A remark on embedding theorems for Banach spaces of distributions
~-~AN S T R I E B E L Universitiit Jena, Jena, D D R
I. Introduction and results
This note is more or less an appendix to the paper [9]. We use the notions of [9] and recall some of them. _R~ is the n-dimensional Euclidean space:
x -~ (x 1 .. . . . xn) E l~,~. S(R,,) is the Schwartz space of rapidly decreasing (complex) infinitely differentiable functions, S'(R,,) is the dual space of tempered distributions (with the strong topology). F is the Fouriertransformation in S'(R,~), p - 1 the inverse Fouriertransformation. We use special systems of functions {~k}~=0 (see [9], 4.2.1) with
1. ~dx) r S(RD, F ~ ( x ) ~ 0; k = o, 1, 2 . . . .
2. ~{ iV; /V = 1, 2 , . . . ; with supp F ~ c {~]2 ~-N =< I}l ~ 2k+N} for k --- 1, 2 , . . . ; supp -Y~0 C {~I ]el <---- 2N}; (supp denotes the support of a function).
3. [~ c 1 > 0 with cl ~ (~i=o oo ~1)(~);
4. ~ c~ > 0 with I(D~'F~k)(~)] <= c~]~l -I~1 for 0 ~ I~1 =< [n/2] + 1; k = 1, 2 , . . . The most important system of functions of this t y p e is the following. We consider a function ~(x) e S(Rn); (Fcf)(x) ~ 0;
s u p p / 7 ~ c {~[2 -N -<
l~[ ~
2N}; (F~0)(~) > 0 for 1/%/2 ~ l~l ~ ~r (1) I t is not difficult to see t h a t the functions ~k(x) with(F~k)(~) = (F~)(2-k~); ]c = 1, 2 . . . . ; (2) b y suitable choice of ~%(x) are a system of the described type.
Now we define the spaces 2~,q=T~,q(R~) and B~,q=.5~,~(R,,), Let -- o o < s < oo; 1 < ~ o < oo; 1 < q < oo; {~k}~~ is a system of the described type. We set
{ ,xl }
F ; , , --- f l f e S'(Rn), [[{f* = ( 2~U~l(f , ~,)(x)l~)u -~ < oo . (3)
R n
66 H A N S T t ~ I E B E L
f * q)k -= (2z)"/~F-x(Fq~k-F.f) is t h e convolution o f f a n d %. I n t h e same w a y we define for - - r < s < oc; 1 < / o < oo; 1 ~ q ~ ooi
B ~ , q = f l f e W ( R = ) , II{f. ~0k}ll,~(L ) -- (k=02"~llf* ~~ < oo , (4) (with t h e usual m o d i f i c a t i o n for q = oo). Le = Lp(R,) is t h e usual space of Lebesgue-measurable complex functions w i t h Ifl P integrable. I n [9], t h e o r e m 4.2.2, it is shown t h a t t h e spaces _~,, (and
B;, 0
w i t h t h e n o r m s[[{f,~}II~(z~ )
(andII{f* ~}llt~%) ) ~
are B a n a e h spaees, a n d i n d e p e n d e n t of t h e choice o f t h e s y s t e m {%}. A t least for s > 0 t h e spaces B;,q are t h e usual Besov spaces i n t r o d u c e d b y Besov [1], see also Nikol'skij [5] a n d Taibleson [8]. The equivalence of t h e u s u a l definitions a n d t h e definition (4) is p r o v e d in [9], see also [10]. The idea o f using definitions of t y p e (4) is due to Nikol'skij [5] a n d P e e t r e [6, 7]. The spaces x~;, ~ are i n t r o d u c e d b y t h e a u t h o r in [9]. Special cases are t h e well-known Lebesgue spaces$
w
/~,e = H~ = { f i f e W(R.), F-~(1 -1- l~eIe)"Ff E Lp(R.)} 9 (5) (See [9], t h e o r e m 4.2.6). F u r t h e r we define t h e spaces C t = C'(R,); t ~ 0. C = C ~ is t h e set o f all complex continuous f u n c t i o n s f(x) in R n w i t h f(x)---> 0 for
Ix]--> oo. L e t t be a n integer. T h e n is
W e
C' =
{flDogf
e C for use t h e usual n o t a t i o nDOg =
[0r =< t}.
OxO~ 1 Oxen, o~ = (~1 . . . ~n); [~1 = 09, ~j integers ~ 0 .
* ' " j = l
C ~ w i t h t h e n o r m
[Ifllo = ~ m a x
IDogf(x)l
[ogl ~ t x e R a
becomes a B a n a c h space. L e t be t # integer. W e set t = [ t ] - t - { t } ; [t] integer; 0 < { t } < 1;
a n d define
flY
ID~ -- Dogf(y)] }~.C ['1, sup [x--y]('~ < oo for all 0c w i t h 1o~[ = [ t ] .
x C = y x, y E R n
O w i t h t h e n o r m
A I:~EIVIA.RK O1~ : E M B E D D I N G T H E O I ~ E 3 I S F O t t BAIN-AOI:I S P A O E S OF D I S T R I B U T I O N S 67
[D~f(x) -- D~f(y)]
[Ifllo = llfllc{,l + ~ sup
I~t=[,] ~ y Ix - - y[{'}
x, y E R n
becomes a Banach space.
The aim of this paper is the proof of the following theorem.
THEORW~. (a) Let o o > q > p > 1; 1 _<r_< oo; -- oo < t _ < s < 0o; and
s - - n i p = t - - n / q . (6)
T h e n holds
BSp, r c ~tq, r 9
(b) Let o o > q > p > l ; l < r < oo; -- o o < t = < s < oo; and
s - n i p = t - n / q .
T h e n holds
(c) Let 1 < p < 00; t >=0;
a n d
(d) Let 1 < p < oo;
(7)
F;,, c ~'q.r 9 (8)
1 ~ r < ~ 00. Then holds
n + '
, C (9)
rt
B Y +' p , r C C' for t r integer.
1 < r < oo; O < t r Then holds
(lo)
/-/;,ca'q; a < p _ _ < q < ~ ;
s - n / p = t - n / q .This relation is also known [5]. The embedding theorems (9) and (10) are also known.
A special case of (11) is
n + t
H~ C Ct; 1 < p < ~ ; 0 < t # integer.
The first part is well known, see for instance [5]. We give two independent proofs of (7). The first proof is v e r y short and uses the definition (4). A similar proof is given b y Peetre [6]. The second proof is inspired b y a paper of Yoshikawa [11].
Perhaps it will be interesting from the methodical point of view. A special case o f theorem (b) is (see (5))
n + t
F ~ r C C ' . (11)
6 8 H A ~ S T I ~ I E B E L
2. First proof of t h e o r e m (a)
W e choose two systems of functions {~Vk(X)}k~o a n d {~(x)}k~o of t y p e (1), (2) w i t h
( F ~ ) ( ~ ) = 1 for ~ e s u p p F ~ . L e t be f e/~p,,. W i t h
1/(~ = 1 - l i p - f - llq follows f r o m Y o u n g ' s i n e q u a l i t y for convolutions t h a t
]If * cfk]lLq = [If~ * ~ * QkIILu < I]~kllLollf * ~kl[% 9 We h a v e
a n d
~k(x) = 2k"~(2~x); k = 1, 2 . . . . ;
II~kllL~
= 2[1QI[Lo;
k = 1, 2 . . . W e o b t a i n f r o m (12), (13) a n d t h e definition (4)IlfilB, < cxll{f* ~k}llr < c21l{f* ~k}llr
q, rx 7)%)
This proves t h e o r e m (a).
(12)
(13)
p~ r
3. P r o o f of t h e o r e m (b)
L e t {~k(x)}~=0 a n d {~k(x))~=0 be the same systems of f u n c t i o n s as in Section 2. We consider t h e m a t r i x {Kk, j(x)}_~<k,i< ~ w i t h
(FKk, k)(x) = [xl'2-k'(Fok)(x); k = 1, 2 , . . . ; Kk, i(x ) = 0 otherwise.
I t is n o t difficult to see t h a t the a s s u m p t i o n s of t h e multiplier t h e o r e m 3.5 (b) of [9] hold. This shows
1 1
I1[ ~ IF-l[lxlW(f * ~k)]]'JTIILq = ]][ ~ ]F-l[[X]t2-k'F~k2k"F(f * ~k)]l']T[ILq
k = l k ~ l
oo 1
k = l
F o r the >>inverses) multiplier {~:k,i(X)}_~<k,j<~
F.K~,~(x) = Ixl-'2kt(Fek)(x); k = 1,2 . . . . ; _~k,j(x)= 0 otherwise;
2k RE1VfARI~ O N E M B E D D I N G T H E O R E M S F O R B A N A C H S P A C E S O F D I S T R I B U T I O N S 69
t h e assumptions of t h e o r e m 3.5 (b) of [9] are also true. So we can p r o v e t h e opposite direction of t h e last inequality. I t follows
;oo 1
[[fl[~,, ~ fff * %ILL,
+ [t~=1 [F-X[ fx I'F(f 9
%)] l'] "t{L, 9 (14) W i t h t h e aid o f this equivMent n o r m in Ft~,, it is n o t difficult t o p r o v e t h e o r e m (b). I t is k n o w n t h a t~4]xl " = r "'; 1 < ~ < o o ; 7 + ~ = 1 ;
see [3]. L e t be
f e S(R,,)
a n d q > ~o. The last relation a n d (6) show F-XEIxI'-F(f *~D](x) = c f (F-Xl~l-('-0)(x
- y ) . _F-I[I~I'_F( f 9q~k)](y)dy
R n
f --n I 1
= c' Ix -- Yl (~-7+~)_F-x[l~i~2'(f,
%)](y)dy.
R a
W i t h t h e aid o f t h e generalized triangle i n e q u a l i t y we f i n d
1
( ~ IF-X[Ixl'F(f, ~0](x)l~) -;
k = l
c Ix - - Yl ~
IF-*[l~l'~(f
* 9~)](y)1")"dy.k = l Rn
W i t h the aid of the t t a r d y - L i t t l e w o o d - S o b o l e v inequality, see [3], follows
1 oo' 1
II( ~ IF-X[lxl'F(f * e~)]l')" IILq =< ell( ~
I F - ~ E I z I * F ( f *~0] 1')71l~p 9
k = l k = l
T o g e t h e r with (14) this shows
llf][ F' _-< cllfl[~
, f e S(R~) .q,r p,r
S(_R.) is dense in ~ , ~ , [9]. This proves t h e o r e m (b).
4. Proof ot theorem (c), (d)
__n+m
F i r s t we p r o v e (9). L e t m be an integer; m ~ 0 . Let, f E B ~ i . W e c h o o s e a s y s t e m {~)~--0 of t y p e (1), (2) w i t h
k=0
~ 0 H A N S T R I E B E L
Qk(x) has t h e same m e a n i n g as in Section 2. T h e n holds for [~] ~ m D~'f ~ Z ? ~ f * cfk ~ k~( ~ * qk * l ~ e k .
( ~ : convergence in S', see [9]). We h a v e
(Da~k)(x) --~ 2kn+l~lk(naQ)(2~x) "
W i t h t h e aid of u i n e q u a l i t y follows in t h e same w a y as in t h e second section,
(lip + l/lo'
= 1 ) ,IlD~'f * ~kliL~ <= [ID'e~llLp IIf * ~kllL~ =< ~ 2k~-- P + I~lkllf * ~ull',p =< dllfll _~ + .
k = O k=O k = O B P
p , 1
T h e last e s t i m a t e shows t h e convergence o f ~ D~f 9 ~k in L,~(R,,). On t h e o t h e r k=0
h a n d t h e s u m converges in S'(Rn) to Daf. So we o b t a i n sup [D~f(x) l < cllfll & + .
Is[ ~ m x E R n B P l
(9) w i t h t = m follows n o w f r o m t h e fact t h a t G~(R~) (the set o f all complex
n _ _ + m
i n f i n i t e l y differentiable f u n c t i o n s w i t h c o m p a c t s u p p o r t in R~) is dense in Bff, ~ ,
[9].
N e x t we prove (10). L e t be t < m. T h e n holds
0 < t ~ integer. W e choose a n integer c' = (c, c~),
m ,oo
m w i t h (15) (', ")o,~ d e n o t e s t h e i n t e r p o l a t i o n spaces in t h e sense of Lions-Peetre [4], see also [2]. W e s k e t c h a short p r o o f o f t h e last relation. The operator
& / -
%a/
w i t h t h e d o m a i n o f d e f i n i t i o n
D ( A j ) -= f I f e C, ~ e C is t h e infinitesimal g e n e r a t o r o f t h e semigroup in C
Gi(v)f = f(Xx, . . . , Xj_l, :9 + v, x1+1 . . . x,); j --- 1 . . . n .
(15) follows n o w f r o m t h e i n t e r p o l a t i o n t h e o r y for c o m m u t a t i v e semigroups [2, 4]
a n d t h e t h e o r y o f e q u i v a l e n t n o r m s in these spaces, [10]. (10) follows n o w from (9), (15), a n d t h e general i n t e r p o l a t i o n t h e o r y [4, 9],
A R E I ~ A R ] ~ O1~ ] ~ I ~ I B E D D I I ~ G T H E O R E I ~ I S F O R B A N A C H S P A C E S O F D I S T R I B U T I O N S 71
t - - - - + m
B ~ , = ( B ~ , B ~ I ), c (C, C"~), ~ (C, Cm), = C'.
9 r r~,~ t
W e o b t a i n (11) f r o m (10) a n d t h e inclusion p r o p e r t y , [9], t h e o r e m 5.2.3, F:,, : P
5. Second proof of theorem (a)
5.1. A special semigroup of operators
W e consider t h e h o m o g e n e o u s p o l y n o m i a l o f degree 2m w i t h real coefficients a ( x ) = ~_, a . : ; x = (xl . . . . , x~) e tr ;
c~ --~ (~1 . . . . , ~ ) m u l t i i n d e x ; x ~ = x~ 1 . . . x~ - . L e t for a suitable n u m b e r c ~ 0
( - - 1 ) " ~ a~x ~ ~ l x [ 2~, x e R ~ . (16) la]=2~
I t is easy t o see t h a t G(T); ~ 0 ;
(G(~)f)(x) = e(-~)(~+l)~'(~)f(x); f e LI,(R,, ) ;
is a s t r o n g c o n t i n u o u s s e m i g r o u p o f o p e r a t o r s in L~(R~); 1 < p < ~ ; w i t h t h e i n f i n i t e s i m a l g e n e r a t o r A,
(Af)(x) = (-- 1)'+~a(x)f; D(A) = {f](1 § ]a(x)])f C Lp}.
^
(D(A) d e n o t e s t h e d o m a i n o f d e f i n i t i o n o f t h e o p e r a t o r A). W e d e f i n e G(~) b y
^
G(v)f -~ F-~G(v)Ff; 0 ~ ~ < oo; f E S(R~) . (17) W e set
T h e n holds
h(~) = (F-le(-D~+I"(~))(~) e S(R~) .
(18)
n 1
hJ~) = F-~(d-l:+l~'~))(~) = ~ ~ h ( ~ 2~) e S(R~). (19)
1
W e u s e d ~a(x) -= a(~:mx) a n d a well k n o w n (and easily p r o v e d ) t r a n s f o r m a t i o n f o r m u l a for t h e F o u r i e r t r a n s f o r m a t i o n . (17) a n d (19) show
I
( 0 . > : > . ) = (..>-.,.
f
= ( . . > - . , .f
( , 0 )R n R n
7 2 H A N S T R I E B E L
A
The last formula is also meaningful for f E Ls(.Rn). G(z) is a continuous semigroup of operators in Lp(R.); 1 ~ ~o ~ oo: That the operators G(T) are linear and bounded follows from (20). The semigroup property follows from (17). The continuity of the semigroup follows from (20),
(2~) -~12 f h(y)dy = (Fh)(O) = e ~ = 1 ,
1:r n
2 ,
and the usual estimation technique. We want to show t h a t A,
( A f ) ( x ) - - - ~, a,~D~f, D ( A ) = W~,~(Rn), (21) lal =2,~
is the infinitesimal generator of G(~). (W~ ~ is the usual Sobolev space). Let f E S(Rn) and b(x) = 1 ~ [x] 21, where j is a sufficiently large positive integer.
A denotes the Laplacian, E is the identity. Then we have
: Lb--~ T + ( - 1)~a'(x) L1
{ )l}
< c' F-~ (1 + ( - 1)~z~ ~) + ( - 1)=a(x) ~+
-~- (-- 1)nAn) [b(x)\ T + ( - 1)'~a( ) q - + 0 for z ~ 0 .
We used u inequality for convolutions and known estimates for Fourier- transformations. T h a t A is the infinitesimal generator of G(z) follows now from:
(a) the last estimate, (b) S(/~n) is dense in W~ (Rn), (c) ~ is closed operator with 2~
non e m p t y resolvent set.
We notice an interesting special case. Let be a(x) = -- [xl*]2. I n this case holds
h ( ~ ) = c e 2
h~($) are the Gauss-Weierstrass kernels. ^
For the further considerations we need an estimate for the operators G(~).
Let
1 1 1
~ q ~ p > 1; - - = 1 - - - - + - - . (22)
z p q
A R E M A R K O N ~ E I ~ B E D D I N G T H E O R E I g S F O R ] 3 A N A C H S P A C E S O F D I S T R I B U T I O N S 73
Young's inequality for convolutions and (19), (20) show
^ ~ ( • _ •
[IG(r)fI[Lq g Ith~IILJ[flIL e ~- vv 2.~-p q'llfllLp; f E Lp(R.) .
v is independent of T. F r o m the t h e o r y of semigroups of operators follows
oo
-- z E ) - l f = e-~"G(a)fda; z > % ;
0
see [12]. We obtain with theaid of (23) for sufficiently large m
f ~ (__I _ • ~ ( i - • - x
I](A -- "rE)-lf[[Lq < c e-,O(~-2~.p
a'llfilL~d~
= c']lf[]%~ z ~ ' ~ u"0
and
(23)
II(E -- ~A)-lfI]L, <= cjlflIL~ff :'~'P q "; 0 < ~ _< ~1; f e Lp(/~,). (24) The idea of using inequalities of such t y p e for the proof of embedding theorems is due to Yoshikawa [11].
5.2 P r o o f of theorem (a).
The last estimate gives the possibility of a new proof of theorem (a). s, t, to, q, r have the same meaning as in the theorem. We choose an integer m with
2m > s -- t . (25)
W i t h o u t loss of generality we m a y assume t > 2m. Otherwise we would use the lifting p r o p e r t y of the spaces Btq,, and B~,,, see [9]. Finally we choose an integer k with
2 k m > s > t > 2 m . (26)
F r o m the interpolation t h e o r y of the spaces B~,r, [9], and the known fact D ( : l = f o n o w s
~ . , = (LF, n(Ak)) s = (D(.A), D(ff4k)) s-2m (27)
2 ~ m ' 9 2 m ( k _ l ) ' r
and a similar formula for Bt+,. The interpolation t h e o r y for semigroups of operators, [4], shows
8 1
[[f4l B, ~" z - ~ 9 -- E)kfH'Lq + I[f[[z, (28)
q, 9 0
~4 HAI~S TRIEBEL
(with t h e usual modification for r -~ ~ ) . ~ ~ 0 is a suitable n u m b e r . Using (24) we f i n d for f E B~,, C D(A)
I](G(~) - -
E)k, fHL, ~
I[(E - - ~+])-X(G(T) -- ka n d
IIf[lLq ~
]](E - - :~)-lf]lL q ~- [](E --.A)-I.AflIL q ~ C(]If}]L~, +
IlAf]lLp) ~ C'Hfi]~, 9 The last two relations t o g e t h e r w i t h (27), (28), a n d (6) showA s-2m
[llf[IB' < c(l]fl[~,, + l[
:liB,,, ) ~ c'llfl]~, , This proves t h e o r e m (a).R e f e r e n c e s
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Pares A p p l . 45 (1966), 143--290.
3. I~6RMA-~DE~, L., Estimates for t r a n s l a t i o n i n v a r i a n t operators in Lp spaces. Acta Math.
104 (1960), 93--140.
4. L i o n s , J. L. a n d P E E ~ E , J., Sur u n e classe d'dspaces d'interpolation. Inst. Hautes Etudes Sci. Publ. Math. 19 (1964), 5--68.
5. NrKoT.'SKIJ, S. M., Approximation of functions of several eariables and embedding theorems.
Nauka, Moskva, 1969. (Russian.)
6. PEETRE, J., 2'underingar om Besov-rum. Unpublished lecture notes, Lurid, 1966.
7. --~)-- Sur les espaces de Besov. C. R. Acad. Sci. Paris 264 (1967), 281--283.
8. TAIBLESO~, M. i . , On the theory of Lipschitz spaces of distributions on Euclidean n-space I. J . Math. Mech. 13 (1964), 407--479.
9. TRIEBEL, i . , Spaces of distributions of Besov type on Euclidean n-space. D u a l i t y , inter- polation. A r k . Mat. (next issue).
10. --~)-- I n t e r p o l a t i o n theory for function spaces of Besov type defined in domains. I. Math.
Naehr. (to appear).
11. YOSHIKAWA, A., Remarks on the theory of interpolation spaces. Journ. Fae. Sci. Univ.
Tokyo, 15 (1968), 209--251.
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Received February 2, 1972. H a n s Triebel
Sektion Mathematik der Universit~t 69 J e n a
Helmholtzweg 1 D D R
