Peetre, Jaak

English

Peetre Jaak

University of Szeged

On interpolation of Lp spaces with weight functions By J A A K P E E T R E in Lund (Sweden) 0. Introduction According to a classical theorem of BERNSTEIN ( 1 9 1 4 ) a sufficient condition for a func t ion/ on the real line (— «.) to be given by an absolutely convergent Fourier integral. № = ^ f e * i f t Q d Z , KL,, ' is that / € £ 2 a n d satisfies a Lipschitz condition of exponent in L2, i .e . | | / ( X + 0 - / ( . V ) | | L 2 = for some E > 0 and as t-0. A more precise condition in this sense reads: ( 0 . 1 ) j | | / ( x + i ) -/(x)||l2 J 5 7 3 < o Now the r equ i r emen t /€¿2 ' s inessential. (Usually this theorem is given for Fourier series but the change to integrals is immediate; see [9], vol. 1, pp. 2 4 0 — 2 4 1 ; see also [8].) More recently BEURLING (see [2]) has shown that condition (0 . 1) on / is equivalent to the following one on / : (0.2) J | /'(c)[2(x>(|i|) dc < f o r some non-decreasing co with J o This is of importance in some questions of Spectral Synthesis (see [1], [2]). The purpose of this note is to show how this as well as some other results of [2] can be interpreted from the point of view of the theory of interpolation spaces. The plan is as follows. We first (Section 1) briefly summarize some general notions on interpolation spaces (A0,Al)0q which will be needed in what follows. Then (Section 2) we specialize to the case of spaces Lp(w), Lp space with weight function >v. In particular we characterize (Z,p(ir0), Lp{wi))e q when q = 1 or q = This should be contrasted to known results when q=p (see [7]). The applications to Fourier integrals finally are given Section 3. 62 J. Peetre 1. General notions on interpolation spaces Let A0 and AL be two Banach spaces both continuously imbedded in the same topological vector space 21. If 0<i<°=>, f e A o + A ^ set K(t,f) = K{t,f-, A0,A,)= inf (H/oLo+ill/ilL,) /=/o+/i and, if 0 < i J ( t J ) = J(t,f; A o m a x (jl/H^, / \\f\\Al). There are basically two ways of obtaining interpolation spaces: 1° We impose a "growth condition" on K(t,f) of the form W , / ) ] < - • , • ' .. where <P is a suitable functional. 2° Upon representing / in the form / = j / ( 0 - j - (non unique!) u we impose a "growth condition" on J(t, f(t)) of the form The most important special case is when = <¡>e,№ = [ J (t-e<p(t))"~ 1/9 , 1, In this case the two constructions 1° and 2° lead to the same spaces (up to an equivalence of norm) which we shall denote by (A 0 ,A, ) 9 ^. For more details about these spaces, see e. g. [6], see also [3], [4], [5]. 2. The case of Lp spaces with weight functions Let Z be a locally compact space provided with a positive measure ¡x. Let w be a positive ^-measurable function (weight function). We denote by Lp(w), I p -< the space of /(-measurable function / such that \wf\p is /(-integrable and endow it with the norm n M w ) = W \ \ L p = ( S W \ ' d v ) U " . Interpolation of spaces with weight function 69 Let now u'o and w, be any two fixed such functions and take A0 = XpOo), A, = Lp(iv,), Thus in what follows K(t,f) = K{t,f- L„( w0), ¿-p(vvj)), J(t,f)=J(t,f; Lp(u'0), £,(«',)). By [7] we have (2.1) (2.2) Put further * ( ' , / ) - l l / l l i . ll/llz. p(niax(H'o, iwi))* | = w0 Uo 0 (vv0</vw1), K d n f ) = \\f\\LpM, = (lV0<iVV!), (Wo^tWi), IIM«)' w = u ' (2) = /11'! (Wo^tW,), • (vt'o< t\\>i <2w0).. - W ) = 1 1 / 1 1 W = 0 ( e l s e w h e r e ) . Then holds T h e o r e m 2. 1. The following inequalities are valid: (2.3) * ( / , / ) : £ C „ ( I J A ^ / F } " ' ( / - 1 , 2 , 3 ) where = 0 ( a . 1 ) , ^ ( f f ) l = l ( a - l j , It follows that if and only if (p3(a) = min (i , f ) | _ j ML'(w0),L"(Wl))Piq (q =p) = a (CT<1), J [(-"KiitJY — ( < = 1 , 2 , 3) . P r o o f . For simplicity we consider the case / = 1 only; the other two cases cart-he treated in a similar fashion. The inequality to the left in (2. 3) KJt,f)^CK{t,f) 64 J. Peetre is trivial, by (2. 1), since clearly iv(1) Smin (iv0, / t e j . Remains the inequality to the right. But we get, again by (2. 1), {K(t,f))p ^C'J |min O o , tWi)f\' dp z f \w0f\Pdn + 2-r f \w0f\'dfi + »•0<2<tt'i 2t»'iS»'o<22Iivi • +2~2p f \w0f\'dn+..]* 2!WiS»'o<23rwi 3S C'[[Kl(2t,f)]' + 2-p[Ki(2*t,f)]' + 2-*'[Kl{2*t,f№+...] ^ ^K^taJ) a and this inequality too follows. If /€(£p(w0) , Lp(wx)\q the trivial half of (2. 3) shows that ©e / dt HP •Conversely, if this condition holds and moreover q~.p, upon writing the other half of (2. 3) as [t-°K(t,f)]p =s C' foi°- »'[{otyoKt (<x t j ) ] ' — . o • we get by MINKOWSKI 'S inequality [t-°K{t,f)Y> dt pli [t-0KL(t,f)]< dt HI • which shows f€(Lp(w0), Lp(wx))0q. Next we prove — this is our main new contribution T h e o r e m 2 .2 We have ••(2.4) f£(Lp(w0\Lp(wx))9A Jf and only if ..(2.5) ^ ^ ( " ^ f e ) ) for some non-decreasing <p such that i UJ± J UwJ > 1 + 1 = 1 p p Interpolation of spaces with weight function 65 P r o o f i) Assume that (2. 5) holds true. Put f / (w0<2«w1^2w0), 10 (elsewhere). UP w0f\'dn\ wo<2'*>VI3 2M'O Then, by (2. 2), J \Wof\>d»]l% S c 1 <p{ 2~k) / bfek wq< 2k wi ̂ 2wo dfi UP the last estimate because cp is non-decreasing. It follows by HOLDER'S inequality that ^ 2~ke J(2k,fk) ^ C| | / | | ^ C ii/II. Z [cp(2-x) / HP1 If we put this gives / ( 0 = ( l o g 2 ) - 1 / dt UP' r j(t,m) J t° t and, since also / = Jf(t) — , we have established (2. 4). o { ii) Conversely assume (2. 4) holds true. Then dt or, in view of (2. 1), with f K(t,f) iJ te t o <p(A) oo = f [ dt ~ [J t9[K{t,f)]"-1 t . UP which is obviously non-decreasing. But K(t,f) s max( l , tX)k\-^, f 5 A 66 J. Peetre Hence (<P(V)P S xe r H P K \ j , f [min (1, ?A)]P dt (AO®[max (1, tX)]p T C A® K h ' f \ p - 1 or A° <p(A) and it follows at once the crucial condition: [ ( A" Y J U(A)J dX_ A Thus we have shown that / satisfies (2. 5). We conclude by mentioning a sort of dual result. T h e o r e m 2. 3. We have (2-6) / £ ( L p K ) , £>•,)),,, if and only if f I w w0(p (2.7) Ï» for all non-decreasing <p such that Prof , i) Assume that (2. 7) holds true. We claim that then (2.8) sup 11/11, Indeed if this is not the case there is a sequence of non-decreasing functions such that f H - / , , , f i v M P d t 1 J W o ( P v - ] / d,= l, x o Jf we set (<p(0)" = 2(<PM)Y v then <p is obviously non-decreasing too and q>(t)Y dt_ M < y Interpolation of spaces with weight function 67 but I\Wo(p^r}f\dfl = ?1 = co-Thus we get a contradiction1)-Upon taking <p(i) = min (1 ,1t 0 ) in (2.8) we get f\mm(w0,t0Wl)f\Pdn C't'o", z so by (2. 1) we get (2. 6). ii) Conversely asume (2. 6) holds true. Then by (2. 1) f\m'm(w0, twJf¡"dp S Ct9". z Multiply by |r/> and integrate! We get / [ M h min(vv0, iWj) L t 1 J z o But, since (p in non^decreasing, dt dt № 1»0 »1 min(iv0, nvj) p dl t - /['few WO 2u>i Thus we have established (2. 7). R e m a r k 2. 1. Theorem 2. 2 and 2. 3 should be compared to the known result ([see [7]) It would be inteiesting to have a characterization of (Lp(w„), Lp(ir i))0j9 which covers the remaining case 1<<?<°°, q ^ p too. 3. Applications to Fourier integrals We consider the space Wpq of functions / on ( - » , <») such that J | f(x + Nt) - ]/(.v + (N- 1)t) + ... + (-1 y f{x) d t LvtS"+ 1 1/9 ') We owe the above argument to N I L S - O L O F WALLIN. 68 J. Peetre where 1 S i / 5 » , 0 < s < N . This definition is essentially independent of N. In view of [3], [5], [6] we have iv;-" = H»)s where Hp is the space of functions whose JVth order derivative is in Lp (with respect to Haar measure). Now take p = 2. By PARSEVAL'S formula we then get (3.1) fcwr if and only if / e [ M l ) ; L 2 ( m ] . v W" Thus we may apply the result of Section 2. Let us write • Ly = [Lp{\),L,Xm]s . W" On applying theorem 2. 1 (</ = °°, / = 1) we get: if and only if (3.2) [ / 1/(^)1" Ct'. lilai/i In view of the duality theorem of LIONS (see [4] or [5], chap. Ill) this is clearly related to [2], theorem II. On applying theorem 2.2 we get: f^L*; 1 if and only if (3.3) for some non-decreasing q> such that t> )" dt : co. MO) ' 0 In particular (p = 2) we get by (3. 1): / £ fV^' if and only if (3.4) f i for some non-decreasing (p such that ¡ [ h o ) 2 dt (pit)) t If the condition on (p reads simply / . ^ < °° and setting (o(t) = (q>(t))2 2 o (<P(0) we get (0. 2). This is essentially [2], theorem III. As in [2] similar results hold if is replaced by (1 + |£|)'v. It is also clear that we in the above fashion can treat the case of Fourier integrals in any number of variables. Interpolation of spaces with weight function 69 References [1] A. BEURLING, On the spectral synthesis of bounded functions, Acta Math., 81 (1949), 225—228. [2 ] Construction and analysis of some convolution algebras, Ann. Inst. Fourier, 14 (1964), 1—32. [3] J . L . LIONS, Théorèmes de traces et d'interpolation, Ann. Scaola Normale Sup. Pisa, 1 3 ( 1 9 5 9 ) , 389—403. [4] Sur les espaces d'interpolation; dualité, Math. Scand., 9 (1961), 147—177. [5] — and J. PEETRE, Sur Une classe d'espaces d'interpolation, Inst. Hautes Etudes Sci. Publ. Math., 19 (1964), 5—68. [6] J. PEETRE, Esjpaces d'interpolation, généralisations, applications, Rend. Sem. Mat. Fis. Milano, 34 (1964), 133—161. [7] — On an interpolation theorem of Foiaç and Lions, Acta Sci. Math., 25 (1964), 255—261. [8] Applications de la théorie des espaces d'interpolation dans l'Analyse Harmonique, Ricerche Mat., 15 (1966), 3—36. [9] A . ZYGMUND, Trigonometrical series (Cambridge, 1 9 5 8 ) . (Received March 5, 1966) 

On interpolation of Lp spaces with weight functions

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On interpolation of Lp spaces with weight functions

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