aa r X i v : . [ m a t h . DG ] A ug COMPLETE SOBOLEV TYPE INEQUALITIES
HAOJIAN LI
Abstract.
We establish Sobolev type inequalities in the noncommutative settings by generalizingmonotone metrics in the space of quantum states, such as matrix-valued Beckner inequalities. Wealso discuss examples such as random transpositions and Bernoulli-Laplace models. Introduction
Poincar´e inequalities (PIs) and log Sobolev inequalities (LSIs) have been well developed in thelast few decades. See [Led99, GZ03] for properties, applications and criterion of PIs and LSIs.Gross showed that log Sobolev inequalities and hypercontractivity are equivalent for Dirichlet formoperators, see [Gro75]. Beckner inequalities (BIs), as an interpolation between PIs and LSIs, wereintroduced by Beckner ([Bec89]) in 1989 for the canonical Gaussian measures on R n . Later Ledoux([Led97]) introduced a family of inequalities with the same pattern of LSIs and PIs, which alsosolved the regularity issues of porous medium equations ([Dem05, V´az07, BGL13]).Log Sobolev inequalities in the quantum (noncommutative) settings have been studied recently,see [CM17, CM20, LJL20, GJL18, DR20, BCR20]. The idea of characterizing matrix-valued Sobolevtype inequalities is still absent from the literature. Surprisingly, we explore a vast variety of Sobolevtype inequalities by introducing the generalized monotone metrics in the space of quantum states.Let us first recall that an ergodic system T t = e − t ∆ on a probability space (Ω , µ ) satisfies the λ -LSI if there exists λ > Z ρ ln( ρ ) dµ − Z ρ dµ ln( Z ρ dµ ) ≤ λ E ∆ ( ρ, ρ )(1)for any function ρ , where E ∆ ( ρ, σ ) = R Ω ∆( ρ ) σdµ is the energy form. We now use the notationEnt( ρ ) = R ρ ln( ρ ) dµ − R ρdµ ln( R ρdµ ) for the relative entropy. An equivalent formulation of LSIsis the exponential decay of the relative entropy:Ent( T t ( ρ )) ≤ e − λt Ent( ρ )(2)for any positive ρ , see [BGL13, LJL20]. For a convex function f , let us consider the relative entropyfunctional Ent f ( ρ ) = Z f ( ρ ) − f ( σ ) − ( ρ − σ ) f ′ ( σ ) dµ, where σ = R ρdµ . We observe that Ent f = Ent for f ( x ) = x ln( x ). Then T t satisfies the generalizedSobolev inequality associated to f if there exists λ > f ( T t ( ρ )) ≤ e − λt Ent f ( ρ )(3)for any positive ρ . Again f ( x ) = x ln( x ) returns the classical LSIs. Let p ∈ (1 ,
2) and f ( x ) = x p ,then (3) is equivalent to k ρ k pp − k ρ k p ≤ pλ ( p ) E ∆ ( ρ, ρ p − ) , where p E ∆ ( ρ, ρ p − ) is the analogue of the Fisher information associated to ∆. The limiting cases p → + and p → + reduce to LSIs and PIs, respectively ([BT06]). Setting q = p and g = ρ /q , weobtain BIs k g k − k g k q ≤ − qλ (2 /q ) E ∆ ( g, g ) , (4)which was first introduced Beckner ([Bec89]) in 1989 for the canonical Gaussian measure on R n with optimal constants λ ( q ) = 2.We aim at extending (3) to a finite von Neumann algebra ( N , τ ) equipped with a normal faithfultracial state τ . We consider the semigroup T t = e − tA : N → N of completely positive self-adjointunital maps. Let N fix = { ρ | T t ( ρ ) = ρ } be the fixed point algebra of T t , which admits the conditionalexpectation. Then the generator A is said to satisfy the λ -modified f -Sobolev inequality (M f SI) if d f ( T t ( ρ ) k E ( ρ )) ≤ e − λt d f ( ρ k E ( ρ )) , ∀ ρ ∈ N + , where d f ( ρ k σ ) = τ ( f ( ρ ) − f ( σ ) − ( ρ − σ ) f ′ ( σ )) for ρ, σ ∈ N + . We say A satisfies λ -complete f -Sobolev inequality (C f SI) if the above inequality remains true for A ⊗ id M , where M is anyfinite von Neumann algebra. The case f ( x ) = x ln( x ) has been studied in a series of paper, see[GJL18, BGJ20, LJL20]. By imposing more conditions on f , we would recover most properties ofCLSIs such as stability under tensorization and change of measure.Intriguingly, the study of generalized monotone metrics in the space of quantum states shedslight on C f SIs and the Bregman relative entropy. The monotone metric was anticipated by Moro-zova and Chenstov ([MC89]) to transfer the geometric techniques to the noncommutative settings.Motivated by Morozova and Chenstov, Petz ([Pet96]) introduced monotone metrics systematicallyusing the relative modular operators and discovered the equivalent relation between operator mono-tone functions and the monotone metrics. Later on, Hiai and Petz ([HP12]) extended the monotonemetrics to two parameters. Continuing Petz’ study, we define define the generalized monotone met-rics associated to two-variable functions via the double operator integral. By this new definition ofgeneralized monotone metrics, we explore a wide range of Sobolev type inequalities.The paper is organized as follows. In section 2, we introduce the generalized monotone metrics.In section 3, we define C f SIs and establish C f SIs for derivation triples. In section 4, we discussexamples and applications such as complete Beckner inequalities and random transpositions andBernoulli-Laplace models. 2.
Generalized Monotone Metrics
Monotone metrics.
Let N be a finite von Neumann algebra equipped with a normal faithfultracial state τ and β : N → N be a completely positive trace preserving (CPTP) map. The setof positive elements in N is denoted by N + . Let L p ( N , τ ) denote the noncommutative L p space,written as L p ( N ) if the trace τ is clear from the context. Let R + = (0 , ∞ ) in the sequel. The leftand right multiplications by ρ ∈ N are defined by L ρ ( a ) = ρa and R ρ ( a ) = aρ, ∀ a ∈ N . Note L ρ and R σ commute for any ρ, σ ∈ N . For ρ, σ ∈ N + and f : R + → R + , we define J fρ,σ : N → N by(5) J fρ,σ = f ( L ρ R − σ ) R σ , OMPLETE SOBOLEV TYPE INEQUALITIES 3 where L ρ R − σ is the relative modular operator , see [Pet07] for more information. We use J fρ if ρ = σ .The inverse of J fρ,σ is given by (cid:16) J fρ,σ (cid:17) − = f − ( L ρ R − σ ) R − σ . Let ρ, σ ∈ N + and f : R + → R + , then the following conditions are equivalent ([HP12]): β ∗ ( J fβ ( ρ ) ,β ( σ ) ) − β ≤ ( J fρ,σ ) − ;(6) β J fρ,σ β ∗ ≤ J fβ ( ρ ) ,β ( σ ) . (7)Let us recall the following generalized Lieb’s concavity theorem ([Pet85, HP12, HP13]). Theorem 2.1.
Let β : N → N be a CPTP map and f : R + → R + be an operator monotonefunction. Assume that ρ, σ ∈ N + , then β ∗ (cid:16) J fβ ( ρ ) ,β ( σ ) (cid:17) − β ≤ (cid:16) J fρ,σ (cid:17) − . Hiai and Petz usually require that ρ, σ, β ( ρ ) , β ( σ ) are invertible. As we pointed out in [LJL20] thatit is enough to assume the positivity by perturbation argument ρ + ǫI for ǫ → + . ConsequentlyHiai and Petz defined the monotone metrics with two parameters γ fρ,σ by γ fρ,σ ( a, b ) = h a, (cid:16) J fρ,σ (cid:17) − ( b ) i , ∀ a, b ∈ N , (8)where h a, b i = τ ( a ∗ b ) is the Hilbert-Schmidt inner product. For ρ, σ ∈ N + and an operatormonotone function f : R + → R + , we have γ fβ ( ρ ) ,β ( σ ) ( β ( a ) , β ( a )) ≤ γ fρ,σ ( a, a ) , a ∈ N . Corollary 2.2.
For an operator monotone function f , the monotone metric γ fρ,σ ( a, a ) is a jointlyconvex function for ( ρ, σ, a ) for ρ, σ ∈ N + and a ∈ N . Generalized monotone metrics.
Let us recall that for F : R + × R + → R + and ρ, σ ∈ N + the double operator integral is defined by Q ρ,σF ( a ) = Z ∞ Z ∞ F ( s, t ) dE ρ ( s ) adE σ ( t ) , where E ρ (( s, t ]) = 1 ( s,t ] ( ρ ) is the spectral projection of ρ . We denote it by Q ρF if ρ = σ . For acomprehensive account of the double operator integral, see [DK51, kre56, dPS04, dPS07, BS03,PS10]. For operators ρ = P ki =1 s i p i and σ = P lj =1 t j q j with discrete specrtum, this simplifies to aSchur multiplier Q ρ,σF ( y ) = k X i =1 l X j =1 F ( s i , t j ) p i yq j , ∀ y ∈ N . Note that (cid:0) Q ρ,σF (cid:1) − = Q ρ,σF − . Let f [0] ( x, y ) = f ( xy ) y for f : R + → R + , then Q ρ,σf [0] = J fρ,σ . Let usintroduce two families of functions: C − = { F ; βQ ρ,σF β ∗ ≤ Q β ( ρ ) ,β ( σ ) F , ∀ ρ, σ ∈ N + and CPTP β } , (9) C + = { F ; β ∗ Q β ( ρ ) ,β ( σ ) F β ≤ Q ρ,σF , ∀ ρ, σ ∈ N + and CPTP β } . (10) Definition 2.3.
Let F ∈ C + and ρ, σ ∈ N + . We define the ( two-variable ) generalized monotonemetric γ Fρ,σ : N → N by γ Fρ,σ ( a, b ) = h a, Q ρ,σF ( b ) i . (11) H. LI
It follows from the definition that γ Fβ ( ρ ) ,β ( σ ) ( β ( a ) , β ( a )) ≤ γ Fρ,σ ( a, a ) , ∀ a ∈ N . (12)We use the same notation as (8) defined by [HP12], but we only refer to (8) if the superscriptfunction f is one-variable. Let f be operator monotone, then we identify γ fρ,σ = γ Fρ,σ with F = f − . Theorem 2.4.
Let F ∈ C + satisfying λF ( λx, λy ) ≤ F ( x, y ) for any λ ∈ [0 , . Then the generalizedmonotone metric γ Fρ,σ ( a, a ) is a convex function for ( ρ, σ, a ) of ρ, σ ∈ N + and a ∈ N .Proof. We use the standard trick and consider β : M ⊗N → N defined by (cid:18) x x x x (cid:19) x + x . Then β is CPTP. Let ρ = (cid:16) λρ
00 (1 − λ ) ρ (cid:17) , σ = (cid:16) λσ
00 (1 − λ ) σ (cid:17) , and a = (cid:16) λa
00 (1 − λ ) a (cid:17) for some λ ∈ [0 , γ Fλρ +(1 − λ ) ρ ,λσ +(1 − λ ) σ ( λa + (1 − λ ) a , λa + (1 − λ ) a ) ≤ γ Fλρ ,λσ ( λa , λa ) + γ F (1 − λ ) ρ , (1 − λ ) σ ((1 − λ ) a , (1 − λ ) a ) . We further have γ Fλρ,λσ ( λa, λa ) ≤ λγ Fρ,σ ( a, a ) . (13)Indeed γ Fλρ,λσ ( λa, λa ) = λ h a, Z ∞ Z ∞ F ( x, y ) dE λρ ( x ) adE λσ ( y ) i = λ h a, Z ∞ Z ∞ F ( λx, λy ) dE ρ ( x ) adE σ ( y ) i≤ λγ Fρ,σ ( a, a ) . Applying (13) completes the proof. (cid:3)
The monotonicity of γ Fρ,σ ( a, a ) does not necessarily imply the joint convexity for F ∈ C + since thecondition λF ( λx, λy ) ≤ F ( x, y ) sometimes fails. Let f be operator monotone and F = f − , weactually have the equality. Proposition 2.5.
We have the following properties. (1)
The sets C + and C − are positive cones. (2) Let F ∈ C + and F ∈ C − , let F ′ ( x, y ) = F ( x + t, y + s ) and F ′ ( x, y ) = F ( x + t, y + s ) forany fixed t, s ≥ . Then F ′ ∈ C + and F ′ ∈ C − . (3) If F ∈ C + , then F ∈ C − . Similarly if F ∈ C − , then F ∈ C + . (4) Let f : R + → R + be operator monotone, then f [0] ∈ C − and f − ∈ C + .Proof. We only give proofs for (3) and (4). The equivalence between β ∗ ( Q β ( ρ ) ,β ( σ ) F ) − β ≤ ( Q ρ,σF ) − (14)and βQ ρ,σF β ∗ ≤ Q β ( ρ ) ,β ( σ ) F . (15)yields (3). (4) follows directly from Theorem 2.1 and (3). (cid:3) OMPLETE SOBOLEV TYPE INEQUALITIES 5
Example 2.6.
Let f ( x ) = x − x ) , then f is operator monotone. Indeed, f ( x ) = R x r dr and x r isoperator monotone for r ∈ [0 , . Then we have f [0] ( x, y ) = x − y ln( x ) − ln( y ) ∈ C − and f − ( x, y ) = ln( x ) − ln( y ) x − y ∈ C + . Let f [1] ( x, y ) = f ( x ) − f ( y ) x − y denote the difference quotient of f . We consider the following set c + = { f ′ | f [1] ∈ C + } . (16) Proposition 2.7.
We have the following properties. (1)
The set c + is a positive cone. (2) The set c + is invariant under right translation. (3) Let f ( x ) = x + k with k ≥ , then f ∈ c + .Proof. We only give the proof of (3). Let g ( x ) = ln( x + k ). By Example 2.6 and Proposition 2.5,we have g [1] ∈ C + . If follows from the definition that f = g ′ ∈ c + . (cid:3) Example 2.8.
Let F ( x, y ) = x p − y p x − y for p ∈ (0 , . Let us recall that f ( x ) = x p = sin ( pπ ) xπ Z ∞ r p − r + x dr. By Proposition 2.7, we have f ′ ∈ c + and F = f [1] ∈ C + . We cannot find a function f such that F − = f [0] . Thus C + is a strict extension of the operator monotone functions. Remark 2.9.
In a discussion, we noticed that Haonan Zhang also gave a proof of the exampleabove separately. Haonan Zhang was trying to develop the matrix-valued Beckner inequalities usingthe geodesic convexity techniques in [CM20] and [CM17] . Complete Sobolev type inequality
Derivations.
Let N be a finite von Neumann algebra equipped with a normal faithful tracialstate τ . Let N H N be a self-adjoint Hilbert N - N bimodule with the antilinear form J . A derivationof a von Neumann algebra N is a densely defined linear operator δ : L ( N , τ ) → H such that(1) dom( δ ) is a weakly dense ∗ -subalgebra in N ;(2) the identity element 1 ∈ dom( δ );(3) δ ( xy ) = xδ ( y ) + δ ( x ) y , for any x, y ∈ dom( δ ).We always work with a closable derivation and denote the closure by ¯ δ . A derivation δ is said tobe ∗ -preserving if J ( δ ( x )) = δ ( x ∗ ). Every closable ∗ -preserving derivation δ determines a positiveoperator δ ∗ ¯ δ on L ( N , τ ). It was shown in [Sau90] that T t = e − tδ ∗ ¯ δ : N → N is a stronglycontinuous semigroup of CPTP maps. See[HN95], [IO80], [Pet09], [Kap53], and [BR76] for moredetails. The functional calculus of a derivation δ is given by δ ( f ( ρ )) = Q ρf [1] ( δ ( ρ )) = Z ∞ Z ∞ f ( s ) − f ( t ) s − t dE ρ ( s ) δ ( ρ ) dE ρ ( t ) . (17)Now let T t = e − tA : N → N be a strongly continuous semigroup of completely positive unitalself-adjoint maps on L ( N , τ ). The generator A is a positive operator on L ( N , τ ) given by A ( x ) = lim t → + t ( T t ( x ) − x ) , ∀ x ∈ dom( A ) . It was pointed out in [Sau90] that dom( δ ) = { x ∈ N |k A / x k < ∞} is indeed a ∗ -algebra andinvariant under the semigroup. The weak gradient form of A is defined byΓ A ( x, y )( z ) = 12 ( τ ( A ( x ) ∗ yz ) + τ ( x ∗ A ( y ) z ) − τ ( x ∗ yA ( z ))) . H. LI
If the weak gradient form Γ A ( x, y ) ∈ L ( N ) for all x, y ∈ dom( A / ), we say the generator A (or T t ) satisfies Γ -regularity . Theorem 3.1 ([JRS14]) . If A satisfies Γ -regularity, then there exists a finite von Neumann algebra ( M , τ ) containing N and a ∗ -preserving derivation δ A : dom( A / ) → L ( M ) such that (18) τ (Γ A ( x, y ) z ) = τ ( δ A ( x ) ∗ δ A ( y ) z ) . Equivalently Γ A ( x, y ) = E N ( δ A ( x ) ∗ δ A ( y )) , where E N : M → N is the conditional expectation.
Throughout the paper, we always work with a closable ∗ -preserving derivation δ and a stronglycontinuous semigroup T t = e − tA of completely positive unital self-adjoint maps on L ( N , τ ) satis-fying Γ-regularity.3.2. Generalized Fisher information.
The f -Fisher information I f,τA of A is defined as I f,τA ( ρ ) = τ ( A ( ρ ) f ′ ( ρ )) , ∀ ρ ∈ dom( A / ) ∩ L ( N ) and f ′ ( ρ ) ∈ L ∞ ( N ) . Equivalently I f,τA ( ρ ) = lim ǫ → + τ ( A ( ρ ) f ′ ( ρ + ǫ . For a derivation δ , the Fisher information is defined as I f,τδ ( ρ ) = τ (cid:16) δ ( ρ ) Q ρf [2] ( δ ( ρ )) (cid:17) , ∀ ρ ∈ dom( δ ) ⊂ N , (19)where f [2] ( x, y ) = f ′ ( x ) − f ′ ( y ) x − y . Then I fδ ( ρ ) = I fδ ∗ ¯ δ ( ρ ) . We use I fA or I fδ if the trace is clear from the context. In the rest of thissection, we always consider convex and continuously differentiable f : R + → R + such that f [2] ∈ C + .By Theorem 3.1, for any A satisfying Γ-regularity, there exists a closable ∗ -preserving derivation δ A : dom( A / ) → L ( M ) such that Γ A ( x, y ) = E N ( δ A ( x ) ∗ δ A ( y )) where E N : M → N . Thus I fA ( ρ ) = I fδ A ( ρ ) . The choice of δ A is not necessarily unique, but I fA is uniquely determined. We recapture the widelyused Fisher information I A ( ρ ) = τ ( A ( ρ ) ln( ρ )) by choosing f ( x ) = x ln( x ). We shall also observethe relation between the f -Fisher information and the generalized monotone metric I fδ ( ρ ) = γ f [2] ρ,ρ ( δ ( ρ ) , δ ( ρ )) . Lemma 3.2 (non-negativity) . The f -Fisher information is nonnegative.Proof. The convexity of f implies that f [2] ≥
0. Set w = ( Q ρf [2] ) / ( δ A ( ρ )) with( Q ρf [2] ) / ( y ) = Z ∞ Z ∞ (cid:18) f ′ ( s ) − f ′ ( t ) s − t (cid:19) / dE ρ ( s ) ydE ρ ( t ) . Thus I fA ( ρ ) = τ ( δ A ( ρ ) Q ρf [2] ( δ A ( ρ )))(20) = τ ( E N ( ww )) ≥ . (21)Similarly I δ is also nonnegative. (cid:3) An important example is f ( x ) = x p for p ∈ (1 , p -Fisher information by I pA or I pδ . As an application of Theorem 2.4, we get the following result. Corollary 3.3.
The p -Fisher information is convex. OMPLETE SOBOLEV TYPE INEQUALITIES 7
Recall that for any finite von Neumann algebra N , there exists a σ -finite measure space ( X, µ )such that Z ( N ) ∼ = L ∞ ( X, µ ) and N = R X N x dµ ( x ), where Z ( N ) is the center of N and N x is afactor for any x ∈ X . Now we rewrite the f -Fisher information by using the direct integral I f,τδ ( ρ ) = Z X I f,τ x δ ( ρ x ) dµ ( x ) . Lemma 3.4.
Let τ and τ be normal faithful traces over N and dτ dτ ≥ c for some c > . Then forany ρ ∈ N + , cI f,τ A ( ρ ) ≤ I f,τ A ( ρ ) . The result remains true for I fδ .Proof. Two traces only differ by two measures µ and µ over the center L ∞ ( X, µ ) ∼ = L ∞ ( X, µ ) ∼ = Z ( N ) . Note that dτ dτ ≥ c if and only dµ dµ ≥ c . Setting the pointwise differential form w x =( Q ρ x f [2] ) / ( δ A ( ρ x )), we infer that cI f,τ A ( ρ ) = c Z X τ ( E N ( w x w x )) dµ ( x ) ≤ Z X τ ( E N ( w x w x )) µ ( x ) = I f,τ A ( ρ ) . (cid:3) Bregman relative entropy.
Let us recall the definition of f -Bregman relative entropy d f,τ ( ρ k σ ) = τ ( f ( ρ ) − f ( σ ) − ( ρ − σ ) f ′ ( σ )) , for ρ, σ ∈ N + . For simplicity, we would call it as f -relative entropy. Equivalently d f ( ρ k σ ) = lim ǫ → + d f ( ρ k σ + ǫ . We write d f ( ρ k σ ) if the trace τ is clear from the context. For a comprehensive study of Bregmanrelative entropy, see [MPV16, PV15, Vir16]. It follows from the definition that d f ( ρ k σ ) ≥ ρ = σ . Note that we identify the Lindblad relative entropy d f ( ρ k σ ) = τ ( ρ ln ρ − ρ ln σ − ρ + σ ) = D Lind ( ρ k σ )with the choice f ( x ) = x ln( x ). The f -relative entropy admits an integral representation ([PV15]) d f ( ρ k σ ) = Z s =0 τ (cid:18) ( ρ − σ ) ddt f ( σ + ( s + t )( ρ − σ )) | t =0 (cid:19) ds. Let
K ⊂ N be a von Neumann subalgebra of N and E K be the conditional expectation onto K .The relative entropy with respect to K is defined by d f K ( ρ ) = d f ( ρ k E K ( ρ )) . Noting τ (( ρ − E K ( ρ )) f ′ ( E K ( ρ ))) = 0, then d f K ( ρ ) = τ ( f ( ρ ) − f ( E K ( ρ ))) . (22) Lemma 3.5.
The following equality remains true d f ( ρ k σ ) = d f K ( ρ ) + d f ( E K ( ρ ) k σ ) for any σ ∈ K . Proof.
Note τ ( ρf ′ ( σ )) = τ ( E K ( ρ ) f ′ ( σ )). Then we have d f ( ρ k σ ) = tr (cid:0) f ( ρ ) − f ( E K ( ρ )) + f ( E K ( ρ )) − f ( σ ) + ( ρ − σ ) f ′ ( σ ) (cid:1) = d f K ( ρ ) + τ (cid:0) f ( E K ( ρ )) − f ( σ ) + ( E K ( ρ ) − σ ) f ′ ( σ ) (cid:1) = d f K ( ρ ) + d f ( E K ( ρ ) k σ ) . (cid:3) H. LI
Together with nonnegativity of f -relative entropy, Lemma 3.5 implies that d f K ( ρ ) = inf σ ∈K d f ( ρ k σ ) . (23)Let N fix ⊂ N be the fixed point algebra of the semigroup T t = e − tA and E be the conditionalexpectation onto N fix , then ET t = T t E = E. Lemma 3.6 (gradient form) . The semigroup T t relates the f -relative entropy and the f -Fisherinformation .The f -Fisher information is the negative derivative of d f N fix ( T t ( ρ )) , ddt d f N fix ( T t ( ρ )) = − I fA ( T t ( ρ )) . Proof.
Let g ( t ) = d f N fix ( ρ t ) = τ ( f ( ρ t ) − f ( E ( ρ t ))). By the chain rule, we obtain that g ′ ( t ) = ddt τ ( f ( ρ t ) − f ( E ( ρ )) = τ (cid:0) − A ( T t ( ρ ))) f ′ ( T t ( ρ )) (cid:1) = − I fA ( T t ( ρ )) . (cid:3) Lemma 3.7.
Let τ and τ be two normal faithful traces over a finite von Neumann algebra N such that dτ dτ ≤ c for c > . For any ρ, σ ∈ N + , d f,τ ( ρ k σ ) ≤ cd f,τ ( ρ k σ ) . In particular, we have d f,τ K ( ρ ) ≤ cd f,τ K ( ρ ) . Proof.
We follow the notations and idea in the proof of Lemma 3.4. Also note that dτ dτ ≤ c if andonly if dµ dµ ≤ c . Again by the non-negativity of the f -relative entropy, we have d f,τ ( ρ k σ ) = Z X d f,τ ( ρ x k σ x ) dµ ( x ) ≤ c Z X d f,τ ( ρ x k σ x ) dµ ( x ) = cd f,τ ( ρ k σ ) . The second assertion follows from (23). (cid:3)
Theorem 3.8 (Data Processing Inequality) . Let
Φ :
N → N a quantum channel (CPTP), then d f (Φ( ρ ) k Φ( σ )) ≤ d f ( ρ k σ ) ∀ ρ ∈ N + , σ ∈ N fix . Proof.
Let a ( t ) = (1 − t ) σ + tρ for t ∈ [0 , H ρ,σ ( t ) = τ ( f ( a ( t ))) . By chain rule H ′ ( t ) = τ ( f ′ ( a ( t ))( ρ − σ )). It follows from the integration by parts that Z (1 − t ) H ′′ ρ,σ ( t ) dt =(1 − t ) H ′′ ρ,σ ( t ) | − Z ( − H ′ ρ,σ ( t ) dt (24) = − H ′ ρ,σ (0) + H ρ,σ (1) − H ρ,σ (0) = d f ( ρ k σ ) . (25)Recall that lim t → + g ( ρ + tσ ) − g ( ρ ) t = Q ρg [1] ( σ ), then we have H ′′ ρ,σ ( t ) = τ (cid:18) lim ǫ → + f ′ ( a ( t + ǫ )) − f ′ ( a ( t )) ǫ ( ρ − σ ) (cid:19) = τ (cid:18) lim ǫ → + f ′ ( a ( t ) + ǫ ( ρ − σ )) − f ′ ( a ( t )) ǫ ( ρ − σ ) (cid:19) = τ (cid:16) ( ρ − σ ) Q a ( t ) f [2] ( ρ − σ ) (cid:17) = γ f [2] a ( t ) ,a ( t ) ( ρ − σ, ρ − σ ) . OMPLETE SOBOLEV TYPE INEQUALITIES 9
Then f [2] ∈ C + implies that H ′′ Φ( ρ ) , Φ( σ ) ( t ) ≤ H ′′ ρ,σ ( t ) . Together with (24) it yields the assertion. (cid:3)
Let f ( x ) = x p for p ∈ (1 , p -relative entropy d p ( ρ k σ ) = τ ( ρ p − σ p − p ( ρ − σ ) σ p − ) . (26)Thus d p K ( ρ ) = τ ( ρ p − ( E K ( ρ )) p ) . (27)It shall be noted that p -relative entropy is different from the (sandwiched) R´enyi entropy. The p -relative entropy with respect the conditional expectation (27) appeared in [BT06], where theystudied the classical (commutative) situations and ergodic systems. The general properties ofBregman relative entropy are systematically studied in [PV15, Vir16, MPV16]. Again we use thestandard argument as in Theorem 2.4 and obtain the joint convexity. Corollary 3.9.
The f -relative entropy d p ( ρ k σ ) is a jointly convex function for ( ρ, σ ) for ρ, σ ∈ N + if d f ( λρ k λσ ) ≤ λd f ( ρ k σ ) for any λ ∈ [0 , . Thus D Lind and d p are jointly convex. f SI .Definition 3.10. The semigroup T t = e − tA or the generator A with the fixed-point algebra N fix issaid to satisfy: (1) the modified f -Sobolev inequality λ - M f SI ( with respect to the trace τ ) if there exists a constant λ > such that λd f N fix ( ρ ) ≤ I fA ( ρ ) , ∀ ρ ∈ dom( δ ) ∩ N + ;( or equivalently d f N fix ( T t ( ρ )) ≤ e − λt d f N fix ( ρ ) , ∀ ρ ∈ N + . )(2) the complete f -Sobolev inequality λ - M f SI ( with respect to the trace τ ) if A ⊗ id F satisfies λ - M f SI for any finite von Neumann algebra F .Let C f SI(
A, τ ) be the supremum of λ such that A satisfies λ - C f SI , or denoted by C f SI( A ) if thereis no ambiguity. Sometimes we use C f SI( T t ) for convenience. C f SI is a generalization of the complete logarithmic Sobolev inequality. An important example is f ( x ) = x p for p ∈ (1 , p SI and CLSI + [LJL20]. Lemma 3.11.
Let E K : N → K be a conditional expectation. Then I fI − E K ( ρ ) = d f ( ρ k E K ( ρ )) + d f ( E K ( ρ ) k ρ ) and hence C f SI( I − E K ) ≥ . Proof.
We have d f ( E K ( ρ ) k ρ ) = τ (cid:0) f ( E K ( ρ )) − f ( ρ ) − ( E K ( ρ ) − ρ ) f ′ ( ρ ) (cid:1) = τ ( f ( E K ( ρ ) − f ( ρ ))) + τ (cid:0) ( I − E K )( ρ ) f ′ ( ρ ) (cid:1) = − d f ( ρ k E K ( ρ )) + I fI − E K ( ρ ) . Thus C f SI( I − E K ) follows from the nonnegativity of f -relative entropy. (cid:3) This result for CLSI was given by [DPR17]. We can obtain a better constant for C p SI, see [LJL20].Applying Lemma 3.6, we have the following equivalence:
Proposition 3.12.
The following conditions are equivalent:( λd f N fix ρ ≤ I fA ( ρ ) for any ρ ∈ N + ; (2) d f N fix ( T t ( ρ )) ≤ e − λt d f N fix ( ρ ) for any ρ ∈ N + . Combining Lemma 3.7 and Lemma 3.4, we obtain the following change of measure principle . Theorem 3.13 (change of measure principle) . Let τ and τ be normal faithful traces over N and c ≤ dτ dτ ≤ c for some c , c > . Then C f SI(
A, τ ) ≥ c c C f SI(
A, τ ) . The change of measure principle is often referred to as Holley and Stroock ([HS86]) argument.They proved that LSIs are stable under change of measures. This remains true for CLSIs [LJL20].C f SIs are also stable under tensorization as an application of the data processing inequality.
Theorem 3.14 (tensorization stability) . Let T jt : N j → N j be a family of semigroups with fixed-point algebras N fix ,j ⊂ N j for ≤ j ≤ k . Then the tensor semigroup T t = ⊗ kj =1 T jt has thefixed-point algebra N fix = ⊗ kj =1 N fix ,j . Moreover, we have C f SI( T t ) ≥ inf ≤ j ≤ k C f SI( T jt ) . Proof.
It suffices to prove for the 2-fold tensor product. For the n -fold tensor product, we may usethe standard induction argument. Let E , E , and E be the conditional expectations onto N fix , N fix , and N fix , respectively. Applying (3.5) and Theorem 3.8 gives d f N fix ( ρ ) = d f ( ρ k E ⊗ id ( ρ )) + d f ( E ⊗ id ( ρ ) k E ( ρ )) ≤ d f ( ρ k E ⊗ id ( ρ )) + d f ( ρ k id ⊗ E ( ρ )) . For the n -fold tensor product, we may use the standard induction argument. (cid:3) The following result is motivated by [Spo78] and [GJL18] (lemma 2.6).
Theorem 3.15.
Let T t = e − tA be a semigroup of completely positive self-adjoint unital maps on L ( N , τ ) . Suppose there exists some positive constant λ such that I fA ( T t ( ρ )) ≤ e − λt I fA ( ρ ) , ∀ ρ ∈ N + , then C f SI( A ) ≥ λ. Proof.
We use Lemma 3.6 again. Define g ( t ) = d f N fix ( T t ( ρ )), then − g ′ ( t ) ≤ − e − λt g ′ (0) . Integrating both sides from [0 , ∞ ) yields C f SI( A ) ≥ λ . (cid:3) Remark 3.16.
In many situations we do not need the condition f [2] ∈ C + , such as Lemma 3.2,3.4,3.5, 3.6, 3.7, 3.11, Prop 3.12, Theorem 3.13 and 3.15. However, this condition is necessary toobtain the date processing inequality. f SI of derivation triple. Let us recall the definition of a derivation triple in [LJL20]. Let N be a finite von Neumann algebra equipped with a normal faithful tracial state τ , and δ be aclosable ∗ -preserving derivation on N . Suppose there exists a larger finite von Neumann algebra( M , τ ) containing N and a weakly dense ∗ -subalgebra A ⊂ N such that(1)
A ⊂ dom( δ );(2) δ ∗ ¯ δ : A → A ;(3) δ : A → L ( M , τ ).We define π δ : Ω ( A ) → M by π δ ( a ⊗ b − ⊗ ab ) = δ ( a ) b, where Ω ( A ) = { P j ( a j ⊗ b j − ⊗ a j b j ) | a j , b j ⊗ A} ⊂ A ⊗ A . Thus Ω δ ( A ) is Hilbert A -bimodulewith inner product ( δ ( a ) b , δ ( a ) b ) A = b ∗ E N ( δ ( a ∗ ) δ ( a )) b , where E N : M → N is the conditional expectation and ( · , · ) A is the N -valued inner product. Alinear operator Rc : Ω δ ( A ) → M is called the Ricci operator of ( N ⊂M , δ, τ ) provided that
OMPLETE SOBOLEV TYPE INEQUALITIES 11 (1) Rc ( aρb ) = aRc ( ρ ) b, ∀ a, b ∈ A , ρ ∈ Ω δ ( A );(2) there exists a strongly continuous semigroup ˆ T t = e − tL : M → M of completely positive tracepreserving maps such that Γ L ( a, b ) = E N ( δ ( a ∗ ) δ ( b )) and δ ( δ ∗ ¯ δa ) − L ( δ ( a )) = Rc ( δ ( a )) for any a, b ∈ A .The derivation δ is said to admit a Ricci curvature Rc ≥ λ bounded below by a constant λ , if( Rc ( ρ ) , ρ ) A ≥ λE N ( ρ ∗ ρ ) for any ρ ∈ Ω δ ( A ). We say the generator A of T t = e − tA admits Rc ≥ λ if there exists a derivation triple ( N ⊂M , δ, τ ) such thatΓ A ( a, b ) = E N ( δ ( a ∗ ) δ ( b )) , ∀ a, b ∈ A and δ admits Rc ≥ λ. It shall be noted that the choice of δ is not unique, thus we may find a largerRicci lower bound of A by choosing a good δ . Lemma 3.17.
Let ( N ⊂M , δ, τ ) be a derivation triple with a Ricci curvature Rc ≥ λ > . Then I fδ ( e − tδ ∗ ¯ δ ( ρ )) ≤ e − λt I fδ ( ρ ) , ∀ ρ ∈ N + . Proof.
In the proof, we use the following notations A = δ ∗ ¯ δ , T t = e − tL , and T t ( ρ ) = ρ t . Letˆ T t = e − tL be the semigroup given in the definition of the Ricci curvature. Let us consider twofunctions h ( t ) = I fδ ( ρ t ) = τ (cid:18)Z ∞ Z ∞ δ ( ρ t ) f ′ ( s ) − f ′ ( l ) s − l dE ρ t ( s ) δ ( ρ t ) dE ρ t ( l ) (cid:19) and k ( t ) = γ f [2] ρ t ,ρ t ( ˆ T t ( δ ( ρ )) , ˆ T t ( δ ( ρ )))= τ (cid:18)Z ∞ Z ∞ ˆ T t ( δ ( ρ )) f ′ ( s ) − f ′ ( l ) s − l dE ρ t ( s ) ˆ T t ( δ ( ρ )) dE ρ t ( l ) (cid:19) . By Γ L ( a, b ) = E N ( δ ( a ∗ ) δ ( b )), then k ( t ) = γ f [2] ˆ T t ( ρ ) , ˆ T t ( ρ ) ( ˆ T t ( δ ( ρ )) , ˆ T t ( δ ( ρ ))) . Noting f [2] ∈ C + , we deduce that k ( t ) ≤ k (0) and k ′ (0) ≤ . (28)By the product rule, we have h ′ ( t ) = − τ ( Z ∞ Z ∞ δ ( A ( ρ t )) f ′ ( s ) − f ′ ( l ) s − l dE ρ t ( s ) δ ( ρ t ) dE ρ t ( l ))) + h ′ r ( t )and k ′ ( t ) = − τ ( Z ∞ Z ∞ L ( ˆ T t ( δ ( ρ ))) f ′ ( s ) − f ′ ( l ) s − l dE ρ t ( s ) ˆ T t ( δ ( ρ )) dE ρ t ( l ))) + k ′ r ( t ) , where h ′ r and k ′ r are the derivatives corresponding to dE ρ t . We make an important observation h ′ r (0) = k ′ r (0) . (29)Together with δ ( δ ∗ ¯ δa ) − L ( δ ( a )) = Rc ( δ ( a )), then h ′ (0) − k ′ (0) = − τ ( Z ∞ Z ∞ Rc ( δ ( ρ ))) f ′ ( s ) − f ′ ( l ) s − l dE ρ ( s ) δ ( ρ ) dE ρ ( l )))= − τ (cid:16) Rc (cid:16) Q ρf [2] ) / ( δ ( ρ )) (cid:17) ( Q ρf [2] ) / ( δ ( ρ )) (cid:17) . Using that Rc ≥ λ , we infer that h ′ (0) − k ′ (0) ≤ − λh (0) . By (28), then h ′ (0) ≤ − λh (0) . (30)Setting h s ( t ) = I fδ ( ρ t + s ), then h ′ s (0) = h ′ ( s ). Inequality (30) remains true for h s . Hence h ′ ( s ) = h ′ s (0) ≤ − λh s (0) = − λh ( s )completes the proof. (cid:3) Applying Theorem 3.15, we get the following complete Sobolev inequality.
Theorem 3.18.
Let ( N ⊂M , δ, τ ) be a derivation triple with a Ricci curvature Rc ≥ λ > . Let f : R + → R + be continuously differentiable and f [2] ∈ C + . Then we have C f SI (
N ⊂M , δ, τ ) ≥ λ. Applications p norm estimate.Theorem 4.1. Let ( N ⊂M , δ, τ ) be a derivation triple with a Ricci curvature Rc ≥ λ > and T t = e − tδ ∗ ¯ δ . Then we have k T t ( ρ ) − E ( ρ ) k p ≤ e − λt s p ( p − k ρ k − p/ p ( k ρ k pp − k E ( ρ ) k pp ) / , ∀ ρ ∈ N + . Proof.
This proof is inspired by [RX16]. For self-adjoint a, b ∈ N , we define G a,b ( s ) = k a + sb k pp − p ( p − s k a + sb k p − p k b k p . It follows from the definition that G a,b is convex over R if G ′′ x,y (0) ≥ x, y ∈ N .In [RX16], they considered the following function ψ ( s ) = k a + sb k pp with an additional condition that a is invertible. They proved that ψ ′′ (0) ≥ p ( p − k a k p − p k b k p . (31)It implies that G ′′ a,b (0) ≥
0. This remains true if a is not invertible, see the proof of Theorem 2 in[RX16]. Let a = E ( ρ t ) = E ( ρ ) and b = ρ t − E ( ρ ), then k a + sb k pp ≥ k E ( a + sb ) k pp = k a k pp . Hence the right derivative of G a,b at 0 is nonnegative, and convexity implies G ′ a,b ( s ) ≥ s ≥
0. In particular G (1) ≥ G (0), then k E ( ρ ) k pp + p ( p − k ρ t k p − p k ρ t − E ( ρ ) k p ≤ k ρ t k pp . By C p SI, we have k ρ t k pp ≤ k E ( ρ ) k pp + e − λt (cid:0) k ρ k pp − k E ( ρ ) k pp (cid:1) . Chaining the two inequalities gives k ρ t − E ( ρ ) k p ≤ e − λt p ( p − k ρ t k − pp (cid:0) k ρ k pp − k E ( ρ ) k pp (cid:1) . Noting k ρ t k p ≤ k ρ k p and taking the square root of the inequality above complete the proof. (cid:3) OMPLETE SOBOLEV TYPE INEQUALITIES 13
Bakry-´Emery criterion.
Let (
M, g ) be a smooth n -dimensional Riemannian manifold with-out boundary. For a smooth function U ∈ C ∞ ( M ), we define a probability measure µ by dµ = 1 Z U e − U dvol with the normalization factor Z U = R M e − U dvol and a Bakry-´Emery Ricci curvature Rc U by Rc U = Rc + Hess( U ) . Applying Theorem 3.18 , we obtain the modified Laplace operator∆ U = ∆ + ∇ U · ∇ , where ∆ is the Laplace-Beltrami operator. See [LJL20] for the definition of a derivation triple of aRiemannian manifold. Theorem 4.2.
Let ( M, g, µ ) be a smooth Riemannian manifold with the measure µ defined by dµ = Z U e − U dvol with Z U = R M e − U dvol for U ∈ C ∞ ( M ) . Given that Rc U ≥ κ > and f [2] ∈ C + ,then C f SI(∆ U ) ≥ κ. Let f ( x ) = x p , we obtain the complete Beckner inequalities. Corollary 4.3 (complete Beckner Inequalities) . Let ( M, g, µ ) be a smooth Riemannian manifoldwith the measure µ defined by dµ = Z U e − U dvol with Z U = R M e − U dvol for U ∈ C ∞ ( M ) . Giventhat Rc U ≥ κ > , then C p SI(∆ U ) ≥ κ. By Theorem 3.13, we have the following result.
Theorem 4.4.
Let ν be the probability measure defined by dν = Z V e − V dvol with V ∈ C ∞ ( M ) ,where Z V is the normalization factor. If k U − V k ∞ ≤ C and f [2] ∈ C + , then e C C f SI(∆ U ) ≥ C f SI(∆ V ) . Random Transpositions.
Let S n be the permutation group on { , . . . , n } , and we considerthe Laplace operator ∆ n given by(∆ n f )( σ ) = 1 n n X i,j =1 (cid:2) f ( σ ) − f ( σ ij ) (cid:3) , where σ ij denotes the configuration of σ after swapping the elements on i -th site and j -th site. Forexample let σ = (1 3 2) and i = 1 and j = 2, then σ ij = (3 1 2). It is well-known that ∆ n isergodic, and thus E n ( f ) = n ! P σ ∈ S n f ( σ ) . The lower bound of Ricci curvature Rc of the randomtransposition on the Symmetric group S n is defined using the geodesic convexity, and Rc ≥ n ([EMT15] and [FM16]). MLSIs were studied in [Goe04], [BT06], and [GQ03] using the martingalemethods in [LY98], and they proved that1 ≤ MLSI(∆ n ) ≤ . The upper was given by the spectral gap λ (∆ n ) = 2 ([DS87]). We also apply the martingalemethods and establish a similar relation between C p SI(∆ n +1 ) and C p SI(∆ n ): Theorem 4.5.
Let p ∈ (1 , , then p ≤ C p SI(∆ n ) ≤ and ≤ CLSI(∆ n ) ≤ for any n ≥ . As pointed out by [BT06] (Section 4), the upper bound of M p SI(∆) is also given by the spectralgap. It is sufficient to give the lower bound. Let M be a finite von Neumann algebra equippedwith a normal faithful trace τ , and we consider M -valued function. For f ∈ ℓ m ∞ , let τ ( f ) = m P mj =1 τ ( f ( j )). Lemma 4.6.
For any M -valued function f defined over { , . . . , n } and n ≥ , we have τ ( f p − E ( f )) ≤ n n X i,j =1 τ h ( f ( i ) − f ( j )) (cid:16) f ( i ) p − − f ( j ) p − (cid:17)i , where E ( f ) = n P ni =1 f ( i ) . This lemma is an immediate application of C p SI( I − E ) ≥ p ([LJL20]). The scalar case of thefollowing lemma was proven in [BT06], and here we give an operator valued version. Lemma 4.7.
Let p ∈ (1 , and F ( ρ, σ ) = τ (cid:2) ( ρ − σ )( ρ p − − σ p − ) (cid:3) , then F is jointly convex for x, y ∈ M + .Proof. Let f = ( ρ, σ ) and E ( f ) = ( ρ + σ ), then δ ( f ) = ( f (2) − f (1) , f (1) − f (2)) and δ ∗ δ = I − E .We can rewrite F as F ( ρ, σ ) = p I pδ ( f ). By Corollary 3.3 F is jointly convex. (cid:3) Now we prove Theorem 4.5.
Proof.
Let N i ⊂ L ∞ ( S n +1 , M ) be a von Neumann subalgebra generated by { e ji } n +1 j =1 satisfying e ji ( σ ) = ( , if σ i = j ;0 , otherwise . We denote the corresponding conditional expectation by E N i . By martingale equality, we have d p ( f k E ( f )) = d p ( f k E N i ( f )) + d p ( E N i ( f ) k E ( f )) . Since i is also uniformly chosen from from the n + 1 sites, then d p ( f k E ( f )) = 1 n + 1 n +1 X i =1 h d p ( f k E N i ( f )) + d p ( E N i ( f ) k E ( f )) i . (32)For any fixed i , we define f i ( j ) = X σ i = j f ( σ ) . Let E ij ( f )( σ ) = 1 n ! ( f i ( j ) , if σ i = j ;0 , otherwise , then E N i ( f ) = 1 n + 1 n +1 X j =1 E ij ( f ) . Let us define a projection map P ij ( f )( σ ) = ( f ( σ ) , if σ i = j ;0 , otherwise , then d p ( f k E N i ( f )) = 1( n + 1)! n +1 X j =1 τ ( P ij ( f ) p − E ij ( f ) p ) OMPLETE SOBOLEV TYPE INEQUALITIES 15
Also by our assumptionC p SI(∆ n )( n + 1)! τ ( P ij ( f ) p − E ij ( f ) p ) ≤ p ( n + 1)!2 n X { ( σ,k,l ) | σ i = σ kli = j } τ h(cid:16) f ( σ kl ) − f ( σ ) (cid:17) (cid:16) f ( σ kl ) p − − f ( σ ) p − (cid:17)i . It is important to observe that n +1 X i =1 n +1 X j =1 X { ( σ,k,l ) | σ i = σ kli = j } τ h(cid:16) f ( σ kl ) − f ( σ ) (cid:17) (cid:16) f ( σ kl ) p − − f ( σ ) p − (cid:17)i =( n − X σ,k,l τ h(cid:16) f ( σ kl ) − f ( σ ) (cid:17) (cid:16) f ( σ kl ) p − − f ( σ ) p − (cid:17)i . Thus 1 n + 1 n +1 X i =1 d p ( f k E N i ( f )) ≤ n − n C p SI(∆ n ) I p ∆ n +1 ( f ) . (33)Applying Lemma 4.6, we obtain that d p ( E N i ( f ) k E ( f )) ≤ n + 1) n +1 X k,l =1 τ h ( E ik ( f ) − E il ( f )) (cid:0) E ij ( f ) p − − E il ( f ) p − (cid:1) i . By the definition of E ik , we have( E ik ( f ) − E il ( f )) (cid:0) E ij ( f ) p − − E il ( f ) p − (cid:1) = n ! X σ i = l f ( σ kl ) − n ! X σ i = l f ( σ ) n ! X σ i = l f ( σ kl ) p − − n ! X σ i = l f ( σ ) p − . Together with Lemma 4.7, it implies that τ (cid:2) ( E ik ( f ) − E il ( f )) (cid:0) E ij ( f ) p − − E il ( f ) p − (cid:1)(cid:3) ≤ n ! X σ i = l τ h ( f ( σ kl ) − f ( σ ))( f ( σ kl ) p − − f ( σ ) p − ) i . Then we have d p ( E N i ( f ) k E ( f )) ≤ n + 1) n ! n +1 X k,l =1 X σ i = l τ h ( f ( σ kl ) − f ( σ )) (cid:16) f ( σ kl ) p − − f ( σ ) p − (cid:17)i . Thus 1 n + 1 n +1 X i =1 d p ( E N i ( f ) k E ( f )) ≤ n + 1) p I p ∆ n +1 ( f ) . (34)Combining (33) and (34), we obtain1C p SI(∆ n +1 ) ≤ n − n C p SI(∆ n ) + 1 p ( n + 1) . Lemma 4.6 implies that C p SI(∆ ) ≥ p. By induction method, we haveC p SI(∆ n +1 ) ≥ p. Indeed, by assuming that C p SI(∆ n ) ≥ p we obtain that1C p SI(∆ n +1 ) ≤ p (cid:18) n − n + 1 n + 1 (cid:19) ≤ p . We only prove the estimate for C p SI and the argument remains true for CLSI. (cid:3)
Bernoulli-Laplace Model.
We consider the Bernoulli-Laplace model with n distinct sites { , . . . , n } and r identical particles, where n ≥ ≤ r ≤ n −
1. Each site can be occupiedby at most 1 particle. Let C n,r be the state space of the configurations of r elements occupying n sites. The Laplace operator ∆ n,r : L ∞ ( C n,r ) → L ∞ is defined by(∆ n,r f )( σ ) = 1 n X i Let p ∈ (1 , , then p/ ≤ C p SI(∆ n,r ) ≤ and / ≤ CLSI(∆ n,r ) ≤ for any n ≥ and ≤ r ≤ n − . Again the upper bound of M p SI(∆) is also given by the spectral gap, see [DS87]. Let M be a finitevon Neumann algebra equipped with a normal faithful trace τ , and we consider M -valued function.For f ∈ ℓ m ∞ , let τ ( f ) = m P mj =1 τ ( f ( j )). The proof of 4.8 is quite similar to the proof of 4.5. Proof. Let N i ⊂ L ∞ ( C n +1 ,r , M ) be a von Neumann subalgebra generated by { e i , e i } defined by e ji ( σ ) = ( , if σ i = j ;0 , otherwise , j = 0 , . Let E N i be the corresponding conditional expectation onto N i . By martingale equality, we havethat d p ( f k E ( f )) = d p ( f k E N i ( f )) + d p ( E N i ( f ) k E ( f )) . Since i is uniformly chosen from from the n + 1 sites, then d p ( f k E ( f )) = 1 n + 1 n +1 X i =1 h d p ( f k E N i ( f )) + d p ( E N i ( f ) k E ( f )) i . (35)For any fixed i , we define f i ( j ) = X σ i = j f ( σ ) , j = 0 , . Let E i,j ( f )( σ ) = 1 a j ( f i ( j ) , if σ i = j ;0 , otherwise , where a = (cid:0) n − r (cid:1) and a = (cid:0) n − r − (cid:1) . Let us define projections P i,j ( f )( σ ) = ( f ( σ ) , if σ i = j ;0 , otherwise , j = 0 , . OMPLETE SOBOLEV TYPE INEQUALITIES 17 Then d p ( f k E N i ( f )) = 1 (cid:0) n +1 r (cid:1) X j =0 , τ [ P i,j ( f ) p − E i,j ( f ) p ] . By the definition of C p SI, then we haveC p SI(∆ n,r − j ) τ [ P i,j ( f ) p − E i,j ( f ) p ] ≤ p n X { ( σ,k,l ) | σ i = σ kli = j } τ h(cid:16) f ( σ kl ) − f ( σ ) (cid:17) (cid:16) f ( σ kl ) p − − f ( σ ) p − (cid:17)i . An important observation is that n +1 X i =1 X j =0 , X { ( σ,k,l ) | σ i = σ kli = j } τ h(cid:16) f ( σ kl ) − f ( σ ) (cid:17) (cid:16) f ( σ kl ) p − − f ( σ ) p − (cid:17)i =( n − X σ,k,l τ h(cid:16) f ( σ kl ) − f ( σ ) (cid:17) (cid:16) f ( σ kl ) p − − f ( σ ) p − (cid:17)i . Thus 1 n + 1 n +1 X i =1 d p ( f k E N i ( f )) ≤ n − n j =0 , { C p SI(∆ n,r − j ) } I ∆ n +1 ,r ( f ) . (36)Applying Lemma 4.6, we obtain that d p ( E N i ( f ) k E ( f )) ≤ a a ( a + a ) τ h (cid:18) f i (1) a − f i (0) a (cid:19) (cid:18) f i (1) a (cid:19) p − − (cid:18) f i (0) a (cid:19) p − ! i . The definition of f i ( j ) infers that f i (1) = X σ i =1 f ( σ ) = 1( n − r + 1) n +1 X k =1 X σ k =1 ,σ i =0 f ( σ ki ) ,f i (0) = X σ i =0 f ( σ ) = 1 r n +1 X k =1 X σ k =1 ,σ i =0 f ( σ ) . Together with Lemma 4.7, it implies that τ (cid:2) ( E i, ( f ) − E i, ( f )) (cid:0) E i, ( f ) p − − E i, ( f ) p − (cid:1)(cid:3) ≤ ( n − r )!( r − n ! n +1 X k =1 X σ i =0 ,σ k =1 τ h (cid:16) f ( σ ik ) − f ( σ ) (cid:17) (cid:16) f ( σ ik ) p − − f ( σ ) p − (cid:17) i . Similarly τ (cid:2) ( E i, ( f ) − E i, ( f )) (cid:0) E i, ( f ) p − − E i, ( f ) p − (cid:1)(cid:3) ≤ ( n − r )!( r − n ! n +1 X k =1 X σ i =1 ,σ k =0 τ h (cid:16) f ( σ ik ) − f ( σ ) (cid:17) (cid:16) f ( σ ik ) p − − f ( σ ) p − (cid:17) i . Thus d p ( E N i ( f ) k E ( f )) ≤ a a a + a ) ( n − r )!( r − n ! n +1 X k =1 X σ i = σ k τ h (cid:16) f ( σ ik ) − f ( σ ) (cid:17) (cid:16) f ( σ ik ) p − − f ( σ ) p − (cid:17) i . Then we have n +1 X i =1 d p ( E N i ( f ) k E ( f )) ≤ p I p ∆ n +1 ,r ( f ) . Together with (36), it implies that1C p SI ∆ n +1 ,r ≤ n − n j =0 , { C p SI(∆ n,r − j ) } + 2( n + 1) p . Noting ∆ n, = ∆ n,n − = I − E , we obtain that C p SI(∆ n, ) ≥ p and C p SI(∆ n,n − ) ≥ p . 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Letters in Mathe-matical Physics , 106(9):1217–1234, 2016. Department of Mathematics, University of Illinois, Urbana, IL 61801, USA E-mail address , Haojian Li:, Haojian Li: