A generalization oî the Riesz-Thorin Theorem

A N N A L E S S O C I E T A T I S M A T H E M A T I C A E P O L O N A E S e r i e s I : C O M M E N T A T I O N E S M A T H E M A T I C A E X V I I ( 1 9 7 3 )

R O C Z N I K I P O L S K I E G O T O W A R Z Y S T W A M A T E M A T Y C Z N E G O S é r i a I : P R A C E M A T E M A T Y C Z N E X V I I ( 1 9 7 3 )

M agdalena J aroszewska (Poznan)

A generalization oî the Riesz-Thorin Theorem

1. Introduction, notation. The aim of this paper is to prove a gener­

alization of the Riesz-Thorin Theorem on interpolation of linear operators {for the Riesz-Thorin Theorem see [7]), where T is an operator from LP(Q) to L (q,X)(û) (both spaces with mixed norm). We also formulate here a theorem which gives an application of this generalization. The results contain those of Campanato and Murthy [4], and Benedek and Panzone [

].

The index i =

, n everywhere, unless otherwise stated. Let R be the set of real numbers, m{ >

, integer,

< pt- < со, 1 < < oo, p 1 > ... ^ p n, q 1 > ... > qn, 0 < < mi Jr p i . In the following we shall apply vector notations, i.e. p = (px, ..., p n), x = (x1} ..., xn) etc. Let Q{ be open, connected, bounded subsets of the real Euclidean space Rmi. By d(Qi) = $ we denote the diameter of Q{ and we write Ü = Pi=i &i, & = P rl=lQi . Let I (Xe-, q {) c= Rmi be the ball with centre at x°te Q{ and radius q ,-> 0 and let Sf = T(x\, gt) д(х0, g) = P?=

I(®?, &'), S = I (x0, g)nQ.

We suppose that the boundary дй{ of is of Lebesgue measure zero in Rmi and that satisfy condition (A) (see [5]). To simplify the notations, we shall write, for example:

\f(x)\dx = J . .. J I f(x)\dx1 . . . d x n,

/ \f(x)\»dx = H/|ÇÎ(S) = / 1 ... ( / I/ ( *)[”' .

Ъ $> Ьп

In this paper we use the definitions of L iP,X)(Q) and M^P,X)(Q) spaces from [5] and [

]. The definitions are slightly modified so that numbers Аг- related e.g. to M (P,X)(Q) should be substituted, in the definition and elsewhere, by numbers Яг-— P Again, if we refer to results of those papers,

Рг p n

we accept this modification. Then e.g. condition A* > —- mi should be substituted by Яг- > etc.

/

2. Definition. We shall denote by L^P,^(Q) the space of functions / locally integrable in Q, for which there exists a positive constant M depending on f such that for every x0, q with x\ eQ{) 0 < q { < there holds the inequality

( 1 ⁾ Then

inf f Jf ( x ) - c \ pd x < M ( f ) П f= i

( 2 )

П

\\\f\\\L(p>*)(0) = sup { / 7 QihPn,n inf / \ f{x)-c\ pdccy/Pn = in f[iif(/)]1/J?«

*=1 ceRbS

0 <6{<еЧ

is a seminorm in l t p,^(Q). It is easy to show the fact: if / e L iP,^(Q)f then for every x 0 , q with e , 0 < q { < q *- there exists only one constant о depending on x0, g , f such that

(3) inf j \f { x ) - c \ pdx = J \ f ( x ) - c { x 0, Q,f,p)\pdx.

ctRbs ь«

The existence of the only one constant c follows from the uniform con­

vexity of the space I^(S). It follows from (2) and (3) that the seminorm in L M (D) can be expressed as:

(4) lll/llli(P,V) = su? { f j e î hPnlPi f I f { x ) - c { x Q, e J , p ) \ pd x y IPn.

x V e â i i=

1

o

The space L (P,*](Q) is a normed space with three equivalent norms (6), (6) and (7):

(б) ||/|1ьМ,0, = { ll / f ô ( 0) + lll/lll2h>.4(fi)}lft’",

( 6 ) = Шта) + ШЯ 1 Ы ы,

(7) п

= \]f\\mQ) + sup { f ] Qï*ipnlpi J \f(x) - c { x ü, Q J , 2)1 pdx}llPn.

xi eûi i==1 0< Q i«ï\

The equivalence of norms (5) and (6) is obvious, but the equivalence of norms (6) and (7) we shall prove in Lemma 1, separately. Let %{x0, q ) be the characteristic function of the set S. Then we observe that X'C(x0, Q,f, 2) is the projection of %- f , f e L 2 (Q) on the line {%•<?: ceR}

c L 2 (Q) (where L 2 (Q) denotes the space LP(Q) with p = (p 19 . . . , p n)t р { = 2 ) and then, among others the correspondence f X'G(xoi

is linear and continuous from L 2 (Q) to {%'C\ ce R}.

Biesz-Thorin theorem 89

L emma 1. I f e L {p,^{Q), 2 < < oo and condition (A) is satisfied, then

(8) ll/llz,(p>A)(fl) II/II i ,( p * a )(/3) ^ cill/z,iP>A)(«)ll •

P roof. It is sufficient to prove the right-hand side of inequalities (

), because the left-hand side one is obvious. Let us fix x\e û i , 0 < < q

Let f e L iP^{Q), pi ^ 2 . It is easy to see from the Holder’s inequality that L (P,X){Q) <=. L(

,A)(i3). It is clear that

И^о, Q’f ’ P ) - c (xo? 2 )12

< 2 2|c(a?0, Q , f , p ) - f ( s o ) \ 2 -\-2*\c(x0, 6 , f , 2 )- f{x) \K Integrating both sides of this inequality with respect to x{ on St, succes­

sively, for i — 1 , n, applying condition (A) we get (9) 1 ф 0 , в , / , р ) - с ( ! 1 ; 1 „ в , / , 2 ) \ ^ 2 [ ] к Г 112 еТт* 12 У~

г'= 1

x{( J \c(x0, e , f , p ) - f ( x ) \ 2dx)ill2 + ( J \ f { x ) - C( x 0, Q,f,

)|

<L?)1/2}

bs . b$

П

2 / 7 K 112 Qimi'2 {( f \o{æ0, Q , f , p ) - f ( x ) \ 2d x y 2+ .

*=i

+ ( / \f(oo)-c{xQ, e , / , p ) |

<&r)1/2}

ft®

22f j K m Qrmil2{ j \Фо, Qj,p)-f(v)\Pdæ}llPn-fl[p(Si)fpi-2^

i —l bS i = l

n

C z f l

Next, it is also clear, that

\ f ( x ) - c ( x o, Q, f, 2)\Pl < 2pi \ f ( x ) - c ( x 0, Q, f , p ) l Pl +

+ 2р1 |с(я?о, Q, f , P ) - c ( x o , e , f , 2 ) Г . Integrating both sides of this inequality with respect to x{ on suc­

cessively, for i =

, ..., n and rising to the suitable powers, we obtain (10) { / \ f ( x ) - c { x 0, g , f , 2 ) \ pd x y IPn

bs

< 2 { f l f ( x ) - c ( x 0, f , p ) \ pdx}llPn +

b$ П

+ 2 \c(x0, Q , f , p ) - c { x 0, Q, f, 2 ) \ [ J [ p { S i ) f lpi.

From (9) and (10) we get

{ / 1 f ( x ) - c ( x 0, e J , W d x } llp« ^ C s f l & ' Vil \ m LiP’*Ha>,

foS i=

hence

n

sup { П QÏhPnlPi / If ( x ) ~ G(xo, Q,f, 2)1 pdxy,Pn < C 3 \\\f\\\L(p,x){Q)

i= 1 bS

о<в{<е\

and hence the thesis follows. _

We shall introduce the product space Ca(0) and we shall prove the isomorphism of L^P,X)(Q) and Ca{Q) under some assumptions, applying the method of Campanato [3].

D efinition . We shall denote by C(Q) the space of continuous func­

tions / in Q.

It is a Banach space with the norm ll/llc(5j =* sup //(я?) I.

xeSi

D efinition . We shall denote by Ca(Q) the space of functions / which, for every , yi e Qi , satisfy the Holder’s condition in Q with the exponent a = K , ..., a j

П

\ f ( v) - f ( y) \ < K [ ] \xi - V i \ ai,

< аг <

. г=1

The space Ca{Q) is a Banach space with the norm

(

11

The term

ll/llo»(5) sup \f{%)\ + sup xeü xi>VieSii

xi*Vi

\fix) - f { y ) \ n

П \xi-Vi\ai

is

11 l/l I lca(fi) sup xi,Vi‘Qi

\f(x ) - f ( y ) i n

П \xi-Vi\ai

г= 1

a seminorm in Ca{Q). It follows from Theorem 2 [5], that

_ } , __ /kb) ,

L (P,X){Ü) c= Ca{Q), where щ = —--- -.

Pi

T heorem 1. If condition (A) is satisfied, mt < A* < Щ + Ри

■at = (Аг- —т г-)/рг-, then spaces L(V,X)(Q) and Ca(Q) are isomorphic.

Biesz-Thorin theorem 91

Proof, (a) Let х г , у { е й { , let f e l S p,^( Q) . It is easily to find from (23) [5] that

sup 11 l/l 11ьОМ)(я) •

xi>Viepi /7 \x. — y.\ai 1 1 i=l

We have from (16) [

]

П

N®, eJ) < см\и»мт -[J ep-m‘)№ <

г=

n и гг-

^5ll/lli(P>A)(fl) j f j &i Pi ^ ll/llz,(P>A)(0)-*

Applying Lemma 3 [5] and Theorem 1 [5] we get Sup|/(a?)| Cfe||/Hi (p,A)(lî) ctiid W

deduce next

ZcQ

\ ~f ( qq \ _f {'ll} I

snp I/(x) I + sup_ —---< C 7 (||/||х(р,А)(Я) + M l/l I\ l № \ q )) xeQ х%уф U \ xi - y i \ ai

an<i 11/11^“ (^) < 11/11ь(г»>А)(Я) •

(b) Let fe Ca( ü ), х{, у{ e $ t- c It is clear that П

{inf j \ f ( x ) - c \ pdx y iPn < J

l/l I lca(«) • ] 7 l ^ -

/ilai

СеЛЬ«

n n

< c.111Л ! /7 f>? / to < c10 ■ • 111/II ic.(Bj • /7 е.-л

» = l 51S i = l

hence

I I l/l I li(P»A)(fi) ^ ^10 I I l/l I ^ “(Й) and

||/|1х(г>.А)(0) ^ ^п11/11са(й)- Prom (a) and (b) the thesis follows.

3. We shall denote by I a linear operation defined for all simple functions / on X — Pi= 1 Rli into the class of measurable functions on Г = Р » =1ЕЧ Let 8 = Р г?=1#г- = Р£=

[

(жг-, д{) п й {] and let (7) denote the norm in the space L(P,X)(Q).

T heorem

. Let the linear operation T be continuous simultaneously

from L p 1 (Q) to l } ,l>’V){Q), where f = (p[, = («}, •••? £«)•

V = (А{, A^), j = 1 , 2 , such that for every fe L p7 (O) there holds the ine­

quality

hen for every 0 < t < 1, T is a linear operation from LP(Q) to where p , q, A are defined in the following way :

1 l - t t

— H--- 2 » % = 1,

Pi p\ Pi

1 l —t t

—1 H--- if. г = 1,

9% 9i 9i

Af l - t t „

A + — AJ, i = 1,

«< 9 a 9i

and there holds the inequality (13)

Proof. We introduce the notations

Pj

(Pi, --- j PL)

P

(Pi, --',Pm)

' 1 i \

9j = (9i, ■ / 1

" V r

J 5

ii

' i i \ II ч / 1

■ , • ■- , — , ~ ( л ,

, «1 /

i =

,

. We define the functions:

at (z) = ( l —z)ai+za*j i = l , . . . , m , Pi(%) = ( 1 - * ) $ + * $ , i = 1, .. . , w,

? 4 {t)pi{t) =

—t)A}# + zA?/Sj, i = 1 , z = Ç + iy.

We have, in particular : a^tf) = аг-, /1г-(/) = /Зг-, а*(0) = aj, /?г-(0) = $•

аг(1) = < $ , Ш = fi , АЛО)А(О) = А}Д, A, (l)^(l) = AJ/?J.

Let ns fix 0 < t < 1. At first, we suppose that qi > 1. Let «я/c ,Q,

• j / — measurable subset. We denote by the vector space of simple

functions on s/, which are finite, linear combinations of characteristic

functions of measurable subsets of We know from [1] that SF{sé)

is dense in Lp{s4) for p { > 1. It is sufficient to prove (13) for / е ^ ( ^ )

Let ns take z in the strip 0 < £ < 1. Let f e ^ r(Q), g € ^( S) and \\f\\jj>ia) = 1*

Biesz-Thorin theorem 93

ilflUii-fys) — 1* We define, as in [1], the functions:

Vi = Vi(xi+ 17 •••:> Я'то)

...7® to )II£(^ .... ÿ (pi

<pj = v A V i + i , ••• ^{, У п )}

/ i ^! » l - f y + i(*) 1-^(в)

ii i— i FT T| ■,Iz^/(p|=1sfc)] 1-^+l 1-^'

1 < г < га — 1, 1 < j < n — 1,

ai+i(*) ai(z)

1 ° i+ l ai ,

“i W

Fz{xi, . . . , x m) = |/| “i signf - f j ip{,

i — 1 n — 1

1 - W

GziVi, - = \g\ -signg f j v j .

We see that for z = i!: Ft = / , ^ = g and we have (see [

])

= K ,!l (

-....,

ато(°))(Д)

|Я|ато(°)/ато =

’

= ii ? 1+,-, ii ( * ....

\

“ to O)/^) = =

and ||Gy _L_ = 1 , ll^l+ij - i - =

.

l ' - p H s ) Let ns consider the function

»(*,

,

) = / / f T F , - G , d y + f [ I F , - c ( x , e, TFS, 2)]G,dy.

i = l S S

Let ns fix x , q . The function h(x, q , z ) satisfies, in the strip

< £ <

,

* = £ + assumptions of the three-lines theorem (see [7], p. 93). If /€#"(£), TFZ is of the form

S т о -1

T F Z - V [ J é ÿ ^ I i x , ) , 7 = 1 »'=1

where ci;- are constants ^

, li} are linear expressions in z, Xj are character­

istic functions of some parallelepipeds. Simultaneously

S

Ф , e , t f „

) = у f f 4 №-с(х, e , t x„

),

7 = 1 i = l

7

= 1 г=1

Substituting this in h(x, g, z ), we see that A (a?, g, г) is a finite linear

combination, with constant coefficients, of exponentials az with a >

.

Then we apply the Holder’s inequality to the function h(x, q , z ), when

0

= щ. From the previous assumptions and considerations, we get

n i i I

IЛ(®, e , » ) l ' < / 7 e V *| / TFin-Qin-dy\ + \ $ [TFirj — c(x, e , T F irj,2)]Giridy\

k = I S S

n 1 X

< 1 П e / * \\TFiv\\ + \ \ T I \ - c ( x , в, I F in, 2)11 . Ив,.,II -L .

L (l ( S ) ^ L 1~P {S

» 1 1 n l 1

= f ] e ^ { l |I ’J ’iJ i5l(s,+ [ J e l V 4 T F in- c ( x , в, TF^, 2 )IIl , !(J

k = 1 *=l

n , x W 1 1 n l l

< П _{ft— X} ^< _{ft —} П _{- x} = п _ft—

In an analogous way we deduce that

Щх, e, 1 +»4)l < f j e t l \\TF1+ij L i M m < f j а ¥ * - м , .

i =l &= 1

Applying the three-lines theorem we get for 0 < t < 1 П

(14) Ih(x, в, «)| < И ^ - м Щ е ^ к i= 1 From the definition of || • it follows that

П П

(15) f ] L m + П ^{eT 4} fi\ \ Tf - c( x, e, Tf, 2)||i3(S)}

г'= 1 i — l

= sup \h(x, Qy t)l for g e ^ ( S ) and ||g|| _j_ = 1 .

9 l}~PV3)

From (14) and (15) for every x, g with x ^ ü ^ 0 < g{ < g% there hold»

the inequality

П

ll2ynW(sl+ / 7 еГ>лт - с ( х , e , T f , n \ L 4 m < M \ ~ ‘Ml г=1

and hence

т ы ^ х щ < м \- 'м \

for every f^ ^ { û ) with ||/||хд>(л> = 1- At last we get the thesis:

№f^Lq,k{Q) ^ M\ 1 M^jWjjp^Qy

If ft = 1, we set Fz as previously in the proof and Gs — g.

Riesz-Thorin theorem 9 5

We know from [

] and from 2. of this note the following facts:

(a) L^>X\Q ) = M M (Q), ¹ < pi < o o , 0 < < mif

2 ,_ /yyi.

(b) L{P'X){Q) = Ca{Q), 1 < Pi < oo, щ < At <

- Pi Let T be a linear, continuous operation from LP(Q) to some space with the norm equivalent to the norm || 'llL{p,x)^Qy We shall denote by M ( p , A) the norm of the operation T. We shall formulate the next theorem, taking into consideration (a) and (b).

T

3. Let condition (A) be satisfied, let the linear operation T be continuous, simultaneously, from Lp (Q) to L 9 (Ü) and from Lp (Ü) to C° 2 (0), 0 < af < 1, such that

\\Tf\\Lql(Q) < M , ( q \ 0)11Л1ьЛо),

\\тпс >ъ ^ м м , т \ \ ^ ау We set for

< t <

1 Pi ~

1 — t t

Pi + Pi ’ i = 1, •••, m, ^{1 1} — t t

— ! + 2 > ^ — 1) •

% 4i Ф • • , n ,

then

A = 4i

t - p A = t 4

1 2 , mA

Г + 1 Г ) , i = 1 , 2 , n,

Tïb • / (P

1 ° if 0 < i < —-—-—— -, then T is a linear operation from LP(Q)

a i + щ /О л

to L( 9 ,X\Q ) and there holds the inequality

m-jqf

2° if —^ —-——г < t < 1 , then T is a linear operation from LP(Q)T

<4 + q\

_ r¥Yb* 77Ъ’

to Ca(Q), = t ~ --- - = ta\ —

— t ) — and there holds the inequality

Q .% Q.i

w \ \ ^ B) = < ^ ; - v , о )M‘ 2 ( q \ m n m a y

References

[1] A. B e n e d e k and R. P a n z o n e , The spaces L p with m ixed norm , Duke M ath.

J. 28 (1961), p. 301-324.

[2] S. C a m p an a t o, Proprieta di H ôlderianita di alcune classi di fu n zio n i, Ann. Scuola.

Norm. Sup. di Pisa 17 (1963), p. 175-188.

{3] — Proprieta di una fam iglia di spazi funzionali, ibidem 18, 1964, 137-160.

[4] —, M. К . V. M u r t h y , TJna generalizzazione del teorema di Riesz-Thorin, ibidem 19 (1965), 87-100.

[5] M. J a r o s z e w s k a , Holder's condition of some function spaces , Prace Mat. 15 (1971), p. 75-86.

(6] — Przestrzenie L(p’X)(Q) i Pasc. Math. (1972), in print.

(7] A. Z y g m u n d , Trigonometric series, vol. II, Cambridge U niversity Press, 1959.