Generalized Orlicz Spaces and Wasserstein Distances for Convex-Concave Scale Functions

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1 Generalized Orlicz Space and Waerein Diance for Convex-Concave Scale Funcion Karl-Theodor Surm Abrac Given a ricly increaing, coninuou funcion ϑ : R + R +, baed on he co funcional ϑ (d(x, y dq(x, y, we define he Lϑ -Waerein diance W ϑ (µ, ν beween probabiliy meaure µ, ν on ome meric pace (, d. The funcion ϑ will be aumed o admi a repreenaion ϑ = ψ a a compoiion of a convex and a concave funcion and ψ, rep. Beide convex funcion and concave funcion hi include all C 2 funcion. For uch funcion ϑ we exend he concep of Orlicz pace, defining he meric pace L ϑ (, m of meaurable funcion f : R uch ha, for inance, d ϑ (f, g ϑ( f(x g(x dµ(x. Convex-Concave Compoiion Throughou hi paper, ϑ will be a ricly increaing, coninuou funcion from R + o R + wih ϑ( =. Definiion.. ϑ will be called ccc funcion ( convex-concave compoiion iff here exi wo ricly increaing coninuou funcion, ψ : R + R + wih ( = ψ( =.. i convex, ψ i concave and ϑ = ψ. The pair (, ψ will be called convex-concave facorizaion of ϑ. The facorizaion i called minimal (or non-redundan if for any oher facorizaion (, ψ he funcion i convex. Two minimal facorizaion of a given funcion ϑ differ only by a linear change of variable. Indeed, if i convex and alo i convex hen here exi a λ (,.. ( = (λ and ψ( = λ ψ(. For each convex, concave or ccc funcion f : R + R + pu f ( := f (+ := lim h h [f( + h f(]. Lemma.2. (i For any ccc funcion ϑ, he funcion log ϑ i locally of bounded variaion and he diribuion (log ϑ define a igned Radon meaure on (,, henceforh denoed by d(log ϑ. (ii A pair (, ψ of ricly increaing convex or concave, rep., coninuou funcion wih ( = ψ( = i a facorizaion of ϑ iff d(log ϑ = ψ d(log + d(log ψ ( in he ene of igned Radon meaure. (iii The facorizaion (, ψ i minimal iff for any oher facorizaion (, ψ d(log ψ d(log ψ in he ene of nonnegaive Radon meaure on (,. (iv Every ccc funcion ϑ admi a minimal facorizaion ( ˇϑ, ˆϑ given by ˇϑ := ϑ ˆϑ and x ( y ˆϑ(x := exp dν (z dy where dν (z denoe he negaive par of he Radon meaure dν(z = d(log ϑ (z.

2 Proof. (i, (ii: The chain rule for convex/concave funcion yield ϑ ( = (ψ( ψ ( for each facorizaion (, ψ of a ccc funcion ϑ. Taking logarihm i implie ha log ϑ locally i a BV funcion (a a difference of wo increaing funcion and, hence, ha he aociaed Radon meaure aify d(log ϑ = d(log ψ + d(log ψ = ψ d(log + d(log ψ. (iii: The facorizaion (, ψ i minimal if and only if for any oher facorizaion (, ψ he funcion u = = ψ ψ i convex. Since log ψ = log u ( ψ + log ψ, he laer i equivalen o d(log ψ d(log ψ which i he claim. (iv: Define ˆϑ a above. I remain o verify ha ˆϑ <. Le (, ψ be any convex-concave facorizaion of ϑ. Wihou rericion aume ψ ( =. Then he Hahn decompoiion of ( yield dν d(log ψ. (2 Hence, for all x ˆϑ(x = x x ( exp ( exp y dν (z dy y d(log ψ (z dy = ψ(x <. Thi already implie ha ˆϑ i finie, ricly increaing and coninuou on [,. (For inance, for x > i follow ˆϑ(x ˆϑ( + x. Moreover, one eaily verifie ha ˆϑ i concave. Since ν +, ν are he minimal nonnegaive meaure in he ( Hahn or Jordan decompoiion of ν = ν + ν, i follow ha ( ˇϑ, ˆϑ i a minimal cc decompoiion of ϑ. Example.3. Each convex funcion ϑ i a ccc funcion. A minimal facorizaion i given by (ϑ, Id. Each concave funcion ϑ i a ccc funcion. A minimal facorizaion i given by (Id, ϑ. Each C 2 funcion ϑ wih ϑ (+ > i a ccc funcion. The minimal facorizaion i given by x ( y ϑ ˆϑ(x (z := exp ϑ dz dy (z and ˇϑ := ϑ ˆϑ. (The condiion ϑ (+ > can be replaced by he ricly weaker requiremen ha he previou inegral defining ˆϑ i finie. 2 The Meric Space L ϑ (, µ Le (, Ξ, µ be a σ-finie meaure pace and (, ψ a minimal ccc facorizaion of a given funcion ϑ. Then L ϑ (, µ will denoe he pace of all meaurable funcion f : R uch ha ( ψ( f dµ < for ome (, where a uual funcion which agree almo everywhere are idenified. Noe ha due o he fac ha r (r for large r grow a lea linearly he previou condiion i equivalen o he condiion ( ψ( f dµ for ome (,. Theorem 2.. L ϑ (, µ i a complee meric pace wih he meric { ( d ϑ (f, g = inf (, : ψ( f g } dµ. 2

3 The definiion of hi meric doe no depend on he choice of he minimal ccc facorizaion of he funcion ϑ. However, chooing an arbirary convex-concave facorizaion of ϑ migh change he value of d ϑ. Noe ha alway d ϑ (f, g = d ϑ (f g,. Proof. Le f, g, h L ϑ (, µ be given and chooe r, > wih d ϑ (f, g < r and d ϑ (g, h <. The laer implie ( ( ψ( f g dµ, ψ( g h dµ. r Concaviy of ψ yield ψ( f h ψ( f g + ψ( g h. Pu = r +. Then convexiy of implie ( ( r ψ( f g ψ( f h + ψ( g h r ( ψ( f g r + ( ψ( g h r. Hence, ( ψ( f h dµ r ( ψ( f g r dµ + ( ψ( g h dµ r + = and hu d ϑ (f, h. Thi prove ha d ϑ (f, h d ϑ (f, g + d ϑ (g, h. In order o prove he compleene of he meric, le (f n n be a Cauchy equence in L ϑ. Then d ϑ (f n, f m < for all n, m wih m n and uiable. Chooe an increaing equence of meaurable e k, k N, wih µ( k < and k k =. Then ( ψ( f n f m dµ k for all k, m, n wih m n. Jenen inequaliy implie ( ψ( f n f m dµ µ( k k and hu k ψ(f n ψ(f m dµ µ( k µ( k (. µ( k In oher word, (ψ(f n n i a Cauchy equence in L ( k, µ. I follow ha i ha a ubequence (ψ(f ni i which converge µ-almo everywhere on k. In paricular, (f ni i converge almo everywhere on k oward ome limiing funcion f (which eaily i hown o be independen of k. Finally, Faou lemma now implie k for each k and n N. Hence, ha i, ( ψ( f n f which prove he claim. Finally, i remain o verify ha ( dµ lim inf ψ( f n f m dµ m k ( ψ( f n f dµ, d ϑ (f n, f d ϑ (f, g = f = g µ-a.e. on. The implicaion i rivial. For he revere implicaion, we may argue a in he previou compleene proof: d ϑ (f, g = will yield k ( ψ( f g dµ for all k N and all > which in urn implie k ψ(f ψ(g dµ =. The laer prove f = g µ-a.e. on which i he claim. Example 2.2. If ϑ(r = r p for ome p (, hen wih p := p if p and p := if p. ( d ϑ (f, g = /p f g p dµ 3

4 Propoiion 2.3. (i If ϑ i convex hen f Lϑ (,µ := d ϑ (f, i indeed a norm and L ϑ (, µ i a Banach pace, called Orlicz pace. The norm i called Luxemburg norm. (ii If ϑ i concave hen d ϑ (f, g = ϑ( f g dµ ϑ(f ϑ(g L (,µ. (iii For general ccc funcion ϑ = ψ d ϑ (f, g = ψ( f g L (,µ. (iv If µ(m = hen for each ricly increaing, convex funcion Φ : R + R + wih Φ ( = ( Jenen inequaliy. d Φ ϑ (f, g d ϑ (f, g Proof. (i If ψ(r = cr hen obviouly d ϑ (f, = d ϑ (f,. See alo andard lieraure [2]. (ii Concaviy of ϑ implie ϑ( f g ϑ(f ϑ(g. (iv Aume ha d Φ ϑ (f, g < for ome (,. I implie ( ( Φ ψ( f g dµ. Claical Jenen inequaliy for inegral yield ( ( Φ ψ( f g dµ which due o he fac ha Φ ( = in urn implie d ϑ (f, g. 3 The L ϑ -Waerein Space Le (, d be a complee eparable meric pace and ϑ a ccc funcion wih minimal facorizaion (, ψ. The L ϑ -Waerein pace P ϑ ( i defined a he pace of all probabiliy meaure µ on equipped wih i Borel σ-field.. ( ψ(d(x, y dµ(x < for ome y and ome (,. The L ϑ -Waerein diance of wo probabiliy meaure µ, ν P ϑ ( i defined a { ( } W ϑ (µ, ν = inf > : inf ψ(d(x, y dq(x, y q Π(µ,ν where Π(µ, ν denoe he e of all coupling of µ and ν, i.e. he e of all probabiliy meaure q on.. q(a = µ(a, q( A = ν(a for all Borel e A. Given wo probabiliy meaure µ, ν P ϑ (, a coupling q of hem i called opimal iff ( ψ(d(x, y dq(x, y w for w := W ϑ (µ, ν. Propoiion 3.. For each pair of probabiliy meaure µ, ν P ϑ ( here exi an opimal coupling q. Proof. For (, define he co funcion c (x, y = ( ψ(d(x, y. Noe ha c (x, y i coninuou and decreaing. Given µ, ν.. w := W ϑ (µ, ν <. Then for all > w he meaure µ and ν have finie c - ranporaion co. More preciely, c (x, y dq(x, y. inf q Π(µ,ν 4

5 Hence, here exi q n Π(µ, ν.. c w+ n (x, y dq n(x, y + n. In paricular, c w+(x, y dq n (x, y 2 for all n N. Hence, he family (q n n i igh ([3], Lemma 4.4. Therefore, here exi a converging ubequence (q nk k wih limi q Π(µ, ν aifying for all n ([3], Lemma 4.3 and hu c w+ n (x, y dq(x, y + n c w (x, y dq(x, y. Propoiion 3.2. W ϑ i a complee meric on P ϑ (. The riangle inequaliy for W ϑ i valid no only on P ϑ ( bu on he whole pace P( of probabiliy meaure on. The riangle inequaliy implie ha W ϑ (µ, ν < for all µ, ν P ϑ (. Proof. Given hree probabiliy meaure µ, µ 2, µ 3 on and number r, wih W ϑ (µ, µ 2 < r and W ϑ (µ 2, µ 3 <. Then here exi a coupling q 2 of µ and µ 2 and a coupling q 23 of µ 2 and µ 3.. ( ( r ψ d dq 2, ψ d dq 23. Le q 23 be he gluing of he wo coupling q 2 and q 23, ee e.g. [], Lemma.8.3. Tha i, q 23 i a probabiliy meaure on.. he projecion ono he fir wo facor coincide wih q 2 and he projecion ono he la wo facor coincide wih q 23. Le q 3 denoe he projecion of q 23 ono he fir and hird facor. In paricular, hi will be a coupling of µ and µ 3. Then for := r + r ( ψ(d(x, z r + =. dq 3 (x, z ( ψ(d(x, y + d(y, z dq 23 (x, y, z ( r ψ(d(x, y + r ( ψ(d(x, y dq 23 (x, y, z + r ψ(d(y, z dq 23 (x, y, z Hence, W ϑ (µ, µ 3. Thi prove he riangle inequaliy. ( ψ(d(y, z dq 23 (x, y, z To prove compleene, aume ha (µ k k i a W ϑ -Cauchy equence, ay W ϑ (µ n, µ k n for all k n wih n a n. Then here exi coupling q n,k of µ n and µ k.. ( ψ(d(x, y dq n,k (x, y. (3 n Jenen inequaliy implie d(x, y dq n,k (x, y n ( wih d(x, y := ψ(d(x, y. The laer i a complee meric on wih he ame opology a d. Tha i, (µ k k i a Cauchy equence w.r.. he L -Waerein diance on P(, d. Becaue of compleene of P (, d, we hu obian an accumulaion poin µ and a converging ubequence (µ ki i. According o [3], Lemma 4.4, hi alo yield an accumulaion poin q n of he equence (q n,ki i. Coninuiy of he involved co funcion ogeher wih Faou lemma allow o pa o he limi in (3 o derive ( ψ(d(x, y dq n (x, y n 5

6 which prove ha W ϑ (µ, µ n n a n. Wih a imilar argumen, one verifie ha W ϑ (µ, ν = if and only if µ = ν. Remark 3.3. For each pair of probabiliy meaure µ, ν on W ϑ (µ, ν inf q Π(µ,ν ϑ(d(x, y dq(x, y. Reference [] R.M. Dudley: Real analyi and probabiliy. Cambridge Univ Pr, 22. [2] M.M. Rao, Z.D. Ren: Theory of Orlicz Space. Pure and Applied Mahemaic, Marcel Dekker (99. [3] C. Villani: Opimal Tranpor, old and new. Grundlehren der mahemaichen Wienchafen 338 (29, Springer Berlin Heidelberg. 6