Pdf logarithmic sobolev and poincare inequalities for the. We derive the mosertrudingeronofri inequalities on the 2sphere and the 4sphere as the limiting cases of the fractional power sobolev inequalities on the same spaces, and justify our approach as the dimensional continuation argument initiated by thomas p. A capacity approach to the poincare inequality and sobolev. On improved fractional sobolevpoincare inequalities. S p sobolev spaces, poincare inequality, embedding theorems, metric spaces, doubling measures, spaces of homogeneous type, upper. On poincare and logarithmic sobolev inequalities for a class.
May 23, 2017 the rate of change of the sharp constant in the sobolevpoincare or friedrichs inequality is estimated for a euclidean domain that moves outward. Rn can be characterized in terms of the generalized. Subelliptic global high order poincare inequalities in stratified lie. Sobolev inequality of uniform norm proof explanation. Sobolev spaces department of mathematics and systems analysis. A generalization of the classical sobolev space w1,2rn is the newtonian space n1,2x for a metricmeasure spacex. Regarding poincar e inequalities, a combination of the poincar e inequality. The sobolev space is a vector space of functions that have weak derivatives. If n sobolev embedding is a direct consequence of morreys inequality. A consequence of the sobolev type inequality is a partial con. How to get rid of one of the weights in a twoweight poincare. On improved fractional sobolev poincare inequalities. Characterizations of orliczsobolev spaces in terms of. We prove the equivalence of the corresponding weak and strong type inequalities.
In the next section, the analogous questions for discrete gradients are. In this section, we establish several sobolev inequalities in the hyperbolic spaces h n. Discrete sobolevpoincare inequalities 1ex for voronoi. It is possible to achieve this also for variable exponent sobolev spaces, as we prove in. This different behavior of the poincare and log sobolev constant was also observed in the setting of metastability in 2, remark 2. Our characterization of this inequality for p6 2, which is of substantial interest, comes as a relatively straightforward corollary of our main results for the elliptic equations. By continuing to use our website, you are agreeing to our use of cookies. Extremal functions in poincare sobolev inequalities for functions of bounded variation vincent bouchez and jean van schaftingen dedicated to jeanpierre gossez, on the occasion of his 65th birthday arxiv. The case p poincare inequalities, sobolev inequalities, metric spaces, doubling measures. Hardy and poincare inequalities in fractional orliczsobolev. Classical isoperimetric inequality and its applications to integral inequalities 3. Recall that i satis es a poincar e inequality if the energy of any centered function is controlled by the energy of its derivative. Sobolev inequality, poincare inequality, fractional norms, weighted sobolev spaces, john domains, sjohn domains, cusp domains.
On poincare and logarithmic sobolev inequalities for a. Characterizations of orliczsobolev spaces by means of. An effective method to prove poincare or logarithmic sobolev inequalities is the bakryemery curvaturedimension criterion 1. In classical theory the constant of the poincare inequality is c diamd. Theorem 1 gagliardonirenberg sobolev inequality assume 1 p sobolev conjugate of p. Franc 11 i, weighted sobolev poincare inequalities and pointwise ineqzalitzes for a class of degenerate elliptic equations, trans. Sharp sobolev inequalities on the sphere and the mosertrudinger inequality.
Assuming that a given probability measure gives rise to a lsi, we derive generalized poincar. We show that, in contrast with the spectral gap, for bounded degree expander graphs, various log sobolev constants go to zero with the size of the graph. Sobolev inequality and sobolev embeddig theorems theorem 1 sobolev s embedding theorem. Lectures on isoperimetric and isocapacitary inequalities. These inequalities turn out to be weaker than the standard log sobolev inequality, but stronger than the poincare spectral gap inequality. These inequalities generalize the results of mugelli and talenti in h 2 to higher dimensional spaces. We show that, in contrast with the spectral gap, for bounded degree expander graphs, various log sobolev constants go to. All of the previous examples share the common feature of exhibiting a selfimproving property. Wheeden, weighted sobolev poincare inequalities for gru,shin type operators, comm.
Conversely, a parabolic harnack inequality implies a version of the sobolev inequality as shown by salo. We proof the eyringkramers formula for the optimal constant in the poincare pi and logarithmic sobolev inequality lsi for the associated. Lectures on isoperimetric and isocapacitary inequalities in. Poincaretype inequalities for broken sobolev spaces. We study a certain improved fractional sobolev poincare inequality on do.
Jul 06, 2020 we then establish a comparison procedure with the well studied random transposition model in order to obtain the corresponding functional inequalities. Sobolev and isoperimetric inequalities for degenerate. Discrete sobolev poincare inequalities omgdmv congress, september 21, 2009 8 20 discrete poincare inequality lemma 1. Hot network questions reading an old philips radio manual with scr rows. This paper is concerned with intermediate inequalities which interpolate between the logarithmic sobolev lsi and the poincar. Targeting sobolev inequalities and their purposes to research on manifolds and ricci circulate, sobolev inequalities, warmth kernels below ricci circulate, and the poincare conjecture introduces the sphere of study on riemann manifolds and makes use of the. It is classical for the poincare inequality and for logarithmic sobolev in equality as in 8. Introduction it is known that the classical sobolev poincar. Intuitively, this inclusion expresses the fact that the existence of sufficiently many weak derivatives implies some continuity. Capacity minimizing functions and their applications to sobolev. Recall that the standard logarithmic sobolev inequality is of the form.
Pdf extremal functions in poincaresobolev inequalities for. We also show that for many metric spaces the sobolev functions can be identified with functions satisfying poincare inequalities, and this allows us to extend to the metric space setting the fact. There are two other important functional inequalities giving gaussian concentration. Existence and size estimates of greens function for certain differential operators constructed from degenerate vector fields, pdf. The key ingredients are a hadamard variation formula and an inequality that reverses the usual holder inequality. For sobolev functions with zero boundary values the poincar. We begin with a poincare inequality for sobolev functions with zero boundary values in open subsets. Some definitionsthroughout this article x will be a metric space equipped with a regular, borel, doubling measure of dimension d. Characterizations of orlicz sobolev spaces in terms of generalized orliczpoincare inequalities toni heikkinen. Below we give a shorter proof than the above proof.
The proof of the harnack inequality by means of the moser technique can be reduced to verifying a suitable sobolev inequality. Lasis and endre s\uli, year2003 we present two versions of general poincare type inequalities for functions in broken sobolev spaces, providing bounds for the lqnorm of a. By the h older inequality we get 1 jbj z b jf f bjpdx 1 jbj jf f bjp pn pnp z b. We provide sufficient conditions for boundary hardy inequality to hold in bounded lipschitz domains, complement of a point the so called point hardy inequality, domain above the graph of a lipschitz function, complement of a bounded lipschitz domain in fractional orlicz sobolev setting. The sharp poincaresobolev type inequalities in the. While our method falls into a rich class of comparison techniques for markov chains on different state spaces, the crucial feature of the method dealing with chains with a large distortion. This note is devoted to the study of fractional order poincares inequalities on. Poincare and sobolev type inequalities for complex m. From concentration to logarithmic sobolev and poincare. Logarithmic sobolev and poincare inequalities for the. In the special case of n 1, the nash inequality can be extended to the l p case, in which case it is a generalization of the gagliardonirenberg sobolev inequality brezis 2011, comments on chapter 8.
The question is the connection between sobolev inequalities and the classical isoperimetrie inequality for subsets of euclidean spaces. Complex hessian operator, msubharmonic function, integrability, poincar. On poincare and logarithmic sobolev inequalities for a class of singular gibbs measures. Wesaythatu2h2 1 m, u6 0, is an extremal function for 1. On the rate of change of the sharp constant in the sobolev. Poincare sobolev inequalities, muckenhoupt weights. In section 4, we consider a further generalization of 1. Limit of sobolev inequality sunyung alice chang and fang wang abstract. On poincare and logarithmic sobolev inequalities for a class of singular gibbs measures djalil chafai, joseph lehec to cite this version.
Sobolev type inequality for functions with unrestricted boundary values 4. Jun 15, 2018 other sobolev inequalities in the hyperbolic spaces. This is proved along with the characterization in 15. From affine poincare inequalities to affine spectral inequalities.
Apr, 2017 sobolev inequalities, heat kernels under ricci flow, and the poincare conjecture. Sobolevtype spaces from generalized poincare inequalities. At the same time, as for scalar equations, sobolev and poincare inequalities. This result is a direct extension of the p 2 case already studied in mv02a. Download sobolev inequalities, heat kernels under ricci flow. On improved fractional sobolevpoincare inequalities project euclid. We prove a high order poincare inequality for f minus a polynomial of order m. Oct 30, 2015 we study a certain improved fractional sobolev poincare inequality on domains, which can be considered as a fractional counterpart of the classical sobolev poincare inequality.
Translation of sobolev function on strongly included sets. Poincare inequalities 5 of a sobolev function uis, up to a dimensional constant, the minimal that can be inserted to the poincar e inequality. A note on poincare, sobolev, and harnack inequalities, international mathematics research notices, volume 1992, issue 2, 1 january 1992, pages we use cookies to enhance your experience on our website. Weighted poincare and sobolev inequalities for vector fields satisfying hormanders condition and applications, pdf revista matematica iberoamericana, vol. The second one is due to gross and, in view of theorem 16, implies the concentration inequality for. Poincare inequality an overview sciencedirect topics. We examine in this way concentration properties for the product measure of the ex ponential distribution, as well as, more generally, of measures satisfying a poincar. This condition is rather mild it is much weaker than a rh.
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