On equality of minimal and maximal realizationsAbstract. One considers minimal and maximal closed extensions of linear pseudo-differential

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXX (1991)

Jouko Tervo (Jyvâskylà)

On equality of minimal and maximal realizations

Abstract. One considers minimal and maximal closed extensions of linear pseudo-differential operators in appropriate spaces. Specifically, the identity of the extensions in question from the Lebesgue space L2 into L2 (and also from the Hormander space Bp l into the Lebesgue space Lp) is examined. Furthermore, a sufficient condition for the identity of the extensions from Bp k into

is obtained.

1. Introduction. Denote by L(x, D) a linear pseudo-differential operator from С* into the Schwartz class S. The minimal closed extension of L(x, D) in the space BPtk is denoted by L~k. The corresponding maximal closed extension is denoted by L*tk. Here Bp k, p e [ l ,

k e K ', is the linear subspace of the dual S’ of S such that the Fourier transform Fи belongs to l}™(R") and that Fu- k e Lp: = Lp (R") for any

Bp k (К ' denotes an appropriate collection of weight functions R"->R). Bp k is equipped with the norm ||м||р,к:= (27i)~"/p||Fw 'k\\Lp. Similarly one defines minimal and maximal closed extensions L~,q and I/Д from Bp<ko to Lq (here k

eK ', k

= 1). Furthermore, if L(x, D) maps Co into C°°(G) and the formal transpose L'(x,D): Co(G)->-S exists, the extensions L~k(G) and L'*k(G) from Bp k to Bl°ck(G) can be defined. Here G is an open subset of R" and Bl°k(G) is the appropriate local distribution space corresponding to Bpk.

One knows that always L~k <= L'*k. If L~k = L'*k, we say that L(x, D) is essentially maximal in Bp<k. Some classes of operators are known to be essentially maximal in appropriate spaces (cf. [6], [4], [9], [3], and [7]).

We establish a decomposition for the convolution iJ/*(0L(x, D)(p) (cf.

Lemma 3.1, Theorem 3.5). This decomposition is applied to show a sufficient criterion for the essential maximality of L(x, D) in L2, where L(x, D) belongs to the Beals and Fefferman class of operators (cf. Corollary 3.6). Further­

more, we prove that L~k(G) = L'*k(G) if

sup|(DJBfL)(x, «I ^ С,,„л M £ ):= C.j,iJE(l + |{|2)1/2,

x e K

where K <è G, p e [ l ,

and k e K ' (cf. Corollary 4.8). Finally, the relation Lpp, = L'pfp-, p e ] l , 2], is obtained if

sup I(D*xDlL)(x, 0 | ^ C ^ k ^ ) .

xeR "

We remark that the operator L(x, D) defined (in R2) by (L(x, D)cp)(x) = [eXl(l + \x\2)(Dl + l)(eXl(l + M2)<?)](x) is not essentially maximal in L2: By partial integration one sees that

Re(L(x, D)(p, (p)b2 = Re((Dl + l)(exl(l + \x\2)(p), eXl(l + \x\2)q>)L2

^ \\eXi(l + \x\2)(p\\L2 ^ \\eXl(p\\L2, for all (peC$(R2), and so one easily sees that the kernel N(L

,ko) = {0} (note that B2,ko = L2). Oil the other hand, observe that the function и defined by u(x) = 1/(1 + |x|2) belongs to L2 and that (L(x, D)u)(x) = 0 for any x e R 2. Hence ueN(L'2fko)\{0}, which shows that L~2tko ф L'2*ko.

2. Preliminaries

2.1. Let K' be the collection of positive continuous functions к: R"->R so that

(2.1) Щ + fi) < С к М Щ ) for all £, n eR",

where C and R are positive constants and kR{Ç):= (1 + |£|2)я/2. Replacing in (2.1) £ with Ç + r] and rj with —rj one sees that k _1eK ' whenever k e K '. More generally, kse K ' for s e R if k e K '. In addition, the function fcv defined by k'/ (Ç) = k( — Ç) lies in K' for any k e K '. Let p be in the interval [1, oo[.

Denote by BPfk the completion of Co := C

(R”) with respect to the norm (2.2) M \ p,k = ((2

) - f \ ( F № № ) \ pdi)1"’,

R "

where F is the Fourier transform from the Schwartz class S into S. Let S' be the dual of S (that is, S' is the space of all tempered distributions). Then Bp<k is essentially that subspace of S' whose elements и satisfy: The Fourier transform F u e L l°c(Rn) and Fu ■ k e L p. In addition,

(2.3) IMU = ((2 я)"” J \{Fu-k){(rdi)11-.

R "

Choose 0from C

such that 0(x) — 1 for all x e B ( 0, 1):= (

R"| |x| < 1}.

Define functions 9', 9t and 0J

C

by

(2.4) '9' = (2к)~п9 \ 9t(x) = 9(x/l), 0J(x) = 9'(x/l), where le N. Furthermore, let

(2.5) = F9'l.

Then F[j/l = (2n)n(9’iy = 9t and so |(Fi/ff)(£)| < C and (Fij/^Ç)-* 1 as /-►

. Hence by the Lebesgue Dominated Convergence Theorem

(2.6)

\\ф,*и-и\\р,к =

\ (W im F u )(0 -(F u )(t;))k (l;fd l;y " ’

R n

— > 0 as / ->

.

Furthermore,

(2.7)

— w||p,fc — > 0 as l-rco,

since ||0,и||р,* ^ С \Щ

гкл\\и\\Ргк < C||0||1>fcJ u ||p,fc (cf. [5], p. 42).

Let G be an open subset of R" and let {Gj} be a sequence of relatively compact open subsets of G such that Gj a G

and (Jj°=i G} = G. Denote by BlpC k(G) the completion of C

(G) with respect to the quasinorm

00

b,k(<p)=

I

7=1

where {6L} c Ccf(G) is a sequence such that supp 0 . c= Gj+l and 0^(х) = 1 for xeGj. Then Bpj^G) is essentially that subspace of D'(G) whose elements и obey: F(\J/u)-keLp for any ^

C

(G). In addition, the quasinorm

00

(2.8) qp,k(u) = X (l/

j)\\&ju\\P'k/(l + \\&jU\\P'k)

7 = 1

defines the topology in Blpf(G) and so the topology of Blpf(G) is equivalent with the topology defined by the seminorms pPxj( u)'- = 11®,-м|1р,ь j e N.

2.2. Let L be a linear operator from Co into S such that the formal transpose L : C

-+S exists, that is, there exists L' such that

(2.9) (Ь(р)(ф): = J (L(p)(x)i//(x)dx = (p(L'\l/) for (p, феСо.

R "

Define dense operators Lpk. Врк-+Вр>к and L *k: Bpk-^Bpk as follows:

|D (L M) = C

,

\ L Pjk(p = L(p for (p e C

,

rD(L'*k) = {ueBp>fc| there exists / e B pk such that

J u(L'(p) = f((p) for all (peCg},

l L'*ku = /.

НФ)\ < IM U M Ip, 1/kv for u e B Ptk, феБ

(here p 'e ] l ,

l/p + 1/p' = 1), LPtk is closable and L'*k is closed. In addition, Lp k c= LPtk. Let LPtk: BPtk^ B Pfk be the smallest closed extension of LPtk. Then

T ~ r- T'*

■L'p, k Eip<k.

Let ре [1,

[ and denote (as above) by Lp the Lebesgue space in R". We find that ||u||p = ||Fuv||P)jto for u e L p and so

ll<Ai*M-w||P = IIFuv - W 'FMv||p>Jto = ||F

-0? -Ft/v||Mo

\\u — 0

||p->O as /-►

(2.10)

(

.

)

Since (

.

)

0 as

l

For p,qe[_ 1, oo[ we can define a dense closable (resp. dense closed) operator LPtq: Bp<ko-*Lq (resp. L*q: BPtko-+Lq) by

(2.13) \ DiL^ ~ >

[Lp<q(p = L(p for (p

C

,

(2.14)

D (L’p*q) = {u e B p<k01 there exists / e L q such that u(Lq>) = f((p) for (peCo}, L’

> = /•

Denote by L~e: BpM-+Lq the smallest closed extension of Lp>9. One sees that L c L*

'

Suppose that L is a linear operator Co -►C00(G) such that the formal transpose L': Co^G)-+S exists. Define dense operators L pk(G): Bpk->Blpk(G) and L*k(G): BpJt->Bft{G) by

(2.15) D(Lp>k(G)) = C%,

Lp,k(G)(p = Lq> for (peCo,

j D(Lp*k(G)) = {u e BPtko I there exists / e B lp k(G) such that

(2-16) < u{Lcp) = f(<p) for (peCo(G)},

lL'p*k(G)u = /.

Then Lpk(G) is closable (cf. [10], p. 77), L'p* (G) is closed and Lpk(G) c L'*k(G).

Let Lpk(G) be the smallest closed extension of Lpk(G). One has

L~Ptk(G ) ^ L'p*k (G ).

2.3. Let (Ф, ф) form a pair of weight functions in the sense of [1]. Choose M and m from R. We say that the smooth function L(-, •)eC 00(R" x R") belongs to the class S if

(2.17) KDÏDfO(*, «I « 9Ф ” - М(х, a

for all x, { e R ‘ . Define the linear pseudo-differential operator L(x, D) by (2.18) (L(x, D)<p)(x) = (2я)-« f L(x, <peS,

R "

P

where L(-, ')eS%£. The class of pseudo-differential operators is defined as follows:

Ljfr" = {L(x, D)\L(x, D) is defined by (2.18), where L(-,

Let L(x, D) be in . Then L(x, D) maps S into itself and the formal

transpose L (x , D): S-+S exists. Furthermore, many rules of computation are

valid in {JM,mL%$ (cf. [1]). We recall

Th e o r e m

2.1. Let sé be a subset of S%’§, such that

(2.19) p°>°(L(-f •)):= sup £)</>'“'(.x , (DJDfL)(x, 01}

x ,(*e R "

< C«., < oo for all L{-, -)estf, a, /leNô.

Then there exists a constant C > 0 such that

(2.20) ||L(x, D)<p\\:= ||L(x, D)(p\\Ll ^ C\\<p\\ for all (peS,

where C is independent of L(x, D) on the set {L(x, D)\L(‘, -)es/}. ■ For the proof, cf. the proof of Theorem 2 in [1].

Let L(*, •) in C°°(R"xR") be such that

(2.21) sup I(

;

?

)(

, 01 « C ^ , KkKt + m (()

x e K

for any K C G (here G is an open set in R") and (a, f ) e N l n. Then L(x, D) maps S into C°°(G) and the formal transpose L'(x, D): C

(G)->S exists (cf. [8]; the formal transpose can be defined by L'(x, D)<p = (фЬ)'(х, D)<p for (peCo (G), where i^eCo(G), ф(х) = 1 in supp<p).

3. On minimal and maximal realizations in L2 3.1. Define a seminorm p ^ f

on by

РЙГ(£<-> •)) = sup {ф|«-м (х, ()ф'-'~т(ху 01 (DÎDÏD(x, 01}.

x ,£ e R "

Then S$f;” becomes a Fréchet space with the locally convex topology defined by the seminorms p ^ f 1, a, /J

N

. If феБ we denote the pseudo-differential operator generated by the symbol (Fi/r)(-) by \f(D).

Let Oj, j e N, be as in Section 2.1. Furthermore, let Lj( -, •) = 0; L(-, •). We have

Le m m a

3.1. Suppose that L(*, ‘)

C°°(R"

R") such that for some Ca>/? > 0

and Na>fie R

(3.1) sup \ m D lU x , O K C .jk H .J t) f o r t e R".

x e s u p p fy

Then for each N e N and for any ф е$

(3.2) ф*Ь](х, D)q> = X ( ( -i)M/y'){DyxLj)(x, 0)((хуф)*(р)

\ y \ < N

+ R j^ tN(x, D)(p, for all (peS, where

(3.3) R ^ .iï( x , 0 = (2 it)-" } lV (l-ï)''-1 £ (1/y!) J J S W ) ( i + «(4-{))

0 \ y \ = N R " R "

x (DlLjiy, £)ei{x~y,n~®dydr\.

P ro o f. In virtue of the Fourier inversion formula we obtain (3.4) (ф * Lj(x, D)<p)(x) = (2ic)" :" f F(\j/ * Lj{x, D) <p)(ij)ei("'x) dr\

R "

= (2n)~n J (F\l/)(rj)F(Lj(x, D)(p){rj)ei(n'x)dri

R "

—

(F\lf){ri)(

(Lj{y, D)(p)(y)e~i{y',,)dy)ei(,t’x)dri Rn R"

= (2it)-" { W M 1 {((2it)-" J Lj(y, i)(F<p)(i)ei™ d t ) e - ,<™'dy}ei"-1‘4r,.

Rn R» Rn

Since F\/f and F(peS and since suppL^-, rj) c supp^-, the order of integration in (3.4) may be changed to obtain

(3.5) (i//*Lj(x, D)<p)(x)

= (2*)-" f ((2it)-" f J (Рф)(пЩ(У, i)e‘<*-™-ï’dyd4)(F<p)(Ç)e,™ d ( .

R " R " R "

In virtue of the Taylor formula we get (3.6) (2 к ) - J J W M L j f y , i)e**-™-l4ydr,

R " R "

= ( 2 к ) - Y ( W ) f J ду(Р ф )а )((-ч У ^ (у , Ç)ePx~’*~adydri + RK(x, Ç)

| y | < N R " R "

= (2

) - Y ( W ) f f 0’(*'Л Ю (В Д О '. i)e«*-™-«dydr,+R„(x, 0 .

\ y \ < N R " R "

where

(3.7)

0

(271)-" J J

X Z (VyW4Fÿ)(Z + t(q-Ç))(DyLj)(y> Q e^-M -^dydri =:Rjt^ N(xt £).

M = N

Furthermore,

(3.8) ( 2 к ) - J (2

) - £ (1/y!) f f dy(F>l/)(i)(DyyLj)(y, i)

R " |y| <iV R " R "

x e‘(JC - ™ ~ « dy drj • (F<p)(É) ei M dil

= (2k) - y

( W ) s (S (ПЩЬМУ’ i)e‘<x~™~l)dy)d>i)

[ y | < N ^{R " R " R ”}

x ( — i)171F ((xy ф) * (p) (£) ei(4,x) dÇ, where

J (f (D}Ly)(y, £)е-«™-*ЫуУх«-*Чг1

R " R "

= f F((DlLj}(-, Q){ti-№*«-*>dri = (2n)n(DyxLj)(x, 0 .

Hence

imply the assertion. ■

Furthermore, we need the following lemma:

Lemma 3.2.

Suppose that L(-, •) belongs to S a n d that is the subset of S such that

(3.9) p8S((FM -)):= s u p W F M Q \ * m {x, É) ^ E, for all фе®.

X , i

Let •) be defined by (3.3). Then •)eS® JN,M-JV and

(3.10)

0) < C for all \J/e£%,jeN,

where C >

is independent of j and ф.

P ro o f. In virtue of (3.9), M is a bounded subset in 5 ^ . Furthermore,

|(Z>ÏZ>!Lj)(x,

= | Z 0 (D * -”^)(x)(D;Df L)(x, £)|

u<a

= I Z 0 ( i

а |

« £ O sup|(D «-“0)(x)|C^<fM-l'"(x, 0

x

«S Z C)sup|(D“-"0)(x)|CM Cl* -“l^ M- |'l|(x, а < Г “ |“|(х, 0 and thus p l’J(Lj( - , - ) H Q „ that is, {ТД-, -)}j6N is a bounded set in

In virtue of Theorem 1 in [1] (cf. also the proof),

(ifi(D)oLj(x, D))(x, ()

= z (wmimamLjXx, а + а д , a

\ y \ < N

= Z

\ y \ < N

= Z ( ( -/)м/у!)((1)3;^)(х, D )o(xV)A(D))(x, ^ ) + ^ , N(x, ^),

\ y \ < N

where {Rj^,it}j.^ is bounded in S

^ N,m~N. Since

$(D)oLj{x, D)(p){x) = (ф*Ь]{х, D)(p)(x)

one gets the assertion from (3.12) (note that one has (L(x, D)o\j/{D))(p = L(x, D)№*(p)). л

Lemma 3.3.

Let фг be defined by

Then

PoliHWiM')) < Е/ for all l eN .

(3.13)

P ro o f. We have = F(F9'i) = (2

)"(09

= fy. Since 6

C

we find constants Cp > 0 such that

KD^Kfll *5 C^l + lfD-1' 1

and so (recall that Ф(х, fl ^ C(l + |{|))

Po:?(e,(-)) = sup |(а<'0,)(й|Ф'<,'(х, 0 = sup(l/ll,l|) |(дев){(/1)\Ф|д|(х, fl

x,£ e R n

^ C,sup(l//W)(/W/(/ + | ^ l ) C ^ ( l + | ^ ' ^ C^C^I, which completes the proof. ■

3.2. In the sequel we shall assume that there exists N e N such that (3.14) Фм ~"{х, £)фт~*{х, £) ^ С .

The inequality (3.14) is always valid with M = N when M ^ m:

Фм " м(х, £)фм ~т{х, <*) ^ CM“m.

It is also valid for any M, m eR (with iV large enough) when Фф ^ ckx with some c > 0, x > 0.

The following lemma is easily seen:

Le m m a

3.4. Let L(-, •) be in S%;™. Then there exist re N and C > 0 such

that

(3.15) ||L(x, D)(p\\ ^ C\\(p\\kr,

(3.16) ||L(x, D)q>\\k_r < C||<p|| for all (peS.

P ro o f. Due to (i) (cf. the properties (i)-(iv) of (Ф, ф) in [1]), k - r(-)eS%;™

n S ^ ,ф’~т for r large enough. Hence the application of Theorem 2.1 with L(x, D )ok-r(D) and /c_r(D)oL(x, D) yields the assertion. ■

Since L{x, D): S-y Sis continuous we can define the continuous extension L: S ' S ' of L(x, D) by

(3.17) (Lu)((p) = u(L'(x, D)q>).

We show

Th e o r e m

3.5. Suppose that (3.14) is valid and that L(-, feS^;™. Then there

exists a constant C > 0 such that

(3.18) ||ih * L u - £ ( ( - 0 |у|/7!)(^Ь)((^«Аг)*м)||^С |М |

\ y\ <N

for all u e L 2.

P ro o f. A. First we establish that

(3.19) ||«^*L(x,Z))<p- X ( ( - 0 |у|/у !)(^ Ь )(х ,/))((х ^ г)*Ф)||

\ У \ < И

^ C\\(p\\ for all (peS and le N.

For any le N one gets by (3.13), (3.14) and (3.10)

(3.20) p%?(RM ,.s C> ')) = sup<p№(x, £№'*'(x, й |D i D l R j ^ H(x, {)|

*■!

< C 'su p $ N- A, + ''’l(x, + W(x, O lD lD lR j^ x , f)I

= С'р«<Г№ -” - ,'(Л ^ „ я(-> •)) ^ C'C.

Due to Theorem 2.1 there exists a constant C > 0 such that (3.21) № j.* A x , B)<p\\ < C|MI for all

where C is independent of j, le N. Hence we obtain by (3.2) (3.22) | |

|y|<N

^ C||<p|| for all <peS.

Furthermore,

(3.23) ||^*L ;(x, D)(p — ij/l*L(x, D)(p\\

^ IM oo.i||(b/x, D ) - L { x , D))<jo||

= Ih M c o .ill^ x , D)(p — L(x, D)(p\\->0 as j -ю о, where we have set

(3.24) IMIoojk = sup|(F^)(£)fc(£)| for (peS and к e K ' . In addition,

(3.25)

= (2

)-" J D’((0;L)(

,

R"

= (2я)-" j X © (1//“') f (В"0)(х//)(/)Г“Ц(х,

R " « ^ y R n

м < у

and so

(3.26) ||д а ь ;)(х, D)((xyiA{)*(p)-(Z)];L)(x, jD)((xy^j)*ç))||->0 as ;->oo.

Hence we* obtain (3.19) from (3.22) by letting j-* oo.

B. Let и be in L2. Choose a sequence {(pn} c C

such that

||ç>„ —u II -»0 as n-* oo. Let r be in N such that (3.16) holds. Then

\\L(x, D)q>n — Lu\\k_r-+Q as oo and so (3.27) Цф^Цх, D)(pn-il/l*Lu\\

< \\^i\\oo,kr\\L{x, D)(pn-Lu\\k_r-*0 as n->co.

Furthermore, (хуф1)* и е Н к/.= B2)ks and

(3.28) W(xyil/i)*(pn- ( x y\J/l)*u\\ks < ||xy^rI||00,Jk,||<

t>B —

|| — *>0 as n — > со, where seN such that

(3.29) ||(В Д (х , 2>MI ^ C\\<p\\ks

(note that (DyxL)(-, and use Lemma 3.4). From (3.29) we get (3.30) ||(B ÏL )((x W * u )-(D ’xL){x, B )((x'«*V ,)||

^ C||(xV,)*M -(xvi/q)*(pJ|*s-->0 as n — > oo.

Since by (3.19) for any N

N one has

||./q*L(x, £)(/>„- £ ((-O w/y!)(l)iL)(x,D )((x^I) * ç » J ^ C M ,

|у|<ЛГ

we obtain (3.18) from (3.27) and (3.30) by letting n-»oo. This completes the proof. ■

We show the following criterion for L~ := L~2M = L'# : = L'2^0.

Co r o l l a r y

3.6. Suppose that L ( -,-)

S$;™ and that there exists

such that

(3.31) ^ ) ^ с Ф м - 1( х , а г _1( х , а

(3.32) D(L'*)czHq: = B 2<q.

Then

(3.33) L~ = L'* .

P ro o f. The operator (Log-1)(x, D):= L{x, D)oq~1(D) belongs to Ьф\ф (note that by (3.31), q~l ( ‘) lies in S®,Jf + 1,-m + 1). In the inequality (3.14) is always valid and so by Theorem 3.5 (with N = 1) we obtain for any u e b 2

(3.34) \\i//l*(Loq~1)u — L o q ~ 1(\l/l*u)\\^: C\\u\\.

Let и be in D(L’*). Then by (3.32), q u e L 2. Furthermore,

(3.35) (Loq

(qu))((p) = (qu)((Loq ■ 1)'(x, D)q>)

= (qu)((#vГ 1 (D)oL'(x, D)<p) =

(«

(^)

(«

) - 1(D)

L'(

, D)<p)

= (L'*u)(cp) and

(3.36) F(Ÿi*qu) = (Fil/JFiqu) = F(q№t*u)).

As in (3.35) one sees by (3.36) that

(3.37) L o q ~

(\j/l*qu) = L o q ~

(q({f/l*u)) = L(^,*n).

Since ф ^ и е Н ^ (where re N such that (3.15) holds) we see that ^,*ue£)(L~).

Hence by (3.34), after replacing и with qu and applying (3.35) and (3.37) we obtain

and then (3.38) where

||^ * (L '# M)- L ~ ( ^ * M)||^ C |N |,

||L~(i/f|*u)|| ^ C ||m||9 + ||i^i||00>1||L,#uH,

M U

= sup |(Fty,)(£)| = sup|0j(£)| ^ sup |0(£)| <

.

? « « ■

Using (3.38) and the Banach-Saks Theorem we observe that there exists a subsequence {ф^*и}

{ф^и} such that for some g e b

II(IA) z L~('l'ii * u )-g \\^ Q as s —* со.

j = i

Since

and

(1/s) £ Ь~{ф

л*и) = L~((l/s) £ №ij*u))

J

=i

=i

||(l/s) Y

Ф^*и-и\\->0 as s —> ■ со,

j = i

we see that ueD(LT) and L~ и = g (= L * u). This finishes the proof of (3.33). ■ R em ark 3.7. A. For M = m = 1, we can choose q = 1. The inclusion (3.32) is also trivially valid since H Q = L

for q = 1. Hence L~ = L '# when

' ) е ^Ф,^'

В. Corollary 3.6 shows that any operator L(x, D)el%>\™ from B

q to L2 is essentially maximal (the definitions of minimal and maximal realizations are as in 2.2). Specifically, any operator L(x, D): B

tkm_(

_e)->L

*s essentially maxi­

mal when L(-, •) belongs to the Hormander class of symbols.

4. On essential maximally in BPtk

4.1. Let L(-, •) be in C°°(R"xR") such that

(4.1) sup № Щ Щ х , «I «

■ xesuppOj

where > 0 and

R. Then

defined by (2.16) maps S into S and the formal transpose Ц(х, D): exists. Furthermore, one shows as in

that for

N

(4.2) ^i*Lj(x, D)(p = Ь;#,*<р) + Ял,(х, D)q>, where

(4.3) K,,,(x,{) = } X

! (M M e + t t i - t y i D ’LjHy, ()e‘<*-™-«dyd

dt

O |y| = l R " R"

= f X i (dWl)(i + t4)(i(DyyL1) ( y ^ ) e - ‘^ d y ) e i^">dt,dt

O Ivl = 1 R " R"

= } I | ( ^ 0 г) ( ^ + ^ ) т ( ( ^ ^ ) ( - л ) ) ( / / ) ^ ^ ^ .

О |y| = 1 R "

Lemma 4.1.

Suppose that L(-, •)

C00(R"

R") obeys

and that Rjyi{',

is defined by

Then

-)

C°°(R" x R") and

(4.4) m D l R j ^ x , 0\ = Ca,PJkNfi(t) for x , £

R", where Caypyj is independent of

N and Np : = maxu<cp{Nu — 1}.

P ro o f. A. Since

i))(n) = J (D’LjKy, ()e~‘™ d y

supp Oj

one sees that

(4.5) DI[(F(D;L

)(-, = F((DyyD^Lj}(-, 0)(rj).

We recall that (cf. the proof of Lemma 3.3)

(4.6) \ ( 1 г е № + щ ) \ ^ с ак - м в + щ ) . Furthermore, since (for t > 0 and for a fixed £0

R")

l (8-’e,)(( + tn)F((DlLj)(-, t y W ^ d q

R "

J (d'0№ + trj)F((DyLj)(-, t y W e ^ d r i

(l/ t)(is u p P0 + B (£ o ,l))

for all (x, <^)

R "

B(<^0, 1) we see that -)e C°°(R" x R") and (4.7) (DlD^Rjj)(x, i)

= î Z J

DlDillSW M + triF liD lL jH ^ ty W ^ -id q d t.

О I у I — 1 R "

B. In view of (4.1), (4.5) and (4.6) we obtain for any a, jfreNô and

tg

N

, |

| ^ n +

,

\ Ч ' В Щ W Ш + t4)F((D’ L j)( -, « ) W e '(« )]|

= I Z (Î) ( - i ) l“l(5"+ '0,)(« + t '! ) f ( ( B r " + , C f-" L J.)(-> 0 )fo K < * '1

U*Z0

Z Z © C +; +,,)C I, + yC„,(,- „ J ||D '+“+' - ' ’0J.||I.l J:. l„+rl({ + tr,)/cK„ . B« )

U^0 V^x + tx + y and so

(4.8) ID>iD\[(d4i(t + t

)F((DlL)(-, O)0f)e'(JU,,]|

^ X

We decompose the integration over R" in (4.7) into the two parts

f DaxD ll № = J E&Dtl ]<fy + J D iD il Ш

R" Ul<2|i»| |«|>2|ir|

Bt . If |£| ^ 2|f/|, one has

к - м -

в + »

)к*'-ы( № - ь + г М <

and so we obtain by (4.8)

(4.9) I J />“/>?[ ] ^ | < 2 С ; (/,)Уа / с _ (п+1)( ^ » / ) ^ ( а .

1«1^2|Ч| R"

B2.

Suppose that |£| ^

Then

(4 .1 0 ) \ z + t n \ > |£| - ф | ^ № - \ ч \ > ( 1 /2 ) |£ |

and so

(4.11) k - M-i ( ^ + tri)kNll_u(^)k-(n+2)(ri) ^ k -i( ^ + tti)kNf_u{ ^ k - (n + 1){ri)

<

к _ ! (£)kNp_u(Ç)k-{n +

)(r]).

This implies by (4.8) that

I J

Q w ( J

\S\Z2\t,\ R "

Thus the estimate (4.4) follows from (4.7).

4.2. Since Rjtl( -, •) obeys the estimate (4.4) we know that the formal transpose R'jti(x, D): S-+S of RJti(x, D) exists. Furthermore, we have for p e ] l , oo[

Le m m a 4 .2 .

Suppose that

is a subset of

such that for any

*(•, •) € s / and (a, (1) € N

" there exist constants Ca P > 0 and Nlt e N so that (4.12) sup ID%D\R(x, 0 | « CKtks ,(0-

jceR"

Furthermore, assume that

(4.13) I \F(e,RC, -Ч ))(Ч -Ф (Ч )А ! МЩ ),

R "

for £

R",

N and (4.14) j и м о ,

«

R "

for rjeR",

N and R(-, -)

<

/ . 77zen t/iere exists a constant C > 0 such that

(4.15) HR'(x, D)(p\\Ptk ^ C\\(p\\Ptk for (peS and R ( v ) e ^ >

where

(4.16) C ^ (2n)-nK 1/p'M llp ^ (2n)~nm<ix{K, M}.

For the proof, cf. the proof of Theorem 4.2 in [8].

From Lemma 4.2 we obtain for р с[1 , oo[ (recall that Rm( - ,- ) : = e mR(-,-))

Th e o r e m

4.3. Suppose that

is a subset of C°°(R"xR") such that (4.12)

holds and

(4.17) sup \(D%R)(x, 01 < Ca,m for |a| ^ N k + n + 1, R(-, -) e s t,

x e s u p p6rn

where N ke N such that

(4.18) kK + ^ C t y i |) k ( f ) .

Then there exists a constant Cm > 0 such that

(4.19) \\Rm(x, D)(p\\Ptk^ Cm\\(p\\p>k for (peS and R{-, -)ej t f (where Cm is also independent of p).

P ro o f. A. First we shall verify that

(4.20) J H W . - ч ) ) ( ч - 0 \ / к ' ' Ш ч ^ М т/к ''(0 for ÉeR*.

R "

(4.21) S № * „ ( - . - Ч ) ) ( П ~ Ф ' ' Ш < K »*v(4) for 4 eR-,

R "

where Mm and K m are independent of / and R(-, )

/.

From (4.18) one gets k( — Ç) = k( — rj — (Ç — rj)) ^ CkkNk(Ç — ?/)/c( —

) and so Ck/c)vk(^ —^/)//cv(0- For any

N

, |

| <JVk + w + l, we obtain (4.22) |(>/-{)'F(0IRm(-, ->1))(>1-{)| = |F(DJ(0,R„(-, - ч)))(ч - {)|

« £ © |l/iM |sup|(P“0)(x)| J |(С Г “Л„)(х, -

)|dx

u — t x R "

« £ © s u p |( B “0)(x)| E С 7 )(1 /т|'’|ДО',0 Д 1 sup |Р Г “- '« ( * , -ч)1

и $ т x v ^ T — u x e s u p p 0 m

and so one has with C'm := max {СТ)Ш }

|x| ^ Nic + n+ 1

|F(e(R„(-, -4))(4-9|/fcv(l) < C'mCk(feWl(4-0/fcv(«)fc-W,+.+i1('î-{)

= c mc t k_(„+1,(f/-i)/k v( a . Thus

J № «„,(•, -4))(4-0|/fcv(»i)<i4 « c^ct(J fc-„+I)(4))/fc''(fl =-.M/k4i),

R " R "

where M does not depend on R(-,-)estf.

Since fcv (£) ^ ChkNk(rj — £)kv {rj), the validity of (4.21) can be verified similarly.

B. Due to Lemma 4.2 one has for p' e ] 1, oo [ (4.23) \\K (x, D)q>\\

where Cm = (27i)""max{Km, Mm}. For all (p,\j/eS we obtain

\(Rm(x, D)if/)((p)\ = |il/(R'm(x, D)(p)I ^ \\ф\\р,к\\К(х> D)<P\lPM/fcv

< с т\ЩР,кМ\рл1к^

and thus for any p e ] l , oo[

(4.24) \\Rm(x,D)il/\\P'k ^ C mm \ p,k.

By letting p -> l and noting that ||<p||p>k-4Mli,k as p-> 1 one sees from (4.24) that (4.19) is valid also for p = 1. This finishes the proof. ■

R em ark 4.4. From the above proof one sees that the constant Cm in (4.19) satisfies

(4.25) Cm«(2it)-*Ct { I I ( 9 ( ';“)sup|(Z>;0)(x)|

H < r v = z — u X

x\\Dvx0 J Ll max {Ca>m}}.

|a|$iVfc + n + l

Combining (4.2), Lemma 4.1 and Theorem 4.3 we obtain

Co r o l l a r y

4.5. Suppose that L(-,-)eC °°(R "

R") such that

(4.26) sup \(DlD>t L)(x, £)| < C„,WM 9

xesuppfl j

Then for any j , f e N and k e K '

(4.27) I I

)<P\\

* < Cjj\\(p\\Ptk

for all (peS, where Cjty is independent of le N and p e [ l , oo[.

14 — Comment. Math. 30.2

P ro o f. Due to Lemma 4.1 we have

(4.28) I (D*DiRj,d(x, 0 | ^ for all x, £eR", le N

(since by (4.26), N u = 1 for any и ^ ft). Hence we can apply Theorem 4.3 with stf = •) |/e N } and with m = f and so

||

<

for all

l e N, as desired. ■

4.3. Suppose that L(*, •) obeys (4.26). Then (4.29) \\Lj(x,D)<p\\Ptk^Cj\\<p\\pMl, (4.30)

(cf. [8], p. 49). Since the continuous formal transpose Ц (х, D): S->S exists we are able to define the continuous extensions Lf. S' -+S' of Lj(x, D) (cf. (3.21)).

Furthermore, we obtain (here f e N so that OjOf = Oj) (4.31) L , # , * u) = 0r (ij/t* Lju) + (Ки )г и

for all u e B Ptk: Let и be in BPtk and let {<p„} c S be a sequence such that

I I T h e n

IIt l / ^ L j i x , D)(pn — ij/l * L j ( x , D)(pm\\Ptk

^

< IW oo.^C^I^-tpJlp.fc^O as n, m-MX) and

II

D)(^*(pn) - L 7.(x, ÆX^pJllp,* <

< Cj i w oo>kl||(pn-< pj|p)k-^0 as n, m->oo.

Since also

\ \ ' l ' i * < P n - ' l ' i * u \\P, k ^ Q

as n->oo,

(4.27) shows that (4.31) is valid (cf. the proof of Theorem 3.5).

Let j be in N. Choose f e N so that Ofif = 9 у Then IIOr liJ/i*Lj(x,

\\Gj'\\ukNkU i*Lj(x,

where

= (2я)-” f « щ а и т =

*)-"(/)" f

R " R "

= (2я)"" f |(Т0)(^)|^к( ^ //) ^ < ll^lli,kNk < °o.

R n

We are now ready to establish

Theorem 4.6.

Suppose that L(-, •)e C 00(R"

R") such that

(4.32) sup I(D ^ L )(x , Д К

x e s u p p O j

Then for je N , p e [ l , oo[ and k s K '

(4.33) (L/ Д =

P ro o f. Let и be in D((Lj)'p%). Since ф ^ и е

<=

(4.29) shows that ф ^ и е D({Lj)~k). Let e be a positive number. Choose tpeS such that

\W~u\\p,k < £• Then by (4.31) (4.34) \\(Lj);^i*u)-(L jyp%u\\p,k

< 11 (Lj)p,k («Ai * (w - <?)) “ (Lj)'p*k (м - <P)\\p,k

^ \\6f &А1* е д > - (p) — {Lj)'*k( u - Ф)]I U

+ ||(R;>0/(u-<p)||p,k + ||LJ.(x, Д)(«Аг*ф )-Т ,(х, D)ç>||p,fc

^ P IIi.*wJI«A,* C(L/Pî(t t - <p)] - (L / pJ(ii - <p)IU 4” Cjtf \\u ф||р,к "b ^jll^AI * Ф «Pllp.kfci •

Choosing /0 e N so that for I ^ /0

ll«Ai*[(TjK(w-<p)]-(LJ.);#fc(u-(p)||p>fc < e, ||«A/*<P-<PlUit1 < e, one finds that for l ^ l0

11 (Lj);,k(«Ai *u) - (Lf^uWp' k ^ \ \ 9 \ \ l t k N k s + C j j £ + CjE

and so (L^p^OAi*!*)-*^)^ and фг*и-^и in

This proves that

and (Lj)~ku - (Lj)’p*ku, as required. ■

R em ark 4.7. As the proof of Theorem 4.6 shows, (4.35) 11 (Ь])~к(ф1 * и) - (L;);#k и I Ip,к + 1 |*A,* и - u\Ip,fc -> 0 as Z-MX> (for any ueD((Lj)'*k)).

Co r o l l a r y

4.8. Suppose that L(-,-)eC ® (G x R") such that for any

compact subset K a G and (a, /?)eNo"

(4.36) sup\ (DxD^L)(x, 0 | ^ Св.,.* М 0 for ^eR".

x e K

Then (for p e [ l , oo[ and keK ')

(4.37) L~k(G) = L'p#k(G).

P ro o f. Recall that L'(x, Z)): Co (G)->S exists (cf. Section 2.3). Let ф be in C

(G) and let Ьф(х,

фЦх,

Choose ; e N so that в-ф = ф. Then by (4.36) one has

sup K ^ D f L ^ x , 0 | < C e.,.,M 0

xeR"

and so by (4.35)

(4.38) 11((L*),)~k( ф ^ и ) - ((L^)j)'*kи |\Ptk + 11фг* и - u\\Pyk -► 0 as l

со. One finds that

= U(<LM X’

)<

) = «(£*(*. D)Wjip))

= «(!/(*, D)(iA^ç>)) = (L&(G)

)(<M = фир\( в ) и and

( ( ^ ) ; , к(Ф1*и) = ({Ьф)])'р*к{Ф1*и) =

ь'*к(С)(ф t*u) = 11/ь~к(в)(ф1*и).

Thus, by (4.38), Lp>fc(G)(^I*M)->Lpjt(G)M in jBp,k(G) and iJ/t* u ^ u in BPyk, which completes the proof, ш

5. On the equality L~y = Lp*p'.

5.1. Finally, we shall show a sufficient condition for the identity L~ p' = Lp*p'. Our considerations are based on the following lemma:

Lemma

5.1. Suppose that

•)

C 00(R"

R") such that (5.1) sup\(D%D^L)(x, 0\ < Q A ( £ ) for all £eR".

x e R "

Then the symbols

•) defined by (3.4) obey

(5.2) sup \{Da xD^Rjj){x, £)| < CM for all j, le N.

x.^ e R ”

P ro o f. A. In view of (4.7) we obtain (5.3) (DlDlRjM x, ()

0 |y| = l R "

= } I I ( f)(-o M ! (&'+сж + ‘ ч т 1 > ;+‘ о { - 'Ц ( - ,

0 |y| = 1 R"

x el{x,n)dr}dt.

We consider the integral

/ :=

f):= J

R ”

One sees that

1 = j S (dy*vei)(i + tti)(Dl*‘D l-'‘LJ)(y, i ) ^ - ^ d r , d y ,

R " R "

where

J (а' + |’0,)({ + г1/)№ Г ‘ О Г -Lj)(y, dq

= f ( i / ( i + |x - y |2"))(e'+”ei« + t ),)(Dj;+* B f-’LJ))(y> о

R "

x ( l + ( £

= J (i/d + |x -y |2"))-((i + ( - t 2) " ( i

R " 7 = 1

x(D l+aD l - vLJ)(y, Q e^-w'dri and so

(5.4) ; = J (i + ( - t 2)" (£ д1у)д’**в,в+щ)

R n 7 = 1

x (J ( W l + \ x - y \ 2a))(Dl+°Dllt - ’Ll)(y, i)e«*-’-»dy)dri.

R "

B. For any

^ n + 2 one has

(5.5) |, ' j (1/(1 + |x - y |2"))(£>ï+“£>f—’Z,i)(y> i)e‘^ - ^ d y \

R "

« J |0 '((l/(l + |x - y |2"))BJ+,,D f-,’Lj)(y, ()\dy

R ”

< Z (У J |в;(1/(1+|х-у|2”))||(В5+*+*-»в|-'^)(у,

w ^ i R n

= I (У J С„(1/(1 + |х-у|2”))С;+„+,_^_Д 1(07у=:См,„,„/с1(«).

w < T R "

where we have used the fact that

|D ;(l/(l + |x - y | 2 "))|«C „(l/(l + |x - y |2”)) for x . y e R ”

\ W D f Lj){y, |)| ^ Z e')|(D » 6 jm D Ï - D l'L ) ( y , 01 u^a'

=S I (;')(1//I“l)l№“0)(y//)I I W - “D(L)(y, 01

^ X (Ï) sup

(Я“

u ^ a y e R n

Due to (5.4) and (5.5) one gets

(5.6) |/| =s cM,7,„ J(i + (- t 2)"(Z a?)")(S'+“<w+n,)M0k

R " 7 = 1

C. We have the estimate (cf. (4.6))

|(i + ( - t 2)"(Z <5?)")(<з'+ч ) « + » / ) I

j=l

^ Kd’^ M É + t i r t i + K d a?)”a '+-e,)«+ t7)|

J= 1

^ Cy + 0k - \ y + v\(Ç + tri) + CytV'ttk-\y + vi-2n(Ç + tri) ^ C y ^ k - ^ + trj)

and so by (5.6) one gets

1Л < CatPt7tV J k - 1(^ + tti)k-{n + 2)(t])k1^)dri.

R "

As in the proof of Lemma 4.1, Parts B1-B 2, one sees that |/| ^ and so

№ D iR j,d (x , £ )K J z tt/vO I (01 l\dt ^ CM ,

0 |y| = l v ^ p

which finishes the proof. ■

From Lemma 5.1 we obtain (recall that p 'e ] 1, oo], 1/p+ l/p' = 1 and

* o = l) \

Theorem

5.2. Suppose that L (-,

e C00(R" x R") obeys the estimate (5.1) and let ■) be defined by (4.3). Then for p e [ l , 2]

(5.7) \\Rjfix, D)(p\\P' ^ C\\(p\\pM for all (peS and j, le N.

P ro o f. A. Let

R jfix, D)oF~l . Then by (5.2) we find that WT

M I

= sup

= sup|[Rj-,(x, Z))((F-1 <p))](x)|

JceR” xeR"

= (2n)_"su p |J Rjj(x, Q F (F -1 </>)(№**>di\

x R "

^ C1

\<p(£)\d£ = c t l\(p\\Ll

(peS.

R "

B. Let Ф(х, £) = ф(х, Ç) = 1. Then by (5.8) one sees that Rjfi-, - )eS%’^

and that Pa’,p(RjA*» ')) ^ Cx,p- Hence by Theorem 2.1

\\TjM l2 = WRjAx, W » ||La ^

C 2||F-VII

2

for all (peS.

In virtue of the Riesz-Thorin interpolation theorem (cf. [2]) one has

\\Tj,i<p\\P’ < C l~ 2lp'C llp,\\(p\\p ^ ( C 1 + C2)\\(p\\p for all <peS and thus

II

D)<p\\P' = \\TjAF(p)\\p ^ (Cl + C2)\\F(p\\p = (C1 + C2)||<3£>||p,Jk0,

as desired. ■

Finally we establish

Th e o r e m

5.3. Suppose that L(-, •)eC Q0(R"xR") so that (5.1) holds. Let

pe~\ 1, 2[. Then

P ro o f. Let и be in D(L'*P'). Similarly to the proof of Theorem 5.2 we observe that \\L(x, D)(p\\p> ^ C\\(p\\Ptkl and then iJ/^ueDiL'*^). Furthermore, for any j, le N (cf. (4.31))

Since ||

^

* w — w||p' — ► 0 as l-> oo, the assertion follows by the same kind of argument as in the proof of Theorem 4.6. ■

[1] R. B e a ls and C. F e ffe r m a n , Spatially inhomogeneous pseudo-differential operators I, Comm.

Pure Appl. Math. 27 (1974), 1-24.

[2] J. B erg h and J. L ô fs tr ô m , Interpolation Spaces, Grundlehren Math. Wiss. 223, Springer, Berlin 1976.

[3] K. O. F r ie d r ic h s , The identity of weak and strong extensions of differential operators, Trans.

Amer. Math. Soc. 55 (1944), 132-155.

[4] P. H e ss, Über wesentliche Maximalitat gleichmassig stark elliptischer Operatoren in L2(R”), Math. Z. 107 (1968), 67-70.

[5] L. H o r m a n d e r , Linear Partial Differential Operators, Grundlehren Math. Wiss. 116, Springer, Berlin 1963.

[6] M. S c h e c h te r , Spectra o f Partial Differential Operators, North-Holland Ser. Appl. Math.

Mech., North-Holland, Amsterdam-London 1971.

[7] J. T e r v o , On maximal and minimal realizations o f linear pseudo-differential operators, Comment. Phys.-Math. 75 (1986).

[8] —, On Bpk-boundedness and compactness o f linear pseudo-differential operators, Z. Anal.

Anwendungen 7 (1) (1988), 41-56.

[9] R. A. W ed er, Spectral analysis of pseudo-differential operators, J. Funct. Anal. 20 (1975), 319-337.

[10] K. Y o s id a , Functional Analysis, Grundlehren Math. Wiss. 123, Springer, Berlin 1974.

(5.8)

References

d e p a r t m e n t o f m a t h e m a t i c s, u n i v e r s it y O F JYVÂSKYLÂ SEMINAARINKATU 15, SF-40100 JYVÂSKYLÂ, FINLAND