Analog of modulus of convexity for Grand Lebesgue Spaces
aa r X i v : . [ m a t h . F A ] F e b Analog of modulus of convexity for Grand Lebesgue Spaces.M.R.Formica, E.Ostrovsky, L.Sirota.
Universit`a degli Studi di Napoli Parthenope, via Generale Parisi 13, PalazzoPacanowsky, 80132, Napoli, Italy.e-mail: [email protected], Bar - Ilan University, department of Mathematic and Statistics, 59200,e-mails: [email protected]
Abstract
We introduce and evaluate the degree of convexity of an unit ball, so - called,characteristic of convexity (COC) for the Grand Lebesgue Spaces, (GLS), which isa slight analog of the classical notion of the modulus of convexity (MOC).
Key words and phrases.
Banach, Lebesgue - Riesz and Grand Lebesgue Spaces(GLS) and norms, triangle inequality, unit ball, embedding, modulus of convexity(MOC), weak characteristic of convexity (WCOC).
Let ( X, || · || ) be Banach space, S be its unit sphere: S = { x, x ∈ X, || x || = 1 } and B be its unit ball with the center in origin: B = { x, x ∈ X, || x || ≤ . } Let also ǫ be arbitrary number from the segment [0 ,
2] : 0 ≤ ǫ ≤ . Recall thefollowing very important in the geometrical theory of Banach spaces notion ModulusOf Convexity (MOC) δ X ( ǫ ) for the space ( X, || · || ) : Definition 1.1.
The Modulus Of Convexity (MOC) for the space X = ( X, || · || ) = ( X, || · || X ) , which is denoted by δ X ( ǫ ) is defined as follows δ X ( ǫ ) def = inf ( − || x + y || x, y ∈ B ; || x − y || ≥ ǫ ) ; (1)the Ball definition ; or equally 1 X ( ǫ ) def = inf ( − || x + y || x, y ∈ S ; || x − y || ≥ ǫ ) ; (2)spherical definition.This important for Functional Analysis notion was introduced by O.Hanner(1956), see [22], and was investigated in many works, see e.g. [6], [8], [9], [17], [22],[34], [35] and so one.The other application, indeed, in the theory of random fields, may be found in[29], chapter 3.For example, let X be the classical Lebesgue - Riesz space L p , builded oncertain atomless measure space; the correspondent Module Of Convexity will bedenoted by δ p ( ǫ ); p ∈ (1 , ∞ ) . If p ∈ (1 , , then the function δ p ( ǫ ) is an uniquepositive solution of an equation( 1 − δ p ( ǫ ) + 0 . ǫ ) p + ( 1 − δ p ( ǫ ) − . ǫ ) p = 2 , so that when ǫ ∈ [0 , δ p ( ǫ ) ≥ p − ǫ . (3)If now p ∈ (2 , ∞ ) , then δ p ( ǫ ) = 1 − ( 1 − (0 . ǫ ) p ) /p , (4)and when again ǫ ∈ [0 , δ p ( ǫ ) ≥ ǫ p p p ; (5)see e.g. [8], [9], [17]. We intent in this short report to introduce some weak analog ofmodulus of convexity for Grand Lebesgue Spaces (GLS) and derive someits properties.
It follows from the definition 1.1. that for x, y ∈ B || x + y || ≤ − δ X ( || x − y || ) , (6)a refined triangle inequality.Leu us give some generalization of this notion MOC. Definition 1.2.
Let again ( X, || · || X = || · || ) be the Banach space. Supposethat there exists an another Banach space ( Y, ||| · ||| Y = ||| · ||| ) such that X is embedded in Y : X ⊂ Y, and a non - negative numerical valued function2functional!) ∆[ X, Y ] = ∆[ X ]( u ) , u ∈ X, which is named as a weak characteristicof convexity, (WCOC), such that∆[ X ]( u ) = ∆[ X, Y ]( u ) = 0 ⇔ u = 0 , and for which ∀ x, y ∈ B ⇒ || x + y || ≤ − X, Y ]( ||| x − y ||| ) . (7)It is this inequality (7) for the Grand Lebesgue Spaces X, that was appliedin particular in the theory if random fields, see for example [29], chapter 3, sections3.3 - 3.6.To be more precisely, we want to prove the existence ∆[ Gψ ]( · ) for the so -called Grand Lebesgue Spaces Gψ and derive some its estimations. Brief note about Grand Lebesgue Spaces (GLS).
We recall here for reader convenience some known definitions and facts aboutthe theory of Grand Lebesgue Spaces (GLS) using in this article. Let (
Z, M, µ ) bemeasurable space with non - trivial atomless measure µ. The ordinary Lebesgue- Riesz norm || f || p for the numerical valued measurable function f : Z → R isdefined as ordinary || f || p := (cid:20) Z Z | f ( z ) | p µ ( dz ) (cid:21) /p , ≤ p < ∞ ;and as ordinary L p = L p ( Z, µ ) = { f : Z → R, || f || p < ∞ } . Further, let the numbers ( a, b ) be constants such that 1 ≤ a < b ≤ ∞ ; andlet ψ = ψ ( p ) = ψ [ a, b ]( p ) , p ∈ ( a, b ) , be numerical valued strictly positive functionnot necessary to be finite in every point:inf p ∈ ( a,b ) ψ [ a, b ]( p ) > . (8)In the case when b < ∞ one can assume sometimes p ∈ ( a, b ] . For instance ψ ( m ) ( p ) := p /m , m = const > , p ∈ [1 , ∞ )or ψ ( b ; β ) ( p ) := ( p − a ) − β · ( b − p ) − β , p ∈ ( a, b ) , β , β = const ≥ . The set of all such a functions will be denoted by Ψ = Ψ[ a, b ] = { ψ ( · ) } .
3y definition, the (Banach) Grand Lebesgue Space (GLS) Gψ = Gψ [ a, b ] , consists on all the real (or complex) numerical valued measurable functions f : Z → R defined on the whole our space Z and having a finite norm || f || Gψ = || f || Gψ [ a, b ] def = sup p ∈ ( a,b ) " | f | p ψ ( p ) . (9)The function ψ = ψ ( p ) = ψ [ a, b ]( p ) is named as the generating function forthis space.If for instance ψ ( p ) = ψ ( r ) ( p ) = 1 , p = r ; ψ ( r ) ( p ) = + ∞ , p = r, where r = const ∈ [1 , ∞ ) , C / ∞ := 0 , C ∈ R , (an extremal case), then thecorrespondent Gψ ( r ) ( p ) space coincides with the classical Lebesgue - Riesz space L r = L r ( Z, µ ) . Note that the introduced in [16], [18] etc. norms || f || b,θ,G def = sup <ǫ ≤ b − h ǫ θ/ ( b − ǫ ) | f | b − ǫ i , θ ≥ || f || Gψ ( b ; β ) , β = θ/b. Let f : Z → R be certain measurable function such that ∃ a, b, ≤ a < b ≤ ∞ ⇒ ∀ p ∈ ( a, b ) | f | p < ∞ . The so - called natural function ψ ( f ) ( p ) , p ∈ ( a, b ) for this function f is definedas follows ψ ( f ) ( p ) def = | f | p . Obviously, || f || Gψ ( f ) = 1 . These spaces are investigated in many works, e.g. in [10], [12], [13], [23], [24],[25], [26], [28], [29] - [33] etc. They are applied for example in the theory of PartialDifferential Equations [12], [13], in the theory of Probability [7],[31] - [33], in Statis-tics [29], chapter 5, theory of random fields, [25], [26], [29], [32], in the FunctionalAnalysis [29], [30], [32] and so one.These spaces are rearrangement invariant (r.i.) Banach functional spaces; itsfundamental function is considered in [32]. They do not coincides in general casewith the classical rearrangement invariant spaces: Orlicz, Lorentz, Marcinkiewiczetc., see [28], [30]. 4
Main result. The case of small values of param-eters.
We consider in this section the case of Grand Lebesgue Spaces Gψ a,b , where1 < a < b ≤ . As before, the measure µ is presumed to be atomless.Introduce the following auxiliary function κ [ Gψ [ a, b ]]( u ) := inf p ∈ ( a,b ) ( || u || p ψ ( p ) ) , u ∈ Gψ [ a, b ] . Evidently, this definition is correct also for the arbitrary elements from the space L p : u ∈ L p , ≤ p < ∞ , but we will apply this notion only for the suitable GLS. Theorem 2.1.
In this case, i.e. when 1 < a < b ≤ , the weak characteristic ofconvexity (WCOC) for the space Gψ = Gψ [ a, b ] allows a following lower estimate:∆[ Gψ [ a, b ]]( x − y ) ≥ a − · κ [ Gψ [ a, b ]]( x − y ) , x, y ∈ B [ Gψ [ a, b ]] , (10)so that || x + y || Gψ ≤ − a − κ [ Gψ [ a, b ]]( x − y ) , x, y ∈ B [ Gψ [ a, b ]] . Proof.
Let x, y ∈ B [ Gψ [ a, b ]] . We have from the direct definition of the normin GLS || x || p ψ ( p ) ≤ , || y || p ψ ( p ) ≤ , p ∈ ( a, b ) . It follows on the basis of definition 1.1 being applied to the space L p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) xψ ( p ) + yψ ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ − δ L p || x − y || p ψ ( p ) ! . We derive taking supremum over p using the estimate || x + y || Gψ [ a, b ] ≤ − p ∈ ( a,b ) δ L p || x − y || p ψ ( p ) ! . (11)Further, we will use in particular the relation (3); from one and (11) it follows δ p ( ǫ ) ≥ a − ǫ , ǫ ∈ [0 , . (12)Therefore || x + y || p ψ ( p ) ≤ − a − · || x − y || p ψ ( p ) .
5t remains to take the supremum over p ∈ ( a, b ) , using the direct definition ofthe norm in the Grand Lebesgue Spaces: || x + y || Gψ = sup p ∈ ( a,b ) ( || x + y || p ψ ( p ) ) ≤≤ − a −
14 inf p ∈ ( a,b ) ( || x − y || p ψ ( p ) ) = 2 − a − κ [ Gψ [ a, b ]]( x − y ) , Q.E.D.
We consider in this section the opposite case of Grand Lebesgue Spaces Gψ a,b , where 2 < a < b < ∞ . The measure µ is again presumed to be atomless.Introduce the following auxiliary function θ [ Gψ [ a, b ]]( u ) := inf p ∈ ( a,b ) ( || u || pp p · p · ψ p ( p ) ) , u ∈ Gψ [ a, b ] . Obviously, the last definition is correct also for the space L p : u ∈ L p , ≤ p < ∞ . Theorem 3.1.
In this case the weak characteristic of convexity (WCOC) forthe space Gψ = Gψ [ a, b ] , where 2 < a < b < ∞ obeys a following lower estimate:∆[ Gψ [ a, b ]]( x − y ) ≥ θ [ Gψ [ a, b ]]( x − y ) , (13)so that || x + y || Gψ ≤ − θ [ Gψ [ a, b ]]( x − y ) , x, y ∈ B [ Gψ [ a, b ]] . Proof is quite alike as before in the previous section, as well. We will use therelation (5) for the values x, y ∈ B [ Gψ [ a, b ]] : || x + y || p ψ ( p ) ≤ − || x − y || pp p · p · ψ p ( p ) , x, y ∈ B [ Gψ [ a, b ]] . It remains to take the supremum over p ∈ ( a, b ) , using the direct definition ofthe norm in the Grand Lebesgue Spaces: || x + y || Gψ = sup p ∈ ( a,b ) ( || x + y || p ψ ( p ) ) ≤ − inf p ∈ ( a,b ) ( || x − y || pp p · p · ψ p ( p ) ) = 2 − θ [ Gψ [ a, b ]]( x − y ) , Q.E.D.
Example 1.
Suppose in addition to the of theorem 2.1., i.e. to the case ofGrand Lebesgue Spaces Gψ a,b , where 1 < a < b ≤ , that the measure µ is bounded: µ ( Z ) = 1 . Then one can apply the famous Liyapunov’s inequality: || x || p ≥ || x || a , p ∈ [ a, b ] . Assume yet that the generating function ψ ( · ) is upper bounded: ψ ( p ) ≤ d =const ∈ (0 , ∞ ) . It follows from the proposition of theorem 2.1 || x + y || Gψ [ a, b ] ≤ − a − || x − y || a d , x, y ∈ B [ Gψ [ a, b ]] . Example 2.
We retain the restrictions µ ( Z ) = 1 and ψ ( p ) ≤ d = const ∈ (0 , ∞ ); but let here 2 ≤ a < b < ∞ . Then || x || p ≤ || x || b , p ≤ b ; and we deduceunder conditions and assertions of theorem 2.1 || x + y || Gψ [ a, b ] ≤ − || x − y || aa b · b d b , x, y ∈ B [ Gψ [ a, b ]] . It is interest in our opinion to study the Modulus Of Convexity for other GrandLebesgue Spaces, in particular, which are mentioned in the first section; especiallyin the case when b = ∞ . Open question: are the GLS Gψ [ a, ∞ ] , for instance, the Subgaussian Spaceswith ψ ( p ) = √ p, p ∈ (1 , ∞ )modulative convex? Acknowledgement.
The first author has been partially supported by the Gruppo Nazionaleper l’Analisi Matematica, la Probabilit`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionaledi Alta Matematica (INdAM) and by Universit`a degli Studi di Napoli Parthenope through theproject “sostegno alla Ricerca individuale”(triennio 2015 - 2017) .7 eferences [1] C.Bennett and R.Sharpley.
Interpolation of Operators.
Academic Press,New York, 1988.[2]
Buldygin V.V., Mushtary D.I., Ostrovsky E.I, Pushalsky M.I.
NewTrends in Probability Theory and Statistics.
Mokslas, 1992, Amsterdam, NewYork, Tokyo.[3]
Capone C, Formica M.R, Giova R.
Grand Lebesgue spaces with respect tomeasurable functions.
Nonlinear Analysis 2013; 85: 125 - 131.[4]
Capone C, and Fiorenza A.
On small Lebesgue spaces. Journal of functionspaces and applications.
C. Capone, M. R. Formica and
R. Giova.
Grand Lebesgue spaces withrespect to measurable functions . Nonlinear Anal. (2013), 125–131.[6] Clarkson, James. (1936).
Uniformly convex spaces . Trans. Amer. Math. Soc.,American Mathematical Society, , (3): 396 - 414, doi:10.2307/1989630,JSTOR 1989630[7] V. Ermakov, and E. I. Ostrovsky.
Continuity Conditions, Exponential Es-timates, and the Central Limit Theorem for Random Fields.
Moscow, VINITY,(1986), (in Russian).[8]
T. Figiel.
Uniformly convex norms in spaces with unconditional basis. in: Sem-inaire Maurey-Schwartz (19741975), Espaces
Lp, applications radonifiantesetgeometrie des espaces de Banach, Exp. No. XXIV, 975, pp. 11 pp. (erratum, p.3).[9]
T.Figel.
On the moduli of convexity and smoothness.
Studia Math., , (1976),pp. 121 - 155.[10] Fiorenza A, and Karadzhov G.E.
Grand and small Lebesgue spaces andtheir analogs.
Journal for Analysis and its Applications 2004; 23 (4) : 657 -681.[11]
A. Fiorenza.
Duality and reflexivity in grand Lebesgue spaces.
Collect. Math. (2000), no. 2, 131–148.[12] A. Fiorenza, M. R. Formica, A. Gogatishvili, T. Kopaliani and
J. M.Rakotoson.
Characterization of interpolation between grand, small orclassical Lebesgue spaces . Preprint arXiv:1709.05892, Nonlinear Anal., to ap-pear. 813]
Fiorenza A., and Karadzhov G.E.
Grand and small Lebesgue spaces andtheir analogs.
Consiglio Nationale Delle Ricerche, Instituto per le Applicazionidel Calcoto Mauro Picine, Sezione di Napoli, Rapporto tecnico n. 272/03,(2005).[14]
A.Fiorenza, M.R.Formica, T.Roskovec and F.Soudsky.
Gagliardo -Nirenberg inequality for rearrangement invariant Banach functional spaces. arXiv:1812.04295v1 [math.FA] 11 Dec 2018[15]
A.Fiorenza, M.R.Formica, T.Roskovec and F.Soudsky.
Detailed proof ofclassical Gagliardo - Nirenberg interpolation inequality with historical remarks. arXiv:1812.04281v1 [math.FA] 11 Dec 2018[16]
M. R. Formica and
R. Giova.
Boyd indices in generalized grand Lebesguespaces and applications . Mediterr. J. Math. (2015), no. 3, 987–995.[17] Gao J.
Modulus of Convexity in Banach Spaces.
PERGAMON Applied Math-ematics Letters,
Greco L, Iwaniec T, Sbordone C.
Inverting the p-harmonic operator.
Manuscripta Math.1997; , 259 - 272.[19]
Gurkanli A.T.
Inclusions and the approximate identities of the generalizedgrand Lebesgue spaces.
Turk J Math.2018; , 3195 - 3203.[20]
Gurkanli A.T.
Multipliers of some Banach ideals and Wiener - Ditkin sets.
Mathematica Slovacia, 2005; 55: (2), 237 - 248.[21]
A.Turan G¨urkanli.
Multipliers of grand and small Lebesgue Spaces. arXiv:1903.06743v1 [math.FA] 15 Mar 2019[22]
Hanner O.
On the uniform convexity of Lp and lp,
Ark. Mat., , (1956), 239- 244.[23] Iwaniec T., and Sbordone C.
On the integrability of the Jacobian underminimal hypotheses.
Arch. Rat.Mech. Anal., 119, (1992), 129 - 143.[24]
Iwaniec T., P. Koskela P., and Onninen J.
Mapping of finite distortion:Monotonicity and Continuity.
Invent. Math. 144 (2001), 507 - 531.[25]
Kozachenko Yu. V., Ostrovsky E.I. (1985).
The Banach Spaces of randomVariables of subgaussian type.
Theory of Probab. and Math. Stat. (in Russian).Kiev, KSU, 32, 43 - 57.[26]
Kozachenko Yu.V., Ostrovsky E., Sirota L.
Relations between exponen-tial tails, moments and moment generating functions for random variables andvectors. arXiv:1701.01901v1 [math.FA] 8 Jan 2017927]
Krasnoselsky M.A., Routisky Ya. B.
Convex Functions and Orlicz Spaces.
P. Noordhoff Ltd, (1961), Groningen.[28]
Liflyand E., Ostrovsky E., Sirota L.
Structural Properties of BilateralGrand Lebesgue Spaces.
Turk. J. Math.; 34 (2010), 207 - 219.[29]
Ostrovsky E.I.
Exponential estimations for random fields.
OINPE, Moscow -Obninsk, 1999, (in Russian).[30]
Ostrovsky E. and Sirota L.
Moment Banach spaces: theory and applications.
HIAT Journal of Science and Engineering, C, Volume 4, Issues 1 - 2, pp. 233 -262, (2007).[31]
E.Ostrovsky, L.Sirota.
Multidimensional Dilation Operators, Boyd and Shi-mogaki indices of Bilateral Weight Grand Lebesgue Spaces. arXiv:0809.3011[math.FA] 17 Sep 2008.[32]
E.Ostrovsky, L.Sirota.
Fundamental functions for Grand Lebesgue Spaces. arXiv:1509.03644v1 [math.FA] 11 Sep 2015[33]
E.Ostrovsky, L.Sirota.
Boundedness of operators in bilateral Grand LebesgueSpaces, with exact and weakly exact constant calculation. arXiv:1104.2963v1 [math.FA] 15 Apr 2011[34]
L.Pick, A.Kufner, O.John and S.Fucik.
Function Spaces.
Volume 1, 2ndRevised and Extended Edition, De Gruyter Series in Nonlinear Analysis andApplications 14, De Gruyter, Berlin 2013.[35]
J.Reif.
A characterization of (locally) uniformly convex spaces in terms ofrelative openness of quotient maps on the unit ball.
J. Funct. Anal.,177