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Research Article

Open Access Peer-reviewed

Khalid Boutahir, Ali Hafidi^{ }

Published online: June 09, 2017

heat semigroup legendre operator spectral gap poincaré inequality sobolev inequality logarithmic sobolev inequality φ-entropy inequality

We consider on the interval [-1,1] the heat semigroup generated by the Legendre operator* ** *acting on the Hilbert space with respect to the uniform measure* ** ** *By means of a simple method involving some semigroup techniques, we describe a large family of optimal integral inequalities with the Poincaré and logarithmic Sobolev inequalities as particular cases.

Gross’ logarithmic Sobolev inequality ^{ 2} states that for all smooth functions f on

(1) |

white* ** *denote the normalized Gaussian measure on In this Gaussian context, the Poincaré inequality(spectral gap inequality) is given by:

(2) |

In 1989, W.Bekner ^{ 2} derived a family of generalized Poincaré inequalities that yield a sharp interpolation between Poincaré inequality and logarithmic Sobolev inequality:

(3) |

where *L *is the Ornstein-Uhlenbeck operator: * *Recently, A.Bentaleb, S.Fahlaoui and A.Hafidi proposed in [^{ 3}, Section 2] a generalized of the inequality 3 and obtained the following inequality: for all smooth function *f* on

where and are strictly convex, Similar researches on this kind of inequalities for general probability measure generated by diffusion have been done by many authors (see, for instance, ^{ 4, 10}).

The purpose of this paper is to present a family of integral inequalities on the interval [-1,1] which provide interpolation between the Sobolev and Poincaré inequalities (see Theorem 1 below).

In order to keep the paper reasonably self-contained, we summarize in this section the basic notion that will be used in this work.

We consider the Legendre operator L on the interval *I*:= [-1*, *1] defined by:

acting on acting on functions of class* ** *The Hilbert space * *with respect to the probability measure* ** *The space* ** *admits orthogonal basis for the Legendre polynomials defined by the following generating series:

It is Known that the Legendre polynomials are eigenvectors for the operator −*L*:

In fact, the distribution *μ *is symmetrizing for *L *and the sequence forms the spectral decomposition of the minimal self-adjoint extention of this operator on* ** *With the help of an integration by parts, it is easily seen that

(4) |

where is the positive symmetric bilinear form defined by:

An important consequence of property (4) is:

which expresses the invariance of the measure *μ*.

By means of the above mentioned properties of the operator *L*, essentially the one concerning the symmetry with respect to *μ*, we deduce the existence of a semigroup of operator generated by *L *acting on* ** *by:

(5) |

and such that:

(1) is a contraction in all spaces

(2) is symmetric

(3) is positive and

According to (5),* ** *and* ** *is ergodic:

* *tends to *μ*-almost everywhere as* *

The commutation relation between the action of the operator *L *and the derivation is given as:

where is the operator associated to the family of Jacobi polynomials of second kind:

This commutation formula translates for the semigroup by:

(6) |

where designates the heat semigroup generated by . Notice that is symmetric (and so invariant) with respect to the probability measure The generator *L *satisfies the following dissipativity formula:

(7) |

*f*, *g* being sufficiently smooth on *I*. We emphasize that may be obtained as the projection of the Laplacian on the unit sphere* ** *and is obtained as the projection of the normalized Lebesgue measure on* ** *For* ** *let* ** *denote the domain of the generator *L *of In virtue of density of* ** *in* ** *we may extend formula (4) to* *

Our objective in this section is to establish a family of integral inequalities On which provide interpolation between the Sobolev and Poincaré inequalities. For* ** *we adopt the notation

Let* ** *be a strictly convex function such that* ** *we define the* *-entropy functional of* ** *by:

The quantity is always nonnegative since * *is invariant for the probability measure *μ*. By the ergodic property of the semigroup,

When* * coincides with the classical notion of variance,

When* *

In The sequel, we shall restrict ourself to the following class *C *of real functions* ** *mean that* ** ** *is strictly positive on and

Having in our disposal enough machinery, we are now ready to prove the following estimate of* *-entropy functional :

**Theorem. 1. ***Let** ** Then, for all function** ** **and** *

(8) |

Moreover, the numeric constant at the right hand side of inequality (8) is best. To illustrate this theorem, let analyze some practical applications. The most important examples of the class *C *in our mind are:

and

which corresponds to the limiting case of* ** *as* ** *If* ** *inequality (8), written for* ** *describes the Sobolev inequality: for all* ** *and for all functions* *

(9) |

For* ** *and* ** *inequality (8) is exactly the Sobolev Logarithmic. Replacing f positive by* *, we get

(10) |

Taking into account that

and using the fact that set of bounded functions in* ** *is dense in* ** *we can extend inequalities (9) and (10) to* ** *this last inequality (10) is equivalent to the hypercontractive estimate for the semigroup Whenever and* ** *satisfy* ** *then , for all functions

In other words,* ** *maps* ** *in* ** *with norm one.

*Proof. *By the Fubini theorem it follows from the definition of that for any* *

The last tow equalities follow from the dissipativity property (4), respectively. An integration by part over the time variables yields

Since

we get

Now,

Applying successively (4) and (7), the first integral in this sum is reduced to:

while the second integral is equal to:

Replacing *x* by and invoking again the dissipativity formula (7), the last member in the preceding sum becomes:

As a consequence, after reassembling the terms, we find:

(11) |

with

where we have posed* ** *The of* ** *then allows us to exhibit the desired inequality (8) from (11).

It remains to show that the numeric constant at the right hand side of inequality (8) is optimal. As usual, let us consider* ** *such that* ** *If f is replaced by* ** *in (8), and we pass to limit as* ** *tends to we easily recover the Poincaré inequality with best constant:

which completes the proof.

We close this paper by the following concluding remarks:

Of course letting* * inequality (8) in Theorem 1 gives rise to:

(12) |

Moreover, it’s easy to observe that (8) provides a smooth nonincreasing interpolation for inequality (12):

By (11), we point out at once that, if the equality holds in (8) if and only if f is constant. In particular, inequalities (9) and (10) do not admit nonconstant extremal functions.

**Corollary. 1****.**** ***Let** ** **Then for all nonnegative smooth function** *

(13) |

*Moreover*, *this inequality is optimal*.

*Proof. *We note in the sequel by *μ *the uniform measure on [0,1].

Let* ** *positive function. We consider the function *g** *defined on the interval [-1*, *1] by:

We can apply the theorem 1 to *g *for the measure *μ*, we obtain

(14) |

[1] | D.Bakry, M. Émery, Diffusions hypercontractives. Séminaire de probabilities de Strasbourg 19 (1985): 177-206. | ||

In article | |||

[2] | W. Beckner, A generalized Poincaré inequality for Gaussian measures. Proc. Am. Math. Soc. 1989; 105 (2): 397-400. | ||

In article | View Article | ||

[3] | A.Bentaleb,S. Fahlaoui, A.Hafidi Psi-entropy inequalities for the Ornstein-Uhlenbeck semigroup,Semigroup Forum, 2012; 85 (2): 361-368. | ||

In article | View Article | ||

[4] | A.Bentaleb,S.Fahlaoui, a family integral inequalities on the circle S1, Proc. Jpn. Acad., Ser. A 2010; 86: 55-59. | ||

In article | View Article | ||

[5] | F. Bolley, I. Gentil, Phi-entropy inequalities for diffusion semigroups. J. Math.Pures Appl. 2010; 93 (5): 449-473. | ||

In article | View Article | ||

[6] | J. Doulbeault, I. Gentil, and A. Jüngel, A logarithmic fourth-order parabolic equation and related logarithmic sobolev inequalitiesn Comm. Math. Sci. 2006; 4 (2): 275-290. | ||

In article | View Article | ||

[7] | L.Gross,logarithmic Sobolev inequalities, Amer.J.Math. 1975; 97 (4), 1061-1083. | ||

In article | View Article | ||

[8] | M. Ledoux, The geometry of Markov diffusion generators. Probability theory. Ann. Fac. Sci. Toulouse Math. 2000; 6(9)(2): 305-366. | ||

In article | View Article | ||

[9] | E. Nelson, The free Markov field. J. Funct. Anal. 1973; 12: 211-227. | ||

In article | View Article | ||

[10] | F-Y. Wang, A generalisation of Poincaré and logarithmic Sobolev inequalites, Potential Anal.22(2005) no 1, 1-15. | ||

In article | View Article | ||

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Khalid Boutahir, Ali Hafidi. Family of Functional Inequalities for the Uniform Measure. *Journal of Mathematical Sciences and Applications*. Vol. 5, No. 1, 2017, pp 19-23. http://pubs.sciepub.com/jmsa/5/1/3

Boutahir, Khalid, and Ali Hafidi. "Family of Functional Inequalities for the Uniform Measure." *Journal of Mathematical Sciences and Applications* 5.1 (2017): 19-23.

Boutahir, K. , & Hafidi, A. (2017). Family of Functional Inequalities for the Uniform Measure. *Journal of Mathematical Sciences and Applications*, *5*(1), 19-23.

Boutahir, Khalid, and Ali Hafidi. "Family of Functional Inequalities for the Uniform Measure." *Journal of Mathematical Sciences and Applications* 5, no. 1 (2017): 19-23.

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[1] | D.Bakry, M. Émery, Diffusions hypercontractives. Séminaire de probabilities de Strasbourg 19 (1985): 177-206. | ||

In article | |||

[2] | W. Beckner, A generalized Poincaré inequality for Gaussian measures. Proc. Am. Math. Soc. 1989; 105 (2): 397-400. | ||

In article | View Article | ||

[3] | A.Bentaleb,S. Fahlaoui, A.Hafidi Psi-entropy inequalities for the Ornstein-Uhlenbeck semigroup,Semigroup Forum, 2012; 85 (2): 361-368. | ||

In article | View Article | ||

[4] | A.Bentaleb,S.Fahlaoui, a family integral inequalities on the circle S1, Proc. Jpn. Acad., Ser. A 2010; 86: 55-59. | ||

In article | View Article | ||

[5] | F. Bolley, I. Gentil, Phi-entropy inequalities for diffusion semigroups. J. Math.Pures Appl. 2010; 93 (5): 449-473. | ||

In article | View Article | ||

[6] | J. Doulbeault, I. Gentil, and A. Jüngel, A logarithmic fourth-order parabolic equation and related logarithmic sobolev inequalitiesn Comm. Math. Sci. 2006; 4 (2): 275-290. | ||

In article | View Article | ||

[7] | L.Gross,logarithmic Sobolev inequalities, Amer.J.Math. 1975; 97 (4), 1061-1083. | ||

In article | View Article | ||

[8] | M. Ledoux, The geometry of Markov diffusion generators. Probability theory. Ann. Fac. Sci. Toulouse Math. 2000; 6(9)(2): 305-366. | ||

In article | View Article | ||

[9] | E. Nelson, The free Markov field. J. Funct. Anal. 1973; 12: 211-227. | ||

In article | View Article | ||

[10] | F-Y. Wang, A generalisation of Poincaré and logarithmic Sobolev inequalites, Potential Anal.22(2005) no 1, 1-15. | ||

In article | View Article | ||