ベクトル値函数のローレンツ空間について

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(1)Title. ベクトル値函数のローレンツ空間について. Author(s). 中村, 紘司. Citation. 北海道教育大学紀要. 第二部. A, 数学・物理学・化学・工学編, 25(1) : 20-28. Issue Date. 1974-08. URL. http://s-ir.sap.hokkyodai.ac.jp/dspace/handle/123456789/5978. Rights. Hokkaido University of Education.

(2) Journal of Hokkaido University of Education (Section II A) Vol. 25, No. 1 August 1974. mr^Nas?(^2j!asA)"n?25^ un-^ ffg^49^8^. On the Lorentz Space of Vector-Valued Functions. Koji NAKAMURA Department of Mathematics, Hakodate Branch, Hokkaido University of Education. WM?): -<" hMifiiSKcop - ^-/'•^^FBI^-^^'C. 1. Introduction Let the function <fi(t) be a non-negative, integrable and almost everywhere equivalent to a non-increasing function defined on an interval (0, /), /<oo. We consider the non-increasing (decreasing) rearrangement of a measurable function / on a measure space X and denote by /* which is defined on (0, oo). It is defined as the right-inverse of the Lebesgue measure. ftf(y) of the set {xeX: \f(x)\>y for any positive real number y} [3; pp. 260-299], [7; p. 60]. G. G. Lorentz has introduced the class A (y, p} of all measurable functions so that the integral .(. '0. <f(f)f (t}tldt is finite for K^<co. He has investigated that it is a reflexive Banach function ll \llt>. space equipped with the norm ||/1|=U (p(t)f*(t)tldt) [6]. In this paper, we shall discuss with a class of all vector-valued functions which are Bochner-Lebesgue integrable.. 2. Preliminaries and Propositions. First of all, we establish a number of notions. Let X be a Lebesgue measure space with the Lebesgue measure /<, /<(J^)=/<oo. We denote by R+ the extended non-negative real numbers and Ir the interval (0, r) for any real number r^R+. Now let the function f(x} be the vector-valued Lebesgue measurable function defined on X into a conditionally o'-complete Banach lattice B equipped with the norm |] • ]|. We shall define the antitone [1; p. 3] rearrangement function /* of / the same as numerical case [I], [7], i. e., r<y' implies /*(r)>/*(r') on R+. Let fif(y) be the Lebesgue measure of the set. of all xe X so that |l/(.v)||>lb'|| for an arbitrary vector 3' of the positive cone P in B. Then lif(y) is right-continuous, i. e., lim /Jtf(y,,)=fif(y) where y=:r\y,i. Then we define the function Sn. 4-. S. '. 11. f*(r) as the right-inverse of p,f(y), i. e.,. (2.1) /*(»-) =mt{ye=P: [tf(y)<r, reR+}. We denote by t(x) the step function on X into B which is (2.2) t(x}=j]v,icsi(x), i=>l. ( VH").

(3) On the Lorentz Space of Vector-Valued Functions 21. where the elements fi'eB. The function XB, is the characteristic function of the set E, and A. {E,: f.i(Ei)<oo} is a finite pairwise dis joint sequence so that U Ei=X, Then we have /==!. (2.3) <*(r)=2y,*^,W £3^. where vf^v^^ ... ^v^, v*, eP and {F,: j==l, 2, ..., /;} is the finite pairwise disjoint sequence li. so that || ff ||=||^|| and p{Ej) = f^(Fj) for a certain number j, IJ F!=IINow we shall mention several basic propositions. For a positive coefi&cient ^, we have. (2.4) ^ W)*^/*; (2.5) If f<.g, then f*<.g* for f(x), g(x)eP. Further we able to mention the following proposition without proof, for it is proved the same as a numerical case.. Proposition 2.1. Let f(x) be a measurable function on X i-uith values in the positive cone P in B, and let {f»(x): w=l, 2, ...} be a sequence of measurable functions on X with values. in P also. If Q<.fn T/, then 0<.f<;, \f*. Here, we shall introduce the preorder "<" which is defined as follows; for /, geB^. (the space of all Bochner integrable function [10; p. 217]), f<^g means that \_\\f*\\df.i <, If. J^*||^ for an arbitrary reR+. Since r=fJ.{x: \\f\\>\\f*W^fJi(e}=^{x: H/U>h||} for. I If. any y(=B so that ||/*M||^||^[|, it follows that \ \\f\\dfJi<.\ \^f*\\dfJt. In this case, for convenience, we will denote /<^/*. Then we have the following results :. (2.6) If f<g, then Af<^g for any positive ^ ; (2.7) In the positive cone P of a Banach lattice B, if /(A-X^(.»-) for A-eA"', then f<g; For two vector-valued functions / and g, since f<^f* and g<^g*, we obtain. (2.8) f+g<f*+g*. In fact, since fjt{x: \\f+g\\>\ly\\}<.ft{t: || f*+g* || > II y\\} for any yeP, then we have. f)f+gW<f \\f*+g*W where p.(e)=r for any r(<l). Moreover, we have the following propositions, Proposition 2.2. // / and g are t'wo Bochner integrcible vector-valued functions on X,. then. (2.9) {f+g)*<f*+g*. In view of (2.8) and definition of /*, it follows that. .\\(f+g}*\ld^=s^\\\f+g^: eeX, ê)=r\<.\^\\f*+g*\\d^. Tr. ^. ^. e. ). ^. Ir. Then we have our assertion.. Proposition 2.3. Let f be a vector-valued Bochner integrable function defined on X into a Banach lattice B, and let g on X into the conjugate space of B' ivith the norm ||-||'..

(4) 22. K.. NAKAMURA. Suppose that f and g are Bochner integrable, then. (2.10) f\\g\\'\\f\\dfz<:^\g*\\'\\f*\\d^, ivhere /ji(X)=leR+. For a proof, to begin with, we consider the step function as (2.2). Since Tts, are orthogonal to each other, then. II th 11 = US v^s, ll = 2 II y- II ^. (vi e B). .1=1 • J=l. It is sufficient to prove the numerical case. Next we consider ascending sequences of step functions {!„}, {s,,} so that tn^f and s,, f 5'respectively. Since ^ \\tn-f\\dfi. and \ ||s,,-s'||'rf/( are converged to zero as n—>co for / and g are Bochner integrable, it follows that. ^\\t4\\s,,\\'-\\f\\\\gy\d^ converge to zero, i. e.,. f II <„ II II s,, ^'du.—f (>JI / ]| H g |? (^-). On the other hand, since we have known. n <„ ii M'< n ^n 11 sr, u' already [S], [7], thus it proves our assertin.. 3. The spaces A (y, Bp) In the sequel, let K^<oo and let AT be a finite measure space with the Lebesgue measure p., fJ.{X)=l<<x>. We denote by I the interval (0, I) (=Ii) of R+. Let the function y(r) be a non-negative, non-increasing and not identically zero defined on the real numbers, we shall say. that <p is a iveight function. V will denote a real UMB-lattice [1; p. 371] with the norm \\-\}v, V its conjugate space with the norm |*l|y. Let the vector-valued function f(x) be a Lebesgue measurable and defined on X into V. n. For a vector-valued function /, there is a sequence of step functions t,,{x)=J.] v,T(,E,{x} as (2.2) t'=^l. so that t,, converges relatively uniformly to /, i. e., \tn(x)—f{x]\<^^,iU for some ll and /!„ 4.0. [1; p. 369], and \\t,,(x)\\<. 2\\f(x)\\, In the UMB.lattice V, since. \\t,,W-f(x)\\v<. Un\\ V\\ U\\V, then we have the sequence <„ converging to /. Hence we obtain that feBp. Now, the relative uniform convergence is equivalent to the order convergence in V, i. e.,. /(;v)=o—lim(,,(;r). Then we have. (3.1) /*(./•) =o-lim<;(r) in viw of Proposition 2.1. Therefore since the sequence t!,(r) converges relatively uniformly. to /*(r) and W,(r)\{r <^2|| f*(r)\\v so we have Vj|^(r)-/*(r)U^/^0 as n-^°o and /*e^. I. Since /* is equimeasurable to /, i. e., fJt\\f\\(y) =l^\\f-\\{y) for any y^V, then /* is /(-measurable. We state the following propostions. Proposition 3.1. (Zaanen) For a vector '-valued measurable function f and the weight.

(5) On the Lorentz Space of Vector-Valued Functions 23. function <p {finite numerical'), <f>f* is also measurable. Proposition 3.2. Let </>„ be a sequence of numerical step functions so that <pn(f')= ". .. .. '.. ••'. -I. :•.'. 2 a,'XFi(,r), ff,e7?+ aw^ a\â^... >ffH^>0. Z.e< ;?„ Ae a sequence of vector-valued step functions. i-1. as (2.2). Suppose <pn converges uniformly to </> and t,, converges relatively uniformly to f,. then (3.2) f\\W-v>lH'f*\\^—>0 as nThus (pll"f*<=Bp. Proof. From Minkowski's inequality, we have 1/p. //». :. IIP \^/p. ^<pl,!m-<f>l"'f*Kd/^ ]" ^Wtfn-v^f*Kd^ +[\\W,ltf*-vl"'f*Kd/z}. I/?. = , fv.,\W-f*\\^) + (f\vl.,"'-vll^r\\f*Hd^ . Here, since t*n converges relatively uniformly to /*, so ||<>fS—/*||^<J|'i,i||^||M||^ for some M and ^n 4 0' Then the first term as above converges to zero for. f«.,\W-f*K^<.^V»\\Mpy\Mpvd/Jt—^0. HP The last term is smaller than ( \ s\\f*\\^d/.t) for any e>0. Thus it is converging to zero. Therefore we have our assertion.. Now we shall consider the class of all vector-valued functions / on X into .V so that Vllpf*eBp and denote A(y, Bp), i.e.,. (3.3) A(y, B,}={f: ^f*e=Bi,}=[f: ^\vW-Vl"'f*Hd^—>o], where sequences </>„ and tn are defined on Proposition 3.2. For one such f(eA(<p, Bp)), we put. (3.4) p(/) = (j)WS||^) = (^V\\fUÎP. Theorem 3.3. A(y>, Bp) is a normed linear space 'with respect to the norm (3.4). Proof. For /, geA{<p, Bp), there are two sequences tn and s,, so that these sequences converge relatively uniformly to / and g respectively. Then there is a sequence of step functions (<n+Sn)* which converges relatively uniformly to (f+g)*. Let \t,,-f\<.^,u and |s,,- g\<.p.nV for some u, v and d,,, ,«„ 4. 0 respectively. Here we put £,,=max(^,,, /<„). Since \(t,,+s,,)-(f+g)\ <.e,,('li+v), then the sequence t,,+s,, converges relatively uniformly to f+g. Thus we have (f+g)*=o—lim(t,,+s,,)*. Therefore we obtain f+geA(if>, Bpj the same as Proposition 3.2, The class A((p, Bp) is a linear space.. Now we shall show that p is the norm which is defined by (3.4). Suppose p (/)=0, then we have ||/*||iy=0 almost everywhere. In fact since ||'[|r is the norm in V, then/*=0 is almost everywhere, i. e., /=0 a. e. Conversely, if /=0 a. e., then we have p(/)=0 clearly. Next we can see easily that p(^/)=/ip(/) for positive coefficients A. In the view of the proof of Proposition 2.2, finally, we have.

(6) 24. K.. NAKAMURA. p(f+g)=(fv>\\(f+gnpyd^) <(û*+g*\\^) ill' Ir \ÎP. ^V\\f*Kd^~"+^v\\g^d^~"=p(f)+p(g). Thus we obtain the triangle inequality. Therefore the class A(y, Bp) is the normed linear space. Theorem 3.4. Let f be a vector-valued function into the positive cone P in a real UMB-. lattice V and let /„ be a sequence of A{y, Bp) which is defined the same as the function f. •y/"t/> then (o(/)=lim p(/). ^(p, -B^) ('s fl Banach space ivith the norm (3.4). Proof. If 0<^/,,f/for an ascending sequence /„, then we have /Sf/* by Proposition 2.1. Moreover, since the norm ||-||p satisfies the weak Fatou property [10; p. 446], then. ll/Slli'tll/ l|r- It follows from the Lebesgue convergence theorem, that we obtain p(/)= lim p(/,,) and fe=A(y, Bp}. 4. The reflexivity of A(<p, Bp) Now we consider the dual space of A{<p, Bp). Here we shall define the linear functional F on A[</>, Bp) and it follows that. (4.1) F(f)=\^<g{x),f(x)>d^ }x. for a vector-valued measurable function g(x) defined on X Into the conjugate space V of V, where <•, •> is the bounded linear functional [10; p. 225]. From the above reason, F is the bounded linear functional on A((/>, Bp). Here we shall give some definitions and propositions which will be needed in the sequel.. Let G(c)= \ \\g(r)\\d/jt and <S(c)= \ (p(:r')dp. for 0<c<7. The function G(c) will be called concave TC. '"C. luith respect to <p (in short, (p-concave) on I, if. G(.C)-G^>G^-G^. ®(cFW^(3(6)-(S(c)' for any a and b in I. Moreover if G(c) is y-concave where G(c) is defined with certain rearrangement of g, then g will be called antitone with respect to ip. We obtain the following proposition.. Proposition 4.1. Let g be a Bochner integrable function on X into V and let gr be a rearrangement of g defined on I into V. The function g is antitone with respect to <p if and only if. (4.3) \\gr(r)\\=v{r)\\D{r)\\ a.e. where ||Z)|| is a decreasing function as a real value. Proof. Suppose || gr(r}\\ =<f(r)\\ D (r)|| is almost everywhere. Fo,r an arbitrary c so that 0<a<c<b<il for any a and b, since. G(b)-G(c)= \^{r)\\D(r)\{d^- ^ y(r)||D(r)||^ Ill,'. •. '". •. ". '. -lie. ^(,r}\\D(r)\\d^^\\D(c)\\\^ v{r}dix. l(c,6)' ' " ••--—•• • • ••J(c,(,)-. =||£>(c)H((P(A)-(»(c)), then we have.

(7) On the Lorentz Space of Vector-Valued Functions 25. ^\\>GW~G(C). WH^0(6)_(S(c)-. On the other hand, since. G{c)-G{a)= ^ v(r)\\D(r)\\d^\\D(c)\Wc)-<l>(a)), ' (°, c). then we have. G(c)~G(a}>W(^. (P(c)-(!>(ff)^ll^c;"'. Thus we have the inequality (4.2). Conversely, let the function G be (o-concave. The. following. .^n-G(c+A)-G(c)=l/^c-'^"^. 11^(C)11=^^^=^T^ 1(C',C+<|). is a decreasing of c for any fixed h>0 on (0, l—h). Hence the limit. WC)\\=^\\D,,(C)}\=^ exists almost everywhere. Therefere we obtain our assertion.. In view of (2.10) and Holder's inequality, for g*<'(f>D, we have the following. (4,4) \F(f}\<.^\g\\y.\\f\\yd^<.^\\DV,\\f*\\vdt^ lfp f* \ 1/9 . / /» ^ 1/<?. <.^\\r\\tvd^'\^\D^,d^Y=p{f)(^v\\D\\^di^'\ \{q 1/9. (4.5) sup^ | F(/)| <: jnf^ ^y V\\D\\"y.d^ H D ^. d^'' . (>(/)<!' " g'«l>V •'I (>(/)<!" <?'<y0 ''r. Here we put the right hand side in (4.5) being equal to T(g), Hence we establish the. following. (4.6) \\F\\<:r(g). Now we shall study the reverse inequality of (4.6). To show the result, we need the notion of level function g° of a given vector-valued Bochner integrable function g (truly, g*).. It is the smallest function among the functions satisfying g <^t/>D(\\g*\\<^y\\D\\) with an antitone function D, i. e.,. ga=mt{h: g*< h =<pD}.. Proposition 4.2. (Lorentz) Let g be a Bochner integrable and let D9 be defined as g° =y>2)°. For any G(x)= \ |ls'*(r)!|^, we define that the function G° is also of the form 'Is. Ga{x)=\^\\g°{r)\\dp.. Then G{x)=G°(x) holds a.e. (consequently \\g*(r)\\=\\go(r)\\ a. e.) except 'Is. perhaps for the maximal interval (a, b) of constancy of \\D\\; on each such intrval {a, b),. .|k*(»-)?< JI^WII^.. l(a,6)"- ' " •)(a,6). Thus, for a given Bochner integrable function g, we have to say that the infimum of the right hand side of the inquality (4.5) is attained for D° and hence we may describe that ( 9.R~\.

(8) 26. K.. NAKAMURA 1/9. i. (4.7) r(g)=[\V\\D«\\"v,diJi\".. It is easey to see that r(g) satisfies the norm condition. In fact v{g) is non-negative and T(g}=° is equivalent to g=0 a.e,, for \\'\\v is the norm on V. It is clear that T{^g)=^(g) for positive ^. For two Bochner integrable functions g^ and g^ so that gi<^<pDi and g^yDs. respectively, we have || g, + g, \\< q> (|| D^ \\ +1| D§ H). Therefore, if g,+g,< <f>D, then y \\ D91|< p(]| A0 II +||D§||). Thus we can see the triangle inequality ^(,gi+g2)<.v(gi) Jf'c(gt)- Moreover, since -D,° f.D° where g,,'[ g and g1,=vD^, then r satisfies the Fatou property. From the above reasons, we consider the dual space of A(f, Bp) which is the class of all such vector-valued Bochner integrable function g, denoted by A{y>, Bp), i. e.,. (4.8) ^7^)= {g: r{g)< °°, ^(A-)eF', A-eA-}. The following theorem will show that the Banach conjugate space A*((p, Bp) of A((f>, Bp) is isomorphic to the dual space A((p, Bp) for Kp<oo. Theorem 4.3. Let </> be a weight function and let <p,, be a sequence of step functions. zuhich converges to <p uniformly. If gÂ((p, Bp) and Kp<cx,, 1/^+1/^=1, then \\F\[^r{g). Proof. Firstly, let the element v'e.V be given and, let the non-negative number f and e>0 be given.; Then there exists an element, veV so that |H|y=r and r\\v'\\v'—e<^<y', v>. <r|M|y (This follows from the fact that r\\v'\\r'=sup{\<V', V>\: \\v\\v=r}.). Let s'(x)= ve'v :. ft. S V',XE,(X) be a step function on X into V where the sets E, are dis joint. There is a non1=1. .'. ,. •. *.. .. .. .. •. ft. negative step function c{x')=^fiTt.Ei(x) satisfying pp(c)==l and \ c(;i')]|s'(.r)|^rfiU=r(s'). Let e>0 1=1. ~. •/!. * be given, and we set i?=e/S/<(£',•). For i=l, 2,...., k, we are able to determine the elements 1=1. .. •. Vi^Vsoth&t\\Vi\\v=ri'a.ndr.i\\v',\\r'-T3<.<v'i,v><.ri\\v'i^v'. Then we have a step function *. /•. *. t(x)=J]v,'X,Ei(x) on X into V, satisfying pp(t)=pp(c)=l, so that \,<s', t>d/jt=j]<v',, v> /<(£',) 1=1. •'I. i=l. lies between \ c(x)\\s'(x')\\Y,df^-6 and \ c(x)\\s'[x)\\v'd^. i.e., between •i-(s')-£ and r(s'). Thus, there is a sequence of step functions s,,(() so that r(y,,s,', — (PjD°) converges to zero as n—>-oo for. ipn—xp uniformly. Then. rte)(=T(yZ3»))=limr(y,,s;,). «->.oo. Now we will have to take functions t,, for the above function s;, so that pi,(t,,)=l and T(<f>,,s',i)-lln<.\<v,,s',,,tn>dfJt^T{(f>,,s',,). Then we have r(,g-)=lim \<(fi,,s,,, t»>dp.. On the }I. K->oo. ^l. other hand, we have <g°, t,,>dp.~ ^«0,,S;,, <«>rf/< ^|<,?°-ff,,S;,, <,,>|rf^( =^\«pD°-<p,,s;,, t,,>\dft<.pp(tn)r(g-VKS',,) — >0 (n — >oo). Therefore we have <g, f>dp.-r{g)\<.\\ <g, f>dfi--c(vnS'n. <z ~ ^ • • • "'} • \-}.z •. +\T{tf>,,S',,)-v(g)\.

(9) On the Lorentz Space of Vector-Valued Functions 27. converges to zero, for 5'= .g-0 and/=lim)I» uniformly. Since \<g,f>d^=lim \<g0, tn>dp., 'X. ^1. thus we obtain. (4.9) \_<g, f>di^=-c(g) 'X. for a function / so that pp{f) =1. Then |] FH ^ _<,§•, f>d{Jt. r(g) is finite whenever \ <g, }X. .. *!X. f>dp. is bounded. We have our assertion. Now, for the sequence of fn<=A(y>, Bp) so that f,,(x)->f(x) (i. e , || /„(.<;) -/(.l;)||v-)-0), we set the linear functional on A*(y, Bp) as follows. (4.10) F,,(g)=\_<f,,,g>d^ >z. for a geA*((p, Bp). Since. \F,,(g)\=. <x. <fn, g>dp,. <.Pp{fn)<g),. then we have. (4.11) sup|^,,(^)]<^(/,,). TW<1. Here there is a g so that g*=g°, T(gc')=l and \ </„, g>dp. =Pt(f,i) the same as above. <z. Therefore we have \\F,,\\==pp(f,,)<.T(g) for all n. Since Pi,{fn) satisfies the Fatou property and. \\f'n\\^T(g), then 7'',, is uniformly bounded and/e/((y, Bp). Thus we have F(g)= \ </, ,gr>^ IX. and \\F\\=pi,(f). This shows us that the dual of A*(</>, Bp) is isomorphic to A(p, Bp), i. e.,. (4.12) A(y, Bp)=^(y7W)Then we have the following theorem. Theorem 4.4. The space A{<p, Bp), Kp<°o, is reflexive as a Banach space.. Summary It is well known that Lorentz spaces A(<f, p), Kp<°o, and A(</>, M), where M is Nfunction, are a reflexive Banach function space. Now, let X be a Lebesgue measure space where p.{X)=l<°° and Bp be the Banach space which is composed of Bochner-Lebesgue integrable functions on X. Let the weight function <p{t} be defined on (0, I), and /* be the antitone rearrangement of a Lebesgue measurable function / on X. Then we consider the collection of all function / where <pf*^Bp and denote A{(p, Bp). The purpose of this paper is to show A(if>, Bp), Kp<cxl, is also a reflexive Banach function space.. References 1. Birkhoff, G. (1967), Lattice Theory. A. M. S. Colloquium Publications, Vol. 25. 2. Halperin, I. (1960), Function space. Proc. Interrant Sympos. Linear Space. Jerusalem, pp, 273-288. 3. Hardy, G. H,, Littlewood, J, E. and P61ya, G. (1934), Inequalities, Cambridge. 4. Hill, E. and Phillips, R. S. (1957), Functional analysis and semi-groups. A. M. S. Colloquium Publications, Vol. 31. 5. Krasnosel'skil, M. A. and Rutickii, Ya, B. (1961), Convex functions and Orlicz spaces. Russian.

(10) 28. K.. NAKAMURA. scientific books, Gronigen. 6. Lorentz, G. G. (1951), On the theory of space A, Pacific J. Math. 1, pp. 411-429. 7. Lorentz, G. G. (1953), Bernstein polynomials, Tronto. 8. Nakamura, K. (1970), On A(if, M).spaces. Bull. Fac. ScL, Ibaraki Univ. Math., No. 2-2, pp. 31-39, 9. Nakano, H. (1951), Modulard linear spaces. J. Fac. Sci. Univ. Tokyo 6, pp. 85-131. 10. Zaanen, A. C. (1967), Integration. Amsterdam,.

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