= ft = ft(l Pit,

Transcription

1 TWO ELEMENTARY THEOREMS ON THE INTERPOLATION OF LINEAR OPERATORS1 RICHARD O'NEIL Bth therems have t d with functins satisfying Hlder cnditins. Definitin. Let T be an peratr which takes functins whse dmain is n-space int functins whse dmain is a metric space. T will be said t be f Hlder type (a, /3) nrm TV if fr g = Tf, implies I f(x) ~ f(x - h)\ ^ A h \a fr all x and h, I g(u) ~ i(v) I ^ NA u v Y fr all u and v. (Thrughut this paper, when dealing with a metric space we shall dente the distance between u and i; by [ u v\.) Therem 1. Suppse that 0^a0^aigl, (30^0, ft^o and that T is a linear peratr taking functins whse dmain is n-space int functins whse dmain is a metric space. If T is simultaneusly f Hlder type (a0, (3) nrm TV0 and f Hlder type (ai, /3i) nrm TVi and if O^tgl, then T is f Hlder type (a, (3) nrm TV where a at = a(l t) + ait, = ft = ft(l Pit, N g RnN^'Ni, and where P depends nly n the dimensin f n-space. Prf. Withut lss f generality we may assume \f(x)~f(x-h)\ S A ". We first prve the therem in case the dmain f / is the real line, that is when n = 1. Fr r>0, let Then jl/r- * /H if * <r, I 0 if 5 ^ r. Presented t the Sciety, January 23, 1964 under the title Interplatin f peratrs fr Lip spaces; received by the editrs Octber 16, This research was supprted by the Air Frce Office f Scientific Research. 76

2 THEOREMS ON THE INTERPOLATION OF LINEAR OPERATORS 77 f Kris) ds= \ Kris) ds=\, J J M<r f Kr is) ds = 0, and, J* Ki is) ds = 2/r. Let frix) = //(*- s)kris) ds = j fis)krix - s) ds. Let er(x)=/(x)-/r(x), g=tf, gr=tfr, and nr=re,. Then g = gr+vr by linearity f T. fi ix) = J fis)k; ix-s)ds = J fix - s)k! is) ds = j ifix-s)- fix))k; is) ds. / ' (x) ^ f fix -s)- fix) Ki is) ds = f(2/r) - 2r-». ^ WO Case 1. A <r. /,(«)-/,(*-A) ^ A sup /;(y) g2 /* '-1 Case 2. A =r. = 2 I A I"* I A Ii-ir"^1 ^ 2 h «ir -"i. /r(x) - /,(* - h) = J* (Jix -s)-fix-h- s))kris) ds ^ h " = A» A I"-"1 ^ A «if«-a». In either case,/, satisfies a Hlder cnditin f rder ct\, indeed, /,(x) -/,(*- A) g 2f-^ A ^. I & («) ~ frw I ^ /Vi2r^"i\u-v *.

3 78 RICHARD O'NEIL [February er(x) = fix) - f^x) = j ifix) - fix - s))kris) ds. Case 1. A ^r. e,(*) =g f J l»l<r \s\"kris)ds^r". Case 2. \h\ <r. er(x) er(x A) ^ 2r" ^ 2ra- \h\a. e,(x) -erix-h)\ g /(x) -/(* - A) + /r(x) -/,(* - A) ^ A a + A a ±S 2 A \ar"-". Thus er satisfies a Hlder cnditin f rder a- Therefre, if we set r = (7Vi w-» 77r(«) rjriv) ^ N02ra-a«\ u v I13". ^-V7V0)1/(ai-">, I g(«) - #00 ^ I gr(«) - griv) I + I 17,(«) - -Oriv) \ g 22Vrif"_a' u - n & + 2AV*~a 0 = 4/V TVi u z> 0. This prves the therem when the dmain f / is ne dimensinal. Fr ra> 1, the case ra = 2 is already sufficiently general t illustrate the prf. In this case we let (3/7rr2-3 I 5 I Ar3 if I 5 I < r, K^S) = \ ' ' l0 if I * I r. Fr a given A = (Ai, A2)?^0, let 6/30 dente directinal differentiatin in the directin 0 = h/\h\. Then fid/dff)kris) ds vanishes and J\ id/d9)kris)\ds = Oil/r). id/d6)frix) \ ^ f \fix-s)- fix) id/d8)kris) ds J \»\<r ^ r"oi\/r) Therefre, if 0< A <r,6 = h/\h\, = Oir"-1). frix) - frix - h) I g I h I SUp I id/dd)friy) = A O^1) = 0(1 A 0,ira-"i). The rest f the prf ges thrugh as befre.

4 1966] THEOREMS ON THE INTERPOLATION OF LINEAR OPERATORS 79 Definitin. An peratr T is said t take Lp int Lip a with nrm TV if fr g=tf, I g(u) ~ g(v) I = ^II/IIp Iu ~ v \a f r au u an(iv- If / is a measurable functin and y>0, let m(f> y) = m( I/I > y) = measure f [x: \f(x) > y). It is easily shwn that Furthermre, fr p>0, J f(x) dx = J *»(/, y) dy. m(\f\p> y) = meb\s{x:\f(x) \" > y] = measfx: \f(x) \ > yllv} = m(f, yxlv). (II/IIp)* = J \f(x)\*dx = f m(\f\",v)dv Given k^o, let and let m(/, wi/p) di> = /> I J /* 00 f* CO»?(/, y)yp~i dy.,,. //(*) if /00 *, fk(x) = <. (k sgnf(x) if f(x) > &, f(x) = f(x) - fk(x). Therem 2. Suppse that 0<p^pi^, a0^0, ai^o, ared <Aai P is a linear peratr taking measurable functins n a measure space int functins whse dmain is a metric space. If T simultaneusly takes Lp<> int Lip a0 with nrm TV0 and LPl int Lip ai with nrm TVi and if 0 f t ^ 1, <Ae«P t es Pp i«/ Lip a with nrm N where l/p = l/pt = (1 - I)/p + t/pi, a = at = (1 0«+ tai, TV g TvJ~'tv[/(1 - tj~'t ^ 2NYlN[. (It is t be remarked that 1/(1 ty~hl tends t 1 as t tends t 0 r 1.)

5 80 RICHARD O'NEIL [February Prf. Suppse, withut lss f generality, that / P= 1. Fix k^o, then /=/*+/*. Let g=tf, g0=tf and gi=tfk, then g = g+gi by linearity f T. /> CO y*>-lm(jk, y) dy = p0 I s% OOy*»-lm(J, y + k) dy 0 *^ 0 XOO /» OO (z - ^)p»-1w(/, 2) dz^p j z"«-lmif, z) dz f% 00 ^ />0Aw'-p j z^m/, z) dz ^ ipk* -'>>/p)i\\f\\vy J k = pk^-v/p. Thus /*HPg( /p)1/p &1~p,P; since T takes 7> int Lip a0 with nrm JV, g(«0 - gw S Nip/py^k1-*1** \u- v \"K y^mifk, y) dy = Pi I y^~lm(j, y) dy /> 00 /» jfc *^ ^ ^i/few-" I y^mif, y) dy ^ pik^-^/p. J Thus /4 Plg(pi/^)1/!,i^1_I',pi, and this last equatin is valid even if pi =. giiu) - giiv) ^ Niipi/pyink1-''1* \u - v \"K If we set 4 = 1/p - I/pi, then \/p - 1/pi = Ail-t) and l/p-l/p = At. if we let kva = (//i _ l)ip/pyi">ipi/p)-ll»-in/ni) \u-v «- \ I g(«) - g(?) I ^ I f(«) - f(») I + I gi(«) - gi(») = Nipi/py'^k-**' m - d» + Niipi/pynk'**-1-') «-» «i = ^-Vi(i//(i - t)1-t)(p,/p)1-tl"(pi/p)"pl\«-v\\ Let s = ip/pyi-t)imipi/p)tln, then

6 1966] THEOREMS ON THE INTERPOLATION OF LINEAR OPERATORS 81 lg B = (l/p) \g{x/p) - (I- t)(l/p) lg(l/^0) - t(l/pi) lg(l/px). But x lg x is a cnvex functin f x^o, s that (l/p) \0g(l/p) ^ (1 - t)(l/p) l0g(l//> ) + t(l/pi) 105(1//,!). Thus lg P^O, P^l and the therem is established. Remark. It is pssible t strengthen the result f Therem 2. We shall say that a measurable functin / belngs t weak Lp if there exists a number A such that fr all y>0, (f, y) S (A/y)p. If /GPP then / belngs t weak 7>, since m(f, u)u^ du^ p I m(f, u)u*~x du J ^ pm(f, y) f m"-1 du = m(f, y)yp. J 0 *(/, y) <\\f\\>/y)>- We shall say that a functin /GLip a if fr all u and, We shall say /GLip a if I /(«) /(») I ^ A M D ". /(«) -/(«0 = (\ u - v\") as w v\ tends t zer r infinity. 1. Under the hyptheses f Therem 2, P takes weak 7> int Lip a if p<p<pi- 2. Under the hyptheses f Therem 2, P takes L" int Lip a if p<p<pi- T prve 1, we suppse that m(f, y) g l/yp. Then J* k 2p-iw(/; 2) rfz g (^ _ pt)kf*-p, Similarly, if we let II/4L ^ (/>!//-! - py»»ki->">k

7 82 RICHARD O'NEIL kpa = (TV/TVi) u - v \""-"\ I g(u) ~ g(v) ^ TV"'TVi I u - v f{ (/, //>- P)'"" + (Pi/Pi - p)llp1}. T prve 2, we bserve that /> m(f, 0i/p) *, Since w(/, vllp) is a mntne functin f v, the finiteness f the integral implies Therefre, and Again we may let m(f, vllp) = (l/v) m(f> y) = (l/yp) as ^ tends t zer r infinity. as y tends t zer r infinity. I!/*IIp = (k1-plp«) as k tends t zer r infinity, lk*l pi = (kl~plpi) as k tends t zer r infinity. kpa = (TV/TVi) u v \"-"k I g(u) - g(v)\ = (\ U - t) «). References 1. R. Firenza, Theremi di interplazine per transfrmazini tra spazi di funzini cnderivate hblderiane, Ricerche Mat. 10 (1961), , Ulteriri risultati sulla inter plazine tra spazi di funzini cn derivate hblderiane, Richerche Mat. 11 (1962), Rice University