Specification of Fractional Poincare’ Inequalities for the Sequence Measures Generalization

The Fractional Poincare’ inequalities in R are endowed with a fairly general sequence measure. We show a control of norm by a non–Local quantity. The assumption on the sequence measure is that it satisfies the classical Poincare’ inequality, with general results. We also verify quantity of the tightness at infinity provided by the control on the fractional derivative in terms of a sequence of a weight growing at infinity. The illustration goes to the generator of the OrnsteinUhlenbeck semi group and some estimates of its powers. Keywords: Poincare Inequalities, Non-Local Inequalities, Fractional Powers, Sequence Measure 1. Introduction Fractional diffusions naturally appear in many models (see [1-7]). The theory also appears naturally in mathematics: in probability they appear in the important class of infinitely divisible Markov processes (see [8]); in analysis they naturally appear in the study of singular integral operators (see[3-6]) as well as in the so-called “Dirichlet-to-Neuman” boundary value problem and in [9] (see also [10, 11]). Clément Mouhot, Emmanuel Russ and Yannick Sire [12] prove a Poincaré inequality on R , endowed with a measure M(x)dx, involving non-local quantities on the right-hand side in the spirit of Gagliardo semi-norms for Sobolev spaces W , (R ) with fractional order s ∈ (0, 1) (see [13]). Poincaré inequality for the non-local fractional energy associated with such fractional diffusion is, therefore, a natural and interesting question since this inequality governs the spectral gap of the underlying operator and the speed of (fractional) diffusion towards an equilibrium. The work of [12] is verified, using the same strong methodology, applying the sequence measure of a -function. M denotes a positive weight in L (R ). By L (R , M), the space of measurable functions on R is signified which are square integrable with respect to the sequence measure M(x )dx , by L (R , M) the subspace of functions of L (R , M) such that f(x )M(x ) R dx = 0 , and by H (R , M), the Sobolev space of functions in L (R , M), the weak derivative of which belongs to L (R , M). Finally, for any measurable subset A ⊂ R by L (A,M) the obvious restriction of the definition above to the set A is designated. The assumption is that % is a sequence C function and that this sequence measure M satisfies the usual Poincaré inequality: there exists a constant λ(M) > 0 such that ∀* ∈ + (R , %), |-*(. / )| %(. / ) R0 1. / ≥ 3(%) 4*(. / ) − R0 *(. )%(. )1. R0 4 %(. / )1. / (1) If the sequence measure % can be written % = 678, this inequality is known to hold (see [14-29]), whenever there exists ε > 0, c > 0 and R > 0 so that. ∀|. | ≥ <, (1 − =)|∇?(. )| − ∆? ≥ A (2) The inequality (1) holds(see [12]), for instance, when % = (2C)7 ⁄ exp(− |. | 2 ⁄ )is the Gaussian measure, but also when %(. ) = 67|F0|, and more generally when %(. ) = 67|F0|GHIwith ε > 0. If ?is convex, |?| < ∞,then Hess(V) ≥ cstId 58 Abdelilah Kamal H. Sedeeg and Shawgy H. Abd Alla: Specification of Fractional Poincare’ Inequalities for the Sequence Measures Generalization and the sequence measure M(. )d. satisfies the logSobolev inequality, which in turn implies (1) (see [22], [12]). The Poincaré inequality (1) admits the following selfimprovement for completeness Proposition (1.1): Assume that there exists ε >0 such that ( 7N)|∇8|O − ∇? . → ∞ QQQQQQQQQQQQQQQR + ∞, % = 678 (3) Then there exists 3T(%) > 0 such that, for all functions* ∈ (R , %) ∩ + (R , %): ∫∫ |∇*(. )| %(. ) R0 1. ≥ 3T(%) |*(. )| (1 + R0 |∇In %(. )| )%(. )1. (4) The Authors in [12] generalize the inequality (1) in the strengthened form of Proposition (1.1), replacing the H semi-norm, in the left-hand side, by a non-local expression in the flavour of the Gagliardo semi-norms. The following theorem are shown below (see [12]). Theorem (1.2): Assumes that M = e7W is a C positive L function which satisfies (3). Let ε > 0 . Then there exist λ 7X(M) > 0 and δ(M) (constructive from our proof and the usual Poincaré constant λ(M) such that, for any function f belonging to a dense subspace of L (R , M), the formula: Z |*(. ) − *(. / )| |. − . / | / 7[ %(. )67\(])|F07F0HG| R0×R0 1. 1. / ≥ 3 7[(%) |*(. )| (1 + |∇In %(. )| 7[) R0 %(. )1. (5) In particular (5) is satisfied for any function *with zero average (see [12]) belonging to the domain of the operator = −∆ − -?. -. Functions of this domain with zero integral with respect to %(. )1. are dense in (R , %). Observe that the right-hand side of (5) involves a fractional moment of order (1 + =) related to the homogeneity of the semi-norm appearing in the left-hand side. One could expect (see [12]) in the left-hand side of (5) the Gagliardo semi-norm for the fractional Sobolev space +( 7[) ⁄ (R , %), namely Z |*(. ) − *(. / )| |. − . / | / 7[ %(. )%(. / ) R0×R0 1. 1. / Notice that, instead of this semi-norm, (see [12]) obtain a “non-symmetric” expression. However, our norm is more natural: one should think of the sequence measure over . / as the Lévy measure, and thesequence measure over . as the ambient measure. It is thus emphasized that the sequence measure is rather general (see [12]) and in particular, as a corollary of Theorem (1.2), an automatic improvement of the Poincaré inequality (1) is obtained by: Z |*(. ) − *(. / )| |. − . / | / 7[ %(. )%(. / ) R0×R0 1. 1. / ≥ 3 /[(%) `|*(. )| %(. ) R0 1. . The question of obtaining Poincaré-type inequalities (or more generally entropy inequalities) for Lévy operators was studied in the probability community in the last decades. For instance it was proved by Wu [23] and Chafaï [24] that Entbc(*) ≤ `e (*)∇* ∙ g ∙ ∇* 1h +Zicj*(. ), *(. + k )l 1mb(k ) 1mb(. ) (see also [25]) with Entbc(*) = `e(*)1h − e n`*1ho, And icis the so-called Bregman distance associated to e: ic(p, p − =) = e(p) − e(p − =) − eT(p − =)(=), Where e is some well-suited functional with convexity properties, g the matrix of diffusion of the process, ha rather general measure, and mb the (singular) Lévy measure associated to h . Choosing e(. ) = . and g = 0 yields a Poincaré inequality for this choice of measure h, mb . The improvement of this approach is that it does not impose any link between sequence measure % on . and the singular measure |k |7 7( 7[)on k = . − . / . (see [12]). Remark (1.3): Note that the exponentially decaying factor 67\(])|F0 7 F0HG| in (5) also improvesthe inequality as compared to what is expected from Poincaré inequality for Lévy measures (see [12]). Other extensions in progress are to allow more general singularities than the Martin Riesz kernel |F0 7 F0HG|0HOqI (see [26]) and to develop an rtheory of the previous inequalities. The proof of [12] heavily relies on fractional powers of a (suitable generalization of the) Ornstein-Uhlenbeck operator, which is defined by: * = −%7 div(%∇*) = −∇* − ∇ ln% ∙ ∇*, for all * ∈ i( ∗) ∶= {g ∈ + (R , %); (1/√%)div(%-g) ∈ (R )}. One therefore has, for all * ∈ |( ∗) and g ∈ + (R , %), ` *(. )g(. )%(. ) R0 1. = `∇*(. ) ∙ ∇g(. )%(. ) R0 1. . It is obvious that is symmetric and nonnegative on (R , %), which allows to define the usual power }for any ~ ∈ (0, 1)by means of spectral theory. Note that ( /[) ⁄ is not the symmetric operator associated to the Dirichlet form ∫∫ |(F0)7(F0HG)|O |F0 7 F0HG|0HOqI%(. ) R0×R0 1. 1. / . The Authors in [12] proved Theorem (1.2) by first establishing off-diagonal estimates of Gaffney type on the American Journal of Applied Mathematics 2017; 5(3): 57-67 59 resolvent of L on (R , %). These estimates are needed, since Gaussian pointwise estimates on the kernel of the operator are not available.Then, they bound the quantity, `|*(. )| (1 + |∇ ln%(. )| 7[)%(. ) R0 1. , In terms of  ∗( 7[)  ⁄ *O(R0,]) . This will be obtained by an abstract argument of functional calculus based on rewriting in a suitable way the conclusion of Proposition (1.1). Finally, using the off-diagonal estimates for the kernel of , it is establish that  ∗( 7[)  ⁄ *O(R0,]) ≤ Z |*(. ) − *(. / )| |. − . / | / 7[ %(. ) R0×R0 1. 1. / , Which concludes the proof. They also borrow methods from harmonic analysis. This seems not so common in the field of Poincaré and log-Sobolev inequalities, where standard techniques rely on global functional inequalities, see for instance the powerful so-called  -calculus of Bakry and Émery [27]. 2. Resolvent of ∗ by the Off-Diagonal  Estimates Recall that for every f ∈ |( ∗), it is defined ∗* = −%7 1m (%-*) = −∆* − % ∙ -* (6) From the fact that ∗ is self-adjointand nonnegative on L (R , M)we have: ‖( ∗ − μ)7 ‖O(R ,) ≤ 1 dist (μ, ∑( ∗)) Where Σ( ∗) denotes the spectrum of L , and μ ∉ Σ( ) . Then it can be deduced that ( + ( − 1)∗)7 is bounded with norm less than 1 for all  > 1. Since ( − 1) ∗(I + ( − 1) ∗)7 = I − (I + ( − 1) ∗)7 , the same is true for ( − 1) ∗(I + ( − 1) ∗)7 = I − ( + ( − 1) ∗)7 with a norm less than 2. Moreover, (I + ( − 1) ∗)7 * ∈ + (R , %). Actually, when * ∈ (R , %) is supported in a closed set  ⊂ R and  ⊂ R is a closed subset disjoint from  , a much more precise estimate on the norm of (I + ( − 1) )7 * and ( − 1) ∗(I + ( − 1) ∗)7 * on  can be given. Here are these off-diagonal estimates for the resolventof ∗. Lemma (2.1): There exists = (%) > 0 with the following property: for all compact disjoint subsets ,  ⊂ R ,  bounded, with dist(, ) =:  + ε, ε > 0, all functions * ∈ (R , %) supported in  and all  > 1 ‖(I + ( − 1) ∗)7 *‖O(,]) + ‖( − 1) ∗(I + ( − 1) ∗)7 *‖O(,]) ≤ 867GHI √ ‖*‖O(,]). Note that, in different contexts, this kind of estimate, originating in [28], turns out to be a powerful tool, especially when no pointwise upper estimate on the kernel of the semigroup generated by ∗is available (see [29-31]). Since no reference for these off-diagonal estimates for the resolvent of ∗ could be obtained, here a general proof [12] is provided. Proof of Lemma (2.1): As in [32] it is argued, since (I + ( − 1) ∗)7 is bounded with norm less than 1 for all t > 1 it is clearly enough to restr

The following theorem are shown below (see [12]).
Theorem (1.2): Assumes that M = e 7W is a C positive L function which satisfies (3). Let ϵ > 0 . Then there exist λ 7X (M) > 0 and δ(M) (constructive from our proof and the usual Poincaré constant λ(M) such that, for any function f belonging to a dense subspace of L (ℝ , M), the formula: In particular (5) is satisfied for any function *with zero average (see [12] Observe that the right-hand side of (5) involves a fractional moment of order (1 + =) related to the homogeneity of the semi-norm appearing in the left-hand side. One could expect (see [12]) in the left-hand side of (5) the Gagliardo semi-norm for the fractional Sobolev space + Notice that, instead of this semi-norm, (see [12]) obtain a "non-symmetric" expression. However, our norm is more natural: one should think of the sequence measure over . / as the Lévy measure, and thesequence measure over . as the ambient measure. It is thus emphasized that the sequence measure is rather general (see [12]) and in particular, as a corollary of Theorem (1. The question of obtaining Poincaré-type inequalities (or more generally entropy inequalities) for Lévy operators was studied in the probability community in the last decades. For instance it was proved by Wu [23] and Chafaï [24] that Ent b c (*) ≤`e (*)∇* • g • ∇* 1h + Z i c j*(. ), *(. (see also [25]) with Where e is some well-suited functional with convexity properties, g the matrix of diffusion of the process, ha rather general measure, and m b the (singular) Lévy measure associated to h. Choosing e(. ) = . and g = 0 yields a Poincaré inequality for this choice of measure h, m b . The improvement of this approach is that it does not impose any link between sequence measure % on . and the singular measure |k | 7 7( 7[) on k = . − . / . (see [12]).
Remark (1.3): Note that the exponentially decaying factor 6 7\(])|F 0 7 F 0HG | in (5) also improvesthe inequality as compared to what is expected from Poincaré inequality for Lévy measures (see [12]). Other extensions in progress are to allow more general singularities than the Martin Riesz kernel |F 0 7 F 0HG | 0HOqI (see [26]) and to develop an r theory of the previous inequalities.
The proof of [12] heavily relies on fractional powers of a (suitable generalization of the) Ornstein-Uhlenbeck operator, which is defined by: for all * ∈ i( * ) ∶= {g ∈ + (ℝ , %); (1/√%)div(%-g) ∈ (ℝ )}. One therefore has, for all * ∈ |( * ) and g ∈ + (ℝ , %), It is obvious that is symmetric and nonnegative on (ℝ , %), which allows to define the usual power } for any ~∈ (0, 1)by means of spectral theory. Note that ( /[) ⁄ is not the symmetric operator associated to the Dirichlet form The Authors in [12] proved Theorem (1.2) by first establishing off-diagonal estimates of Gaffney type on the resolvent of L on (ℝ , %). These estimates are needed, since Gaussian pointwise estimates on the kernel of the operator are not available.Then, they bound the quantity, . This will be obtained by an abstract argument of functional calculus based on rewriting in a suitable way the conclusion of Proposition (1.1). Finally, using the off-diagonal estimates for the kernel of , it is establish that Which concludes the proof. They also borrow methods from harmonic analysis. This seems not so common in the field of Poincaré and log-Sobolev inequalities, where standard techniques rely on global functional inequalities, see for instance the powerful so-called † -calculus of Bakry and Émery [27].
Lemma ( Note that, in different contexts, this kind of estimate, originating in [28], turns out to be a powerful tool, especially when no pointwise upper estimate on the kernel of the semigroup generated by * is available (see [29][30][31]). Since no reference for these off-diagonal estimates for the resolvent of * could be obtained, here a general proof [12] is provided. Proof of Lemma (2.1): As in [32] it is argued, since (I + (" − 1) * ) 7 is bounded with norm less than 1 for all t > 1 it is clearly enough to restrictto = > 0.

Control of ƒ ‡ * (ˆ7 §)⁄ ©ƒ ‡ˆ(ℝ ª ,«) of Fractional Powers of ‡ *
This section is devoted to the control of the norm of fractional powers of * . This is the cornerstone of the proof of Theorem (1.2). In the functional calculus theory of sectorial operators * , fractional powers (see [12]) are defined as follows (see [12]): They can also be defined in terms of the resolvent by the Balakrishnan formulation (see instance [12]): As the representations (9) or (10) are redundant; instead reliance shall be on the powerful tool of the so-called "quadratic estimates" obtained in the functional calculus (see [12]).This is the object of the general next lemma. Let µbe a holomorphic function in + ¥ (∑ b H )such that for some , g, ¶ > 0, |µ(k )| ≤ infw|k | · , |k | 7¸{ , for any k ∈ ¹ b H .Since * is positive self-adjoint operator on (ℝ , %)and * is one-to-one on (ℝ , %) by (1), one has by the spectral theorem, Whenever -∈ (ℝ , %) . Choosing Whenever -∈ (ℝ , %).