$N\in \mathrm{N}$
in
$U(X_{0})\cross\Gamma(\xi^{0})$
.
Moreover, we
define
the refinement
$\dot{\nu}VF_{\psi}^{(s)}(f)’‘\iota \mathrm{s}$follows.
Deflnition
3.
$(x_{0}, \xi 0)\not\in WF_{\psi}^{(s})(f)\Leftrightarrow$
$\int\int$
$|\nu V\psi f(x, \xi)|2(1+|\xi|2)^{s}dXd\xi<\infty$
.
$U(x_{0})\mathrm{x}\mathrm{p}(\epsilon^{0})$
It
is
clear that if
$f\in L_{2}(\mathbb{R}^{n})$
,
$l/VF_{\psi}(f)=\mathrm{t}\mathrm{h}\mathrm{e}$
closure of
$\bigcup_{s\geqq 0}WF_{\psi}((_{S})f)$
.
We necd the
following definition to state Theorem
1.
Definition
4.
Let
$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{e}\Omega=[t\xi|\xi\in\Omega,t>0\mathit{1}$
.
$(X_{0}, \xi^{0})\not\in\overline{WF}^{\psi}$
is
defined
as
follows:
$x_{0}\not\in \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}_{x}WF$
and
$\xi^{0}\in \mathbb{R}^{n}$
,
or
$x0\in$
$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}_{x}WF$and
$\mathrm{r}(\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{e}\Omega)$does not
intersect
$\{\xi\in \mathbb{R}^{n};(x0, \xi)\in$
$WF\}$
for
any
$r\in SO(n)$
with
$\mathrm{r}(\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{e}\Omega)$including
$\xi^{0}$.
Here,
$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}_{x}WF$denotes
the projection of
WF onto
Theorem
1.
Let
$f\in L_{2}(\mathbb{R}^{n})$
,
and
$s\geqq 0$
.
When
$n=1,$
$WF_{\psi^{S}}^{()}(f)=$
$WF^{(s)}(f)$
.
When
$n\geqq 2,$
$WF_{\psi^{S}}^{()}(f)\subseteq\overline{WF^{(s)}(f)}^{\psi}$
and
$WF^{(s)}(f)\subseteq$
$\overline{WF_{\psi^{S}}^{()}(f)}^{\psi}$
We have the
same
inclusions between
$WF_{\psi}(f)$
and
$WF(f)$
.
The latter
part
of this
paper
is the
characterization
of
Besov,
Triebel-Lizorkin
spaces
by using
$o\mathrm{t}\mathrm{t}r\mathrm{w}\dot{\mathrm{a}}$velet
transform
$S\cdot$.
We use
continuous decompositiorsnot
only
of the radial direction
but
also
of
the
unit
sphere of the frequency
space.
(See,J.
$\mathrm{P}\mathrm{e}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{e}[4],\mathrm{H}.\mathrm{T}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{b}\mathrm{e}\mathrm{l}\iota 5]$)
$\mathrm{D}\mathrm{e};\mathrm{f}\iota.$A
$\mathrm{i}t\iota.on\prime r$
,
Let
$\phi(x)$
be
a rapidly
$\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{r}\mathrm{e}\dot{\mathrm{a}}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}$function
whose Fourier transform
is compactly supported
in
$\frac{1}{2}\leqq|\xi|\leqq 2$
.
Moreover, suppose that any
half line
starting
from
the
origin intersects
$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}f^{\wedge}\mathrm{t}\zeta$).
Let
$\phi_{r}(X)$
be
$r^{n}\phi(rx)$
.
Then,
$\hat{\phi}_{f}.(\xi)$is equal
to
$\hat{\phi}(_{r}^{\xi})$.
Definition
of
Besov spaces
$\dot{B}_{p,q}^{S}(\mathbb{R}^{\mathfrak{n}})$.
$f\in\dot{B}_{p,q}^{s}(\mathbb{R}^{n})(s>0,1\leqq$
$p,$ $q\leqq\infty)$
is
defined
by the
following:
$( \int(r^{s}1|\phi r)||L))^{q_{\frac{d\mathrm{r}}{r})^{\perp}}}*\dot{f}(X\nu(dxq<\infty$
.
Deflnition
of
Triebel-Lizorkin spaces
$\dot{F}_{P,q}^{S}(\mathbb{R}^{n})$.
$f\in\dot{F}_{\mathrm{p},q}^{s}(\mathbb{R}^{n})$
$(s>0,1\leqq p<\infty,1\leqq q\leqq\infty)$
is
defined
by the
following:
$||( \int(_{\Gamma}s.
\phi r*f(X))q_{\frac{dr}{r}})l1||_{L_{\mathrm{p}}()}dx<\infty$
.
Theorem
2.
$f\in\dot{B}_{\mathrm{p},q}^{s}(\mathbb{R}^{n})(s>0,1\leqq p, q\leqq\infty)$
can be
$chi\iota \mathrm{r}\mathrm{a}\mathrm{C}$ter-ized
by
the followin
$g$
:
$||| \xi|^{s}|||\xi|\frac{\mathfrak{n}}{2}|W\psi f(x, \xi)|||\mathrm{L}\rho^{(d})x||t\mathrm{f}\mathrm{l}(\phi^{d}\mathrm{r})<\infty$
.
Theorem 3.
$f\in\dot{F}_{p_{1}^{S}q}(\mathbb{R}^{\mathfrak{n}})(s>0,1\leqq p<\infty, 1\leqq q\leqq\infty)$
can
be
chara
cterized
by the
following:
I.WAVE
FRONT
SETSDEFINED
BY
$O\iota 4\Gamma$WAVELET
$\mathrm{T}\mathrm{R}\mathrm{A}\mathrm{N}\mathrm{s}\Gamma.\mathrm{o}\mathrm{R}\mathrm{M}\zeta \mathrm{A}\mathrm{N}\mathrm{D}$
H\"ORMANDEn’S
WAVE
$\Gamma \mathrm{R}\mathrm{O}\mathrm{N}\mathrm{T}$SEW
As we have
already defined, the wavelet
$\psi(x)$
is of
essentially
two
parameters
that is
$\mathrm{r}\mathrm{o}\mathrm{t}\dot{\mathrm{a}}$tionally
invariant around
$\ominus$when
$n\geqq 2$
.
For
the
purpose of proving Theorem
$1;\mathrm{w}\mathrm{e}$prepare three propositions.
$(i\sqrt ere_{l}\Phi\overline{-} ‘ p, \cdot\cdot p, \mathit{1}\prime j\mathcal{L}-\mathrm{R}^{\vee}.
)$
Proposition
1(
$\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{S}\mathrm{e}\mathrm{V}\mathrm{a}\mathrm{l}$formula and
inversion
formula).
For
$f,$
$g\in L_{t}(\mathbb{R}^{n})$
,
$\iint W_{\psi}f(x,\xi)\overline{\mathrm{V}V\psi g(x,\xi)}dXd\xi=C\psi\int f(t)\overline{g(t)}dt$
.
Here,
$C \psi=(2\pi)^{n}I\frac{|\hat{\psi}(\xi)|^{2}}{|\xi|^{n}}d\xi$
.
$f\}o\mathrm{m}$
this,
we
als
$0$
have:
$f(t)=C_{\psi}^{-1} \iint W\psi f(X,\xi)\cdot|\xi|^{\mathrm{g}}2\psi(|\xi|r\xi(t-X))dxd\xi$
,
when
$n\geqq 2$
.
When
$n=1,$
$|\xi|\mathrm{r}\zeta(\iota-X)$
is replaced
by
$\xi(t-x)$
.
Proposition
$2(\mathrm{L}_{\mathrm{o}\mathrm{C}}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{t}\mathrm{y})$.
If
$x_{0}\not\in$
suppf, then
there exists
a neighbourhood
$U(x_{0})$
of
$x_{0}$
such
that
$W\psi f(x, \xi)$
is rapidly
decreasing in
$\xi$with
respect
to
$x\in U(x\mathrm{o})$
uniformly.
Proposition
3(
$\mathrm{G}1_{0}\mathrm{b}\mathrm{a}1$Sobolev
property).
$f \in H^{s}(\mathbb{R}^{n})\Leftrightarrow\int\int|W\psi f(X,\xi)|^{2}(1+|\xi|2)^{s}<\infty$
.
Proof
of
Theorem
1.
It suffices
to show
when
$n\geqq 2$
.
Moreover,
by
the fact that
$WF\psi(f)=\mathrm{t}\mathrm{h}\mathrm{e}$
closure
of
$\bigcup_{s\geqq 0}WF_{\psi}^{(s}()f)$
,
it
suffices to
prove the
statement
for any
$s\geqq 0$
fixed.
Step.1
Let
$(0, \xi^{0})\not\in\overline{\nu VF^{(s)}(f)}^{\psi}$
If we take.a conic neighbourhood
$\Gamma(\xi^{0})$
of
$\xi^{0}$as the union
of
all
$\mathrm{r}(\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{e}\Omega)$,
where
$\mathrm{r}$is any rotation with
$\xi^{0}$included
in
$\mathrm{r}(\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{e}\Omega)$, then
there
exists a function
$\phi(x)\in C_{0}^{\infty}(\mathbb{R}^{n})$
which is
always equal
to
1
near
$x=0$
and satisfies
$\int_{\Gamma(\xi)}0|(\phi f)\wedge(\xi)|2(1+$
$|\xi|^{2})^{s}d\xi<\infty$
.
This follows from
the
definition
$\mathrm{o}\mathrm{f}\overline{W}^{\psi}$,
the
definition
of
H\"ormander’s
wave front set.and Heine-Borel’s
lemma.
What we
want to say
is that there exist a conic neighbourhood
$\tilde{\Gamma}(\xi^{0})$of
$\xi^{0}$and
a
neighbourhood
$U(\mathrm{O})$
of
$0$
,
satisfying:
$\int\int_{-}$
.
$|W\psi f(_{X}, \xi-)|2(1+|\xi|^{2})^{s}<\infty$
$U(0)_{\mathrm{X}}\Gamma(\xi 0\rangle$
1
Here,using the
inversion
formula,
we divide
$W\psi f(x,\xi)$
into two parts:
$W \psi f(x, \xi)=|\xi|^{\frac{n}{2}}\int(\phi f)(t)\cdot\overline{\psi(|\xi|\Gamma\epsilon(t-x))}dl$
(1)
$+| \xi|^{\frac{n}{2}}\int((1-\phi)f)(t)\cdot\overline{\psi(|\xi|\Gamma\epsilon^{(}t-X))}dt$
(2)
If
$U(\mathrm{O})\subset\subset\{\phi(x)\equiv 1\},\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}$
, by the argument
of
propotion 2, (2)
is
rapidly
decresing
in
$|\xi|$
with
respect
to
$x\in U(\mathrm{O})$
uniformly.
There-fore,
it
is clear that
$(0, \xi 0)\not\in WF_{\psi^{s}}^{()}((1-\phi)f)$
.
On
the
other
hand, if
we take
$\tilde{\Gamma}(\xi^{0})$sufficiently small, then we get the
following:
$\iint_{-}$
$|W_{\psi}(\phi f)(x, \xi)|^{2}(1+|\xi|^{2})^{s}dxd\xi$
$U(0)\mathrm{x}\Gamma(\epsilon^{0})$
$\leqq\int_{\overline{\Gamma}(\xi^{0})}d\xi\int_{\mathrm{R}_{x}^{\mathfrak{n}}}|W\psi(\phi f)(x, \xi)|2(1+|\xi|2)^{s_{d}}xd\xi$
$=(2 \pi)^{n}\int d\mathcal{T}|(\mathrm{R}^{\mathfrak{n}},\phi f)(\wedge\tau)|^{2}(\zeta^{0})\int_{\tilde{\Gamma}}\frac{d\xi}{|\xi|^{n}}(1+|\xi|2)^{s_{\hat{\emptyset}(\frac{f\xi}{|\xi|}\tau)^{2}}}$
If
we change variables from
$\tau$to
$\omega=\mathrm{H}r\epsilon^{\tau}$as
before
,
$\omega$must be in
$\Omega$.
very small. The inequality above is followed by:
$\leqq(2\pi)^{n}(\epsilon 0)\int_{\Gamma}d\tau|(\phi f)\wedge(\tau)|^{2}\Omega\int\frac{d\omega}{|\omega|^{n}}(1+\frac{|\tau|^{2}}{|\omega|^{2}})^{s}\hat{\psi}(\omega)^{2}$
$\leqq C\int_{\mathrm{t}\Gamma(0)}|(\phi f)(\mathcal{T}\wedge)|^{2}(1+|\tau|^{2})^{s}d_{\mathcal{T}}<\infty$
(
$\mathrm{H}\mathrm{e}\mathrm{r}\mathrm{e},$ $\mathrm{c}$
is
a
constant.)
Therefore,
$(0, \xi^{0})\not\in WF_{\psi^{s}}^{()}(\phi f)$
.
Step.2
Let
$(0, \xi^{0})\not\in\overline{Wp_{\psi}^{(s)}(f)}^{\psi}$
If
we take a
conic
neighbourhood
$\Gamma(\xi^{0})$
of
$\xi^{0}$as the
union
of all
$\mathrm{r}(\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{e}\Omega)$,
where
$\mathrm{r}$is
any
rotation
with
$\xi^{0}$included
in
$\mathrm{r}(\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{e}\Omega)$,
then
there
exists a neighbourhood
$U(\mathrm{O})$
of
$x=0$
and
$\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{s}U(0)_{\mathrm{X}\Gamma}(\iint_{0,\epsilon)}|W_{\psi}f(X, \xi)|^{2}(1+|\xi|^{2})^{s}dXd\xi<\infty$, as
in
Step 1.
Here,
using
the
inversion
formula,
we divide
$\mathrm{f}$into two
parts:
$f=f_{\Gamma}+f_{\Gamma^{\mathrm{c}}}$
,
where
$f_{\Gamma}(t)=C^{-1} \int_{)}\psi\Gamma(\epsilon 0\cross\int W\psi f(x, \xi)\cdot|\mathrm{R}_{*}n\xi|^{\mathfrak{n}}T\psi(|\xi|\Gamma\epsilon(t-x))dXd\xi$
$f_{\Gamma^{e}}(t)=C_{\psi}-1 \int\int W_{\psi}\Gamma(\epsilon 0)\epsilon\cross \mathrm{R}x\mathfrak{n}f(X, \xi)\cdot|\xi|^{\frac{\mathfrak{n}}{2}\psi}(|\xi|_{\Gamma}\epsilon(\iota-x))dXd\xi$
.
Then,
$\overline{f_{\Gamma^{\mathrm{c}}}}(\mathcal{T})=c^{-1}\psi\int_{\Gamma(\epsilon^{0e}})\int_{\mathrm{R}_{x}^{n}}\nu V\psi f(x, \xi)\cdot|\xi|-\frac{n}{2}\hat{\psi}(\frac{\Gamma\xi}{|\xi|}\mathcal{T})e^{-}dir\cdot xdx\xi$
If
we take a sufficiently small conic neighbourhood
$\tilde{\Gamma}(\xi^{0})$of
$\xi^{0}$,
then
we obtain
$\hat{\psi}(\frac{\Gamma\xi}{|\xi|}\tau)\equiv 0$
for
$\mathrm{a}\mathrm{n}\mathrm{y}\tau\in\tilde{\Gamma}(\xi^{0})$and for
any
$\xi\in\Gamma(\xi^{0})^{\mathrm{C}}$
Next,
we choose
$\phi(x)\in C_{0}^{\infty}(\mathrm{R}^{n})$
satisfying that
$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\phi(X)\subset U(0)$
and that
$\phi(x)\equiv 1$
in
some neighbourhood
$U_{1}(0)$
of
$0$
.
Then,
we
further divide
$f_{\Gamma}(t)$
into two
parts:
$f_{\Gamma}=f_{\Gamma,\phi}+f_{\Gamma,1-\emptyset}$
,
where
$f_{\Gamma,\phi}(t)=C_{\psi}-1 \int\int_{\mathrm{x}\mathrm{r}(\epsilon 0)\mathrm{R}_{x}^{\mathfrak{n}}}\emptyset(x)\cdot W_{\psi}f(X,\xi)\cdot|\xi|^{\mathrm{n}}2\psi(|\xi|r_{\zeta(}\iota-x))dXd\xi$
$f \mathrm{r},1-\phi(t)=C^{-}\psi^{1}\mathrm{r}(\int_{\epsilon^{0})\cross}\int_{x}(1-\phi.(x))W\psi f(_{X}, \xi)\cdot|\xi|\frac{\mathfrak{n}}{2}\psi(|\mathrm{n}\hslash\xi|\Gamma\xi(\{-x))dXd\xi$
Let
$U_{2}(0)\subset \mathrm{c}=\{\phi(x)\equiv 1\}$
, then we
can
easily
see that
$f_{\Gamma,1-\phi}(t)$
is
$C^{\infty}$
with respect to
$t\in U_{2}(0)$
,
by
Proposition
2,
and ’the exchange of
order of differentiation and integration’. Therefore,
it
follows
$(0, \xi^{0})\not\in$
$WF^{(s)}(f\Gamma,1-\phi)$
.
Lastly, we want to show
$(0, \xi^{0})\not\in WF^{(s)}(f_{\Gamma},\phi)$
.
This is the heart of
matter in proving Theorem 1. In
fact,
more
strongly,
we
can
show
the
global Sobolev property of
$f_{\Gamma,\phi}$.
$\overline{f_{\Gamma}|\emptyset}(\mathcal{T})=c_{\psi}^{-}1\int_{)\Gamma(\xi 0}\int_{\mathrm{R}_{x}^{n}\cross}\phi(_{X})\cdot\nu V\psi f(X,\xi)\cdot|\xi|^{-}2\mathrm{n}\hat{\psi}(\frac{r_{\xi}}{|\xi|}\mathcal{T})e-:\mathcal{T}\cdot xd_{X}d\xi$
Here,
if we
put
$g(x, \xi)=\phi(x)W_{\psi}f(x, \xi)\cdot(1+|\xi|^{2})^{\frac{}{2}}$
,
then we
can
see
$\int\int$
$|g(x, \xi)|2dXd\xi<\infty$
.
$\Gamma(\xi^{0})\cross \mathrm{R}^{\mathfrak{n}}*$
(This
follows from
the
hypothesis and from the fact that
$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\phi(X)$If we denote
the
Fourier partial transform of
$g(x, \xi)$
from
$\mathrm{x}$to
$\tau$by
$\hat{g}(\tau, \xi)$
,
$\overline{f_{\Gamma,\phi}}(\mathcal{T})(1+|\mathcal{T}|^{2})\frac{}{2}$
.
$=c_{\psi}^{-1} \int \mathrm{r}(\xi^{0})\int_{\cross \mathrm{R}*n}g(_{X}, \xi)e^{-}|ir\cdot x.\xi|^{-\frac{\mathfrak{n}}{2}\cdot\hat{\psi}(}\frac{\Gamma\zeta}{|\xi|}\tau)(\frac{1+|_{\mathcal{T}1^{2}}}{1+|\xi|^{2}})\overline{2}.dxd\xi$
$=C_{\psi}^{-1}(2 \pi)^{\frac{\mathfrak{n}}{2}}\int_{\Gamma(\xi^{0})}\hat{g}(\tau, \xi)\cdot I\zeta(\mathcal{T}, \xi)d\xi$
Here,
$K(\tau, \xi)$
is defined by
$| \xi|^{-\frac{\mathfrak{n}}{2}}\hat{\psi}(r\mathrm{d}\epsilon^{\mathcal{T})}(\frac{1+|\tau 1^{2}}{1+|\xi 12})\dot{\overline{\mathrm{a}}}$.
Because
$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\hat{\psi}$is
a
compact
set
not including the origin
$0$
(by
the
defintion
of
$\psi$),
there
exists
$a$
constant
$\mathrm{C}$such
that
$|K( \tau,\xi)|\leqq c|\xi|^{-\frac{\mathfrak{n}}{2}}\hat{\psi}(\frac{\prime\epsilon}{|\xi|}\mathcal{T})$
.
Therefore, by using the result
in
the
proof of
Proposition
1(
i.e.
the
continuous decomposition of the
unity),
the integral
$\int|IC(\tau,\xi)|2d\xi$
is
bounded
from
above. (the
bound
is
$(2\pi)^{-n}C\psi c^{2}.$
)
After all,
we obtain
the
following
inequality:
$\int|\overline{f_{\Gamma,\phi}}(\tau)|^{2}(1+|\mathcal{T}|^{2})s_{d_{\mathcal{T}}}\leqq c^{-1}c^{2}\psi\int(\int_{\Gamma(\epsilon)}0|\hat{g}(\mathcal{T},\xi)|2d\xi)d\mathcal{T}$
$=C’ \int_{\Gamma(\zeta^{0})}d\xi\int_{\mathrm{R}_{r}^{\mathfrak{n}}}|\hat{g}(\mathcal{T},\xi)|2d\tau$
$=c_{\Gamma(\xi 0)}’ \int\int_{\mathfrak{n},\cross \mathrm{R}}.|g(_{X},\xi)|^{2}dxd\xi<\infty$
.
2.
$\mathrm{C}\mathrm{l}\mathrm{l}\mathrm{A}\mathrm{R}\mathrm{A}\mathrm{c}\mathrm{T}\mathrm{E}\mathrm{R}\mathrm{I}\mathrm{z}\mathrm{A}\mathrm{T}\mathrm{I}\mathrm{O}\mathrm{N}\mathrm{O}\Gamma$BESOV,
$\mathrm{T}\mathrm{R}\mathrm{I}\mathrm{E}\mathrm{B}\Sigma \mathrm{L}-\mathrm{L}\mathrm{I}\mathrm{Z}\mathrm{O}\mathrm{R}\kappa \mathrm{I}\mathrm{N}$SPACES
VIA
$\mathit{0}\iota p_{\Gamma}$CONTINUOUS WAVELET TRANS
$\mathrm{r}\mathrm{o}_{\mathrm{R}\mathrm{M}}S$
Now we prove
Theorem
2
and Theorem
3.
Tlleorenl
2.
$f\in\dot{B}_{\mathrm{p},q}^{s}(\mathbb{R}^{n})(s>0,1\leqq p, q\leqq\infty)$
can
be
$\mathrm{c}h$aracter-ized by the
$fo\mathit{1}l_{o\mathrm{w}}\mathrm{i}\mathrm{n}g$.
$|||\xi|^{s}|||\xi|^{\tau_{1}}W_{\psi}f(x, \xi)||n|L(dx)|’|Lq(\theta^{d}\Gamma)<\infty$
.
Proof.
Sufflciency:
For simplicity, let
$r_{l\}l,\Gamma} \mathrm{s}^{*}.\frac{n}{2}\theta\langle<)=\{\xi|W\psi f(x, \xi)$
,
$\phi_{|\xi|}=\int\psi_{1\zeta \mathrm{I},\mathrm{e}}r\theta d\epsilon$
,
where
$d\theta_{\xi}$is the
Haar
measure on
$S^{n-1}$
.
Then,
$\hat{\phi}_{r}(\xi\backslash )$is conpactly supported
in
$C_{1}\tau\leqq|\xi|\leqq C_{2}\mathrm{r}$
(because
$\hat{\psi}(\xi)$
is compactly
supported.)
and
any
half line
starting from
the
origin
intersects
$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\hat{\phi}_{f}(\xi)$.
$( \int|\int\psi_{\mathrm{I}}\epsilon \mathrm{I}^{r},‘*f(x)d\theta_{\zeta}|^{\mathrm{P}}dx)\mathrm{p}\mathrm{I}\leqq(\int(\int|\emptyset_{\mathrm{I}}\epsilon|,r_{(}f(X)|p*d_{X})\frac{1}{\prime}d\theta_{\zeta})^{q}$
$\leqq C\int(\int|\psi_{1\epsilon 1,r}\mathrm{C}*f(_{X})|pdx)^{\mathrm{I}}’ d\theta\zeta$
The
first inequality
is
due
to the
continuous version
of the
Minkowskii
inequality
and
the
second
one is due to the
H\"older
inequality.
Af-ter
integrating
both hand
sides of
this inequality with
respect
to
$|\xi|^{sq-1}d|\xi|$
,
we can see
that the usual
Besov
norm can
be
bounded
from above by the
Besov
norm
via
the
wavelet transform.
Necessity:
Let,
$\hat{\sigma}_{r}(_{\mathcal{T}})^{2}=(2\pi)^{n}C_{\psi}^{-1}\int\hat{\psi}(\frac{\Gamma\xi}{f}\mathcal{T})^{2}d\theta_{\zeta}$
.
(See
the
proof
of
Proposition 1.) Then,
$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\hat{\sigma}_{r}(\mathcal{T})$is
located
in
$C_{1^{f}}\leqq|\tau|\leqq C_{2}r$
and
$\int.\hat{\sigma}_{r}(\mathcal{T})^{2_{\frac{d\mathrm{r}}{r}=1}}$
,
that is,
By
using
this
continuous
decomposition
of
the unity,
$|| \psi_{1}\epsilon|,r‘*f(x)||L(\nu)dx\leqq\int||\psi_{|\epsilon|},r‘\Gamma||_{L_{1}}(dx)||*\sigma\cdot f*\sigma|r|_{L_{\nu(d}}x)^{\frac{d\mathrm{r}}{f}}(1)$
Because the Fourier transform of
$\psi_{|\zeta|},r‘*\sigma_{r}$
is
not equal
to
$0$
only
when
$C_{3}|\xi|\leqq f\leqq C_{4}|\xi|$
, and because the
$L_{1}$
norm
of
$\psi_{|\xi|},r_{(}$
and
$\sigma_{r}$is bounded,
(1)
$\leqq C\int_{c_{3}|}^{c_{4}|\zeta|}\xi|)^{\frac{dr}{r}}||f*\sigma r||_{L(}\prime dx$
$=C \int_{C_{3}}^{c_{4}}||f*\sigma \mathrm{t}\mathrm{I}\epsilon|(x)||\iota P(dx)^{\frac{dt}{t}}$
The last term above
is independent of the
rotation
$d\theta_{\xi}$,
and
more-over,
$|||\xi|^{s}||f*\sigma t|\epsilon|||L_{\mathrm{p}}(dx)||_{L_{q}(\mathrm{f}\mathrm{f}\mathrm{i}}d)=t-s|||\xi|^{s}||f*\sigma 1\zeta|||_{L_{\mathrm{p}}(d}x)||_{L_{q(}}lk)$
,
we
can conclude that the Besov
norm
via the wavelet transform is
bounded from
above by
the
usual
Besov.
norm.
(Theorem 2) q.e.d.
Theorem 3.
$f\in\dot{F}_{\mathrm{p},q}^{s}(\mathbb{R}^{n})(s>0,1\leqq p<\infty, 1\leqq q\leqq\infty.)c$
an
be
chara
$\mathrm{c}$teriz
$ed$
by the
following:
$||||| \xi|^{S}+\frac{\mathfrak{n}}{2}|W\psi f(X, \xi)|||_{\mathrm{I}q}L(‘ d\phi)||_{L(x}d)\mathrm{p}<\infty$
.
Proof.
Sufflciency:
As
in Theorem.2, let
$\phi_{|\zeta|}=\int\psi\}\mathrm{t}1^{r},‘ d\theta\epsilon$
.
Then,
$|| \xi|^{s}(\phi 1\xi|^{*}f(x))|^{q}=|\int(|\psi_{1\xi}|,r_{\zeta}*f(x)||\xi|^{s})d\theta\epsilon|^{q}$
$\leqq C\int|(\psi_{|\xi|,\Gamma}‘*f(X))|\xi|^{S}|^{q}d\theta\zeta$
.
Hence,
we can
easily
see that the usual biebel-Lizorkin
norm
is
bounded from
above by the
norm via
the
wavelet transform.
Necessity:
This
part needs very
deep
results which are
continuous
versions of the work of
C.Fefferman-E.M.
$\mathrm{S}\mathrm{t}\mathrm{e}\mathrm{i}\mathrm{n}[2]$and
$\mathrm{H}.\mathrm{b}\mathrm{e}\mathrm{b}\mathrm{e}\mathrm{l}[5]$.
First, we state the
results
without proof.( The
proof
is carried out
in
the
same
way
as
in
the discrete
case.
$\mathrm{S}\mathrm{e}\mathrm{e}\mathrm{l}21[51\cdot)$Claim
1.(Continuous
version
of [2]) Let
$f(x,y)$
be
a
function
of
$(x,y)\in \mathbb{R}_{x}^{n}\cross \mathbb{R}_{y}^{n}$
,
and
$Mf(x, y)$
be a maximal function of
$f(x,y)$
with respect to
$x$
.
Then,
$||( \int|Mf(x,y)|q\frac{dy}{|y|^{n}})\frac{1}{\mathrm{r}}||(\int|f(x,y)|q\frac{dy}{|y|^{n}})\frac{1}{q}||_{L}’(dx)$
,
where
$1<p<\infty,$
$1<q\leqq\infty$
.
Claim
$2.$
(
$\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}\mathrm{s}$version
of
maximal inequalities in
[5])
Let p,q,r be
$0<p<\infty,$
$0<q\leqq\infty$
,
and
$0<f< \min(p, q)$
.
Let
$\hat{f}(\xi,y)$
be the
Fourier partial transform of
$f(x,y)$
with respect to
$x$
, and
$\Omega_{|y|}$be
a set including the
support
of
$\hat{f}(\xi, y)$
with respect to
$\xi$
.
Let the diameter
$d_{|y|}$
of
$\Omega_{|y|}$be a continuous
function
of
$|y|$
, and
$d_{|y|}>0$
.
Then the following inequality
holds:
$||( \int(\sup_{z\in \mathrm{R}}\frac{|f(x-z,y)|}{1+|d_{|y|}z|^{\frac{1}{r}}}\mathfrak{n}’)^{q}\frac{dy}{|y|^{\mathfrak{n}}})q1||_{L_{\mathrm{p}}(x}d)$
$\leqq C||(\int|f(_{X},y)|^{q}\frac{dy}{|y|^{n}})\tau\iota||_{L,(dx)}$
Claim
$3.$
(
$\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{l}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}s$version
of
multiplier theorem
in
[5])
Let
$0<p<\infty,$
$0<q\leqq\infty$
,
and
$\kappa>n(\frac{1}{2}+\frac{1}{\min(p,q)})$
.
Let
$\Omega_{|y|},$
$d1\nu|$
be as in
Claim.2.
Then,
the
following inequality
holds:
$||( \int((M(\cdot,y)*f(\cdot,y))(x))^{q}\frac{dy}{|y|^{n}})^{1}\mathrm{f}||_{L,(dx)}$
Claim 1
is
essential
in proving
Claim
2.
In
proving Claim
3,
we
need
Claim
2
and the
following
inequality:
$\sup_{z\in \mathbb{R}}\frac{|(M(\cdot,y)*f(\cdot,y))(_{X}-Z)|}{1+|d_{||}yz|^{\frac{\mathfrak{n}}{r}}}\hslash$
$\leqq C\sup_{z\in \mathrm{R}^{n}}\frac{|f(x-z,y)|}{1+|d_{1^{y}\mathrm{I}}z|r\mathrm{n}}\cdot||\hat{M}(d_{\mathrm{I}y}|.,y)|H_{2}^{\kappa||}$
.
(Here,
$0<r< \min(p,$
$q),$
$\kappa>\frac{n}{2}+\frac{n}{r}.$
)
As in the
proof
$\mathrm{o}\mathrm{f}^{t\downarrow \mathfrak{e}}\mathrm{n}\dot{\vee}_{\mathrm{e}}^{\backslash }\mathrm{c}\mathrm{e}$’ssary
condition of Theorem 2, we use the
continuous
decomposition of the unity:
$\int\sigma_{r}*\sigma_{r}(x)\frac{d\mathrm{r}}{f}=s(_{X})$
.
$|||||\xi|^{s}(\emptyset \mathfrak{l}\xi \mathrm{I},r‘*f(X))||L_{?(}\mathrm{T}^{d}\star)||_{L,(dx})$
$=|||| \int|\xi|^{s}(\psi_{1}\zeta|,r_{\zeta}*\sigma_{r})*(f*\sigma_{r})(X)\frac{d\mathrm{r}}{f}||_{L_{i}(_{\mathrm{T}}\#}d)||_{LP()}dx$
$\leqq\int_{C_{1}}^{C_{2}}\frac{dt}{t}|||||\xi|s(\psi_{\mathrm{I}}\xi \mathrm{I})f\mathrm{e}*\sigma_{t1\zeta|}).*(f*\sigma_{t}\mathrm{I}\epsilon|)||_{L(\phi)}\mathrm{c}d||_{L,(dx)}$
We
apply
Claim
3
to the integrand of the last
term
$\mathrm{a}\mathrm{b}\mathrm{o}\mathrm{v}\mathrm{e}\{$$d_{|\zeta|}=C|\xi|$
, and
$\psi_{|\xi|_{1}}r_{\zeta}\mathrm{t}\sigma \mathrm{I}\epsilon|(\overline{*}\tau)=\hat{\psi}(\frac{\gamma\zeta}{|\xi|}\tau)\cdot\hat{\sigma}(\frac{\tau}{t|\xi|})$
.
Thus,
$\sup_{\xi}||\hat{\psi}(\frac{\Gamma\zeta}{|\xi|}C|\xi|\cdot \mathcal{T})\cdot\hat{\sigma}(\frac{C|\xi|\cdot\tau}{t|\xi|})|H_{2}\kappa||$
is
bounded
from above. Therefore, the
Triebel-Lizorkin
norm
via the
wavelet transform is bounded from above
by the
usual
norm.
(Theorem
$3$
)
$\mathrm{q}.\mathrm{e}.\mathrm{d}$.
Remark?.
Theorem
3. can be
extended
to
the
case
when
$0<$
$p<\infty$
.
The
case when
$0<q\leq\infty$
remains to be proved. Also in
Theorem 2, the case when
$0<p\leqq\infty,$
$0<q\angle=\infty$
remains to
be
proved.
Such
troubles
occur because we used
th\"e H\"older
inequality
and
the
Minkowskii
inequality
in
$\mathrm{t}\acute{\mathrm{h}}\mathrm{e}$$\underline{d.}$ $ik\text{ノ}\ovalbox{\tt\small REJECT}/_{\overline{fl}}\{_{\overline{\mathrm{F}}}Jt7j\kappa \mathscr{L}\wedge\wedge\zeta_{f}^{o},_{J}’(l_{=}^{)\angle J<\mathit{1}})$
$t*/\langle\sigma)\dot{7}_{\mathrm{X}^{-}\text{フ}}.$
.
$b_{\text{ノ^{}|-\cdot j}<\ ^{\cap}\zeta}..i..\eta\beta_{k^{t}}];,\not\simeq_{\backslash }\backslash$$\underline{\mathrm{T}\mathrm{k}_{\mathcal{E}Ore}\nu\wedge\gamma.}$
$a$
1
$S_{J}^{\cdot}2$
)
$e\nearrow\zeta_{/,d^{-}}^{\rho}(\theta_{=}^{\angle}$
.
$</)(_{-\wedge}^{-\lrcorner}.\backslash \mathrm{i}[-(\mathrm{c}$$\prime \mathrm{r}\dot{i}_{\underline{t\prime}\ovalbox{\tt\small REJECT}_{7}\nearrow \mathcal{L}}cfa\backslash \cdot\cdot rF(7$
$\alpha$
$||$
a
$\mathrm{f}^{lS}$)
$||_{\dot{L}_{z}s\cdot)}(d\angle= c||t^{\zeta}t)||_{\angle_{\lambda}}\cdot‘ u()$
$p\backslash ^{1}\cdot\nearrow’’\}_{J}^{\backslash }$
.
$\lrcorner\tilde{L}^{-_{7}}\lrcorner\theta$$-_{\mathrm{E}Gy7}\overline{p}_{A}\sim’\triangleleft\overline{/0}^{\prime\xi}’\dagger$ $\mathrm{r}7]\eta\overline{\nearrow \mathit{0}}\prime irl-- f\acute{\wedge}/z\prime 7$ $\mathrm{f}r\nu_{\int^{F}}Sj\mathrm{t}|.\mathit{0}’\sim \mathit{1}i‘)_{\mathrm{c}}$
$X^{g}{}_{\text{ノ}\mathrm{P}}\wedge^{\vee}| \ell_{\mathit{0}}^{\triangleright}1L\emptyset\pi_{R^{\text{ノ}}^{}\prime\neq}\overline{y}\mathit{4}_{\overline{\mathrm{f}}}\int r\urcorner\ \text{ノ}\mathrm{k}\sigma\}\tau^{\text{ノ}}\mathrm{g}_{\nearrow}x\text{ノ_{}\overline{\eta}}lj_{\mathrm{A}}\mathrm{x}..\backslash S_{C}\mathrm{h}_{\mu V}\triangleleft’\hslash.37:|’\triangleleft \mathrm{L}\underline{\not\in_{\backslash }}$
$\epsilon_{\nu a}:X[\wedge\backslash 7--\mathrm{c}\yen’.\overline{T}7\cdot$
$7^{r_{\overline{\phi}}}.\text{ノ}\mathrm{h}\dot{3}$.
$\mathrm{K}\mathrm{t}z,\zeta_{J}$
.
$z\cdot\prime \mathrm{f}$)
$=/ \int^{\backslash }\int e^{-_{\mathrm{A}}}’\prime S_{i}?$
)
$\gamma,(t)\overline{rz,\zeta^{(SdJs}}?\mathrm{t}\{arrow;[]\cdot?_{a\cdot x\mathrm{I}})dt$
$\downarrow\sim 2^{-\uparrow}\sim\vee\iota‘\cdot(’\zeta \mathrm{c}\text{ノ}\dot{\rho}\mathrm{L}A\dot{\mathrm{x}}f\tau\wedge\cdot‘ r\ovalbox{\tt\small REJECT}_{\triangleleft}’’\sim(1\backslash$
$\int\int|k^{(\mathcal{Z}},$
$\zeta_{\text{ノ^{}\mathfrak{t}}}.\mathrm{c},\mathrm{f})|\ovalbox{\tt\small REJECT} Aj<\mathrm{M}$$\int\int|\kappa(\mathcal{Z},\cdot I_{J}^{\prime;J}\chi:,|AzA\zeta<\mathrm{M}$
$fl\backslash \cdot/\prime 1^{\cdot}|)_{-}^{Z_{\angle}}-\approx-- t\mathrm{S}_{/\mathrm{J}\backslash }’.-7\cdot\backslash \backslash \backslash \cdot\tau..\backslash$
$\gamma_{x},;^{c\chi_{J^{=}/}}\grave{J}/^{\frac{n}{L}}\cdot r’/\mathrm{f}/_{f}r$
(t-Z)
$)$
$\eta-\backslash t7^{\cdot}$
.
%
6
$\mathrm{P}\epsilon- \mathrm{f}|A1\mathrm{t}|\ell)A\zeta$
.
$\overline{6},\mathrm{n}^{o_{7}}\mathrm{n}\prime\prime \mathrm{J}|$Triebe
$|.\kappa_{z\cdot\rho_{0}7(_{-}^{\cap}}^{\backslash }J|--r^{\mathrm{s}}’\tau \mathit{1}^{(-\epsilon\cdot\circ}<s\cdot<\cdot\Leftrightarrow\cdot\circ\nearrow\backslash -$
$/–.r<<\infty \text{
ノ
}/-\leq t\angle=\infty)$
$\mathrm{t}\partial/\not\subset_{*0}^{g}$.
$(_{t_{\backslash }^{- 1-}}\sim..\{, |_{\overline{\mu}}^{-}|--1)_{a}$
$\# C-$
$\mathrm{F}_{\ell,}s^{\backslash }(=\lrcorner\epsilon)\cap\forall l\mathrm{i}$$\vee^{4}\cdot/_{\sim}\mathrm{T}^{\cdot}$
.
$\prime L\triangleleft^{\wedge}\mathrm{w}\mathrm{t}\dot{2}^{A}.\backslash ’\cdot k4,\cdot$
$||(\ell_{\rho}.\kappa f_{lX}J||_{\perp_{\gamma}u_{)}^{+}(}|||\xi|^{S}|||\xi|^{\frac{\prime 1}{\mathrm{J}_{-}}}W+\mathrm{f}(X,t)||_{\mathrm{L}x}\ell\sqrt)||\iota\uparrow^{(\frac{d\}}{/f\mathit{1}}}\sim\{^{\epsilon}./\cdot T^{-\underline{\vee}}/|-<\mathit{0}^{<+\infty}\nearrow l\mathrm{f}/\underline{\mathit{2}}l$
$[_{\overline{\backslash }}f_{\overline{\backslash }}.$
.
$( \mathrm{t} \oint_{p}(\chi)\zeta-.\chi_{(}/\mathcal{R}’\iota)_{\text{ノ}}$
$\oint_{p}^{\wedge}lf\text{ノ}|i$
‘\‘A
$J’\nu \mathrm{L}\gamma^{d}\nearrow\prime m$
$[$
.
$\xi c- R^{\sim}|$
$\mathrm{I}_{\lrcorner}\mathrm{k}|\angle\underline{-^{2f}\cdot}\mathcal{E}f\mathfrak{F}_{7}\prime s$3
$r-\backslash$$\downarrow>\mathrm{t}\eta\gamma_{l\mathrm{A})}^{\theta,}r\lambda cx_{ap\downarrow f,}^{\lrcorner}\nearrow^{-}\nearrow\zeta\emptyset \mathrm{f}_{/*}/\nearrow^{-}\overline{i}\sigma$
&
$’$
\neg
$\zeta \text{ノ}\overline{7}\backslash ((_{\backslash }^{-}$4
$\eta 7\cdot$
.
A 6
$\sigma$
$\mathrm{P}\ll’ \mathrm{n}\iota\alpha r|_{\prec}S^{-}$ $\mathrm{p}_{e}\mathrm{f}\cdot i4*\cdot\dot{\mathrm{c}}\mathrm{i}o_{l}\alpha d$
.
$|P$
$\mathcal{T}te\theta re\nu[]\sim(\mathrm{J}$.
$l\vee\backslash I|/\text{ノ^{}-}\prime l\backslash q7^{g}\not\subset$ $-\tau\backslash \sim \mathrm{k}t-\backslash \perp’\triangleleft’\backslash ‘|\text{ノ}\grave{\beta}^{j}A7^{\cdot}\cdot\hslash\prime 6$.
$Bes_{\theta\prime/_{\check{\prime}}^{\dot{d}}}\ovalbox{\tt\small REJECT} \mathfrak{l}_{\backslash }-_{7}‘ i7*\nearrow|\overline{\mathrm{A}}^{-}<\overline{\dot{\triangleright}}\mathrm{J}$’
$/|_{*_{-l\backslash }}^{y}.-$
A
,
$\#\delta$
,
$\ovalbox{\tt\small REJECT}$
$a(x, \zeta)\epsilon’\zeta_{/_{J}\Delta}^{l\cdot[]}.$
.
$(.\phi‘\sim>\rho\supset\prime \mathit{0}\backslash \cdot$.
$’$
$|p_{f}^{y}‘ d^{p}‘ aLCx,\mathit{5}y|\angle=C_{d,\beta}\langle_{l+l\mathrm{f})}|.|_{\frac{\xi}{/\zeta \mathit{1}}}-’\not\in_{\mathrm{I}^{\dagger}}f\}s|‘\cap \text{ノ}--\lrcorner|<8$
,
$\xi\nu^{\backslash }.\mathrm{f}\eta\prime_{f_{-}- 7}\mathrm{t}-74$
.
$\mathcal{E},$
$<\epsilon.\partial<\mathcal{E}\gamma_{d}-\}$
.
$\downarrow i.$.
$\mathrm{f}^{c-}\Gamma_{\Gamma’ 7}-(s^{\backslash }<t)s’’\cap r_{;,\mathrm{t}}^{5^{\backslash }}-(.-\cup\cdot\epsilon)\partial$
$[_{\backslash }^{\wedge}\gamma^{\prime\tau}(7$ $\sim^{1’}-\otimes\dot{q}\mathrm{c}p\backslash \cdot\cdot \mathrm{r}_{\delta}’\chi_{x-}l^{-}\mathrm{C}\iota$
$||a\not\simeq||,\cdot\cdot\tau-F_{\rho}^{s_{9\mathrm{t}}}(_{\overline{\overline{\mu}}\epsilon)}||$
a
$f||_{F^{\sigma,\triangleright},} \frac{\prime}{7},.\angle C(=||_{\# 1}|S||,\backslash \rho_{\gamma\prime \mathrm{t}\prime}(^{\cap}\overline{\mu}\mathcal{E}_{\delta^{+||\mathrm{f}}})f_{F}-)q$
’
弓
“/1
p
$\sqrt$烹裏
(
$t-\backslash$クエ
$-$
グム,
$/-\backslash \hat{z}i$稜
$\mathfrak{x}$周
$\iota$
’ て、
$||A_{-f}||\Gamma_{\rho,}^{\sigma\cap}-\theta(\overline{\mu}\epsilon, )$
1
$\tau\overline{\nearrow}A\mathrm{R}/’/\neq_{g^{-}/}\mathrm{z}\prime \mathrm{k}74$
$\vee- C\iota$
/3
$7\wedge\cdot$.
[
$S\mathrm{J}$ $\epsilon$ノ
F.5
A
$[\mathrm{d}.$.
I
$d$
.
$f||_{t^{arrow}-)t,7}s$
.
$(\ell.-\neg\lrcorner g,\leqq C_{/}(||f||s\cdot\underline{.-}‘’+||\dagger||,\cdot.)f\overline{\gamma}\prime t^{(}-\epsilon)\mathfrak{k}^{\theta’}f7$オ A
$\ovalbox{\tt\small REJECT}_{p\backslash \mathrm{A}}$ろ.
$||$
A
$\#||/-\gamma \mathrm{J}^{\backslash },7^{-\wedge}\text{ノ}\angle=C_{x^{||\not\simeq.,s_{\text{ノ}}}}||/-f’ 7\backslash /$
$1^{\cdot}\phi$
,
1
$\mathrm{t}^{)}\mathrm{J}\nu_{)}*_{t}^{x}\text{ノ}2\ovalbox{\tt\small REJECT}*-\sigma$)
$\theta^{\overline{t}}\eta 7\cdot\cdot h4p$
$\underline{p_{er}n\iota^{\ell}\lambda \mathrm{k}t..}$
$\tau/_{1epre\sim}n\mathrm{f}$
.
$\ell’7$
$\beta e\sigma_{\mathit{0}’}’/’(_{\mathcal{B}}\check{\mathrm{A}}\ovalbox{\tt\small REJECT}\dot{\Gamma}q|\downarrow-\backslash \sim\iota\iota 74’\acute{\mathrm{x}}$く
$|\tilde{p}_{\text{ノ}}\neg \mathrm{R}$}
$\wedge\backslash \text{皮}*$る
,.
$\Gamma_{n}^{\mathrm{A}}\mathrm{J}\backslash \cdot\#\vee d\pi\overline{e}.$
tfl
$l\xi-Ar\infty$
)
$\eta b^{\gamma_{\overline{d}}}\dot{)}.7.$
.
$\mathrm{A}/^{1}z_{-}^{\vee}\dot{\rho}\dot{\tau}\hslash 7(X-\eta a\angle\iota)E*\rho 6\overline{r}rl\sqrt’\cup(|\gamma_{-/}-\dot{\mathrm{a}}_{-}\ovalbox{\tt\small REJECT}|\not\in_{\dot{\mathrm{A}}}\text{ノ}\dot{\mathrm{f}}_{\vee}^{\backslash }\wedge\vee h-,$
$\mathrm{L}$
.
1
上土る
,
$[^{O_{J}}$
$l_{J}$
$:\wedge^{-\mathit{9}}\mathrm{T}^{e}-\prime y,- \mathcal{T}\iota$ $a)\eta$
$’,x(\triangleleft \text{ノ}\circ\dagger^{\mathrm{A}_{1}^{\wedge}}\prime \mathit{1}\overline{\prime,}\mathfrak{l}\prime r;\backslash \varpi \mathrm{t}\dot{\tau}\dot{\not\supset}$
$\mathrm{r}\gamma_{\mathrm{J}}r/\mathit{0}]\ell-\sim q<..\tau\ovalbox{\tt\small REJECT} \mathrm{Z}^{-}{}^{\mathrm{t}}C^{\ulcorner]_{C}}\sigma’-\rho$
$\frac{1}{\prime,\check{\mathrm{b}}}\ovalbox{\tt\small REJECT},\dot{7}\hslash A’[j]\int\backslash \prime t_{\#}.\infty(7t_{\overline{\mathrm{f}}}\mathrm{r}\nearrow_{lk\acute{\mathrm{T}}}\backslash \mathrm{c}\pi 7fi\backslash \cdot t_{\wedge}-Al^{Z_{47}^{\mathrm{g}}}q7\frac{\nearrow}{/-}\ulcorner_{47}/\underline{\cdot\neq}$
$l_{\sim}^{-},$
Acknowledgements.
The auther would like to
express
his
sincere
gratitude
to Prof. H. Komatsu, Prof. K. Kataoka, Prof. K. Asada
and
Dr.
S. Tanabe
for
many valuable suggestions and encouragement.
He would like to express his
sincere
gratitude also to
Mr.
S.
Yamazaki
for
$\mathrm{h}.\mathrm{e}\mathrm{l}\mathrm{p}\mathrm{f}\mathrm{u}\mathrm{l}$discussions.
$t\mathit{7}^{-}\backslash$ $\Gamma_{7}$
司
$\eta$ $|^{-}\sim^{{}^{t}d}$A.
補
$\mathrm{t}$-//|
放タオ才
\nu
A
」
$\tau’\tilde{l}^{\vee}\acute{\mathrm{r}}\prime\prime- 7$不幾
)4
$\mathrm{L}$
$+\check{L}7\mathcal{T}^{\wedge}\backslash \dagger$
,
r–
$\backslash \backslash$ $\succeq 1_{\backslash }^{-}$$\mathrm{v}:_{\mathrm{A}}^{\sigma}<,n’.\cdot\dot{u}^{\acute{\tau}}\triangleleft^{\mathrm{a}_{\mathrm{X}}}l\mathrm{C}$
$\mathrm{f}7v$
REFERENCES
1.
Daubechies
I.,
Ten Lectures on
Wavelets,
SIAM
CBMS-61,
1992.
2.
Fefferman
$\mathrm{C}_{1}.\mathrm{S}\mathrm{t}\mathrm{e}\mathrm{i}\mathrm{n}\Sigma l$M., Some
maximal inequalities,
Amer.J.Math.
93 (1971),
107-115.
3.
H\"ormander L.,
$p_{our}$
;
in
$\mathrm{t}egrdo_{P}$
erators.I,
Acta.Math. 127
(1971),
79-183.
4. Peetre
J.,
New Thoughts on Besov Spaces, Duke Univ.,Durham, 1976.
5. Triebel
H., Theory
of
Function
Spaces,
Birkh\"auser, 1983.
$\zeta,$ $\}\sqrt$
a
$\mathrm{r}\mathrm{e}_{\nu\backslash \mathrm{Z}}i\mathrm{R}_{)}.U‘ l\wedge\prime \mathrm{e}4tX^{\prime\iota}-i\theta’\triangleright|\vee\Phi$ $\ovalbox{\tt\small REJECT} AAp\mathrm{c}\wedge\cdot\cdot\nu \mathrm{a}l$
h
$*,:-”\dot{\mathrm{A}}\mathrm{t}$
’
$\mathrm{f}.\iota clA-r^{\mu}7l4\prime t^{J}4\mathrm{m}:A\dot{4}r^{A}\wedge\cdot 4\cdot[]\prime \mathrm{t}\mathrm{L}$
$d_{\mathrm{t}\sim}$
‘
ne
$e\angle-\cdot"\triangleright \text{ノ}1\downarrow\sim W_{a\nu}\mathrm{c}l\mathrm{e}\mathrm{t}\iota,\cdot$J.
$t4\cdot\alpha^{1}\iota_{\mathrm{L}}\iota es\backslash J$
A.
$G^{\backslash }\mathrm{I}’\circ s..\Gamma 4\sim e\iota_{l}\sim ml‘ \mathrm{c}\infty 4Ik$.
$\tau_{c}\iota_{\ h\vee}|.\mathrm{e}_{\mathrm{C}}\mathrm{t}_{t\mathrm{A}}.‘\sim,$
$eds_{\text{
.
ノ}}$
$s_{\dot{\gamma}\downarrow_{1}^{\prime\sim_{s}}} \cdot\not\subset \mathrm{r}- V_{e\mathrm{r}}\int_{C\iota}$
?
ノ
$\theta er/l.\ .$
(/??7),
$\wedge’.\mathit{3}‘ f^{-}92r\zeta$
.
7.
爪
4
$‘ l_{\wedge}$$/\langle.J$ $\sqrt.\cdot\kappa t_{\ddot{\mathrm{x}}_{-}}^{J}.\cdot\cdot.\tau\Phi$
し
$\not\in_{\wedge}b\neq_{R^{-}}^{x}/\nearrow^{\sim}9\angle_{2}.\eta_{\sqrt}\nearrow p_{J}\kappa$’
ノ
$\frac{\iota}{\wedge l}\mathrm{x}.k\not\in\pi \mathcal{X}^{\mathrm{J}}\overline{\mathit{1}}’\# f\mathrm{a}1^{\mathit{3}}3^{-l\sim}d\backslash \dot{\prime}$’
$\gamma\nearrow f(./\text{
〃
}\mathit{1}^{\cdot})\text{
ノ
}///-/^{z/}$
$\mathrm{f}p_{\dot{\alpha}\mathrm{i}_{V\ddot{a}}\{_{d\iota}}.V^{\backslash }\iota A$
L.,
$\ _{A}A$
)
$d\star" m\dot{\sim}/\not\simeq_{u\supset\iota}4-\tau \text{ノ}lA^{\cdot}\lrcorner 4\mathit{4}$
$\sim\gamma^{\ ae\mathrm{c}_{\text{ノ}}}$
Ze
$iCs\mathrm{C}\Lambda t_{t}.fC\mathrm{f}_{\dot{\grave{\alpha}}}r$
$A\gamma\iota \mathrm{a}_{\nearrow}/.S\dot{|}s\iota\iota’|d.\prime \mathrm{A}_{k}\epsilon A\prime 1i\nu en\mathrm{J}u-fe,\iota$
2
$(/\cdot?\mathrm{f}f)$
ノ
13S
$-$
ユタ
1
7.
$S^{\neg}$.
$’\forall \mathit{0}\Gamma \mathrm{i}$
