Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

Family of Functional Inequalities for the Uniform Measure

Khalid Boutahir, Ali Hafidi*

Faculté des Sciences et Techniques, B.P.509, Boutalamine Errachidia, MAROC

Abstract: 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.

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

Citation: Khalid Boutahir, Ali Hafidi, Family of Functional Inequalities for the Uniform Measure. Journal of Mathematical Sciences and Applications, 5 (1). 19-23. January 2017. doi:10.12691/jmsa-5-1-3

1. Introduction

Gross’ logarithmic Sobolev inequality [2] states that for all smooth functions f on

[picture] | (1)

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

[picture] | (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:

[picture] | (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).

2. Preliminaries

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

[picture] | (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:

[picture] | (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:

[picture] | (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:

[picture] | (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

3. The Main Result

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

[picture] | (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

[picture] | (9)

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

[picture] | (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:

[picture] | (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:

[picture] | (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

[picture] | (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

[picture] | (14)

References

[1]   | D.Bakry, M. Émery, Diffusions hypercontractives. Séminaire de probabilities de Strasbourg 19 (1985): 177-206.
[2]   | W. Beckner, A generalized Poincaré inequality for Gaussian measures. Proc. Am. Math. Soc. 1989; 105 (2): 397-400.
[3]   | A.Bentaleb,S. Fahlaoui, A.Hafidi Psi-entropy inequalities for the Ornstein-Uhlenbeck semigroup,Semigroup Forum, 2012; 85 (2): 361-368.
[4]   | A.Bentaleb,S.Fahlaoui, a family integral inequalities on the circle S1, Proc. Jpn. Acad., Ser. A 2010; 86: 55-59.
[5]   | F. Bolley, I. Gentil, Phi-entropy inequalities for diffusion semigroups. J. Math.Pures Appl. 2010; 93 (5): 449-473.
[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.
[7]   | L.Gross,logarithmic Sobolev inequalities, Amer.J.Math. 1975; 97 (4), 1061-1083.
[8]   | M. Ledoux, The geometry of Markov diffusion generators. Probability theory. Ann. Fac. Sci. Toulouse Math. 2000; 6(9)(2): 305-366.
[9]   | E. Nelson, The free Markov field. J. Funct. Anal. 1973; 12: 211-227.
[10]   | F-Y. Wang, A generalisation of Poincaré and logarithmic Sobolev inequalites, Potential Anal.22(2005) no 1, 1-15.