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0
b. u
0,
if t
f
i n o the r words the Po i s son inte gral o f f :
t
u ( x, t )
f (y)
dy .
Then we have q 1/ q dt ) < T o < s < 2.
g.
Now holds
=
sq B p
h.
{ f \ fE L p 0 < s.
}
00
&
( f;
a2u l \ 2 1 \ t 2 L Cl t p { s t
q 1/ q dt < ) T
00
}
Exten s ion o f the proce dure be gun in f .
For a l l the s e case s we have a t least
s > 0.
and g .
Howeve r it
is e a sy to mo di fy the above appro ach so as to cove r the case of negative s i.
s < l.
( an d s = 0 )
too .
Con s ide r in pl ace o f u the solution v= v ( x , t )
o f the boundary p rob lem
11
if t > 0 if t =
v= f
0
Then holds {
j.
f l f E:
LI &
s real .
( -6
00
I It
�� ts
I I .L p
q dt ) 1/ q < t
00
}
Analogous .
We are now faced with the problem o f see ing what i s common in all the se case s .
First let us cons ide r a smal l variant o f
a . , the case s b . -e . being analogous : a' .
O < s < l. &
Bps q = { f I fs Lp where
ej
=
(O,
•
•
One can show that
•
(
,l,
I l li te . fj I L q p ) dt 1/q < oo ( . =l, J J t) ts
!0
00
.
•
•
• • •
,n) }
, O ) i s the j th b as i s ve cto r o f En .
I f we compare a ' . with f . say , we see that the integral s are bui l t up in same fashion .
We have thus to con front the integrands only , i . e . the expres sion s li te . f and t� at J re spectively . I t i s now readily seen that they both are the e ffe ct of a trans lation invariant operator depending on t acting on f , i . e . o f the form ¢ t * f where ¢ t are " te s t functions " depending on t . The dependence on t i s now particularly simple :
12
0) .
Let us con s i de r the
be s t approximat ion of f in L by e xpone n t i a l fun c t ion of type p < r
(9)
:
E (t 1 f)
Then holds
=
inf
iI
f-g l
lL
p
w he re
s upp g c {
I t;. l
0 f (x) = 0 ( I x I s ) , I x l > 0
(13) Then holds
c\) = O (b -v s ) and
holds with 0 replaced by Proof .
f E: B soo 00
•
An analogous statement
o.
Let us take Fourier tran s forms in ( 12 ) :
We get
17
where
8
i s the de lta function .
¢
(1) Cv 8
Using ( 8 ' ) it now fol lows
( � -b } where one take s t \!
With no los s o f general ity we may assume that
¢ ( 1}
b=
\!
1.
There fore taking the inverse Fourier trans form we end up with Cv
\)
e ib
X
In particular hol ds thus
On the o ther hand , s ince
J cp t (-y) ( Note that n
=
1! )
-we
1
f (y) dy
t
!(- z) f (y ) dy t
obtain using ( 13)
The p roof o f f E Bsoo is s imilar. 00
Having e s tabl i shed the propo sition it i s easy to prove the non-differentiabil ity o f the Weie rstrass function .
Take
18 thus c = av with a < 1 and ab � 1 and assume f is differentiable at some point x 0 . With no loss of generality we may assume that x 0 = 0 (by translation , if necessary) and that f ( O ) = f ' ( O ) = 0 (by subtracting a finite number of terms , if necessary ) . Thus ( 1 3 ) holds with s = 1 and in place of 0 . We conclude that av = (b - v) . But this clearly implie s ab < 1 , thus contradicting our hypothesis . Example 2 . Riemann ' s first theorem on trigonometric series . In his famous memoir on trigonometric serie s from 185 9 Riemann considered functions or , better, distributions of the form \)
o
0
f (x)
00
n=-oo
with em = 0 ( 1 ) as j m j �oo and ( for convenience ) c 0 = 0 . In order to study the summability of the serie s he considered the ( formal ) second integral F (x)
00
l:
m=- oo
(Notice that -F ' = f ( in distributional sense , of course ! ) . ) The " first theorem" referred to above now simply says in our language that F sB :oo (which is the same as the Zygmund class) . We leave the particulars of the verification to the reader.
19
Notes For a modern treatment of the variational approach to Dirichlet ' s problem see Lions [ l ] or Lion s-Magene s [ 2 ] .
;
In
partial d i f fe rential e q uations the space w {Q ) is al so o ften den oted H 1 {Q ) . One o f the c lass ical papers by Sobolev i s [3] .
See a l so his book [ 4 ] .
The first systematic treatment
o f Bpsq (Q ) of s > 0 with de fin it ions o f the type a . -e . using finite d i f fe rence s is Be sov [ 5 ] . The space s Bps q (Q ) , s f
intege r are o ften denoted by w; {Q ) , known a s Slobode cki j space s. The space s Bps oo (Q ) are o ften denote d by Hps {Q ) , known as Niko l sk space s . For other work s o f the Sovie t ( = Nikolski j ) School ( Nikol ski j , S lobodecki j , I lin , Kudrj avcev, Lizorkin , Besov,
Burenko v , etc . ) see the book by Niko l ski j [ 6 ] and also the survey articles [ 7 ] and [ 8 ] .
Somewhat outdated but s t i l l read-
able are f urther the survey article s by Magenes-Stampachia [ 9 ] and Magenes [ 1 0 ] where a lso the applications to partial dif ferential e q uations are given .
In the case p = q = 2 see Peetre
[ ll ] , Hormander [ 12 ] , Vo leviv-Pane j ah [ 1 3 ] .
The t �e atment of
nLipschitz space s " in Stein [ 14 ] , Chap . 5 i s based on Taibleson ' s approach [ 1 5 ] .
All o f the relevant works of Hardy and Littlewood
ca n be found in vol . 3 o f Hardy ' s collected works [ 1 6 ] .
In thi s
context see a lso the relevant portions of Zygmund ' s treatise [17] .
The se authors are concerned with the periodic 1-dimensional ca se ( T 1 rather than llin ) . The first systematic treatment of Be so space s using the definition with general
¢
was given in [ 18 ]
20
( c f . also [ 19 ] ) .
But the spe cial case p
=
q
=
2 appears
al re ady in Hormander ' s book [ 1 2 ] whe re a l so the Tauberian condition i s stated ( see notably op . cit . p . 4 6 ) .
The l atter
was later , apparently independently , rediscove red by H. S . Shapiro who made app l i cations of it to approximation theory ( see his lecture note s [ 2 0 ] , [ 2 1 ] ) .
The constructive charac
terization via approximation theory is uti l i zed in Niko l skij ' s book [ 7 ] .
( C f . a l so forthcoming book by Triebel [ 2 9 ] ) .
Concerning classical approximation theory see moreover e . g . Akhie ser [ 2 3 ] o r Timan [ 1 4 ] .
The characte riz ation via
interpolation originate s from Lions ( see e . g . Lion s-Pee tre [25 ] ) .
The t re atment o f the We ierstrass non-di ffe rentiable
function given here goe s back to a p aper by Freud [ 2 6 ] ( see also Kahane [ 2 7 ] ) .
Riemann ' s theory o f trigonometric seri e s
can be found in Zygmund [ 1 7 ] , chap . 9 .
Quotation :
Le s auteurs ont et e soutenus par In te rpo l . J.
Chapter
2.
L . Lions and
J.
Peetre
Pre l i minarie s on int er po l ati on spac es .
Thi s chapter i s e s sentially a digre s s ion .
We want to
give a rapid survey o f those portions o f the theory o f inte r polation space s which wil l be use d in the se q ue l . F i rs t we review howeve r some notions connected with topological vector space s . The mos t important class of topo logical vector space s are the locally convex space s .
In a locally convex space
E
there
exi s ts a base o f ne ighborhoods o f 0 consi sting o f symmetric , balance d , convex sets I , i . e . ( 1- T) U
( 1)
+
T
UC U
if
a UCU
i f J a/ .2_ 1 . and
0< T
.2_ 1 .
A subclass o f the locally conve x space s are the normed space s .
In a normed space E the topo logy come s from a norm ,
i . e . a re alvalue d functional l l x / I de fined on E such that
( 2)
// x + Y 11.2. I I x I I + I I Y I I
( triangle ine q ual ity)
( homogeneous ) / I ex I I = l c I 1 /x / I // x / I > 0 i f x 1- 0 , //OJ / = 0 ( po s itive de finite ) A complete normed space i s t .e rmed a Banach space . In the type o f analysis we are heading for , howeve r , a somewhat larger class of topo logical space s i s neede d , namely 21
22
Thi s me an s that we rep l ace
the locally q uas i - convex one s .
(1) (1
I
by
( 1-T ) U + T UC A U
)
where
A
i s a con s tant 2_
1
if 0 < ' 2 1
whi ch may depend on
U.
I n the s ame
way we arr i ve at the concept o f qu a s i -n orme d sp ace and q ua s i norm i f we rep l ace
(2 1 )
I I x + Yl l
Note that
(2 " )
(2 1 )
l l x + Yl l
x 13 o f E S and these are , moreover, continuous a operations . The dual space S' = S ' ( JRn) is called the space of tempered distributions . By abuse of notation the duality between s • and S i s generally written as an integral : < f , g> e . g. if
8
f n f ( X ) g ( X) dX i f f E S JR
1 1
g ES
is the "de lta function" then we have
•
I I '
45
= g ( O ) = JlRo ( x ) g ( x ) dx By duality D and
v.
extend to
if
g E:
s
S' .
In dealing with the Fourier trans form it i s often convenient to have in mind two space s lRn , one " latin " space lRn = lRnX with the gene ral e lement
x = (x 1 , . . . , xn ) and the dual " greek " space with the general element � = ( � 1 , , � n ) , the dual ity
R�
•
•
•
+ x n E;, n . Thi s i s al so natural from the point o f view of phys ics where x o ften is " time " ( se c ) and t;, " fre quency" ( se c - 1 ) so that x E;, i s " dimensionle s s " . I f f E:
s
= S X its Fourier trans form is an e lement
given by
A
That
J n e -ixE;, f ( x ) dx . ::IR X
Ff ( t;, )
f ( t;, A
f E: S t;, can be seen from the basi c formulas ( i t;, ) f . a
( D af )
(1)
(x s f )
(2)
( - iD
E;,
) S Ff .
More gene rally (1' )
F (a
(2 ' )
F (b f )
*
f)
Fa F f , 1
Fb
*
Ff,
A
f = Ff of
46
under suitable assumption on a and b. We will also need the formula "'
f ( t t;, ) where
(3)
The inverse Fourier trans form is given by
This Fourier inversion formula has as a simple conseq uence Plancherel ' s formula ( for S ) : J
2 n I f ( x) I dx
JR. X
1
Since F and F -l are continuous operations F : Sx + St;, , F - 1 : S t;, + Sx , they extend the duality to tempered distributions S ' . Formulas ( 1) and ( 2 ) (or ( 1 ' ) and ( 2 ' ) remain valid for tempered distributions . Remark. Using instead duality and Plancherel ' s formula , F has an extension to an (essentially) isometric mapping F : L2 L2 • This is the classical Plancherel ' s theorem in modern language . We have also F : L1 + L oo or even r : L1 + c 0 ( the space o f continuouiil functions tending to 0 at oo ) which is Riemann-Lebesgue ' s lemma. By interpolation we get F : Lp+ Lp ' or even F : Lp + Lp ' p if 1 < p < 2 . These are the theorems of Hausdorff-Young and Paley. -+
47
If
f
i s a function o r a ( tempe red) distribution we
denote i t s support by supp f , i . e . the smal le s t closed set such that
f
van i she s in the complement .
more to a phys ical language supp
A
f
con s i s ts o f those
fre q uenc ie s whi ch are neede d to build up b ination s o f characters e ix s
f
from l inear com-
•
:
that
Appealing once
Now let {
v(S) 'I 0 i f f E;,s int 1\ whe re 1\= {2V-1� I s I �2 +l }
(5)
( Tauberian condit ion ) v
v
1 _:_cs>o i f s;s�s={(2-s)-1 2 .::_lsiS2-€)2
(6)
l;v(O
(7)
v S I o6¢v ( 01 � c 2- I I 6
S
for e ve ry
Sometime s we shal l al so re q uire that 1
(8) Let al so
cil
be
an
(9)
cp €
(10)
�(0
( or
00
L: -oo o if t,; s Ro € then we can define v by v � ( f,; /2 ) . If we in addition assume that (E,; ) > 0 v ( E,;) then we get ( 8 ) , upon replacing v (f,; ) by A
A
I
=
A
if necessary. In this case we can take l:
-1
v=
- oo
A
< U . v
This special type of test function , we encountered already in Chap . 1 , except that there we used a discrete parameter t ( roughly t ::: 2 - v ) We are now in a position to formulate our basic definitions Then Definition 1 . Let s be real , 1 � p � oo , 0 < q � oo we set •
•
{fl f
s
S'
l . Definition 2 . Let s real , 1 � P � oo Then we set (with J �) •
0 then oo Bpk l -+ �p -+ Bpk Moreove r �p = Ppk i f 1 < p < oo (or k =
Al so
Proo f :
=
0) .
As was already stated in Chap . 2 , thi s can be
prove d using interpolation and the theorem below. dire ct proof re sul ts e a s i l y i f we notice that nSq N
Howeve r a s q C n l l N
under the said condition s relating the parameters . The proof o f the s tatement invo lving Pps is left to the reade r . The proo f o f the last s tatement concerning Wpk wil l be po stponed to Chap . 4 .
I t i s based on the Mihklin mul tipl ie r theorem.
Much more interesting i s the fol lowing Theorem 5 (Be sov embedding theorem) . We have the ems q bedding Bps q -+ Bp 1 provided l Proo f : After al l the se preparations , the proof can almost be reduced to a trivial ity .
Then by
Let
( 12 ) o f Lemma 1
in the said conditions on the parameters .
p , s 1 < s and 1 < p < oo . It admits the following immediate Corollary ( Sobolev embedding theorem) . We have �p + Lp np - k , p 2:, p , k integer 2:,0 and l < p < oo . l provided 1 Remark . As we know the corollary remains true for p = 1 too but this calls for a special proof (c. f . Chap . 1 ) . Before proving Thm. 6 we first settle the q uestion of real interpolating Besov and potential spaces , for the proof re quires interpolation . The result is already known to us from Chap . 2 . Theorem 7 . We have Bpsq i f s
=
It has several important corollaries Corollary 1 . We have
Proo f : use the reiteration theorem (Chap. 2 ) . Corollary 2 . Bps q does not depend on { o. hl ( 8 )
Using ( 1 7 ) we see that J s f , sup l l h l ( iy ) I l L ,;S c ( 1 p sup I I h 1 ( 1 + i y ) I I L ,;S C ( l p
+
+
A IY I ) ,
I Y I ) A.
I t i s pos s ib le to show that the three l ine theorem ( see Chap . 2) is s ti l l applicab le . Thus we conclude J sf s Lp and s s s f sP; . This prove s [ Pp o ' Pp 1 ] Pp · For the conve r se we have to f ind an at leas t approximatel y optimal repre sentation f
=
h ( 8) .
A natural choi ce is h (z)
Using ( 1 7 ) one gets then the estimate s
70
so i t i s not pre ci se l y what one wants to but the difficul ty can be easily overcome by replacing
h ( z ) by m ( z ) h ( z ) where
m ( z ) i s a scalar value d holomorphic function , with m ( 6 ) = 1 and whi ch behave s as 0 ( I Y I -A > as z -+ oo in 0 < Re z < 1 . We won ' t enter into the details since we shal l later on give another proo f ( Chap . 5 ) . k Coro l l ary 1 . [Wp 0 , l < p < oo . Corol lary 2 .
The space s
change of coordinate s i f
PROBLEM. case
p
=
s
1 0 , 0 < q .::_
oo
Then we have dr ) l/q < r
Proof : 0 < q 2 = max (p , 2 ) There fore i f
£2
�
Q,q which implie s
•
For the proof o f ( 1 7 ) we con sider an
f
s uch that
n f ( �) = I � � p' ( log i � I > - T in a neighborhood o f C
()()
oo ,
e l sewhere
I t i s pos s ib le to demonstrate the asymptotic deve lopment f ( x ) - C l xl
T >
1/q.
p'
3
o f Chap .
There fore i f
Remark.
1 T ( log -jxl )
There fore
with a suitab le C . other hand by th.
- n
3
sp0q� Lp
, x
�o
f E: Lp i f f T > 1/p. (with o = n/p ' ) we mus t have
q ,;S p
On the
0
S imilar techniq ue s c an be used to show that
92
th . 5 o f Chap . 3 cannot b e improved upon .
93
Note s . Th . 1 i s e xp l i ci t ly s tated in Be sov [ 5 ] and Taibleson [ 1 5 ] but its roots lie much deepe r ( I f n 14 . )
=
1 c f . [ 1 7 ] , Chap .
Th . 3 goe s b ack to Calderon - Zygmund [ 7 5 ] ( scalar value d
case , dilation invariant ope rators ) .
They thereby e xtende d
M . Rie s z theorem - whi ch was first proved by complex variable te chni q ue s - to the case o f several vari able s .
The ir result
has important applications to e l liptic p artial di f fe rential e q uations ( c f . e . g . Arkeryd [ 76 ] ) .
A conside r ab le simp l i fi
cation and c lari fication of the proof in [ 75 ] was obt.ained by Hormander [ 7 7 ]
who also e xp l i citly s tated condition
(5' ) .
The ve ctor value d c a se was first clearly conce ive d by
J . Schwartz [ 7 8 ] who used it pre c i se ly for proving theorems of the Paley-Littlewood t ype .
Le t us further mention Benedek
Calde r6n-Panzone [ 79 ] , Littman-McCarthy-Rivie re [ 80 ] , Riviere [ 81 ] and for a general introduction S te in [ 14 ] .
The
Paley-Littlewood theory arose from the work o f the se authors in the 3 0 ' s .
Again original ly complex variab le techni q ue s ,
notoriously comp l icated by the way , we re use d . Chap . 1 3 .
See [ 1 7 ] ,
For the Paley and Littlewood theory in a rathe r
general abstract situation ( di f fus ion semi-group s ) see Ste in [ 82 ] .
I t i s inte re s ting to note that the Mikhlin or
Marcinkiewicz theorem historical ly was p rove d using Paley Littlewood theory . First by Marcinkiewic z ( 1 9 3 9 ) ( see [ 8 3 ] ) for T l and then , using his resul t , by Mikhlin [ 8 4 ] ( 1 9 5 7 )
94
for :JRn . [ 14 ] ) .
Th . 4 goe s back to the work o f Cotlar [ 85 ] ( c f . Concerning lacunary Fourier serie s see [ 1 7 ] .
Quotation :
S ame as for Chap . 2 .
Chapter 5 .
More on interpolation .
We know already several re sults on interpo lation o f Be sov and potential space s ( see Chap . 2 and Chap , 3 , in particular th . 7 and th . 10 of the latter) .
But in these
re sul t s the e xponent p was fixed all the time (except in the cor . to th . 8 where p varied but the o ther parameter s was kept fixed ) .
Now we wish to see what happen s i f al l para-
meters are varied at the s ame time . Taking into account remark 2 in Chap . 4 we see that the interpol ation of Be sov and potential space s can be reduced to the interpolation of the space s t s q (A ) and Lp ( A ) , i . e . , vector value d se q uence and function space s . We there fore begin by reviewing what i s known to be true about thi s . Let us recal l the de finitions o f the above space s . Let A be any q uas i -Banach space . We denote by t s q (A) , where s rea l o < q .::_ oo , the space o f se q uences a = { a v Soo= O with value s in A such that
S]
being any mea sure space c arrying the positive mea sure
ll ,
we denote by Lp (A) , where 0 < p � oo , the space of l-1-mea surable a = a (x ) ( x E S] ) with val ue s in A such that fun ctions
95
96
I I a I I Lp (A)
( J � < I I a ( x ) l l i ) d ]J (x) ) l /p
< oo .
In an analogous way we introduce the Lorentz space Lpr (A) where 0 < p , r � oo More generally , w being a positive ].l -measurable function in � , we define the space of ].l -measurable functions a = a (x) such that •
The space £ s q (A) is really a special instance of Lp (A,w) . Indeed take : �
=
{ o ,l , 2 , ( {v } ) = 1 w (v ) = 2v s q=p
J.l
•
.
•
}
(discrete measure)
We there fore start with Lp (A,w) . The following results are wel l-known and completely understood . For the proofs we refer to the literature . We separate the complex and the real case . Theorem 1 . (vector valued analogue of Thorin ) . Let A = { A ,A1 } be any Banach couple . Let 1 � p , . p 1� oo Then 0 0 holds
97
l w) provided
1 p
W
1
-+
Let
Theorem 2 .
A
(0 < 8 < 1) .
be a q ua s i-Banach couples .
valued analogue o f M. Rie s z ) .
Let
0 < p0
= 1 p
p rovide d
W
I
1
p1 �
-+ Lp ( (A) 8p
oo
1
•
( i ) vector Then holds
w)
w o 1- 8 w l 8 ( 0 < 8 < 1 )
More gene rally we have
and the reve rsed embedding i f analogue o f Marcink iewi cz ) . 0 < Po
again
1
p1 �
(L
Pa r a
1 p
provided Remark .
=
=
=
1- 8 Po
oo
•
(A , w) , + _j_
r � P·
( i i ) ( ve ctor value d
Let A be any B an ach space .
Let
Then hol ds L
plrl
(A , w) ) r 8
pl ( 0 < 8 < 1 )
Lpr ( A 1 w )
•
Notice that in part ( i i ) o f th. 2 we take
and w 0 = w 1 = w . Thus we do not have a ful l analogue o f the Marcinkiewicz theorem in the s calar case .
A0
A1
A
98
Let us now turn our attention to the space s 3.
Theorem
+
Let
A
=
£ 5 q (A) .
{ A 0 , AJ! be any Banach couple .
Le t
Then holds
provide d
, s
=
( 1 - 8 ) s 0 + es 1 ( o < e < 1 )
In view o f the above ob servation that £ s q (A)
Proo f :
i s but a spe cial case of
Lp ( A , w ) thi s is j us t a re cast o f
th . 1 . ( i ) Let A = { A 0 1 A1 } be any q uasi-Ban ach 0 < q 0 1 q 1 � oo then holds :
Theorem 4 . coup le .
Let
(1)
5 q £ o o ( Ao )
provide d
1 q
I
=
5 q £ 1 1 (A1 ) ) eq I
5
< 1 - 8) s 0 + e s 1 < o < e < 1 )
•
More generally holds : (2) The e xponents ( ii )
Le t
min ( q , r ) and A
max ( q 1 r) are the bes t possible .
be any q uasi - Banach space .
Let
Then holds
99
Jl.
(3)
s0q
provide d ( ii i )
0 (A) ,
Jl.
s 1q 1
(A) ) r = 8
Jl.
sr (A)
s = ( 1- 8) s 0 + 8s 1 ( O < 8 < 1 ) .
Let
A
Take further
Let
be any q uasi-Banach space . s0 = s1 = s .
0 < q 0 , q 1 � oo
Then holds
1 q
provide d
Proo f :
( i ) Again
Jl.
s q (A) being a spe c i al instan ce o f
Lp ( A , w ) , ( 1 ) i s a s traight forward con se q uence o f p art ( i ) o f th . 2. Let us next fix attention to the first -+ in (2) . If
we can again make appe al to part ( i ) o f th. 2 .
r �q
r �q.
us there fore assume
For any se q uen ce
Let
a = { a v � =O
let us write a =
00
L:
\) =0
a
E
\) \)
whe re
E
\)
= ( 0 , • • • , 0 , 1 , 0 , • • • ) (with the 1 in the v -th position )
ain that for e ach v v I I Jl. s J.. q J.. (A . ) < 2 ].
\) S .
].
From thi s we obtain by inte rpolation
i
0 ,1)
•
100
whe re we have wri tten Q,
T = Ass ume now that
r
s 1 q1
(A1 ) ) r e
i s s o small that
T
•
is
r -normable .
We
know that thi s i s po ssible in view o f the Aoki-Ro lewicz lemma . There fore tak ing
r-th powers and forming the sum we ge t
v
r . For the proo f of p � q we take inste ad 00
A
f (s ) wher
00
= I: \) =0
thi s time g (x) \)
v v v 2 - s 2 - n n e ix2 g ( 2 - v nx) .
We re q uire that 1 ° supp . g v C. R v and with c pr independent o f v , with p
cpr 2 - vs 2 0 I 19v I I L pr and s re l ated as in ( 8)
.
llO
1
Th i s leave s us with the con di t i on s °
!!. p i condi t ion
2
t;+
n
s. l
(10 )
(i
0,1) .
=
0
n
�
1
and
Upon e l iminat ing , we find p re ci se l y
I f th i s i s s o we see that
•
( 11)
Oq q a E: Q, = £ . In view o f q q l o -+ £ p • ) By p art ( £ , £ 6 r
f E: B
sq pr
i ff
again this le ads to ( i i i ) o f th .
4 we mus t the re fore
P2: q .
ne ce s s ar i l y have ( ii i )
Use part
(ii)
of th .
( i v)
Use part
(iii)
4.
o f th . 4 .
PROBLEM . To find a pre ci se de s c ription o f s q s q l l ' o o lf r � q. ( I t i s thus n o t a Be sov space . ) ) B (B 6 r P0 P1
1
A fte r thus having te rmin a ted our di scus s ion o f inte rpola -
tion o f potential and Be s ov sp ace s le t us indi cate a few app l i cations of the re sul ts obt aine d .
( Other app l i cation s
wi l l be give n l ate r . ) We be gin wi th the fo l lowing impo rtant co ro l l arie s . Coro l l ary =
5
( 1- 6 ) s
o
1
+
1.
Le t
6s ' l
1 p
1 < P 1 p < ()() an d l e t l o 6 1- 6 + (0 < 6 < p Po l
--
I I£ I I
c s < p p
( 2)
( 13)
I If I I
Proo f :
B sp p
< c
1)
-
I I£
1 1 1 s- o6 p
I If I I
1 6 1 �0 p
p
p
I If I I
l
6 p
Then ho lds
51
p
Po
I If1
6
•
s
p
l
l l
We make use o f the fo l lowing ge ne ral re s u l t for
interpolat ion sp ace s :
(*)
Let A
=
{
A0 1 A 1 }
be any q u a s i - Banach
111
1i. 8q
couple and let A be a space such that q
>
0.
Then holds :
+
A for some
I f one applies ( * ) , ( 1 2 ) and ( 1 3 ) readily fol low , mak ing also appeal t o part ( i ) and ( i i ) o f th . 5 re spe ctive l y . s
=
Corol lary 2 .
Le t
( 1 -8 ) s o + 8 s 1 , p1
=
I I f I
I�
- iI
By ( 4 ) and th . 3 o f Chap . 6 we have F
Also ( 3 ) give s
:
B. 1
n ,1 2
-+
130
F
:
L 00
-+
c
2
Thus by appl ication o f ( 7 ) o f Lemma 1 we get -+
c
p if
1 p
1- e + 1-
2.
e
On the otherhand by th . 6 o f Chap . 5 1
l5
Thus ( 9 ) fol lows .
==
e
00
•
To get ( 1 2 ) we have to invoke the ( obvious )
Lorentz analogue o f th. 5 o f Chap . 3 . Remark .
1- e
=r-+
The p roof i s complete .
Th . 1 shoul d be compared with the JIIJ.ikhlin
mul tip l ie r theorem.
If
p
==
1 ( Be rn s te in ' s case ) the condi tions
on a is almost the s ame but not rea l ly .
In fact th . 1 doe s
not apply to the Hilbe rt trans form o r the Calde r6n- Zygmund ope rators and of course the concl usion i s not true e ithe r in
On i f 1 < p < oo ) P the other hand i t i s possible to prove analogous interpolation
thi s c ase .
( Mikhlin says on ly
as C
•
analogue s of Mikhlin . Vve now give a simple ne ce s s ary condi tion for in Cp Theorem 2 .
We have c
p
-+
nn • -p- I 00 •B - p 00 Bp (i. p '
a
to be
131
Proof :
Let
Then i n particular w e mus t have
v *
\)
I
p
a t: C p where
< ()()
In other words holds :B 0 P c
(13)
Q oo l p -+ C p -+ B Cp where p
I� - l I · 2
=
In view of that we at once get ( using also th l) the following Corollary . Assume that ()()
where l = l pl p
v= -ao
- 2l 1
.
Then a t: Cp Be fore giving the proof of th. 3 it is convenient to It is remarkable settle the analogous q uestion for that we now have a much sharper result. Theorem 4 We have •
•
a t:
C
Bpsq < => sup \)
I I ¢ \) *
aI
IC p
nP aa 21 " We next P ask if this is a sufficient condi tion . ( This i s (was ) the 1 ( Ste in) . mul tipl i e r problem) . Th . 1 re adi ly give s a > n -P-
I�
:
I
-
n-1- > n P p
. S lnce
-
2
p < .:!:.
or
2
we are l e ft wi th a gap .
has begun to be f ' lled up on ly re cently .
F i r s t Fe ffe rman
displayed a counte -example showing that
a 0 cannot b e a
He al so showed a positive re sul t , 1 1 + 1 and in addition > 4n then inde e d 4 2 p p L ater Carle son-S j 6lin and S j � lin i n the spe cial
a
case
n
dition .
=
=
p
multiplie r unle ss 1 that i f > n
It
2
and
2.
--
n
=
3
In particular i f
ple tely settled.
re spective ly re laxed the l atte r conn
=
2
the problem has been com-
The fol lowing figure s i l l ustrate the case :
140
We now give a simple proof o f the original re sul t by Fe ffe rman pre sumably due to Stein .
141
The k e y to the p roo f i s provided b y the fo llowing : Lemma ( Ste in ) .
Let
be the uni t sphere in
S
R1 .
Then
we have Lp -+ L 2 (S ) provide d
F
1
I supp
E
c ""
cp i
2 I
I
if
X E:
I
I
for e ach We also construct a second such famil y
a
•
{ � I } such that
1
(20 )
We next de fine l inear mappings
In view of ( 2 0 ) we have S o T
id
I t i s easy to ve r i fy that
Here generally speaking i f
i s a "bundle " o f space s A I (ove r our net o f cube s ) we denote by � p ( { A I } ) the space o f { AI }
150
By interpolation we obtain (s > 0 ) -> .
For S we have to reve rse the
In particular holds
By dualitie s we can include the case interpolation the case
s
=
0.
s
sup I I cp i b I I < MBpsp I
00
151 We leave it to the reader to supply the particulars . In parti cular we get : Coroll ary . E
b
PROBLEM.
If
s > p!:.. then
M Bpsp < = >
< sup l l b l l sp I Bp ( I )
00
To e xtend the above to the case of Bps q with
general q . Remark .
The above , notably th . 1 2 and its corollary , genera l i ze to the case of P ps 1 < p < oo We conclude this chapte r by giving an app l i cation to ,
•
part ial di ffe rential e q uation s . Example . degree
m
Le t A be a partial d i f fe rential operator of in lRn whi ch i s uni formly e l l iptic in the sense
that the fol lowing a priori e stimate holds
He re I I · I I s
I I·I I
cons tant C depends on
l m
and i f again
n (0 ) f. 0
m)
16 3
0 * f r
r -m 11r * A f
There fore we ge t at once I I 0r * f I I L ..':. r -m I I \ I I A I lA f I I ..':. C r -m Lp p
(ii)
By Wiene r ' s theorem there exists "'
n ( t; )
"'
w
l
( t; )
w
c:
A such that
in a neighborhood o f
Let h E: F ( i . e . , F is a " q uasi-Banach Then we de fine the corre sponding maximal
170
ope rator by M f (x)
H
Then
sup I T r f ( x ) I f E r >0
=
i s no longer line a r .
E
B u t i f we assume that
H:
E +F
at least i s continuous and i f we know that
lim T r f ( x ) exists a . e . when f be longs to some dense subspace E 0 of E then we may conclude that
lim T r f ( x ) r >O Inde e d to this end let us set
e xi s ts a . e . for e ve ry
f EE .
N f (x)
Then
N: E +F =
i s continuous a t 0 ( since N f < 2 M f ) and
f E E 0 • Our claim i s that N f (x ) 0 a . e . for fEE . Indeed for any E > o choose f 0 E E 0 so that I I f - f 0 I I < E • Then i t fol lows that I I N f I I 2_ C ( I I N ( f- f 0 ) I I + I I N f 0 I I ) 2_ C I If - f 0 I I + 0 < CE and
N f (x)
N f
=
0.
0
a . e . for
The re i s al s o a converse o f the above re sult ( the
Banach-Saks theorem) . Taking
Example 1 1 . f (x)
lRn
X
1 me a s . K ( x , r )
,
fl
Lebe sgue measure , let
fK ( , r ) f ( y ) dy • X
Then M is ( e s sential ly) the Hardy-Li ttlewood maxima l operator . The Hardy-Littlewood maximal theorem s imply says that M : L1 + L1 oo and
M: Lp+ L P
if
1 < p < oo •
As a conseq uence
171
we obtain
T r f (x )
-+
f (x) a . e .
E:
Lp , l -< p < oo , whi ch i s Lebe sgue ' s ce lebrated theorem on d i f fe rentiation o f the in f
for
-
de finite integral . Example 12 .
Another famous maximal ope rator i s the
Carle son maximal operator related to the a . e . conve rgence o f . s on T l . the F our1er ser1e .
Afte r thi s general remark let us return to our Besov and potential space s .
Using Paley-Littlewood type theorems
i t is not hard to prove that f E: Pps => sup l 0 r * f l /r s r> O unde r suitable as sumptions on 0 s tronger results whe re the of te st functions 0 Theorem 4 .
Let
f E:
PPs =>
Then
holds where
0
Remark .
Lp
We now wish to prove
•
range s ove r a whole fami ly
He re is a resul t in thi s sense : s l l < p < n De fine p by pl p n s
sup r>O
sup l 0 r * f I Ir s t: Lp oo 0
runs through the set o f all functions
l 0 E: L p ' ( !_p + p' fx a. 0 ( x ) dx = 0 re sult with
•
sup
E:
p = 00
l ) with l l 0 l l L , l , supp 0 C. K ( l ) and p n there i s a similar i f l a. l � s . I f p -> s =
•
By the conve rse o f the argument o f e xample 1 0
we can al so e xpre s s the re sul t a s fol lows :
172
• f E Pps => sup r> O
l lf (y ) - n (y ) I P dy ) P /r sE L p oo
f
in f TI
K ( x , r)
•
I t fol lows re adi ly that we have the Tay lor e xpansion : ( mea s . l K ( x , r ) Proo f :
f
K (x , r )
C ( y -x )a D f ( x ) ) a a
( f (y) -
We want to e stimate
=
=
I t suffice s to take x 0 0, r l . S ince we have f I - s g with g E: Lp . Thus we are face d with the e xpre ss ion =
f 0 ( -x )
f a( - x) f
f ( x ) dx
f a( -x )
f (
l
I X - Yl
1
l x - Y l n-s
C ( -x ) a I a I� s a
y.
D
/..,
"
n s _
He re we have used the Tay lor expansion for po int
g ( Y ) dy dx 1 ) g (y ) dy J a I Y l n-s
l
n s at the lx - Y l We divide the integral into two p arts . First we
integrate ove r the set { l x I
l l lzl g (y)
dy < C I x i s M g ( O )
is the Hardy-Littlewood maximal operator ( see
example 1 0 ) and
k
the intege r p art o f s .
Thus we get the
�
173
bound C J I x I s I a ( x) I dx M g ( 0 ) In orde r to e stimate the integral ove r the complementary set { l x l
�
i I yI }
we con s ide r e ach term by i tsel f .
First
come s the term J a ( -x)
f1
g (y) n -s l x i >2 1YI I x - yl
=
dy ) dx
J a ( -x) h ( x ) dx
He re by the theorem of Hardy-Littlewood ( see chap . 2 ) h E L
p
and we get the e stimate l l o i i L • l l h l l L 2_ C I I oi ! L ( f K (2 ) P P P 1 p p (M I g I ( 0 ) ) ,
1 P P l g (y ) ! dy )
•
The re remain the terms corning from the Taylor e xpans ion . long as l a l < s
there arise no comp l i cations and we re adily C ! l o l l L 11 g ( 0 ) Howeve r 1 i f 1 we must use an auxi l i ary fact from
get e st imate s o f the type ! a l = s and thus
As
k
=
s
•
the Cal de ron- Zygrnund theory not mentioned in chap . that unde r suitable as sumptions on J a ( y ) f ( x-y) dy IY I� E
-+ J
a
holds
a ( y ) f ( x-y ) dy
for eve ry
f E Lp
1
a.e . 1 2_ p < oo
4 1
name ly
1 74
and that we have good e stimate s for the corre sponding maximal functions .
Espe cial ly if
p > l ( and we are in thi s
situation ) this i s n o t ve ry difficult . to the reade r .
We leave the details
1 75
No te s In writing thi s chapter we have obtained much inspi ration from the beauti ful work o f Shapiro [ 2 0 ] , [2 1 ] ( see al so Boman-Shapiro [ 6 0 ] ) . (We are awful l y sorry that we have not been able to write in an e q ually lucid manner ! )
The use o f
Wiene r ' s theo rem i n particul ar i n connection with theorem 3 stems from Shapiro .
We re frain howeve r from making a more
detailed comparison . [ 1 12 ] .
We mention also Lofstrom [ 1 1 1 ] , [ 5 9 ] ,
Regarding the problem o f s aturation we re fe r to the
work of Butzer and h i s associate s ( " die Butzer Knaben " ) , see e . g . Butzer-Ne ssel [ 1 1 3 ] . due to Spanne [ 1 1 4 ] .
The counter -example in remark 2 i s
I t has the following app l i cation to
partial di f fe rential e q uations :
If
u is the bounde d so lution
of an e l liptic e q uation of orde r > 2 then its po intwise boundary value s nee d not e x i s t .
Thi s should b e contrasted to
the case o f se cond orde r e q uation ( Fatou ' s theorem e t c . ) . C f . Spanne [ 1 15 ] .
In th i s context see al so S trichart z [ 1 16 ] .
Conce rn ing the use o f d - dimensional measure s see Adams [ 1 1 7 ] , [ 1 1 8 ] ( c f . Pee t re [ 5 6 ] ) . Stampacchia in [ 1 19 ] .
The space s
LPA
we re introduced by
The pre sent treatment fol lows the
survey article by Peetre [ 1 2 0 ] .
The space B . M . O . was first
treated by John -Nire nbe rg [ 1 2 1 ] , whose p aper al so contains the independence o f
p
( th i s i s the John-Nirenberg lemma ) .
He re
" B . M . O . " i s usua l ly interpre ted as ( functions o f ) "bounded mean oscil lation " but it real ly s tands for my children Ben j amin , Mikae l a and Oppi .
The fame o f B . M . O . rose enormously
176
when Fe f fe rman-Ste in [ 36 ] a few years ago identified B . M. O. as the dual o f the Hardy space H 1 { mn+ + l ) { see Chap . 2 and l l ) Theorem 4 i s mode l led on a re s ul t by Calderon and Zygmund [ 1 2 2 ] { see also Ste in [ 14 ] ) .
The mi s s ing e stimate for maximal
functions can also be found in Pee t re [ 1 2 3 ] .
A good introduc-
tion to the entire sub j e ct o f a . e . conve rgen ce { Banach-Sak s etc . ) are al so Cot lar ' s note s [ 1 2 4 ] .
We further mention Garsia ' s
little book [ 12 5 ] which a l so contains a di scussion o f Carle son ' s work { re fe rred to in e x . 1 2 ) .
Chapter 9 .
S tructure o f Be sov Space s .
In thi s chapter we will con side r our space s from the point o f view o f topological vector space s .
Mo re pre ci se ly
we wish to determine their i somorphism classe s . We shall deal not only wi th space s de fined in the whole o f mn but al so with space s defined in an arbi trary open subset It of mn • General ly speak ing i f X i s a q uasi - Banach space o f fun ct ions or di s tributions in 1Rn we de fine
as the space o f the re strictions to i.e.
f sX ( It) i f f there exists
re striction o f
g
to
It
•
It
g sX
X ( lt )
o f the e l emen ts i n X , such that f i s the
The corre sponding q uas i - norm we
de fine by setting I I f I I X ( It )
In other words
= in f I I g I I X
•
X ( rt) gets identi fied to a q uotient o f X : X (rt ) :::
where
XF denote s the subspace o f X cons i s ting o f tho se
fun ct ions or distributions in X the support of which is con X = �p ( Sobo lev space ) thi s was done already in chapter l . h'e obtain the space Wkp ( It ) . Similarl y taking X = Pps or Bps q we obtain the space s Pps ( rt ) tained in F .
In the case
In what fol lows we shal l mostly take
l < p < oo
•
Our
re sults in the extremal case s p = l and p = oo will be rather 177
178
I t i s maybe intere s ting to mention the
incomplete .
fol lowing . I f n = 1 Borsuk proved that al l the space s C k ( I ) , k > 1 , are isomorphic to each other and thus to C ( I ) (= C0 ( I ) ) the space of continuous functions on the closed unit inte rval I .
On the o ther hand , i f n > 1 Henkin proved that the space s Ck ( I n ) , k � 1 , are not even uniform retracts of C ( I n ) whi ch space i s known to be i somorphic to C ( I ) . The proof i s not very difficul t . Fir s t one rep l ace s I n by S n ( the n - dimensional uni t sphere ) . I f Ck ( I ) we re a uni form retract o f C ( I ) i t mus t be injective . I f one con . . s�ders th e mapp�ng grad : Ck ( s n ) -+ C k- 1 ( TSn ) , wh ere TSn � s the tangent bundle o f S n , then there mus t e xi s t a uni form .,_
retraction
M : Ck - l ( TS n ) -+ Ck ( S n ) , i . e . we have
M qgrad
id.
O n the other hand b y a theorem o f Lindenstrauss one can arrange that M i s linear and using invariant inte gration al so invariant ( for the group S 0 ( n ) ) .
But such an invariant
We have a contradiction . A s imilar re sult holds al so for the space s ( I n ) ( S typinsk i ) .
l inear M cannot be continuous .
�
I t i s there fore intere s ting to note that one neve rthe le s s has constructed a Schauder bas is in ( I n ) as wel l as in Ck ( I n )
�
(Cie sie l ski-Domsta and Schone fe l d ) . After the se remarks conce rning Ck ( Q) and fix attention to the case take Q = lRn •
1
0
-
�) � I I
< oo .
We al so say that A i s of e xponent for eve ry S > a
>a
if A i s of e xponent S
etc.
We give a l i s t o f operators o f Be rn s te in type . Example l . exponent 0 .
If
Obvious .
Example 2 .
If
X =
N then A i s of Be rn s te in type o f
X
LP
( nP ) 1 � p �
oo
(with N
19 4 and A is given by (2 )
Af (x)
=
(2 n) -n
f
ix!; H ( i;) f ( !; ) d!; , n e
m:
where H ( !;) is a given homogeneous positive sufficiently differentiable (outside { 0 } ) function , then A is of Bernstein type of exponent a > ( A- 1 ) / p where and in what follows 2 It - � I = � . A typical example is H ( !;) = l s l = s i + · · · + I f the set { H ( !; ) � 1 } in addition in which case A = is strictly convex it is conceivable that the bound can be improved to max (n / p- 1 / 2 , 0 ) but this has not yet been proved in all generality ( see Chap . 7 ) . Example 3 . I f again X = Lp ( JRn ) and A is given by (2 ) but with H having a different degree of homogeneity in diffe rent coordinate dire ctions , i . e . 1 mn = c -
�
•
then A is of Bernstein type of exponent a .:_ (n- 1 ) I p A m m typical example now would be H ( !; ) !; 1 1 + + !;nn in m m which case A = D1 l + • • • + Dn n (it is assumed that the m . are even positive integers) . and A is given by Example 4 . I f X = Lp ( � ) , 1 < p < •
=
•
•
•
J
oo ,
Af (x)
( 2 n) -n
195
with H a s in Ex . 2 then A i s o f Bern s te i n type o f e xponent a
> n/ P - 1/2. =
Lp ( rl ) , 1 < p < oo , whe re rl i s an n -dimensional suffi ciently diffe rentiab le man i fold with boundExample 5 .
If
X
=
ary , carrying the measure coordinate s
�
=
de termined in terms of local
( x 1 • • • xn ) by a den s i ty w ( x ) , i . e .
x
d�
=
s ( x ) dx ,
and A a formally sel f-ad j o int (with re spe ct to w ( x ) ) e l l iptic partial di ffe rential ope rator , then A is - unde r sui tab le assumptions on the boundary conditions - o f Be rnstein type , o f e xponent
a
> ( n-k ) I p where
k
i s a constan t .
I n compact
mani folds (no boundary ) thi s was e s tabl i shed with firs t k
=
}
and l ate r
of his work
on
k
1
=
by Hormande r , in fact as a byproduct
the asymptotic behavior of the spectral function .
The above probably a lso extends to the q uasi-e lliptic case ( c f . Ex . 3 ) .
What can be s aid for other part i al diffe r-
ential ope rator ( s ay , formally hypoe l lipti c one s ) i s not clear . I f we in e x . 5 spe cial i ze to n
=
1 (ordinary di ffe rential
ope rators ) but al low ce rtain singulari t ie s at the boundary we obtain a number o f cl assical e xp an sion s . Example 6 . rl
=
Thus
( -1 , 1 ) , w ( x )
=
1,
2 d d dx ( 1-x ) dx ( Le gendre operato r )
A
corre sponds to e xpan sion in Legendre pol ynomial s . o f Bern ste in type o f exponent
a
> max ( 2/ P - 1/2 , 0 ) .
Here A i s
196
Example
7.
More generally
( - 1 , 1 ) , w (x) d (Gegenbauer operator)
dX
corre sponds to expansion in Gegenbauer (ultra-spherical) polynomials . If v = 1/ 2 we get back Legendre polynomials . Now A i s of Bernstein type of exponent a > max ( (2V + 1 ) / P - l / 2 , 0 ) . Note that if v = n; l (n integer) then A comes by separation of variable s from the Laplace Beltrami operator in sn ( the unit sphere in lRn + l ) . To some extent the above resul t for Gegenbauer polynomials extends to the case of Jacobi polynomials (with ( 1 -x) A (l+x) in place of ( 1 -x 2 ) V-l/ 2 ) Example 8 . I f )1
rl =
A
(0 , 1 ) , w (x)
•
= X
2V
d X 2 V d (Bessel operator) -x- 2 'J dX dX
now restricted with a boundary condition f ( l ) or more generally f ' ( 1 ) + H f ( l ) = 0 , we get expansion in FourierBessel respectively more generally Dini serie s . Here A is again of Bernstein type of exponent > max ( (2 V +l ) / P - 1/ 2 , 0 ) . If v = n--2-1 (n integer) then A comes by separation o f variable s from Laplace operator in the unit ball Kn of lRn ( restricted by suitable boundary conditions
197
on the boundary Example 9 .
s n-l ) . If
S"l
=
( 0 1 00 ) but w and A are the s ame as in
Ex . 8 (no boundary condition s ) analogous re sults hol d .
Now
we have to de al with the Hankel tran s fo rm . Remark .
In mos t o f the above examples i t i s po s s ib le to
modi fy the origin al ( n atural) wei ght w a l i tt le bit without the property o f A being of Bernste in type get ting lost , only the e xponent has to be change d .
E . g . a lre ady in the case o f
ex . 1 i t i s po s s ib le t o replace
w (x)
=
1 by w ( x )
=
lxi
A
I t would be tempting to try to prove a gene ral res ul t in thi s sen se . Re turn ing to the general c ase we now show that for operators A o f Bern s te in type the ope rators are uni formly bounded in
X
u (�) exist and
for q uite a few fun ct ion s
A admits a rather extende d spe ctral calcul us in
X
u.
Thus
( genera l i z ing
the v . Neumann spe ctral calculus in the o riginal Hilbert space N)
•
Theo rem 1 . e xponent (3)
a
•
Suppo se that A i s o f Be rn stein type o f
Then holds
j j u (�) j j �
sup t>O
c j j A �u l l
* B al l
Conve rsely i f ( 3 ) holds then A i s o f Be rns te in type o f e xponent
> a
•
He re general ly spe ak ing Bps q * the mul tipl icative group JR� = ( 0 ,
oo
)
are the Be sov space on .
In the s ame way we
198
denote by P; * , � * , L; being the potential , Sobolev, Lebesgue space s respectively in :IR� Since dA/A is the Haar measure on :IR� thus holds •
£-
Similarly have
A A
+I
being the invariant derivative in :IRx we
df I Ad I
I *+ Lp
••.
We re frain from stating the de finitions in the case o f Pps The groups :IR ( ) being isomorphic ( the canonical isomorphism is provided by the exponential mapping A e A ) , all the previous re sul ts obtained for :IRn can be carried over to the case of :IR+ Proof (outline ) : Because of the expre ssion to the right in ( 3 ) is multiplicative ly invariant, we may take t 1 , i . e . , it suffice s to prove =
- oo , oo
-+
•
=
(3
I
)
l l u < A) I I � c i i A � I I
* Bal l
Consider first the case a integer. We write Taylor' s formula in the form
199
1
-
u ( A)
( t ) dt
aT
/'" ( 1 - � ) a t a +1 u ( ( a +1 ) ( t ) dt 0 t + t
( - 1 ) a+1
I f we ( formally) repl ace
A
by A we the re fore have
u (A ) = ( -l ) a+l
dt
T
Then we get at once using ( 1 )
a .
We now give t>vo simple consequen ce s o f th . 1 . Theorem 2 .
I f A i s of Bernste in type o f e xponent a then
i t i s al so o f e xponent
f3
when
f3 > a
2 00 ( We have already implicitly assumed this in the fore going discussion . ) * B S l * when S > a . Proo f ·. This follows f rom Bal 1 1 Theorem 3 . Let x 0 and x1 be two Banach spaces satisfying our initial assumptions with the same operator A . Suppose A is of Bernstein type of exponent a . in x . ( i=O , l ) . Then A is of Bernstein type of exponent > a = a 0 ( l- 8 ) + a 1 8 in X F ( X) , F being any interpolation functor of exponent In particular we may take X = [ X ] 8 or X = ( X l e q . Here 8 we have put X = { x 0 , x1 } Proof : By th. l we have +
l
l
-+
-+
•
•
l l u ( A) l l x . , x . � c i >- 1 1 � 1 1 l
l
a
Bl
l l*
( i=O , l )
By interpolation (u fixed! ) it follows
which 1n turn implies l l u ( A) l l x , x < But
1 1 >- � 1 1 a l * Bl l
201 and we are through. Next we di s cuss somewhat the role pl ayed by the parti
cular function ( 1 - A ) � .
It turn s out th at in p l ace o f
( 1- A) � we can use say a function u such that v sat i sfies ( � ) I < C ( l + I E.: I ) a
(4)
v ( i; )
He re
du and v = A d1
A
v
its mul tipl icative Fourie r tran s form ,
i n other words the Me llin tran s form , A
A-i i; V ( A) d A / A .
v ( i;) ( 1 - A) �
u ( A)
If
then
A
v ( i; )
( a ) r ( 1- i 0 r ( + l - i E.: ) a r
so that ( 4 ) certainly ho lds in thi s case .
We shal l not give
the detai l s and mention j us t the corre sponding re sul t in the scalar-value d case on
m. ,
the proo f o f which wi ll be l e ft as
an exerci se for the re ade r . Theorem 4 . some
v
(4)
Then
•
Let
f E
S ' ( JR )
s uch that
v * f E L 00 for
whose usual ( additive ) Fourie r tran s form sati s fie s f E B- a oo 00
We mention also a vari an t o f th .
4 ,
o f a somewhat di f fe rent
nature . Theorem 5 . for some
a> 0
Suppose that the condit ion
v * f ( x) for some v , satis fying
202
0
v
u I I a+ 1 1 * B1 I
To get any farthe r we must put stil l more re strictive conditions on A .
A compari son o f ( 3 ) and ( 14 ) sugge sts the
fo l lowing De fin ition 2 .
We s ay that A i s o f Marcinkiewic z type
in X , o f e xponent a , i f sup I I u ( I ) I I < C s up I I X u I I a+ 1 1* \! \! t >O B1
( 15 )
I
Remark .
Why we choo se the n ame o f Marcinkiewi cz shoul d
be pretty obvious , and al so why we previously chose that o f Bern ste in . Cle arly i f A i s o f Marcink iewicz then A i s also of Be rn ste in type , o f some exponent a
•
But the conve rse fail s .
Expre ssed in symbol s we have : Marcinkiewicz
= > Be rn s te in Be rnste in n , . e . a + 1 + n2 1. .
Be cause o f the homogene ity
sup I I a ; I I � c < co r We conclude that
s o th a t A l.. S o f Be rnste 1.n type o f e xponent ·
. - l. n L > n-l -2 1 ( the
Ll6 sphe rical o r more general the strictly convex case ) . view of Planche re l ' s theorem i t i s al so clear that Be rn s te in type of exponent 0 .
A
In is of
There fore we can apply th . 3
and conc lude that A i s of Bern stein type > ( n - 1 ) / p where 1/ p = l l/p - 1/2 1 theorem.
•
In Chap . 7 thi s was done using Hirschman ' s
By t he dis cus sion o f the multip l ie r theorem for
the ball in that chapte r we also know that the above exponent is not t he be st pos s ib le . In thi s case we can also cons ide r po intwise conve rgence ( usually a . e . ) .
( C f . what we said about this is the beginning
of thi s chapte r ) .
With the aid of ( 1 7 ' ) one shows re adily , 1 stil l under the as sumption a > n2 - , that sup I f � ( x ) I < C r
where
M
Chap . 8 ) .
Mf
(x)
i s the maximal ope rator of Hardy-Littlewood ( cf . F rom the maxima l theorem o f the se authors we now
infe r
B y the den sity argument pre sented in Chap . 8 , i t fol lows now n-1 . for any that f ra ( x ) � f ( x ) a . e . , r � and a > 2With other methods (we re turn to thi s point in a few minute s ) oo ,
one can show that
f � ( x ) � f ( x ) a . e . , r�
oo
for eve ry
f s L2
217
and
a > 0.
With a sui tab le modi fication o f the argument of th . 1 1 one can ne xt conclude that f ar ( x ) 7 f ( x ) a . e . r 7 oo , 1 ) . The se 1 - 2 for eve ry f s Lp 1 1 < p < 2 1 and a > (n- l ) ( p re sults ( actually i t s analogue for Tn which case is somewhat harde r) emanate from S tein . n . S o muc h concern1ng JR
Be fore we ente r into the dis-
•
cus sion of the ca se o f Tn (Fourier serie s ) we re cal l the Poi sson summation formula which claims (18)
Y
L:
s zn
f ( x + 2ny)
The most general condi tion fo r the val idi ty of ( 1 8 ) i s that in the notation of L . Schwart z ) . The f s U B s1oo ( i . e . f s L' l convergence ha s to be taken in the distributional sense . For the proo f one has to take the Fourie r trans forms of both members . Con side r now the operator (Af ( x ) =) A f ( x )
( 2 n ) -n
(We use the symbol ' to emphasize that we stay on Tn . ) us write ( 2 n ) -n �a r (x) * f (x)
Let
2 18 with ( 2 1T ) -n We do not have any more the s imple relation ( 1 6 ) so that it is not possible to reduce to the case r 1 . However from Poisson ' s summation formula ( 1 8 ) we conclude that =
� ra (x)
Y
I
E
r n K ( r (x n Z
+
2 1T y) )
if a > n--2-1 ( spherical or more generally strictly convex case ) . Moreover it is not hard to see that
We get thus
1 . -so that A is of Bernstein type of exponent > n2 1n L1 • As be fore using interpolation (th . 3 ) we also get that A i s of Bernstein type of exponent > (n- 1 ) / P where 1/P j l /p - l/ 2 j . It is also easy to carry over the considerations concerning pointwise convergence . The result is that f• ra (x) -+ f ( x) a . e . if and a > (n- 1 ) { 1 /p- 1 / 2 ) . n-1 We say a few words about the limiting case a 2(Bochner ' s critical index) . As Stein has shown the relation =
=
219 n-1
f r --z- ( x )
-+
f ( x ) a . e . doe s not hold in general i f
f E: L 1 • Thi s i s a gene rali zation of the s ame re sult in the real ly =
n > 1 depends on an old re sul t of Bochner ' s whi ch says that f r n-1 2 - ( x ) -+ f ( x ) at points of regul ari ty doe s not hold in gene ral i f f E: L 1 . ( This in con trast to what i s true in :rnn name ly that n- 1 -fr 2 ( x ) ( x ) a t Din i points i f x E: L 1 . ) The se are also po sitive re sul t s . He mention that S te in showed that n-1 f r -r ( x ) -+ f (x) at Dini po ints if f E: L p > 1 or even P f E: L log L . Some simpler re sults in thi s direction can be
more difficult spe cial case
n
1.
The case
•
1
-+
1
E.g.
tre ated by the interpolation techni q ue de ve loped here . we can prove the re sul t j ust s tarte d wi th
f E: L ( log L) 2
•
Up to now we have mos t ly been conce rne d with sphe rical and more generally the s tri ct ly convex case .
Now we s ay
some thing about the gene ra l c ase ( no assumption on the diffe rential geometry of {H ( �)
=
1} !).
We have a lready noted
that we have on ly the we ake r e stimate ( 1 7 ) in p l ace of ( 1 7 ' ) . There fore we must pro ceed diffe ren tly . In the case of :rnn we apply simply dire ctly Hirschman ' s theorem ( th . 1 o f Chap . 7 ) .
Thi s shows that A i s of Bernste in type o f e xponent a > (n-1 ) I P
in the gene ral case too . Thi s can be e x tende d to the case o f Tn . For po intwi se conve rgence a . e . similarly one has to invoke a Paley and Littlewood type re sul t ( see th . 2 of Chap . 4 ) . But the bound obtained in this way i s a bad one : 1 ). 1 - 2 a > n (p Finally we spend some words ( to be e xact a few hundred ) on the q ue stion o f poin twi se conve rgence a . e . i n the L2 case .
22 0 The following may be considered as a modern treatment of some classical topics connected with orthogonal serie s . We take X N L2 ( � ) whe re � is any measure space and we just assume that A is sel f-adjoint positive . Thus we are back in the trivial situation of ex. 1 . We have the following Theorem 1 5 . Assume that Then holds =
f� l u ( � ) f (x) 1 2 d: ) 1/ 2
0 then ce rtain ly
I t fol lows that =
f� ( x )
a ( 1- �) t + f ( x ) � f ( x)
a . e . for
Thi s is a re s tatement of a cl a s sical re sul t o f Zygmund ' s in the case o f orthogonal serie s .
We use d i t alre ady above
in the case o f Fourie r in te grals and Fourie r se rie s . I f in Ex . 16 we take a = 0 , i . e .
Example 1 6 . u ( A)
=
(i lo
if A < 1 if A > 1
2
1 ,1 X and the con c lusion f 0t ( x ) � f (x ) a . e . i s we have u E: B 1 not true e i ther i n genera l . Howeve r in the case when the spectrum i s discre te , say Namely first obse rve that
A = 1,2,
•
•
•
, there is a way out.
u ( E ) obviously i s con s tant if t is
in an interval be tween two conse cutive intege r s . we may as we l l take we see that
t
inte ge r too , thus
The re fore
t = 1,2, . . .
Now
A u (-) t
=
v (A) with v ( A )
=
vt ( A )
=
{
222 A < t- 1 t if t- 1 < A< t if A � t
1 if A
0
-
1 1* 2 Now obviously v E: B 2 and it is possible to show that sup ! ! v i i � c log t. We are thus lead to the conclusion t >o sup j f (x) j < C log t a . e . if f t: L2 ( st) t> O t This is essentially the content of ( the easier side o f ) a classical result by Menchov- Rademacher . Example 1 7 . Let us return to � ( the case of Fourier series ) Then by the same construction as indicated in Ex 1 6 one can show that if n > 1 •
•
! ! :f t ! I L
1
n-1 � C t -2- if f t: L1
=
L 1 (Tn )
(If n 1 the corresponding result holds with log t) . We remark that ft is of course nothing but the partial sum of the Fourier series of f , the spherical one i f =
223
Note s Almost al l the materi al o f thi s chapte r i s taken ove r , in somewhat updated form , though , from my mimeographed note s [ 1 3 7 ] ( 19 6 5 ) . mentione d .
My papers [ 1 3 8 ] , [ 1 39 ] should perhaps a l so be
Re l ate d i de as , i . e . an abstract (ope rator ) setting
for this type of clas sical analysi s , can be trace d e lsewhere in the l ite rature .
Let us mention Littman-McCarthy-Riviere
[ 8 0 ] , Ste in [ 82 ] , Fi she r [ 1 4 0 ] a s we l l a s work by the people of the Butzer School ( re fe rence s may probably be found in [ 11 3 ] .
Regarding the class ical e xpansions ( e x . 6 - 9 ) we have
l i sted already a n umber o f re fe ren ce s in connection with our discussion of the mul tiplier problem for the bal l in Chap . 7 ( see [ 1 0 0 ] , [ 1 0 5 ] - [ 1 0 8 ] ) .
For Hormande r ' s work on the
a symptotic behavior o f the spe ctral fun ct ion see [ 1 4 1 ] , [ 1 42 ] ( c f . al so [ 1 3 9 ] for a l e s s succe s s ful attempt , and Spanne [ 1 4 3 ] . Regarding l e s s pre ci se forms of th . 4 and th . 5 see [ 1 3 8 ] , [ 1 39 ] .
Distribution semi-groups were introduced by Lion s [ 14 4 ]
and have been s tudied b y many authors ( some auxil iary re ference s can be found in [ 1 3 8 ] ) .
He re we mention especially
the pape r by Lars son [ 1 4 5 ] , bec ause he use s Gevrey function s . Th . 1 0 and the appl ic ation to Markov ' s ine q ual i ty ( ex . 14 ) are from S te i n ' s the s i s [ 1 4 6 ] . The tre atment o f Fourie r in te grals and Fourie r serie s i s insp i re d b y the work o f Stein [ 1 0 2 ] , [ 1 4 7 ] ( See als o SteinWei s s [ 3 7 ] , Chap . 6 ) .
Bochner ' s classical p ape r is [ 1 4 8 ] .
See al so the survey arti cle by
v.
Shapiro [ 1 4 9 ] .
That ( 1 7 ' )
doe s not hold in the general not s trictly convex case was
224 noted in [ 1 5 0 ] . Concerning the general case see also [ 9 5 ] .. rom [lll ] . and what concerns Tn Lofst. For the classical theory of orthogonal series see Alexits [ 1 5 1 ] . The sketch given here follows [ 1 5 2 ] . The same type of methods can also be used in the case of the pointwise convergence a . e . of the di ffusion semi-groups of Stein [ 82 ] . Regarding ex . 1 7 see H . Shapiro [15 3 ] . ..
s.
Chapter 1 1 ,
The case
0 < p < 1.
Now we sha l l extend our theory in yet another direction . In the previous treatment o ur Be sov and potential space s were always assumed to be mode lled on Lp with 1 � P < ()() We wish to extend the discus s ion to include the range 0 < p < 1 . First we sha l l answer the q ue stion : genera l i zation ?
Why make such a
Strange ly e nough I myse l f ( circa 1 9 70 ) was 0
lead to con sider the case
o 1 where I f p > nn+k-1 we than have the crucial result : ( * 1 ) I f U satisfie s ( 2 1 ) then ! u ! P is subharmonic . The above de finition of Hp thus was a la Hardy-Littlewood , via harmonic functions . Fe fferman-Stein however managed to obtain a purely "real variable " characterization of Hp , using E: S with cr ( O ) f. 0 . approximative identities only. Let Then 3
a
(4 )
f E:H sup ! or * f ! E: Lp P r >O
holds . (This should perhaps be compared to the Hardy and Littlewood maximal theorem. See our discussion in chapter
8. )
231
Even more , for a s ui table neighborhood (4I )
f
holds .
E
0
Hp sup 0E
r * fI
E
A
L
o f 0 in
S
P
Using ( 4 ' ) i t i s pos s ib le to extend the Calder6n-
Zygmund as well as the Paley and Littlewood theory ( see Chapter 1 ) to the case of Hp . Anothe r re sul t which fo llows from ( 4 ' ) i s the Fe f fe rman-Rivie re -Sagher interpolation theorem for Hp (mentioned in chapte r 2 ) . (5)
(H
Po
,H
)
if
P1 8 p
1 p
1-8 +
( 0 < 8 < 1) .
Another maj or achievement o f Fe ffe rman-Stein not directly re l ated to ( 4 ) or ( 4 ' ) is the identi fi cation of the dual o f H 1 ( mentioned in chapte r s 2 and 8 ) : (6 ) ( To the dual o f Hp when 0 < p < 1 we return be low . ) Thi s ends our review of Hp space s . In orde r to avoid any ri sk o f con fusion le t us al so state e xplici tly that Hp
Lp
if
1< p
I
n-t < C r lal
or
1 f <x > 1
< c
rn-t 1 + (r l x l > k
when ce the de s ired ine q uality.
I
2 36
0
Lemma 5 . I f G i s compact then we have for any < p < the embedding co
S[G]
-+
Lp [ G]
•
Proof : Obvious conse q uence of lemma 4 . So much for embeddings . The next two lemmas describe the dual of Lp [G] . Lemma 6 . Let G be compact . Let M s (Lp [G] ) ' . Then for any G1 with G1 int G there exists g s L 00 [ G ] such that M ( f) if f s Lp [Gl ] . Proof : Again this is a functional analysis exercise . In view of lemma 5 and the Hahn-Banach theorem there exists h S ' such that M ( f) Define g by the formula g ¢ * h where ¢ l on G1 but supp ¢ CG. Set M1 ( f) By lemma 8 , which we have not yet proven , we have M1 s (Lp [G] ) ' too. It suffices now to show that C
=
s
A-
=
A
=
A
A
Choose f so that f g (O )
•
l
1
=
on supp g . Then we get l
--
( 2 TI )
A
A
n / g ( U f ( � ) d�
237
C =I Ifi lL p g E L00 [ G ) . De fine M by M ( f )
Thus ( 7 ) fol lows with Lemma 7 . Then
Let
M E ( Lp [ G ] )
Proo f :
I '
0 < p ..::. 1
.
•
In view of Holde r ' s ine q uality we have
I If
But by lemma 1
I I L ..::_ C I I f I I L 1 p
•
We have also to con s ide r mult ipl ier s .
The following
lemma is a sub stitute for Minkowski ' s ine q uality i f
1� p�
oo
Lemma 8 .
For any G , Lp [ G ] , 0 < p < 1 , i s a q uas i-Banach algebra for convolution . More p reci se ly , considering the special case
G = Q {r)
=
the cube o f s ide 2r and cen ter 0 , a * f E Lp [ Q ( r ) ] and we
a E L [ Q ( r ) ] , f E: Lp [ Q ( r ) ] then P have the ine q ual i ty : if
n ( p1 - 1 ) < C r Proo f :
Let us again take
Then for any g (O)
r
=
1
and wri te
g
=
a * f.
E >0 =
1
Continue a ( � ) and f ( � ) , res tricted to Q ( l+ E ) , to a period i c function wi th period 2 ( l + E ) and expand the re sulting periodic function in a Fourier serie s .
We get
238
f (�) "
and similarly for a ( � ) . By Parseval ' s formula ( for Fourier s eries ) we then get f ( - l TI+Y E ) .
g (0)
i . e . we have "discretized" the convolution . Now it follows readily that (by the p-triangle ine quality) : l p p ! l l g (x) I l Lp :: 2 n o : l a ( l?'s > l l l f ( x- l7Tls > I I �p )
But by the same result by Plancherel-Polya referred to in the proof of Lemma l it can be inferred that l ( L: \ a ( 1?'s > I P ) P
2
c l l a ! I Lp
•
T he proof is complete . It is now easy to prove the counterpart o f the remaining parts of lemma l of chapter 3 .
239 Lemma 9 . a,
D
a
Let
f E Lp [ K ( r ) ]
Proof :
Wri te
Lemma 1 0 .
f E Lp [ K ( r ) ] . Then for any mul ti -index holds and we have the ine q uality
Da
f
=
a * f
with a suitable Then for any
Let
a,
a E L [K ( 2r) ] . P 0 I f s Lp [ R ( r ) ]
and we have the ine q uali ty
Proo f :
Simi l a r .
Finally we note the fol lowing Lemma 1 1 .
We have the embedding
In fact the topo logy induce d in Lp [ R ( r ) ] by Hp agree s with the one induce d by Lp ( i . e . the topology for Lp [ R ( r ) ) whi ch we have been concerned wi th ) and we h ave
I I f I lL p
n-n be the vector fie ld sati s fying the gene ralized Cauchy-Riemann Proo f :
.
•
.
24 0
e q uations ( 2 ) de termined by the boundary condi tion u0 ( 0 , x )
=
f ( x ) , i . e . we have
{;: ( t , 0 /';; ) We have to ve ri fy that sup l l u j ( t , x ) I I L 2 C < oo p t >o To thi s end we wri te again for a fixe d t , u . ( t , x ) wi th sui table a . ( depending on t) . J
J
=
a . * f (x) J
I t suffice s to ver i fy that
sup I I a j I I L 2 C < oo t >o p which can be re adi ly done invoking lemma 4
.
We leave the
de tai l s for the re ade r . I f we in de finition l substitute Hp we obtain the s ame space s . In othe r words : B sq Hp Remark . The s ame in the case o f de finition l i s Coro l lary .
for Lp • Bps q not true . •
Thi s i l lustrate s a po int re fe rred to alre ady , namely that the space s Bps q and Bps q behave q ui te di ffe rently . We h ave ende d our survey of Lp [ G ] . Afte r thi s thorough • background i t i s e asy to develop the theory o f Bps q and Bps q . Since most of the proo fs are entire ly paral le l to the p revious 1 2, P 2 oo ( see notably chapte r 3 and a l so to a le sser extend chap . 4 - 8 ) , we s tate al l results for B; q only and le ave the modifications ne ce s s ary for �J.. pSq to the re ade r .
one s in the case
241
First we inse rt however an example .
Example 1 . Le t f o ( Dirac function ) . Then n ( ! -1 ) , oo and thi s i s the be s t resul t in the sense that f E Bp P 1 - 1) , q < f ¢ Bps , q i f s > n ( ! - 1) o r s n ( pThi s i s seen p exactly in the same way as in the case of e x . 2 of chap . 3 . Notice that the criti cal exponent n ( p1 1) change s its sign 1 . The s i gn i ficance of thi s wil l appear late r . at p Theo rem 2 . B; q is a q uasi-Banach space . I f 1 .2_ p ..:_ oo =
00
=
- -
1 .::_ q .::_
oo
i t i s e ve n a Banach space .
Proo f :
If
1 ..:_ p .::_ oo this i s j us t th . 1 o f chap . 3 .
The
s ame proof goe s through only in one po int we have to invoke
1 ..:_ p .::_ oo was so obvious
lemma 2 ( the corre sponding fact for
that we had no nee d to s tate it on that occasion ) .
Stric t ly
spe akin g , we need also the anal ogue o f lemma 2 of chap . 3 0
n ( p1 - 1 ) .
where , in order to as sure convergence , we need The p roo f i s complete .
Now we turn our attention to the case o f ( 10 ) . Theorem 1 9 .
As sume that
s < 1, 0 < p < l.
Then
( Here u = ut is the solution of Lapl ace e q uation 6 u 0 y , 1 . e . 1n ot h er wor d s 1n JRn+l + Wlth b oun d ary d ata prov1 de d b f the Poi s son integral of f . ) =
·
·
·
Proo f :
·
·
Again i t i s only one dire ction which matters .
As sume thus with Le t us write with
where
\j!
v
t � 2- v
i s given by
v
t au at ·
257
Again we want to discretize . Writing
we get I <Jv
(x)
1 .::_
C
where we have put sup l vt (x+ e ) t
I e 1 .::.
I·
It follows that 1
l l g v I I L -< ( L: n tnp ( ljJ v* (yt) p ) p l l vt* I l L < p y EZ p The proof is thus complete if we can prove the following Lemma 1 2 (Gwil liam) . We have
Again lemma 12 is a :3 imple conse q uence of the following Lemma 1 3 . Let h be any harmonic function defined in an open set (J) C :ffin + l and let p > 0 . Then l h (x l i .::_ C ( n+l K J( x , r ) I h ( Y l I P r 1
1
dy l P
258
holds where at
x
E:
CD.
K (x, r)
i s a ball
c one then has to con sider w ( s ) as a
�;
fractional derivative
l a Riemann-Liouvi lle .
In one case we
l •B s- p' P p
-+
(A) e
a
P
Using the N.ikhlin mul tip l ier theorem ( see chap . 4 ) we see that i f f E Pps and a = T f one can take w de fined by , xn _ 1 ) . Thus ( 2 ) follows but not dire ctly w (t ) = f ( t , x 2 , •
•
•
the stronge r statement embodied in ( 4 ) .
Howeve r an analys i s
o f the general ab stract re sult reveal s that a t least in our particular spe cial case one get s a se ction S sat i s fying the de s ired continuity conditions and which moreover doe s not depend on s .
Whence e f fe ctive ly ( 4 ) .
Method 2 ° ( via a d i f feren tial e q uation ) . i s based on an ide a of Lizork in ' s .
This treatment
We shall base oursel ve s
on the fol lowing lemma which wil l not be prove d . Lemma .
" l
Con s ider the boundary value problem
dt
f
=
+ I f
0
if
t >0 (I
a
if
t
0
.;::::;;:-; t
xl ) .
2 71 Then holds : l . s- p ' p a E: Bp
f E: .pps
l < p < co but s i s arbitrary real .
where
Let us howeve r ve ri fy that the theorem fol lows from the lemma .
In view of the Extension theorem ( see Appendix B ) we may a s wel l replace JRn by the hal f space JRn+ = {x 1 > 0 } Each f E: .pps admi ts then , i f s > pl the uni q ue repre sentati on .
-
with both
f 0 and f 1 in 0
P; if
I
where in addition t
0
0
if
t> 0
Cle arly
1 p s- p ' Bp
ConThere fore follows from the lemma that Tf E: l s- ' P it is clear by the same token that verse ly i f a E: Bp p a = Tf with f E: Pps for some f. This prove s ( 2 ) but we al so .
ge t readily the s tronge r statement ( 4 ) by de fining S with the aid of the formula Sa = f where f i s pre c i sely the solution
2 72 of the boundary problem .
273 B. On -the extension theorem. We begin by proving the Extension theorem as formulated in chap . l . We thus assume that � is bounded with a C 00 (or j ust "sufficiently" differentiable ) boundary and we want to prove every f E: vi
E Bps q for all W EV '. Remark . I f V is finite dimensional then it is seen that ( l ) is equivalent to the following condition (l ' ) I f for some pair (v, U E V x lRd holds <W , v > h t; all (w, h) E A then v = 0 or t; = 0 . Proof : Let
0
� t; E IRnt; and w E V' . Let us write
0
for
2 80 > ,
f = <w,F
f . J
=
>.
<w . , F J
It i s e asy to see that it i s
suf ficient to prove that
I I �*
(2) where
{ � UJ
�
and
f. 0 }
t i l Lp +
{ ¢v }� = O
K 0 and the set
2v s
the bal l wi th radius view o f
( 1)
(3)
L
v =O
are te s t fun ction s such that the set
contains
sufficiently small
()()
(
>
0.
and center
{ ¢ v ( E; ) f. 0 } contains at 2v i; , s be ing
( Use j u st a partition on unity . )
we have
Here k i s any inte ge r
B pO q
ns f s
fs
w i th
E
BpO q
Then we c an prove Lemma 2 . I f ( 4 ) holds then T : Bpkl Bpkoo for any l ,;S P ,;S oo , k integer > o . Proo f :
-+
By Le ibn i tz ' formul a we obtain l:
(5 )
o.
s' + s" =s l:
C sS ' S "
C sa , a n fJ
1-'
1 f e ix E: D a 1 G (x , l: ) ( i l: ) a n f ( l: ) d � n (2 ) n
�-'
�-'
T S ' ( iD ) S " f
where thus the T S ' again are p seudo di f fe rential operators By 1 o we then have D s f E B O q s at i s fy ing ( 4 ) . P The re fore By ( 5 ) and Lemma l i t fol lows that D S Tf E L . P k oo ° Tf E Bp again by 1 -koo Lemma 3 . I f ( 4 ) holds than T : B� kl -+ Bp for any •
l ,;S P
_,;S
oo , k intege r
Proo f : (5
I
)
> 0.
We rewr i te ( 5 ) a s
•
285 By induct ion we obtain (6) where the
s 13 ,.
are p seudo di ffe rential operators sat i s fying ( 4 ) . kl Let f E L: D f Then by 2 ° f i3 i3 with f E B� . By ( 6 ) and Lemma 1 it fo llows that T n 13 f E Lp . There fore again B�kl .
=
From Lemma 2 and Lemma 3 we now e asily get by in terpo lation ( Chap . 3 , th . 7 ) . Theorem 1 . 1 ,;S P ,;S
oo ,
0
2) .
Mat . Sbornik 74 ( 19 6 7 ) , 5 9 5 - 6 0 7 .
( Russian )
z.
Cie s ie l ski and J . Domsta , Cons truction of an ortho normal basi s in Cm ( I d ) and � ( I d ) . Studi a Math . 4 1 ( 1 9 7 2 ) , 2 11 -2 2 4 .
[ 12 8 ] S . Schone fe l d , Schauder base s in space s o f di fferentiab le functions .
Bul l . Amer . Math . Soc . 75 ( 19 6 9 ) , 5 8 6 -5 9 0 .
[ 129 ] H . Triebe l , Uber die Exis tent von Schauderbasen in Sobolev-Be sov Raumen. I somorphie be z iehunge n .
S tudia Math .
4 4 ( 19 7 3 ) ' 8 3 - 10 0 . [ 1 30 ]
s.
Banach , Theorie de s operations lineaire s .
[131]
A.
Pe l c zyn ski , Proje ctions in ce rtain Banach space s .
�'Jarsaw , 1 9 3 2 .
S tudia Math . 19 ( 1 9 6 0 ) , 2 0 9 - 2 2 8 . [132]
z.
Cie s ie l ski , On the i somorphi sms o f the space s H 2 and m. Bul l . Acad . Polan . Sci . Ser . Sci . Math . As tronom . Phys . 8 ( 19 6 0 ) ' 2 1 7-2 2 2 .
301
[133]
-------
' On the orthonormal Franklin system.
Bul l .
Acad . Po lan . Sci . Ser . Sci . Math . Astonom . Phys . 1 2 ( 19 6 4 ) [ 1 34 ]
J.
1
4 6 1-4 6 4 .
Lindenstrauss , On complemented subspace s o f m .
I s rael J . Math . 5 ( 19 6 7 ) , 1 5 3 - 1 5 6 . [135 ]
A.
Pe lczynsk i , On the i somorphism o f the space s M and
m.
Bul l . Acad . Polan . Sci . Se r . Sci . Math . As tronom .
Phys . 8 ( 1 9 6 0 ) , 2 1 7 -2 2 2 . [ 1 3 6 ] P . En flo , A counte r-example to the approximation prob lem in Banach space . [ 1 3 7 ] J . Peetre
I
0 utgava ).
[138]
Acta Math . 1 3 0 ( 19 7 3 ) , 3 0 9 - 3 1 7 .
Ope ratore r av andl ig Rie s z o rdning (provi sorisk Note s , Lund , 19 6 5 . , On the value o f a distributi on at a po int .
Portug . Math . 2 7 ( 19 6 8 ) , 1 4 9 -1 5 9 . [139] [ 14 0 ]
------
, Some estimates for spectral f unctions .
Math .
z.
9 2 ( 19 6 6 )
M.
F i she r , App l ication s o f the theory o f imaginary powe rs
1
of ope rators .
1 4 6 - 15 3 . Rocky Mountain J . Math . 21 ( 1 9 7 2 ) ,
4 6 5 -5 1 1 . [ 14 1 ] L . Hormander , On the Rie s z mean s o f spe ctral functions and e i genfunction expansions for d i f fe rential operators . Recent Advance s in the Bas i c Science s , Ye shiva Unive rsity Con fe rence , 1 9 6 0 , 1 5 5 - 2 0 2 . [ 14 2 ]
------
, The spe ctral function o f an e l liptic ope rato r .
Acta Math . 1 2 1 ( 1 9 6 8 ) , 1 9 0 - 2 18 .
. 302 [143]
s.
Spanne , Proprietes de s developpments en fonctions
p ropre s de s operateurs e l l ipti q ue s et de s solution s d ' un probleme de Dirichle t . 2 6 4 ( 19 6 7 )
1
82 3-825
c.
R. Acad. Sci . Pari s
o
[ 1 4 4 ] J . L . Lion s , Le s semigroup s distributions .
Portugal
Math . 19 ( 1 9 6 0 ) , 1 4 1- 16 4 . [ 1 4 5 ] E . Larss on , Generalized distribution semigroup s o f bounded linear operators .
Ann . Scuola Norm. Sup . Pisa 2 1 ( 19 6 7 ) ,
1 3 7- 15 9 . [ 14 6 ]
E.
Ste in , Inte rpo lation in polynomial classes and
!'1arko ff ' s ine q ual ity . [ 14 7 ]
Duke Math . J. 2 4 ( 19 5 7 ) , 4 6 7-4 76 .
, On ce rtain exponential sums ari s ing in multiple Fourier serie s .
[ 14 8 ]
s.
Ann . Math . 7 3 ( 1 96 1 ) , 8 7- 1 0 9 .
Bochner , Summation o f mul tiple Fourie r serie s by
spherical means .
Tran s . Amer . Math . Soc . 4 0 ( 19 36 ) ,
1 75 - 2 0 7 . [1 4 9 ] V . Shapiro , Fourier se rie s in several variable s .
Bul l .
Ame r . Math . Soc . 7 0 ( 19 6 4 ) , 4 8-9 3 . [ 15 0 ] J . Pee tre , Remark in e i gen function expan sions for e l l iptic operators wi th constants coe fficients .
Math . Scand . 1 5
( 1 9 6 4 ) , 8 3 -9 2 . [ 1 5 1 ] G . Alexits , Theorie de r Orthogona l reihen .
Berl in , 1 9 6 0 .
[ 1 5 2 ] J. Peetre , Appl ications de le theorie de s e space s d ' interpo lation s aux deve 1oppements orthogonaux . Ser s . t1at . Padova 3 7 ( 19 6 7 ) , 1 3 3- 1 4 5 .
Rend.
303 [ 1 5 3 ) H . S . Shapi ro , Lebe sgue cons tants for spherical partial sums .
Te chn. report , KTH , S tockholm , 1 9 7 3 .
[ 1 5 4 ) J . Bergh
and J. Pee tre , On the space Vp ( O < p -< "" ) . Te chn . repo rt , Lund , 1 9 74 .
[ 155 ) [156 J
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0