A restriction theorem on the Folland-Stein spaces

A RESTRICTION THEOREM ON THE FOLLAND-STEIN SPACES

YONNE MI KIM

1. The Heisenberg group

DEFINITION. The heisenberg group Hn is the lie group of real dimension²ⁿ

+

en,

(t, z)(t', z')

=

+

+

+

Its Lie algebra is generated by the left invariant vector fields X^j,Yj,T, j

=

It is easy to verify the following commutation relations:

and all other brackets are zero. On Hⁿ there are a family of dilations lr(t, z) that gives rise to a one parameter group of automorphisms on Hⁿ given by lr(t,z) = (r²t,rz).

The homogeneous dimension of Hⁿ is Q

=

+

Received March 29, 1991. Revised September 4, 1992.

This work was supportedby KNSF, 1990 .

25

26 Yonne Mi Kim

2.The sublaplacian

Let n

£,

= -L(X] +

;=1

The operator£, is homogeneous of degree 2, and r,t = £'.

In the Euclidean space a fundamental solution to D.is given by E = r(n/2)

IxI

21rn / 2 (n - 2) - Analogously the fundamental solution to£, is given by

22-2n1rn+1 -I.

=

⁺

'I' = r(n/2)2 .

(See Folland[3]). Thus £, is locally solvable. The convolution of two functions f and g in Hⁿ is defined by

3. The Bessel potential and the spaces _S~

The definition of the Besse1 potential is from Folland[3]. The prin- cipal tool is the diffusion semigroup Ht generated by -£'.

There is a unique semigroup {Ht, 0 <t < oo} of linear opertors on L1

+

(1) Hd

= ^I *

=

IHn

(2) Ifu E

Co,

(3) Ht is self adjoint, i.e.,HtILP

=

+

=

(4)1 _~ 0 ===? Hd _~ 0, Ht1= 1. Now ifwe extend h(x,t) to be 0 fort ~0, then h is the fundamental solution to £,

+ tt.

Now inspired from the classical case, we define the Bessel potential

If f ELP, 1 <p< 00, then (1+£)-0:/2f= f* la.

So define S~ to be the immage ofLPunder the operator(I

+

We have the following properties of 10:. (See [3])

(1) la. is defined for all x =1= 0and even for x

=

is Coo away from

o.

(2) As x -+ 0,

Ilo:(x)1

=

=

I~I)

⁼

(3) As x -+ 00, Ila.(x)1 = O(lxl-^N) for all N. Hence, la. E L¹ for all a> O.

The spaces A~,q(Hn)is defined to be the space of those functions in LP(Hⁿ⁾ for which the following quantity is finite.

We prove the theorem when the restriction is one of the hyperplanes ofRt X R²ⁿ- 1 ofHⁿ and the case a ~ 1 since the casea :s; 1 is proved by Mekias [1]. First we want to introduce a theorem of Stein which has given the motivation for the study.

28 Yonne Mi.Kim

4. Restriction and Extention theorem for the spaces~(Rn)

4.1. THEOREM (STEIN[7]). (A). The restriction mapR :~(Rn)_

A~'P(Rm) is a bounded linear map aslong as f3 = a - (n - m)/p >

°

and 1 <p < ^00.

(B). Conversely, there existsan extention mapE

R(E(g)) = 9 Vg E A~,p(Rm), f3 > 0, 1 <p< ^00.

Now we introduce our main theorem.

THEOREM (A'). Let 1 :::; a <2+~. The restriction map R' SP. a(HR) _ AP'P (R xa-1. t Hn-1 ),

"

whereRt ^X R²ⁿ- 1 is one of the b.yperplanes{t,Xl, ••• ,Xi = 0, ...,X2n} is a bounded map.

Proof. The proof is divided into two parts. At first, we will show that

But this follows immediately from the inclusion relation S~ C _S~, for a> a' (Folland[3]) once we show that 9= R(J) belongs toV(Rⁿ)

for f E S~(Hn), a < 1. For this we will adopt a proof from Stein[8]

and Mekias[l].

Mter that we willtry to show that 9 E ~:'1.(Rt ^X Hn-l).

"

Let's prove the first claim when a < 1. Let

f

We will restrict ourselves to the case n=l just for simplicity. Put

g(t,y)

= r

+

JR3

To prove that 9EV(R²) it suffices to show that

J

is bounded for all hELq(R²⁾ where;

+ ;

I!gIILJ'

Ig(t,y)1 :::; j 14>(s, x', y')\IJa(t - s

+

Let h(t, y) E Lq(R2). Our claim is to show that

IlgllLP =

f

JR2

f

f

JR2 JR2 JR3

Ja(t - s

+

f

f

JR3 JR2

Let's study the inner integral

f

+

JR2

Fix x' and let

Tz;,h(s,y') =

f

JR2

Ifwe could show that

(1) SUps,y' jIJa(t-S+2_YX',-X',y-y')ldtdy

30 Yonne Mi Kim

and

(2) SUPt,y

J

⁺

are both bounded, independently of(t, y) and (s, y) respectively, then we could use Young's Inequality to conclude

Consider(1) first

SUps,y'

J

⁺

t - s

+

(3) Int=

f

lR2

The estimate for this integral will be made for two separate cases (a)

Ix'l

(lxl

1

ITI¹/2+I:I;'I+I"I<1 ITI¹/2+I:I;'1+1,,1>1

= 11+12 •

Let's look at the main part 11 first. (TheTe is a fast decrease forh)

I1(x') ⁼

f

~

^(I

¹

^{+ Ix'l +}

ITI¹/2+I:I;'I+I"I<1

~

^(I

¹

+

+

\TI^1/2+1,,1<1,I:I;'1<1

~ (i)

+

(i)

= f

+ Ix'l +

:::; f _Ix'l'~-Q

J\TjI/2+1'I1<lx'l

:::; Clx'lo:- Q

f

J 1TI<lx'1 2,I'll<lx'l

= Clx'l':r-

= Clx'ICt- 1

(ii)

= f

+ Ix'l +

:::; C

f ^{f (I}

^I

^{+ Ix'l +}

j=OJITI^1/2+1'I1<2i Ix'l

~ cfl

j=O ITI^1/2+1'I1<2i Ix'l

00

:::; CL(2jlx,\)o:-Q

f

j=O J ITI«2i Ix'1)2,1'I1«2i Ix'l)

00

:::; CL(2jlx'\)Ct-Q(2jlx'\)3

j=O

00

= Clx'I-l+O:

)O:-1

j=O

SinceQ < 1, we see that L:~o(2j)O:-l:::; ^00.

So fl(x') :::;

Clx'I-l+Ct.

Now look at f²(x')

=

Let N be any arbitrarily large numberwhich is chosen as N > 4-^Q.

32

Then

YonneMi Kim

1₂(x')

~

f (Ir1

+ Ix'l + 171I)-N

J 1r I^1/2+lz'I+I'lI>1

~

f

+

+

Jlrll/2+lz'I+I'lI>1,lrll/2+1'l1<lz'l .

+ C

f (/rI

+ Ix'l +

Jlrll/2+lz'I+I'lI>1,lrll/2+1'l1>lz/l

~ ClxTlr-4 +

.f

+

+

Jlrll/2+1'lI>lz/l

By subdividing the range

Ir1

+ 1711> Ix'l

lx'I

Ir

+ Ix'l + 1711

+Ilx'l

and following similar argument as in the previous case(ii) the last in-

tegr~ can be shown to be bounded by C'lx'la-4. So we have showd that

f

+

~ ^Clx'l-Ha

~

JR2 .

Now let's study (2) and estimate the integral, i.e., show

SUPt"

JR2 f

+

~ ^Clx'I-Ha

But by the change of variables,

f

f

h2 h2

So exactly same computation as before yields

Hence, applying Young's inequalitywe see that

From this we can conclude

f

~ f 1~(s,x',y')IITz,h(s,y')ldsdx'dy'

JR2

~ 1I~lIp(J

~ 11~lIp(l°O

- 0 0

JR2

~ 11~lIpllhllf(f Ix'I<-l+a)fdx,)t

if

J:O Ix'I<-l+a)fdx'

+

*.

Next we will estimate the case (b). For this we goback and adopt the estimateof12(since

Ix'l

(-N' +3)q<-1 Following same steps there it can be shown that

Int ~ Clx'I-^N'+3 <

Clx'I-;

Inthis case we can alsoshow that

We have finally showed that 9EV(R²).

From now on we will show that9 E _A~·~.1.(R²),i.e., we want to show

p

that

ff

⁺

lul

ap

u

34 YonneMi Kim

First note that when

lul

f f

J

lul2+

$ C

f I 12~a

=

IIgll~ ^.f

^{.I 1}

and

1

So it remains to show that

1 ^f

⁺

lul<l

J

lul2+

Setu

=

=

=

written as .

get, y)=

f

+

JHl

which can be expressed asg(t,y)

=

gz,(t,y)=

f

JR2

Now look at the quantity

Igz,(t

+

+

+

+

+

$ / 14>(s, x', y')IIJa(t

+

+

+

+

+

+

+

- 2Ja(T - s

+

Let

(4)

KZ',t,y(S, '11', T, z)

=

+

+

+

+

+

+

+

+

+

+

Soifwe could show that

sUPr,z

f

JR2

(5) SUps,y'

JR2 f

are both dominated by a constant C(x't, '11), then we could conclude IIgz,(t

+

+

+

+

+

*)llp

:5 C(x', t,y)II~(

*,

*

and thus by the generalized Young's inequality Ilg(t

+

+

+

+

+

:5 [/O(x',

t,y)II~(*,x',

On making the change of variables

T - ^S

+

expression (5) becomes

f

+

+

+

+

+

+

JR2

- 2JO/(T, -x', z)ldTdz

On the other hand, by the changeof variables T - ^S

+

y'

36

expression (4) becomes

YonneMiKiln

f

+

+

+ y') +

+

+ y') JR2

- 2Ja(s,-x',y')ldsdy' .

Since they are equivalent, let's estimate (5)

f

= f

+

+

+

JR2 JR2

+

+

+

Setting u = (t,O,y), v = (T,-X',Z), the integrand becomes IJa(uv)

+

$lulsuPpE[O,l]IVJa«pu)v) - VJa«pu-1)v)1

= lu IsuppE[O,l] IVJa(p2t

+

+

+

= lulsuPPE[O,l]

I 1

~

⁺

⁺

+

I

=

1

⁺

Ja(T

+

+

+

$lulsUPPE[O,l]I(2p2t

+

1

^v

⁺

⁺

- 2sp2t - 4spyx', -x', z

+

From now on, we willfocus on

Ix'l <

Ix'l

rapid decrease ofJa and hence no problems at 00.

Let A =

I

+

+

When

IW/

+ Iyl

Ix'l,

e.g., say

IW/

+ Iyl

Ix'l,

A =

IC2p2

+

CCIW/2+ Iyl)

Clul,

and the quantity

IT

Ix'l

Iz

IT1

+ Ix'l + Izl,

and so

{ IKz',t"Cs,y',T,Z)ldTdz

JR2

~ ^CCIW/2

Iyl? { CITI

2

Ix'l

JR2

~ CCIW/2

lyl)2 { CITI

2

Ix'l

J

+

CCIW/2

lyl)2 { CITI

2

Ix'l

J

~

C(l

I

+ Iyl? Ix'I-

For the last inequality to hold we need -Q

+

+ ;

When

IW/

+ Iyl

Ix'l,

{ IKz',t,,(s,y',T,z)ldTdz

~ ^Clx'l-

JR2

Hence,

38 Yonne Mi Kim

Now passing to polar coordinates by setting

lul =

= Itl

+ Iyl,

we see that the above inequality is bounded by

Cl (/

_l

^1-

^*,

^*)

lz'l>loor

We see the function K(x',r)

= Ix'I-Har-a

/ .;.;...lIg.;.;...(.;.;...t+_*_,y_+_*);....+_g(..;..,-_t.,."..+..,....*_,_-.;.;...y_+...*.;.;...)_-_2_g.;.;...(*_'.;.;...*)--:....II~^dv.

l/ul<l

lul

ap

$ 'C(AP

+

where

which is :finite sincea -

*

/100 /100

A' =

10

10 Ix'l-l+alx'l-l/pdx'

= 10

Ix'I-H(cr-;>dx'

which is finite since -1

+

>

References

1.M.Mekias, "Restriction Theorem/or Sobolev Spaces", M.I.T.thesis, 1987.

2. G.B. FolIand and E.M. Stein, "E.timatufortheH" Comple:z:ies and AnalllN' on the Heisenberg Group", Comm. Pure Applied Math. Vol. XXVII (1974) pp.42G-522.

3. G.B. FolIand,"6elliptic utimatu and JiUnction Spaces on Nilpotent Lie Group.".

4. N.Aronszajn and K.T.Smith,"Theory0/Bu.e! PotentiabI",Ann.Inst.Fourier 11(1961) pp.385-475.

5. G.H.Hardy, J.E.Littlewood and G.Polya, "Inequalitiu", Cambridge University Press Cambridge, UK, 1934.

6. P.!. Lizorkin,"Ckaracteridica o/the Boundery Valuu0/JiUnctiomfrom Lt(Eⁿ)

on HwerP'anu", Dok Akac. Nauk. SSSR 150 (1963) 986-989.

7. E.M.Stein,"The Characterization0/Functiom ANingaIPotentials", Bull.Amer.

Math. Soc. VoU, 76 (1961), 102-104, VoU,68 (1962), 577-582.

8.E.M. Stein, "Singular Integrals and DiJTenmtiability Propertiu0/Functiom", Princeton Univ.Press (1970).

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