BACHELOR THESIS Dalimil Peša Integal opeatos on function spaces Depatment of Mathematical Analysis Supeviso of the bachelo thesis: Study pogamme: Study banch: pof. RND. Luboš Pick, CSc., DSc. Mathematics Geneal Mathematics Pague 27
I declae that I caied out this bachelo thesis independently, and only with the cited souces, liteatue and othe pofessional souces. I undestand that my wok elates to the ights and obligations unde the Act No. 2/2 Sb., the Copyight Act, as amended, in paticula the fact that the Chales Univesity has the ight to conclude a license ageement on the use of this wok as a school wok pusuant to Section 6 subsection of the Copyight Act. In... date... signatue of the autho i
Above all, I would like to thank my supeviso, pofesso Pick, without whom this thesis would neve have been conceivable. I would also like to thank all the teaches of my so called pactical lectues fo having vey easonable cedit euiements, which allowed me enough time to wok on this thesis. And at last, I would like to thank Jana Zemanová fo showing me how to type the apostophe symbol, which sped up my wok noticeably. ii
Title: Integal opeatos on function spaces Autho: Dalimil Peša Depatment: Depatment of Mathematical Analysis Supeviso: pof. RND. Luboš Pick, CSc., DSc., Depatment of Mathematical Analysis Abstact: In this thesis, we conside Loentz-Kaamata spaces with slowly vaying fuctions and study thei popeties. We fist povide simple definition of slowly vaying functions and deive some of thei popeties. We then conside Loentz-Kaamata functionals ove an abitay sigma-finite measue space euipped with a non-atomic measue and coesponding Loentz-Kaamata spaces. We chaacteise non-tiviality of said spaces, then study when they ae euivalent to a Banach function space and obtain multitude of conditions, eithe sufficient o necessay. We futhe study embeddings between Loentz-Kaamata spaces and povide a patial chaacteisation. At last, we ty to descibe the associate spaces of Loentz-Kaamata spaces and succeed even in some of the limiting cases. Ou contibution is mainly the chaacteisation of non-tiviality, the patial chaacteisation of embeddings, the desciption of associate spaces in some limiting cases and all the esults concening Loentz-Kaamata spaces with the seconday paamete smalle than one. Those esults ae, as fa as we ae awae, new. Keywods: integal opeato function space non-inceasing eaangement maximal opeato iii
Contents Intoduction 2 2 Peliminaies 3 2. Non-inceasing eaangement.................... 3 2.2 Function noms and uasinoms................... 4 2.3 Associate spaces............................ 6 2.4 Fundamental function........................ 8 2.5 Hady-type ineualities........................ 3 Loentz Kaamata spaces 3. Slowly vaying functions....................... 3.2 Loentz Kaamata functionals................... 7 3.3 Relations between L p,,b and L (p,,b)................. 22 3.4 Embeddings between Loentz Kaamata spaces.......... 27 3.5 Associate spaces of Loentz Kaamata spaces........... 3 3.6 Loentz Kaamata spaces as Banach function spaces....... 38 Bibliogaphy 39
Intoduction In this thesis we focus on one paticula scale of function spaces, called Loentz Kaamata spaces. These function spaces wee intoduced in 2 by Edmunds, Keman and Pick in [6], and thei name eflects the fact that thei constuction encapsules both the Loentz-type stuctue of fine tuning of function spaces and the concept of the so-called slowly-vaying functions that had been studied by Kaamata. The oiginal motivation fo the intoduction of these spaces was connected with the investigation of the vey impotant poblem of nailing down optimal patne function spaces in Sobolev embeddings on egula domains in the Euclidean space. Duing the last two decades the Loentz Kaamata spaces have extended thei field of applications fo example to Gauss-Sobolev embeddings ([3]), boundedness of opeatos on pobability spaces ([5]), taces of Sobolev functions ([4]), taget spaces fo opeatos involving Bessel-type kenels and moe. Thei upmost impotance seems to consist of the fact that they povide a class of vey good and useful examples fo vaious tasks in functional analysis and its applications, which is on one hand uite vesatile (note that Loentz Kaamata spaces contain Lebesgue spaces, Loentz spaces, Zygmund classes, Loentz Zygmund spaces, a good deal of Olicz spaces and Macinkiewicz endpoint spaces, the space of Bézis and Wainge and many moe), on the othe hand they ae elatively easily manageable, which makes them extemely handy. Duing the last 5 yeas, Loentz Kaamata spaces have been seveal times put unde a detailed scutiny. Fo instance, an altenative chaacteization by noms within noms of them is given in [7]. They ae biefly mentioned also in [2]. A detailed study focused on some of thei basic functional popeties can be found in []. But none of these woks povided a completely satisfactoy esult as they usually contained vaious estictions. One of ou pincipal goals is to fill in the gap and povide a compehensive study of these spaces. We will concentate in paticula on the following uestions: the non-tiviality of the spaces, (uasi-)nomability, thei elationship to the stuctue of the socalled Banach function space, the embedding elationships among them and the chaacteization of thei associate spaces. We emphasize that we do not adopt any estiction concening the finiteness of measue of the undelying measue space on which the functions ae defined. 2
2 Peliminaies The aim of this chapte is to establish the basic famewok fo late wok. We stive to keep the definitions and notation as standad as possible. Fom now on, we will denote by (R, μ), and sometimes (S, ν), some abitay sigma-finite measue space. When E R, we will denote its chaacteistic function by χ E. The set of all extended complex-valued μ-measuable functions defined on R will be denoted by M(R, μ), its subsets of all non-negative functions and functions finite μ-almost eveywhee on R will be denoted by M + (R, μ) and M (R, μ) espectively. As usual, we identify functions that ae eual μ-almost eveywhee. Fo bevity, we will usually abbeviate μ-almost eveywhee to μ-a.e. and simply wite M, M + and M, instead of M(R, μ), M + (R, μ) and M (R, μ) espectively, wheneve thee is no isk of confusion. In some special cases when R = R n we will denote the n-dimensional Lebesgue measue by λ n. When X is a set and f, g : X R ae two maps satisfying that thee is some positive and finite constant C, depending only on f and g, such that f(x) C g(x) fo all x X, we will denote this by f g. We will also wite f g, o sometimes say that f and g ae euivalent, wheneve both f g and g f ae tue at the same time. We choose this geneal definition because we will use the symbols and with both functions and functionals. 2. Non-inceasing eaangement In this section, we define the non-inceasing eaangement of a function and some elated tems. We poceed in accodance with [, Chapte 2]. We fist define the distibution function. Definition 2.. The distibution function μ f of a function f M is defined fo s [, ) by μ f (s) = μ({t R f(t) > s}). We now define the non-inceasing eaangement as the genealised invese of the distibution function. Definition 2.2. The non-inceasing eaangement f * of function f M is defined fo t [, ) by f * (t) = inf{s [, ) μ f (s) t}. Some basic popeties of distibution function and non-inceasing eaangement, with poofs, can be found in [, Chapte 2, Poposition.3] and [, Chapte 2, Poposition.7]. We now define what it means fo functions to be euimeasuable. Definition 2.3. We say that two functions f M(R, μ), g M(S, ν) ae euimeasuable if μ f = μ g. It is an easy execise to pove that f, g M ae euimeasuable if and only if also f * = g *. A vey impotant classical esult is the Hady-Littlewood ineuality which we list below. Fo details, see fo example [, Chapte 2, Theoem 2.2]. That is, functions whose values ae non-negative eal numbes. 3
Theoem 2.4. It holds fo all f, g M that R f g dμ f * g * dλ. As an immediate conseuence, we get that, fo all f, g M, sup f g dμ f * g * dλ. g M, g * =g * R This leads to the definition of esonant measue spaces. 2 Definition 2.5. A sigma-finite measue space (R, μ) is said to be esonant if it holds fo all f, g M(R, μ) that sup f g dμ = f * g * dλ. M(R,μ), * =g * R It has been poven that in ode fo a measue space to be esonant, it suffices fo its measue to be non-atomic. Fo details, see [, Chapte 2, Theoem 2.7]. We now define the maximal function. Definition 2.6. The maximal function f ** of f M is defined fo t [, ) by f ** (t) = t t f * (s) ds. Some popeties of the maximal function can be found in [, Chapte 2, Poposition 3.2] and [, Chapte 2, Theoem 3.4]. 2.2 Function noms and uasinoms The following two definitions ae adapted fom [, Chapte, Definition.] and [, Chapte 2, Definition 4.] espectively. Definition 2.7. Let : M + [, ] be some non-negative functional on M +. We then say that is a Banach function nom if it satisfies the following conditions: (P) is a nom, i.e. (i) it is positively homogeneous, i.e. a C f M + (ii) it satisfies f = f = μ-a.e., (iii) it is subadditive, i.e. f, g M + : f + g f + g. : a f = a f, (P2) has the lattice popety, i.e. if some f, g M + satisfy f g μ-a.e., then also f g. (P3) has the Fatou popety, i.e. if some f n, f M + satisfy f n f μ-a.e., then also f n f. 2 Thee is also a stonge vesion of esonance, which we omit since it will not be used. Fo details, see [, Chapte 2, Definition 2.3]. 4
(P4) χ E < fo all E R satisfying μ(e) <. (P5) Fo evey E R satisfying μ(e) < thee exists some finite constant C E, dependent only on E, such that fo all f M + the ineuality E f dμ C E f holds. Thee is one class of Banach function noms which will be of special inteest fo us, namely the eaangement invaiant Banach function noms defined bellow. Definition 2.8. We say that a Banach function nom is eaangement invaiant, abbeviated.i., if it satisfies the following additional condition: (P6) If two functions f, g M + ae euimeasuable, then f = g. It will be also useful to define somewhat weake vesion of.i. Banach function noms, namely the eaangement-invaiant uasinoms. Definition 2.9. Let : M + [, ] be some non-negative functional on M +. We then say that is an.i. uasinom, if it satisfies axioms (P2), (P3), (P4), (P6) fom the definition of.i. Banach function nom and also weake vesion of (P), namely (Q) is a uasinom, i.e. (i) it is positively homogeneous, i.e. a R f M + (ii) it satisfies f = f = μ-a.e., : a f = a f, (iii) it is subadditive up to constant, i.e. thee is some finite constant C such that f, g M + : f + g C( f + g ). While in it was convenient to define these tems only fo functionals on M +, thei domains can be natually expanded to whole M by taking fist the absolute value of given function. So we may say that is, fo example, Banach function nom on M and mean by it that it is a functional on M + satisfying the definition above whose domain was expanded in this way. We shall do this implicitly fom now on, without futhe efeence. We may also mention the popeties listed in the definitions above when talking about functionals defined on whole M. If we do so, we always mean that those functionals have said popeties when esticted on M +. Now, having expanded the domain of on whole M, we may define (.i.) Banach function spaces and uasinomed.i. spaces. Definition 2.. Let be Banach function nom on M. Then the set X = {f M; f < } euipped with the nom will be called a Banach function space. Futhe, if is eaangement invaiant, we shall say that X is a eaangement invaiant Banach function space. If is an.i. uasinom, we define uasinomed.i. space in exactly the same manne. 5
A useful popety of uasinomed.i. spaces is, that X Y holds fo any uasinomed.i. spaces X Y that both satisfy (P5). This popety fo Banach function spaces can be found fo example in [, Chapte, Theoem.8], but it is evident fom the poof that it holds fo uasinoms too. We conclude this section by listing one useful classical esult. Theoem 2.. The classical Lebesgue functional defined on M((, ), λ) by ( f = f dλ f = ess sup f (, ) fo (, ), fo = is an.i. Banach function nom on M((, ), λ) fo [, ] and an.i. uasinom on M((, ), λ) fo (, ). Poof. Poof fo [, ] is classical, see fo example [, Chapte, Theoem.2] and [, Chapte 2, Poposition.8], and evey popety of.i. uasinom except the subadditivity up to constant can be poved even in case (, ) in exactly the same way. To veify the one emaining condition, note that t t is concave, while t t is convex and thus a, b [, ] it holds that (a+b) a +b while (a + b) 2 (a + b ). Theefoe, using the definition of fo (, ) we may compute as desied. f + g = f(s) + g(s) ds = f(s) ds + g(s) ds 2 f(s) ds = 2 ( f + g ) 2.3 Associate spaces f(s) + g(s) ds + g(s) ds In this section we define the associate functionals and conseuently the associate spaces. We intentionally make the definitions vey boad and geneal. Fo details on associate spaces of Banach function spaces, see [, Chapte, Section 2]. When talking about associate spaces, it is useful to use the notation fo the numbe defined by fo (, ), = fo =, fo =. which we will do fom now on. 6
Definition 2.2. Let : M [, ] be some non-negative functional and put X = {f M; f < }. Then the set { X = f M sup g M; g R f g dμ < will be called the associate space of X and the functional X by f X = sup f g dμ g M; g R will be called the associate functional of. } defined fo f M As the notation suggests, these tems ae inteesting mainly when the functional is at least a uasinom. We wanted only to show that the definition itself lays no euiements on it. We may now state the Hölde ineuality fo associate spaces, which euies one additional condition. Theoem 2.3. Let : M [, ] be some positively homogeneous functional. Then it holds fo all f, g M that if we intepet as. R f g dμ g f X Poof. Given positive homogeneity of, the conclusion follows immediately fom Definition 2.2. To povide an example, if we take fom Theoem 2. as ou, we obtain that X =. Fo details we again efe to [, Chapte, Section 2]. It is uite obvious fom Definition 2.2 that if two positively homogeneous functionals X and Y satisfy X Y then also Y X. Euivalently, we may say that if X Y then Y X, whee the continuity of the embedding is undestood elatively to the defining functionals, i.e. it means that the identity opeato is bounded. It has been poven, see fo example [, Chapte, Theoem 2.2] and [, Chapte, Theoem 2.7], that the associate functional of a Banach function nom is itself a Banach function nom, and that its associate functional is. This esult has been impoved ecently by Gogatishvili and Soudský in [8]. Since we will use this esult late in the pape, we pesent it below. The symbol X denotes the second associate functional, that is an associate functional of X. Theoem 2.4. Let : M [, ] be a functional that satisfies (P4) and (P5) and which also satisfies fo all f M that f = f. Then the functional X is a Banach function nom. In addition, is euivalent to a Banach function nom if and only if X 7
2.4 Fundamental function In this section, we define the fundamental function of a uasinomed.i. space and state some of its popeties, and then do the same with the endpoint spaces. We poceed in accodance with [2, Section 7.9] and [2, Section 7.]. We note that this topic has been also coveed in [, Chapte 2, Section 5]. We fist define the fundamental function. Definition 2.5. Let (X, ) be a uasinomed.i. space. Then the fundamental function φ X of X is defined fo all t in the ange of μ by φ X (t) = χ E whee E is some subset of R of measue μ(e) = t. Note that the set E in the definition always exists by the assumption that t is in ange of μ, and that the definition does not depend on the choice of E since the nom satisfies (P6) and it holds fo all E R of measue μ(e) = t that χ * E = χ (,t). One impotant esult is the elation of fundamental functions of X and of its associate space X, which we pesent in following theoem. The weake esult fo uasinomed.i. spaces follows diectly fom the Hölde ineuality (Theoem 2.3). Poof of the stonge esult fo Banach function spaces can be found fo example in [2, Theoem 7.9.6] Theoem 2.6. Let (X, ) be a uasinomed.i. space and X its associate space. Then, φ X (t) t (2.) φ X (t) fo all non-zeo t in ange of μ. If futhe X is a Banach function space, then we have euality in (2.) fo all elevant t. We now define what it means fo a function to be uasiconcave. Definition 2.7. Let a (, ]. A non-deceasing function φ : [, a) [, ) is said to be uasiconcave, if it satisfies (i) φ(t) = t = (ii) t φ(t) is non-deceasing on (, a). It has been shown, that fundamental function of any Banach function space is uasiconcave. Fo details, see fo example [2, Remak 7.9.7]. We ae now euipped to define the Macinkiewicz endpoint space. Definition 2.8. Let φ be a uasiconcave function on [, μ(r)). Then the set M φ, defined as { } M φ = f M(R, μ) sup φ(t)f ** (t) <, t (,μ(r)) is called the Macinkiewicz endpoint space. 8
It follows, see fo example [2, Poposition 7..2], that the Macinkiewicz endpoint space euipped with the natually chosen functional f = sup φ(t)f ** (t) t (, ) fo f M(R, μ), is in fact an.i. Banach function space and its fundamental function coincides with φ on the ange of μ. In fact, it is the lagest of such spaces, as is fomalised in the following theoem. Poof can be found fo example in [2, Poposition 7..6]. Theoem 2.9. Let φ be a uasiconcave function on [, μ(r)). If X is an.i. Banach function space the fundamental function of which coincides with φ on the ange of μ, then X M φ. To find the smallest.i. Banach funtion space with given fundamental function, one must fist obseve that any uasiconcave function is euivalent to a concave function. This esult can be found fo example in [2, Poposition 7..]. It can be then shown, as in, fo example, [2, Theoem 7..2], that any.i. Banach function space can be euivalently enomed in such way that its fundamental function is then concave 3. Fo concave φ, we may then define the Loentz endpoint space as follows. Definition 2.2. Let φ be a non-deceasing concave function on [, μ(r)). Then the set Λ φ, defined as { } μ(r) Λ φ = f M(R, μ) f * dφ <, is called the Loentz endpoint space. We note that the Lebesgue-Stieltjes integal in uestion is well-defined since φ is non-deceasing. We futhe get, see fo example [2, Poposition 7..6], that the Loentz endpoint space euipped with the natually chosen functional f = μ(r) f * dφ fo f M(R, μ), is an.i. Banach function space fundamental function of which coincides with φ on the ange of μ. It is the smallest space with this popeties, as is fomalised in the following theoem. Fo poof, see [2, Poposition 7..5]. Theoem 2.2. Let φ be a concave function on [, μ(r)). If X is an.i. Banach function space the fundamental function of which coincides with φ on the ange of μ, then Λ φ X. It follows diectly fom Theoems 2.6, 2.9 and 2.2 that the Loentz and Macinkievicz endpoint spaces ae mutually associate. Theoem 2.22. Let φ be a concave function. Then Λ φ = M φ and M φ = Λ φ, whee φ is defined on (, μ(r)) by φ = t φ(t). and φ() =. In conclusion, we note that the endpoint spaces do not change, up to euivalence of defining functionals, if φ is eplaced with an euivalent function. 3 It is obviously always non-deceasing fom the lattice popety (P2). 9
2.5 Hady-type ineualities Thoughout the pape, we will on seveal occasions use weighted Hady-type ineualities. We now list the exact vesions we will use in ou poofs. Fistly, fo the case [, ], we pesent two theoems, poofs of which, in moe geneal setting, can be found in [9]. Theoem 2.23. Let [, ] and let v, w be two positive weights on (, ). Then thee exists a positive constant C such that the ineuality holds fo all f M((, ), λ) if and only if t w(t) f(s) ds C vf (2.2) sup w χ (, ) v χ (,) < (2.3) (, ) Theoem 2.24. Let [, ] and let v, w be two positive weights on (, ). Then thee exists a positive constant C such that the ineuality holds fo all f M((, ), λ) if and only if w(t) f(s) ds C vf t (2.4) sup w χ (,) v χ (, ) < (2.5) (, ) Secondly, fo the case (, ) we pesent theoem which was obtained by Cao, Pick, Soia and Stepanov in [2]. Theoem 2.25. Let (, ) and let v, w be two positive weights on (, ). Then thee exists a positive constant C such that the ineuality holds fo all f M(R, μ) if and only if both wf ** C vf * (2.6) sup w χ (,) ( v χ (,) ) <, (2.7) (, ) sup t w(t) χ (, ) (t) ( v χ (,) ) <. (2.8) (, )
3 Loentz Kaamata spaces Fom now on we suppose that ou measue space (R, μ) has a non-atomic measue. As noted in Section 2., this means that it is always esonant. We will also, unless stated othewise, assume that μ(r) =. 3. Slowly vaying functions Definition 3.. Let b : (, ) (, ) be a continuous function. Then b is said to be slowly vaying, if fo evey ε > thee exist t, t (, ) such as that t ε b(t) is non-deceasing on both (, t ) and (t, ) while t ε b(t) is on those intevals non-inceasing. We will, fo the sake of bevity, usually abbeviate slowly vaying as s.v. The class of s.v. functions includes, fo example, constant positive functions and, to povide something at least slightly less tivial, the functions t + log(t) and t +log(+ log(t) ) figuing in definition of genealized Loentz-Zygmund spaces studied by Opic and Pick in []. As futhe example of s.v. function, this time non-logaithmic, we pesent a function b defined on (, ) by e log t fo t [, ), b(t) = log t e fo t (, ). Lemma 3.2. Let b be a s.v. function. Then it has following popeies: (SV) The function b is slowly vaying fo evey R. (SV2) Let c >, then: b(ct) lim t + b(t) = lim b(ct) t b(t) =. (SV3) It holds fo evey α that lim t tα b(t) = lim tα, + t + lim t tα b(t) = lim t α. t (SV4) If α then, in ode fo t α b(t) to be integable on some (ight) deleted neighbouhood of zeo, it is necessay and sufficient that t α is integable on said deleted neighbouhood. The same applies fo deleted neighbouhoods of infinity. Poof. To pove (SV), note that if >, then t t is inceasing function and since fo any ε > : t ε b (t) = (t ε b(t)) and t ε b (t) = (t ε b(t)), it suffices to take t, t (, ) that testify that b is slowly vaying fo ε. If <, then the sign of ε is the opposite than that of ε, so if we take t, t (, ) as befoe, we get on (, t ) and (t, ) that t ε b(t) is non-inceasing and t ε b(t) non-deceasing, which, when combined with the fact that t t is deceasing, yields the desied popety. The emaining case = is tivial, just as ae the positiveness and continuousness of the functions in uestion.
As fo (SV2), we may assume without loss of geneality that c, since in that case thee is nothing to pove. We will pove only the case t fo c >, since the case c < is vey simila (only the ineualities ae switched) and the only diffeence in the case t + is that is eplaced by +. So let us assume that c > and fix some abitay ε >. Then thee is some t (, ) such that t ε b(t) is non-deceasing on (t, ) and t ε b(t) is on said inteval non-inceasing. Thus, if we multiply the expession in limit by (ct)ε and (ct) ε, espectively, (c ε t ε ) (c ε t ε ) we get on (t, ) following estimates: b(ct) b(t) = b(ct) b(t) (ct)ε (c ε t ε ) = (ct)ε b(ct) c ε c ε, t ε b(t) b(ct) b(t) = b(ct) b(t) (ct) ε (c ε t ε ) = (ct) ε b(ct) c ε c ε. t ε b(t) Since c >, this means that b(ct) (c ε, c ε ) on (t b(t), ). Because ε was abitay, we may choose it in such way that the inteval (c ε, c ε ) is contained in any given neighbouhood of. Hence, the definition of limit is veified and we have b(ct) lim t =. b(t) The esult (SV3) follows fom (SV2). We will again estict ouselves to one of the cases, namely α > and t. In this case, since t α b(t) is positive and non-deceasing on some neighbouhood of infinity, the limit exist and is stictly geate than zeo. But using (SV2), the aithmetic of limits and a simple change of vaiables, we aive to the following euality: lim t tα b(t) = ( lim t tα b(t) ) ( lim t ) b(ct) = lim t α b(t) b(ct) b(t) t b(t) = lim t α b(ct) = lim t cα t (ct)α b(ct) = lim cα t tα b(t) fo any c >. Hence, the limit must be +, since it is the only element of extended eal numbes with this popety which is also stictly geate than zeo. Remaining cases use the same appoach. The monotonicity of t α on appopiate neighbouhood is always used to show that the limit exists and to eliminate eithe o + and then identical calculation as above is used to eliminate positive numbes, leaving only one option. It emains to pove (SV4). Again, the appoach in all cases is identical so we will pove the statement only fo deleted neighbouhoods of zeo. Fist note that t α b(t) is positive, so its integal is always defined, and continuous on (, ), so its integal ove compact subset of (, ) is finite (this holds even when α = ). Theefoe, integability of t α b(t) on given deleted neighbouhood of zeo is euivalent to integability on any deleted neighbouhood of zeo, which is not deleted neighbouhood of infinity. So let α > and fix ε > such that α ε >. Accoding to (SV3), lim t + t ε b(t) = lim t + t ε = and thus thee exists a deleted neighbouhood of zeo on which t ε b(t) <. Hence, on said deleted neighbouhood, we have t α b(t) = t α ε t ε b(t) < t α ε which is integable thanks to ou choice of ε. In case α <, we find ε > such that α + ε < and deleted neighbouhood of zeo whee t ε b(t) >. We then have t α b(t) = t α+ε t ε b(t) > t α+ε which is by ou choice of ε not integable. This poves the euivalence fo deleted neighbouhoods of zeo. 2
It follows fom the poof of (SV) that this popety can be somewhat stengthened, in the sense that the exponent needs not to emain the same on whole (, ). Indeed, if we take some, 2 R and put b b(t) (t) on some deleted neighbouhood of, = b 2 (t) on some deleted neighbouhood of and on the est of (, ) define b in such way that it will be continuous, we may then use the same appoach as befoe to pove that b is slowly vaying. Thus, we may now add the boken logaithmic function defined in [] to ou set of examples of s.v. functions. It is simple execise to use a simila techniue to pove that poduct and sum of two s.v. function is also s.v. On the othe hand, the estictions α and α in (SV3) and (SV4), espectively, ae necessay. The popety that b is s.v. is simply too weak to detemine the behavio of t α b(t) in those cicumstances. To illustate that the limits of b at both zeo and infinity can be any positive numbe, it suffices to conside all functions identically eual to some positive numbe on (, ), while the function, mentioned befoe, t + log(t) and its ecipocal t ( + log(t) ) ae examples of s.v. function with said limits eual to infinity and zeo, espectively. Similaly, fo b identically eual to one, t b(t) = t is not integable on any deleted neighbouhood of zeo/infinity, while if we put b(t) = (+ log(t) ) 2 then t b(t) is integable on whole (, ). 4. This obsevation leads to the following definition. Definition 3.3. Let b be a s.v. function and (, ]. We say that b is -nice aound infinity, if eithe < and t b(t) is integable on some deleted neighbouhood of infinity, o = and b(t) is bounded on some deleted neighbouhood of infinity. Similaly, we say that b is -nice aound zeo, if eithe < and t b(t) is integable on some deleted neighbouhood of zeo, o = and b(t) is bounded on some deleted neighbouhood of zeo. We now use Lemma 3.2 to pove one athe intuitive esult that will be useful late. Lemma 3.4. Let b be a s.v. function and fix some abitay ε >. Then function f, defined by f(t) = t ε b(t) fo t (, ), satisfies on (, ). f * f Poof. If we ecall Definition 3., we see that thee ae some constants < t t <, such that f is non-inceasing on both (, t ) and (t, ). Because f is also continuous and positive on (, ), we have that thee is some positive 4 Note that, by (SV4), t α b(t) cannot be integable on whole (, ) unless α =. 3
constant K, such that f(t) [K, K] fo all t [t, t ]. Thanks to (SV3), we futhe have that lim f(t) =, t lim t f(t) =. This all togethe yields that thee ae some constants T and T, satisfying < T t t T, such that f ( (K, ) ) = (, T ), f ( (, K ) ) = (T, ). Now, fo any t (, T ) (T, ), we have that λ f (f(t)) = λ ({s (, ) f(s) > f(t)}) = inf{ (, ) f() f(t)}. This means that the distibution function λ f (t) of f is the genealised invese of f on those intevals, and because non-inceasing eaangement of f is, by definition, genealised invese of its distibution function, we get that f(t) = f * (t) fo t (, T ) (T, ). It emains to examine f * on [T, T ], which is simple, since if we define f by f(t) fo t (, T ) (T, ), f (t) = K fo t [T, T ], then clealy f f and thus f * f * = f since f is non-inceasing, while if we define f 2 by f(t) fo t (, T ) (T, ), f 2 (t) = K fo t [T, T ], then clealy f f 2 and thus f * f * 2 = f 2 since f 2 is also non-inceasing and we can combine this to get on [T, T ]. K 2 = K K f 2 f f * f f f K = K2 K The next esult also follows fom Lemma 3.2, but euies little moe wok. Lemma 3.5. Let b be a s.v. and α (, ). Then it holds fo all t (, ) that t s α b(s) ds t α b(t). (3.) Poof. Fist note that the integal on the left-hand side is always finite by ou condition on α and (SV4). Futhemoe, the functions on both sides ae continuous and positive. Theefoe, we have by the classical Bolzano-Weiestass theoem, that thei atio is bounded on any compact subset of (, ), and this holds independently on which function is in nominato and which in denominato. This yields us the desied euivalence fo compact subsets of (, ), and to complete 4
the poof we only need to examine the behavio of those functions on deleted neighbouhoods of zeo and infinity. We pefom only the hade case of deleted neighbouhood of infinity and begin with the estimate. We fix some ε > and exploit the fact that b is s.v. to find some some t (, ) such that t ε b(t) is non-inceasing on (t, ). Fo technical easons, which will be evident fom calculation, we choose (2t, ) as the deleted neighbouhood of infinity on which we obtain the desied estimate. We may now employ the positiveness of t α b(t) and the monotonicity of t ε b(t) on (t, ) to compute fo abitay t (2t, ), t t t s α b(s) ds s α b(s) ds = s α +ε s ε b(s) ds t t t s α +ε ds t ε b(t) = (t α+ε t α+ε )t ε b(t) t α + ε ( ( ) t α+ε t α+ε t 2) ε b(t) α + ε = t α+ε 2α+ε 2 α+ε t ε b(t) = t α b(t) 2α+ε 2 α+ε α + ε α + ε which establishes on (2t, ). We now focus on the estimate. Because α > and b is s.v., we may find some ε > such that α ε > and appopiate t (, ) such that t ε b(t) is non-deceasing on (t, ). Now, since s α b(s) is not integable on deleted neighbouhoods of infinity by ou assumption on α and (SV4), we have that t s α b(s) ds as t and thus we may find some t > t such that t t s p b(s) ds > t s p b(s) ds (3.2) fo all t (t, ). Because of this technical but useful popety, we choose t (t, ) as the deleted neighbouhood of infinity on which we pove. We now use (3.2) and the monotonicity of t ε b(t) on (t, ) to compute fo abitay t (t, ) t s α b(s) ds 2 t s α b(s) ds 2 t t s α ε ds t ε b(t) t = 2t α ε t ε b(t) α ε = 2 tα b(t) α ε, which establishes on (t, ) and completes the poof. Note that we wee vey explicit when caying the calculation. This was done on pupose, because it includes all the majo methods used in calculations thoughout this pape. The following calculations will be simila, so we will povide less detail. We now examine the limit case α =. 5
Lemma 3.6. Let b be a s.v. function that is also -nice aound zeo. Then b, defined on (, ) by t b(t) = s b(s) ds, is also s.v. Futhemoe, we have b(t) b(t) lim = lim =. t b(t) t b(t) Poof. Because we assume that b is -nice, we have that b is finite on (, ). It is also obviously positive and continuous. To pove the emaining popety fom Definition 3., we fix some ε > and note that b is inceasing in itself and thus also when multiplied by t t ε. Theefoe, it emains only to find some deleted neighbouhoods of zeo and infinity whee t ε b(t) is non-inceasing. We will pefom the details only fo deleted neighbouhood of infinity, since the case of deleted neighbouhood of zeo is simple and vey simila. Find the t, fom Definition 3., such that t ε b(t) is non-inceasing on (t, ). To keep the calculation easonably shot, let us denote A = t We now calculate, fo abitay > t > t ( t ε b(t) ε b() = (t ε ε ) ( t s b(s) ds (, ). A + t s b(s) ds t ) ) ε s b(s) ds (t ε ε ) A + t ε b(t) s +ε ds ε t ε b(t) s +ε ds t t ( = (t ε ε ) A + t ε b(t)(t ε t ε ) ) ε t ε b(t)( ε t ε ) ε ε ( ) = (t ε ε ) A tε ε t ε b(t), which is positive on some deleted neighbouhood of infinity, because A is positive and, thanks to (SV3), t ε b(t) conveges to zeo as t appoaches infinity. We now tun ou attention to (3.6). We again pefom only the estimates fo the case of limit at infinity, since the case at zeo is simila and simple. Let us fix some ε > and find some t > such that t ε b(t) is non-inceasing on (t, ). We may then use this popety, togethe with the positiveness of t b(t) to calculate fo abitay t (t, ) b(t) t s b(s) ds b(t) t t s b(s) ds b(t) t ε b(t) t t s +ε b(s) ds = t tε ε, t ε t ε which is smalle than 2ε wheneve t (t 2, ) fo suitably lage t 2. So we have found some deleted neighbouhood of infinity whee b(t) b(t) < 2ε which togethe with the obvious fact b(t) b(t) yields the desied limit in infinity. Simila esult can be obtained fo the uppe-bound integal. analogous and theefoe omitted. The poof is 6
Lemma 3.7. Let b be a s.v. function that is also -nice aound infinity. Then ^b, defined on (, ) by ^b(t) = s b(s) ds, is also s.v. Futhemoe, we have t b(t) b(t) lim = lim =. (3.3) t ^b(t) t ^b(t) We conclude this section by listing, fo futue efeence, an immediate but useful conseuence of Lemma 3.2. Coollay 3.8. Let b be a s.v. function, then fo evey c > thee exists a eal constant K > such that fo all t (, ) the value b(ct) (o euivalently b(t) ) b(t) b(ct) belongs to [K, K]. 3.2 Loentz Kaamata functionals Definition 3.9. Let p (, ], (, ] and let b be a s.v. function. We then define the Loentz Kaamata functionals p,,b and (p,,b), fo f M, as follows: f p,,b = t p b(t)f * (t), f (p,,b) = t p b(t)f ** (t), whee is the classical Lebesgue functional on (, ) as intoduced in Theoem 2.. We futhe define the coesponding Loentz Kaamata spaces L p,,b and L (p,,b) as L p,,b = {f M; f p,,b < }, L (p,,b) = {f M; f (p,,b) < }. Note that the Loentz functionals p, and (p,), defined fo example in [, Chapte 4, Definiton 4.], ae special cases of Loentz Kaamata functionals with b(t) =. If futhe p = then we have that p,,b coincides with the classical Lebesgue functional, as follows fom [, Chapte 2, Poposition.8]. Anothe examples of Loentz Kaamata functionals ae the genealised Loentz-Zygmung functionals studied in []. In poofs of following popositions, we will extensively use Theoem 2.. Fo syntactic easons, we will not uote it explicitly evey time, instead we will only efe to the popeties of which ae stated in it. Poposition 3.. Let p,, b be as above. Then the Loentz Kaamata functional p,,b is a eaangement invaiant uasinom on M, and conseuently (L p,,b, p,,b ) is a uasinomed.i. space, if and only if p <, o p = and b is -nice aound zeo. Futhemoe, if those conditions ae not satisfied, then L p,,b = {}. 7
Poof. Thoughout this poof, the lettes f, g will denote some abitay functions belonging to M. We begin by poving the sufficiency. The eaangement invaiance (P6) is obvious fom definition. The lattice popety (P2) follows immediately fom popeties of non-inceasing eaangement and the lattice popety of. Similaly, the Fatou popety (P3) follows fom anothe popety of non-inceasing eaangement and the Fatou popety of. Futhemoe, since fo evey a R (af) * = a f *, it follows immediately that also af p,,b = a f p,,b. If f is identically eual to zeo, then so is its non-inceasing eaangement and thus also f p,,b. On the othe hand, if f is positive on some set of positive measue, then thee exist some ε, δ >, such that f * ε on (, δ) and thus f p,,b. Fo the subadditivity up to a constant, if we emembe that the non-inceasing eaangement satisfies (f +g) * (t +t 2 ) f * (t )+g * (t 2 ), we may use Coollay 3.8 to get the following estimate: ) ( f + g p,,b = t p b(t)(f + g) * (t) t p b(t) ( f *( t t + g * 2 2) ) t p ( b(t) t p ( ) ( ( ) ( t t t = ( t p 2) b ( ) b f * + g * t 2 2 2 2) ) ( t K 2 2 p b ( t 2 ) ( f * ( t 2 ) + g * ( t 2) ) whee K is the constant fom Coollay 3.8 fo c = ( ) p 2. Now, if we use the change of vaiables s = t, we get 2 f + g p,,b ^K s p b(s)(f * (s) + g * (s)) whee ^K is eual to K if = and to 2K othewise. We may now use the subadditivity (up to constant) of to get the desied esult. We have thus veified that p,,b satisfies (Q). It emains to veify (P4), i.e. that if χ E is a chaacteistic function of some set E R of measue μ(e) < then χ E p,,b, <. Since fo such E it holds that χ * E = χ [,μ(e)], this is euivalent to the uestion, when, whethe t p b (t) is integable on some deleted neighbouhood of, o, when =, whethe t p b(t) is bounded on some deleted neighbouhood of. If p, this is satisfied by (SV4) and (SV3) espectively. In the emaining cases, if we ecall Definition 3.3, we see immediately that we assume that b is such a function that satisfies those euiements. To pove necessity, it suffices to show that p,,b does not satisfy (P4) wheneve those conditions ae not satisfied, i.e. when p = and b is not -nice aound zeo. To this end, let E R satisfy μ(e) < and ewite t b(t)(χe ) * (t) by definition to get μ(e) t b(t)(χe ) * t b (t) dt fo (, ), (t) = sup t (,μ(e)) b(t) fo =, 8
which is infinite since since we assume that b is not -nice. This also means that L p,,b = {}, because, as we ealised when poving that f p,,b is non-zeo wheneve f is non-zeo on some set of positive measue, we have fo such f that f * ε on (, δ), i.e. f * ε(χ E ) * fo some set E of measue μ(e) = δ and thus t p b(t)f * (t) > t p b(t)(χe ) * (t) =. (3.4) Hence, L p,,b = {}, and the poof is complete. It is woth mentioning that fo some special set of paametes p,, b, p,,b may ualify to be a.i. Banach function nom on M. Let us name at least the cases p = [, ] while b(t) = fo all t (, ), when p,,b coincides with the classical Lebesgue functional. In the following lemma we examine fo which choices of p,, b the functional p,,b satisfies (P5) and povide one sufficient and one necessay condition. Poposition 3.. Let b be a s.v. function. (i) Suppose that one of the following condition holds: a) < p, b) = p and thee ae some constants K, t > such that b(t) K fo all t (, t ), then p,,b satisfies (P5). (ii) Suppose that one of the following condition holds: a) p <, b) p = and lim t + b(t) =, then p,,b does not satisfy (P5). Poof. Suppose fist that the assumption of (i) holds. In both allowed cases, we have that thee ae some constants K, t > such that t p b(t) K fo all t (, t ). Indeed, in the fist case p < and we may use (SV3) while in the emaining case p = and so the estimate is assumed. To pove that,,b satisfies (P5), fix some E R of measue μ(e) = < and put K = min{k, inf [t,] t p b(t)}. Then K >, since t p b(t) is continuous and positive. The conclusion follows fom the Hady-Littlewood ineuality (Theoem 2.4) and classical Jensen ineuality 5, which can be found fo example in [2, Theoem 4.2.], because we may use them to obtain f dμ f * (t) dt K t p b(t)f * (t) dt E ( ( ) = K t p b(t)f * (t) dt ( K t p b (t)(f * (t)) dt K f p,,b 5 We use that and theefoe t t is a convex function. 9
whee K is finite and independent of f. Suppose now that the assumption of (ii) holds. We then have, eithe fom (SV) and (SV3) o diectly fom ou assumption, that lim t + t p b (t) =. We fix some R R of finite non-zeo measue and then find, fo abitay n N, some > such that t p b (t) < n fo all t (, ) and some set E R of measue < μ(e) <. We then easily obtain the estimate χ E dμ = R E dμ = μ(e) = μ(e) μ(e) dt > n t p b (t) dt. Fom this, we may obtain the desied estimate by using again the classical Jensen ineuality 6, since it yields ( ( χ E dμ μ(r ) μ(e) χ E dμ > n μ(r ) t p b (t) dt R R = χ E p,,b. Since n N was abitay, we have shown that R is the set testifying that (P5) does not hold. We now tun ou attention to (p,,b). Because diffeent popeties of (p,,b) have diffeent euiements, we will teat them sepaately. We stat by examining the non-tiviality. Poposition 3.2. Let p,, b be as in Definition 3.9. Then (p,,b) satisfies (P4) if and only if p and b satisfy one of the following conditions: (i) p (, ). (ii) p = and b is -nice aound infinity. (iii) p = and and b is -nice aound zeo. Futhemoe, if none of those conditions is satisfied, then L (p,,b) = {}. Poof. The situation is much moe complicated than when we wee examining (P4) in Poposition 3., because, if we fix some E R with measue μ(e) <, then (χ E ) ** is not eual to χ (,μ(e)) as befoe but athe to function ρ defined as follows fo t (, μ(e)), ρ(t) = μ(e) t fo t [μ(e), ). Thus, in ode to veify (P4), it is necessay to concen youself not only with integability/boundedness on deleted neighbouhoods of zeo, but also on deleted neighbouhoods of infinity. To be pecise, if (, ), then (P4) is euivalent to whethe both of the following conditions ae satisfied: (a) t p b (t) is integable on some deleted neighbouhood of zeo. (b) t p b (t) is integable on some deleted neighbouhood of infinity. 6 This time we use that and theefoe t t is a convex function. 2
Just as in Poposition 3., condition (a) is satisfied fo p < by (SV4), and assumed in the emaining case by (iii). Similaly, (b) is satisfied fo p > by (SV4) and assumed in the emaining case by (ii). The method fo = emains unchanged, only instead of (SV4), (SV3) is used. It emains to pove necessity. The method is the same as in Poposition 3.. Ou assumption is that eithe p <, p = and b is not -nice aound infinity o p = and b is not -nice aound zeo, which all yields 7 that (p,,b) applied to the chaacteistic function of any set of non-zeo measue euals infinity. This futhe yields, by the same easoning as befoe, that L (p,,b) = {}. We now show some popeties that ae common fo all paametes p,, b allowed in Definition 3.9. We note that these esults ae not inteesting in the cases when L (p,,b) = {}, but we want to emphasise that it is only the popety (P4) that gets violated in those situations. Poposition 3.3. Let p,, b be as in Definition 3.9. Then (p,,b) satisfies (Q), (P2), (P3), (P5) and (P6). Poof. We omit the poof of (Q), (P2), (P3) and (P6), because it is almost the same as in Poposition 3., only instead of efeencing popeties of noninceasing eaangement one efeences popeties of maximal function, except fo subadditivity up to constant, which is simple since the opeato f f ** is in itself subadditive, and focus on (P5). Fix some E R with measue μ(e) < and abitay f M. Thanks to the Hady-Littlewood ineuality (Theoem 2.4) it suffices to pove that thee is some finite constant C E satisfying μ(e) f * (t) dt C E f (p,,b) (3.5) fo all f M. Using the positive homogeneity and the lattice popety (P2) of we may compute whee t f (p,,b) = t p b(t)f ** (t) = t p b(t) χ (μ(e), ) t p b(t) μ(e) t f * (s) ds f * (s) ds χ (μ(e), ) t p b(t) χ (μ(e), ) t p b(t) >, f * (s) ds since both t p and b ae non-zeo on (, ). Now, we have that eithe χ (μ(e), ) t p b(t) <, which gives us (3.5) with C E = ( χ (μ(e), ) t p b(t) ), o we have χ (μ(e), ) t p b(t) =, 7 Eithe by Definition 3.3 o by (SV3)/(SV4). 2
which, as we ealised when poving Poposition 3.2, leads to the conclusion that f (p,,b) < f = μ-a.e., and theefoe (3.5) holds with any C E >. Now, it follows that if we take [, ] we will have that both f f ** and ae subadditive, which gives us the last popety of.i. Banach function noms. Hence, we have the following poposition. Poposition 3.4. Let p,, b be as in Definition 3.9 and suppose [, ]. Then (p,,b) is eaangement invaiant Banach function nom on M, and conseuently (L (p,,b), (p,,b) ) is.i. Banach function space, if and only if p and b satisfy one of the following conditions: (i) p (, ). (ii) p = and b is -nice aound infinity. (iii) p = and and b is -nice aound zeo. Futhemoe, if none of those conditions is satisfied, then L (p,,b) = {}. 3.3 Relations between L p,,b and L (p,,b) In this section we study when L p,,b = L (p,,b). The fist obsevation follows fom Popositions 3. and 3.4 since they tell us that, fo p <, L (p,,b) is always tivial while L,,b neve is. So we obtain p as a necessay condition. Note that because we always have fo any f M that f * f **, we get tivially 8 fom popeties of that p,,b (p,,b) fo all p,, b allowed by Definition 3.9. Thus, we only have to examine what ae the paametes fo which (p,,b) p,,b. We will show in the next two theoems that p > is sufficient. To this end, we will employ weighted Hady ineualities, which has been pesented in Section 2.5 as Theoems 2.23 and 2.25. Theoem 3.5. Let p,, b be as in Definition 3.9 and let [, ] and p (, ]. Then (p,,b) p,,b. Poof. As hinted, we employ weighted Hady ineuality, moe specifically Theoem 2.23. Remembeing Definition 3.9, we see that the conclusion of the theoem is special case of (2.2) whee w(t) = t p b(t) and v(t) = t p b(t). Thus it suffices to pove (2.3) fo this w, v, i.e. that sup t p b(t)χ (, ) (t) (t p b(t)) χ (,) (t) <. (3.6) (, ) Fistly, note that both t p b(t)χ (, ) (t) and (t p b(t)) χ (,) (t) ae, thanks to (SV3) o (SV4), depending on the value of /, always finite by ou assumption p (, ]. Futhemoe, when undestood as functions of vaiable, these expessions define continuous functions. Indeed, we ae, in dependence 8 See Theoem 2.. 22
on whethe / is finite o infinite, taking eithe supemum o integal of continuous function ove extending/shinking inteval, which always yields continuous function. Hence, to get (3.6) we only have to pove boundedness of the afoementioned function on some appopiate deleted neighbouhoods of zeo and infinity. The method is the same in both cases, so we will do explicitly only the case fo deleted neighbouhood of infinity. Because of the diffeences in definitions of fo = and [, ) we need to do the estimates sepaately fo the cases =, (, ) and =. We stat with =. By definition 9 of and, we may ewite t p b(t)χ (, ) (t) (t p b(t)) χ (,) (t) as t p 2 b(t) dt sup t (,) t p b(t). (3.7) Now, by ou assumption on p, we have < and thus we may choose such p an ε > that + ε <. We may then use the assumption that b is s.v. to p find some such that t ε b(t) is non-inceasing on (, ). Fo technical easons, we now find some > such that t p b(t) > sup t (, ) t p b(t) fo all t (, ). Such a exists thanks to t p b(t) (3.8) being continuous, bounded on some deleted neighbouhood of zeo while inceasing beyond all bounds as t by (SV3). This is the deleted neighbouhood of infinity on which we will show that (3.7) is bounded, so we suppose (, ). Using (3.8) and the monotonicity of t ε b(t) on (, ), we may now compute t p 2 b(t) dt sup t (,) t p b(t) = t p 2 b(t) dt sup t (,) t p b(t) ε b() t p 2+ε dt ε b() sup t (,) = p +ε ε b() ( p + ε) ε b() t p +ε p +ε = ( p + ε) which establishes the boundednes of (3.7) on (, ). We move to (, ). As befoe, we may ewite t p b(t)χ (, ) (t) (t p b(t)) χ (,) (t) as ( t p b (t) dt ( t p b (t) dt. (3.9) Remembeing p >, we see that both < and p >, so we p may choose such ε > that both + ε < and p ε > and p use that by (SV) ae both b and b s.v., to find some such that t ε b (t) is 9 We will use that the tems of supemum and essential supemum coincide fo continuous functions. 23
non-inceasing on (, ) and t ε b (t) is non-deceasing on (, ). We poceed by finding some > such that t p b (t) dt > t p b (t) dt (3.) t fo all (, ), which is possible since p b (t) dt inceases beyond all bounds as. This is the elevant deleted neighbouhood of infinity on which we pove that (3.9) is bounded. We may now compute in simila manne as befoe, using (3.) and the monotonicity on (, ) of both t ε b (t) and t ε b (t), that it holds fo all (, ) that ( t p b (t) dt ε = 2 2 ( ( b() t p +ε dt t p b (t) dt ε b () ε p + ε ε ( p ε ( 2 p ε ) ( ( p + ε)) ( p ε) ε p + ε ε ( p ε ) ( ( + ε)) ( ε) = p p t p ε dt 2 ( ( p + ε)) ( p ε), just as desied. And finally the case =. This time we ewite t p b(t)χ (, ) (t) (t p b(t)) χ (,) (t) as sup t p b(t) t p b (t) dt. (3.) t (, ) Since both < and > we may choose ε > such that both p p + ε < and ε >, and p p such that t ε b ( t) is non-inceasing on (, ) and t ε b(t) is non-deceasing on (, ). As above, we find > such that p b (t) dt > t p b (t) dt (3.2) t fo all (, ). This is the elevant deleted neighbouhood of infinity on which we pove that (3.) is bounded, so we put (, ). We now use (3.2) and the monotonicity on (, ) of both t ε b ( t) and t ε b(t), to get sup t p b(t) t (, ) t p b (t) dt ε b() sup t p +ε 2 ε b () t (, ) which coves the last case and concludes the poof. t p ε dt = 2 ε b() p +ε ε b () p ε p ε ε p 2 ε, p 24
Theoem 3.6. Let p,, b be as in Definition 3.9 and let (, ) and p (, ]. Then (p,,b) p,,b. Poof. This time we utilise the weighted Hady-type ineuality pesented as Theoem 2.25, because the estimate we aim to pove is a special case of (2.6) with v(t) = w(t) = t p b(t), as is eadily veified fom Definition 3.9. Thus it suffices fo us to pove both (2.7) and (2.8) fo this v and w. Rewitten to explicit fom, those conditions ead sup (, ) ( sup (, ) ( t p b (t) dt t p b (t) dt ( ( t p b (t) dt t p b (t) dt ) ) <, (3.3) <. (3.4) Note that t p b (t) dt < by (SV4) because we assume p >. Theefoe, the function we take supemum of in (3.3) is identically eual to on (, ) and thus the condition (3.3) is satisfied. It emains to veify (3.4). Ou situation is analogous to when we wee poving Theoem 3.5 because we ae examining a continuous function which is bounded on compact subsets of (, ). Hence, we need only to pove boundedness on deleted neighbouhoods of zeo and infinity. Since the appoach is the same, we pefom only the calculation fo deleted neighbouhood of infinity. Since we assume p >, we have that < and thus we may p choose such ε > that + ε <. Then we poceed by finding p > such that t ε b (t) is non-inceasing on (, ). We choose (2, ) as the deleted neighbouhood of infinity on which we show that the function in uestion is bounded. We use the monotonicity of t ε b (t) to compute fo abitay (2, ) ( t p b (t) dt ( t p +ε dt ( p +ε = p +ε ( = p +ε p +ε p +ε ( 2 ( 2 p +ε 2 p +ε ( t p b (t) dt ε b() ( ( p + ε ) p +ε p + ε p ε p ε ( p + ε p ε ) t p +ε dt which establishes the desied boundedness on (2, ). ) ε b () In the limiting case p =, the situation is moe complicated. As is obvious fom Popositions 3. and 3.2, one necessay condition fo L (,,b) = L,,b is Such exist because b is s.v. by ou assumption on b and (SV). 25