变指标有界变差空间的准紧集.pdf

第 43 卷第 2 期2023 年 6 月数学理论与应用MATHEMATICAL THEORY AND APPLICATIONSVol.43No.2Jun.2023Precompact Sets in Variable Exponent Bounded Variation SpacesSi YananXu Jingshi*(School of Mathematics and Computing Science,Guilin University of Electronic Technology,Guilin 541004,China)AbstractBy equivariated sets,we consider sufficient conditions for precompact sets in variable exponent Banachspacevalued bounded Wiener variation spaces and bounded Riesz variation spaces in univariate and bivariate cases,respectively.Key wordsPrecompact setBounded variation spaceVariable exponentBanach spacevalued变指标有界变差空间的准紧集司亚楠徐景实*(桂林电子科技大学数学与计算科学学院,桂林,541004)摘要本文利用等度变差集,考虑单变量和双变量情形下变指标 Banach 空间值的有界 Wiener 变差空间和有界 Riesz 变差空间中准紧集的充分条件.关键词准紧集有界变差空间变指标Banach 空间值doi:10.3969/j.issn.1006 8074.2023.02.0081IntroductionIn 1881,Jordan started the study of univariate functions of bounded variation in 1.Subsequently,some generalized bounded variation spaces are investigated,including the bounded Wiener variationspaces 2,the bounded Young variation spaces 3,etc.For univariate functions of bounded variation,a very comprehensive overview about the basic nations and properties of classical and nonclassicalfunctions of bounded variation and their various generalizations may be found in 4,5.The boundedvariation functions have been extensively applied also in other fields,for instance,variations calculus,convergence of Fourier series,geometric measure theory,mathematical physics,etc.Indeed,Chistyakovconsidered superposition operators on bivariate or multivariate functions of bounded variation in 6,7respectively.Verma and Viswanathan studied the Hausdorff dimension and box dimension of bivariatecontinuousfunctionofboundedvariationin8.LindconsideredtheFubinitypepropertiesofthebivariateThis work is supported by National Natural Science Foundation of China(No.12161022)Corresponding author:Xu Jingshi(1966 ),Associate Professor,PhD Email:收稿日期:2023 年 4 月 20 日108数学理论与应用bounded pvariation functions spaces in 9.The application of bivariate regular variation in insurance andfinancial risks was given in 10,11.Variable exponent function spaces have been developed quickly in the last decades due to theirapplications in many fields,such as in nonlinear elastic mechanics 12,electrorheological fluids 13,image restoration 1219 and differential equations in nonstandard growth 20,21,22.In particular,Castillo,Merentes and Rafeiro introduced bounded Wiener variation spaces with pvariable and studiedsome of their basic properties in 23.Then Castillo,Guzmna and Rafeiro obtained some embeddingresults of variable exponent bounded Riesz variation spaces in 24.The compactness problem of bounded variation spaces seems to be far from being fully resolved.Indeed,Ciemnoczolowski and Orlicz gave some results on the relative compactness(especially compactembedding)of the bounded Wiener variation and bounded Young variation space in 25.A weakcompactness criterion for bounded variation space was given in 26.In 27,the authors gavecharacterizations for precompact subsets of the Banach space of continuous functions that are continuousin variation.A characterization of compactness in the Banach space of continuous functions that arecontinuous in variation was also given in 28 by Bernstein polynomials.Bugajewski and Gulgowskiprovided a compactness criterion for bounded Jordan variation spaces in 29.Inspired by 29,we willgive sufficient conditions for precompact sets in variable bounded variation spaces.In Section 2,we willconsiderBanachspacevaluedunivariatevariableboundedvariationspaces,includingtheWienervariationand the Riesz variation spaces.Their bivariate cases will be given in Section 3.In the remainder of this section we state some preliminaries.Definition 1.1Let X be a vector space over K.A function on X is called a convexpseudomodular if(i)(0)=0(ii)(x)=(x),for any x X,and K with|=1(iii)is convex(iv)lim0 0:(f)1.The following lemma is well known,see Lemma 2.1.14 and Corollary 2.1.15 in 20.Lemma 1.1Let X be a vector space,be a convex pseudomodular on X and f X.Then,(i)(f)1 if and only if f 1(ii)if f 1,then(f)f.变指标有界变差空间的准紧集109Definition 1.2A subset in a topological space is precompact if its closure is compact.A subset Wis totally bounded in a metric space X if for each 0 there exist finite elements x1,xm in Wsuch that W mi=1B(xi,),where B(x,r)is a ball in X with center at x and radius.It is well known that precompactness is equivalent to total boundedness in a complete metric spacedue to Hausdorff.In the sequel,we let(E,)be a Banach space.2Univariate variable bounded variation spacesLet us denote 0,1 as I.We call a function p():I (0,)variable exponent.Let p+=p+I:=esssupxIp(x)and p=pI:=essinfxIp(x).We denote the set of all variable exponents with0 p p+as P(I),and the tagged partition of I as :0=t0 t1 tn=1 with a finitesequence x0,x1,xn1,satisfying ti xi ti+1for each i.2.1Variable bounded Wiener variation function spacesFor a bounded function f:I E,we definep()E(f,)=ni=1f(ti)f(ti1)p(xi1),andp()E(f)=supni=1f(ti)f(ti1)p(xi1),where the supremum is taken on all partitions of I.Definition 2.1We define the space of functions of p()bounded variation byBVp()(I,E):=f:I E;f(0)=0 and fBVp()(I,E)0:p()E(f)1.Proposition 2.1Let p()P(I).Then p()Eis a convex pseudomodular.The proposition and proof is similar to properties of the variable bounded variation of scalar settingin 24.Theorem 2.1Let p()P(I).Then BVp()(I,E)is a Banach space.Proof Let fuuNbe a Cauchy sequence in BVp()(I,E).Then for any (0,1),there existsN N,such thatfu fvBVp()(I,E)110数学理论与应用for all u,v N.By Lemma 1.1,we have that p()E(fu fv).We claim that there exists f BVp()(I,E),such that fu fBVp()(I,E).In deed,for fixed t I(fu fv)(t)p(t)max1,2p+1(fu fv)(t)(fu fv)(0)p(t)+(fu fv)(1)(fu fv)(t)p(t)max1,2p+1 p()E(fu fv)max1,2p+1.Since E is a Banach space,then the sequence fu(t)uNhas a pointwise limit,denoted by f(t),i.e.,f(t)=limvfv(t)for each t I.Since for all partitions of Ini=1(fu fv)(ti)(fu fv)(ti1)p(xi1)N,letting v ,we obtainni=1(fu f)(ti)(fu f)(ti1)p(xi1),u N,which implies thatp()E(fu f1/p)1.Then by Definition 2.1,fu fBVp()(I,E)1/p,for u N.ThereforefBVp()(I,E)fu fBVp()(I,E)+fuBVp()(I,E)0 there exists a partition of I,such that for every f Ap()E(f)+p()E(f,).The following set is a nontrivial example which is equivariated.Example 2.1Let p0(1,),E=Lp00,1,and p(x)p0.For any f Lp00,1,let xf(s)=f(s)0,s(s).Then for any partition :0=t0 t1 0,there exists a partition:0=t0 t1 tn=1,and xi ti,ti+1,i=0,1,n 1,such thatp()E(f g)13 2kp+p()E(f g,).Let Yi=f(ti):f A,i=0,1,2,n.Then by the condition(ii),Yiis a precompactsubset of E.Denote Y=Y0 Y1 Yn,then Y is a totally bounded set in En+1,wherefor any =(0,1,n)En+1,En+1=ni=0maxip(xi1),ip(xi).Let W=f(t0),f(t1),f(tn):f A.Then W Y,and W is totally bounded.Hence there existsh1,hN A,such that jNj=1 W,where j=(hj(t0),hj(t1),hj(tn)is a finitenet of W.Thus for each f A,there exists some j such thatni=0maxf(ti)hj(ti)p(xi1),f(ti)hj(ti)p(xi)13 2kp+p+1.Thenp()E(f hj)13 2kp+ni=1(f hj)(ti)(f hj)(ti1)p(xi1)13 2kp+2p+1ni=0(f hj)(ti)p(xi1)+2p+1ni=0(f hj)(ti1)p(xi1)13 2kp+13 2kp+13 2kp+=12kp+.By p()E(2f)2p+p()E(f),we havep()E(f hj)=p()E(2k(f hj)2kp+p()E(f hj)=1.Then by Definition 2.1,we obtain f hjBVp()(I,E).Therefore,A is totally bounded,i.e.,precompact.The proof is complete.112数学理论与应用Remark 2.1The condition(i)of Theorem 2.2 is not necessary.Let p(x)2 and E=R.For eacht 0,1,let x(t)=t,y(t)=t2.Then x,y 20,1.Let A=0,x,y,then A is a precompact set.But the set A A is not equivariated.We know that2(x,I)=|x(1)x(0)|2=1.Because for each partition 0=t0 t1 t2=1,we have|x(t1)x(t0)|2+|x(t2)x(t1)|2|x(1)x(0)|2=1.Then the variation becomes smaller when the partition points are added.Sincex(t)y(t)=t t2,and|x(1)y(1)|=|x(0)y(0)|=0,then for the partition 0:0,1/2,1,we have2(x y,0)=|x(1)y(1)(x(1/2)y(1/2)|2+|x(1/2)y(1/2)(x(0)y(0)|2=18.In order to approximate the value of 2(x,)to that of 2(x,I),the partition can only be 0=t0t1 t2 t3=1,and t1approaches to 0,t3to 1,but 2(x y,)0,therefore,x and x y are notequivariated.Remark 2.2Condition(ii)of Theorem 2.2 is necessary.Fix t I,let(f)=f(t),f BVp()(I,E).Then :BVp()(I,E)E is a continuous function.Because for each f,g BVp()(I,E),(f)(g)=f(t)f(t)=f(0)g(0)+f(t)g(t)(f(0)g(0)=(f(t)g(t)(f(0)g(0)p(x)1p(x)maxp()E(f g)1p,p()E(f g)1p+,where x (0.t).If f gBVp()(I,E)0,then(f)(g)0.In fact,if f gBVp()(I,E)1,then1 p()E(f gf gBVp()(I,E)(1f gBVp()(I,E)p p()E(f g).变指标有界变差空间的准紧集113Thus p()E(f g)f gpBVp()(I,E).Hencemaxp()E(f g)1p,p()E(f g)1p+maxf gBVp()(I,E),f gpp+BVp()(I,E)=f gpp+BVp()(I,E).Therefore,for each t I,f(t):f A is a precompact set.Corollary 2.1Let p()P(I).A set A BVp()(I,K)is precompact,if the following conditionsare satisfied:(i)the set A is bounded(ii)the set A A is equivariated.Since for each t I,A is bounded,then the set f(t):f A is bounded in K,therefore,precompact in K,which is a direct result of Theorem 2.2.Corollary 2.1 is a generalization of results onthe criterion of precompactness for the constant exponent scalar setting in 30.2.2Variable bounded Riesz variation spacesDefinition 2.3Let =0=t0 t1 tn=1 be a partition of I and p()P(I).Wedefine the functional p()R,E()byp()R,E(f)=supp()R,E(f,)=supni=1f(ti)f(ti1)p(xi1)(ti ti1)p(xi1),forf:I E,where the finite sequence x0,xn1satisfies that for each i,ti xi ti+1.Then the space ofbounded p()variation in Riesz sense is defined asRBVp()(I,E):=f:p()R,E(f)0:f(0,0)=0 and p()R,E(f)1.Theorem 2.3Let p()P(I).Then RBVp()(I,E)is a Banach space.The proof is similar to that of Theorem 2.1 or Theorem 3.3 in 24,and is omitted here.Definition 2.4Let p()P(I).A set A RBVp()(I,E)is said to be equivariated,if for each 0 there exists a partition of I,such that for every f Ap()R,E(f)+p()R,E(f,).114数学理论与应用Theorem2.4Letp()P(I).AsetA RBVp()(I,E)isprecompact,ifthefollowingconditionsare satisfied:(i)set A A is equivariated(ii)for every t I,f(t):f A is a precompact set.Proof Let =2k,k N.By condition(i),for every f,g A and 0,there exists a partition:0=t0 t1 tn=1,and xi ti,ti+1,i=0,1,n 1,such thatp()R,E(f g)13 2kp+p()R,E(f g,).Similar to the proof of Theorem 2.2,for each En+1,letEn+1=ni=0imaxip(xi1),ip(xi),i=max(ti ti1)1p(xi1),(ti+1 ti)1p(xi),i=1,n,and 0=(t1 t0)p(x0).Then bycondition(ii),for each f A there exists some j such thatni=0imaxf(ti)hj(ti)p(xi1),f(ti)hj(ti)p(xi)13 2kp+p+1.Thenp()R,E(f hj)13 2kp+ni=1(f hj)(ti)(f hj)(ti1)p(xi1)(ti ti1)p(xi1)113 2kp+2p+1ni=0i(f hj)(ti)p(xi1)+2p+1ni=0i1(f hj)(ti1)p(xi1)13 2kp+13 2kp+13 2kp+=12kp+.By p()R,E(2f)2p+p()R,E(f),we havep()R,E(f hj)=p()R,E(2k(f hj)2kp+p()R,E(f hj)=1.ThenbyDefinition2.4,weobtainfhjRBVp()(I,E).ThusAisprecompact.Theproofiscomplete.Similar to Corollary 2.1,we have the following corollary.变指标有界变差空间的准紧集115Corollary2.2Letp()P(I).AsetA RBVp()(I,K)isprecompact,ifthefollowingconditionsare satisfied:(i)the set A is bounded(ii)the set A A is equivariated.3Bivariate variable bounded variation spacesFor any function p(,):I I (0,),define p+,pand P(I I)similarly to the variate case asin Section 2.By using the same methods in Section 2,we extend results for univariate spaces to bivariatespaces.For the readers convenience,we give their proofs in detail below.3.1Bivariate variable bounded Wiener variation spacesFor any bivariate bounded function f:I I E,and finite partitions :0=t0 t1 tn=1 and:0=s0 s1 sm=1 of interval I,letp(,)E(f,):=mj=1ni=1f(ti,sj)f(ti,sj1)f(ti1,sj)+f(ti1,sj1)p(ti,sj),andp(,)E(f):=supmj=1ni=1f(ti,sj)f(ti,sj1)f(ti1,sj)+f(ti1,sj1)p(ti,sj),where the supremum is taken on all partitions and of I.Definition 3.1We define the space of functions of p(,)bounded variation byBVp(,)(I I,E):=f:I I E:f(0,0)=0 and fBVp(,)(II,E)0:f(0,0)=0 and p(,)E(f)1.Theorem 3.1Let p(,)P(I I).Then BVp(,)(I I,E)is a Banach space.Proof Let fuuNbe a Cauchy sequence in BVp(,)(II,E).Then for any (0,1),there existsN()such that fu fv N(),hence by Lemma 1.1,we have p(,)E(fu fv).Therefore,for fixed t,s,we obtain that(fu fv)(t,s)p(t,s)max1,2p+1(fu fv)(t,s)(fu fv)(0,0)p(t,s)+(fu fv)(1,1)(fu fv)(t,s)p(t,s)max1,2p+1.116数学理论与应用Since E is complete,the limit limufu(t,s)exists for all t,s I.Denote the limit by f(t,s).Thenfor all partitions,mj=1ni=1(fu fv)(ti,sj)(fu fv)(ti,sj1)(fu fv)(ti1,sj)+(fu fv)(ti1,sj1)p(ti,sj)for u,v N.Letting v ,we obtainmj=1ni=1(fu f)(ti,sj)(fu f)(ti,sj1)(fu f)(ti1,sj)+(fu f)(ti1,sj1)p(ti,sj),which implies thatp(,)E(fu f1/p)1.By Definition 3.1,we have fu fBVp(,)(II,E)1/pfor u N.Since fBVp(,)(II,E)fu fBVp(,)(II,E)+fuBVp(,)(II,E)0 there exist partitions and of I,such that for every f Ap(,)E(f)+p(,)E(f,).Theorem 3.2Let p(,)P(I I).A set A BVp(,)(I I,E)is precompact,if the followingconditions are satisfied:(i)the set A A is equivariated(ii)for every t,s I,f(t,s):f A is a precompact set.Proof Let =2k,k N.By the condition(i),for every f,g A and 0,there exist partitions:0=t0 t1 tn=1 and:0=s0 s1 0,p(,)E(f hk)1.So A is totally bounded,i.e.,precompact.The proof is complete.Corollary 3.1Let p(,)P(I I).A set A BVp(,)(I I,K)is precompact,if the followingconditions are satisfied:(i)the A is bounded(ii)the set A A is equivariated.3.2Bivariate variable bounded Riesz variation function spacesDefinition 3.3Let p(,)P(I I).For a function f:I I E,and two partitions :0=t0 t1 tn=1 and:0=s0 s1 0:p(,)R,E(f)1.Theorem 3.3Let p(,)P(I I).Then RBVp(,)(I I,E)is a Banach space.The proof is similar to Theorem 3.3 in 24.Definition 3.4Let p(,)P(I I).A set A RBVp(,)(I I,E)is said to be equivariated,iffor each 0 there exist partitions and of I,such that for every f A，p(,)R,E(f)+p(,)R,E(f,).Theorem 3.4Let p(,)P(II).A set A RBVp(,)(II,E)is precompact,if the followingconditions are satisfied:(i)the set A A is equivariated(ii)for every t,s I,the set f(t,s):f A is a precompact set.Proof By the condition(i),for each x,y A,we havep(,)R,E(f g)15 2p+p(,)R,E(f g,).Let =2k,k R.Let Yi,j=f(ti,sj):f A,i=0,1,n,j=0,1,m,and Y=Y0,0 Y0,1 Yn,m,then by the condition(ii),Y is a precompact set in E(n+1)(m+1).For each=(0,0,0,1,n,m)Y,with E(n+1)(m+1)=mj=0ni=0i,jpi,j,where=max(ti ti1)(sj sj1)p(ti,sj),(ti ti1)(sj+1 sj)p(ti,sj+1),(ti+1 ti)(sj sj1)p(ti+1,sj),(ti+1 ti)(sj+1 sj)p(ti+1,sj+1)andpi,j=minp(ti,sj),p(ti,sj1),p(ti1,sj),p(ti1,sj1),there exists h1,h2,hN A,such that for each f A,mj=0ni=0f(ti,sj)hk(ti,sj)pi,j15 2kp+p+1.Thereforep(,)R,E(f hk)15 2kp+p+1+p(,)R,E(f hk,),变指标有界变差空间的准紧集119wherep(,)R,E(f hk,)=mj=1ni=11(ti ti1)p(ti,sj)(sj sj1)p(ti,sj)(f hk)(ti,sj)(f hk)(ti,sj1)(f hk)(ti1,sj)+(f hk)(ti1,sj1)p(ti,sj)mj=1ni=1(f hk)(ti,sj)(f hk)(ti,sj1)(f hk)(ti1,sj)+(f hk)(ti1,sj1)p(ti,sj)2p+1mj=0ni=0(f hk)(ti,sj)pi,j+2p+1mj=0ni=0(f hk)(ti,sj1)pi,j+2p+1mj=0ni=0(f hk)(ti1,sj)pi,j+2p+1mj=0ni=0(f hk)(ti1,sj1)pi,j45 2kp+,thusp(,)R,E(f hk)15 2kp+45 2kp+=12kp+.By the inequality p(,)R,E(2f)2p+p(,)R,E(f),we havep(,)R,E(f hk)=p(,)R,E(2k(f hk)2kp+p(,)R,E(f hk)1,andf hkRBVp(,)(II,E)=inf:p(,)R,E(f hk)1.So A is totally bounded,i.e.,precompact.The proof is complete.Corollary3.2Letp(,)P(II).AsetA RBVp(,)(II,K)isprecompact,ifthefollowingconditions are satisfied:(i)A is bounded(ii)the set A A is equivariated.References1 Jordan C.Sur la srie de FourierJ.Comptes R