Except for one basic lemma the development is similar to the one dimensional case. The result follows from the same reasoning as the theorem concerningx. Such inequalities contain almost all known norm inequalities for the fourier transform. Sharp weighted sobolev and gagliardonirenberg inequalities on. We show that if a positive absolutely continous measure causes a special relative isoperimetric inequality to hold, then dirichlettype integrals of suffic. Lieb abstract the wellknown duality between the sobolev inequality and the hardylittlewoodsobolev inequality suggests that the nash inequality could also have an interesting dual form, even though. They have some very interesting properties and numerous applications. Kinyin li, rearrangement inequality, mathematical excalibur, volume 4, number 3, january, 1999 march, 1999 yue kwok choy, rearrangement inequality samin riasat, basics of olympiad inequalities. The lorentz spaces, introduced by lorentz in 1, 2, are no exception to this. We present in this paper a necessary and sufficient condition to establish the inequality between generalized weighted means which share the same sequence of numbers but differ in the weights. Many important inequalities can be proved by the rearrangement inequality, such as the arithmetic mean geometric mean inequality, the cauchyschwarz inequality, and chebyshevs sum inequality. The reverse rearrangement inequality allows us to compare the product of sums of terms in an inequality.
We show that if a positive absolutely continous measure causes a special relative isoperimetric inequality to hold, then dirichlettype integrals of sufficiently smooth realvalued functions decrease under an appropriate equimeasurable rearrangement. On a weighted trudingermoser inequality in rn sciencedirect. The rearrangement inequality or the permutation inequality is an elementary inequality and at the same time a powerful inequality. It has an uncanny resemblance to the famous rearrangement inequality, which is about the sum of product of terms, hence its namesake. Pdf a weighted isoperimetric inequality in an orthant. A weighted version of a rearrangement inequality springerlink. Aweighted isoperimetric inequality and applications to symmetrization. A necessary and sufficient condition for the inequality of.
There has been a growing interest in isoperimetric inequalities with weights. Nguyen, a sharp adams inequality in dimension four and its extremal functions, arxiv. Several weighted rearrangement inequalities for uncentered and centered local sharp functions are proved. The rearrangement inequality is a statement about the pairwise products of two sequences. Sharp trudingermoser inequalities with monomial weights.
New p\olyaszeg\ otype inequalities and an alternative approach. Pdf weighted rearrangement inequalities for local sharp. Talenti, a weighted version of a rearrangement inequality, ann. The extremal functions in the weighted sobolev inequalities are fully. Two sequences that are both increasing or both decreasing are said to be similarly ordered. The first version of the weighted lorentz spaces was provided by lorentz himself and was defined as, where is. In particular, weighted norm inequalities for the fourier, hankel, and jacobi transforms are.
Rearrangement inequality yue kwok choy the rearrangement inequality also known as permutation inequality is easy to understand and yet a powerful tool to handle inequality problems. If one is increasing and the other is decreasing they are said to be inversely or, oppositely ordered. We also used rearrangement inequality, chebyshevs sum in equality and a weighted holders inequality to generalize our results to the case. Weighted sobolev norm symmetric decreasing rearrangement. I and a version of 4 which may be called the bilinear hardy inequality for nonincreasing functions and written in the form 2 4 z1 0 zt 0 f sds zt 0 g sds q wt t2q dt 3 5 1 q c z1 0 f p1v 1 1 p1 z1 0 g 2v 2 p2. The weighted version of the amgm inequality follows from the original amgm inequality. Our results were motivated by an inequality, involving harmonic means, found in. However, these estimates do not indicate where the support of tfis located, due to the rearrangementinvariant nature of the lp norms.
In our work we have found that ideas and methods from one. In the following, we are interested in the weighted version of the. Weighted l p, l q inequalities are studied for a variety of integral transforms of fourier type. I and a inequality to hold, then dirichlettype integrals of sufficiently smooth realvalued functions decrease under an appropriate equimeasurable rearrangement. The lower bound follows by applying the upper bound to. Find, read and cite all the research you need on researchgate. By adapting the mass transportation technique of corderoerausquin, nazaret and villani, we obtain a family of sharp sobolev and gagliardonirenberg gn inequalities on the half space r. Pdf we prove a weighted version of the wellknown khintchine inequality for rearrangement invariant norms. The first version of the weighted lorentz spaces was provided by lorentz himself and was defined as, where is the decreasing rearrangement of and is a weight function. Omc 2011 rearrangement inequality lecture 23 clet a x 1 3 y 1 3 z 2 3, b x 3z y 2 3, and c y 1 3 z 1 3 x 2 3. The rearrangement inequality states that, for two sequences. An interpolation of hardy inequality and trundingermoser.
Some scales of equivalent weighted characterizations of hardys inequality. Atomic decomposition of weighted lorentz spaces and operators. But it is worth approaching an inequality rather than solving it. A selfimproving property of sharp function inequalities is established. Pdf some new conditions for weighted fourier inequality. The case q weighted muckenhoupt and gehring type inequalities volume 62 issue 4 s. Wu south china normal university, china andy liu university of alberta, canada we will introduce our subject via an example, taken from a chinese competition in 1978. A dual form of the sharp nash inequality and its weighted generalization eric a. For all these, besides giving the proof, we present several examples that show how to use them in mathematical olympiad problems. A notable difference from the rearrangement inequality is that the variables are now required to be non. These results are applied to obtain new weighted weaktype and strongtype estimates for singular integrals. By adapting the mass transportation technique of corderoerausquin, nazaret and villani, we obtain a family of sharp sobolev and gagliardonirenberg gn inequalities on the half space r n.
Fourier transform maps a given rearrangement invariant banach space into another given space of that type. Sharp isoperimetric inequalities via the abp method uzh. Among other things, beckners paper records the the optimal constant cp in 4 for p. Also it is known that the exact growth trudingermoser inequality 1. More general versions of the inequality compare an elliptic. Weighted spaces are studied in most cases as a generalization of a special case. Weighted norm inequalities for integral transforms. We thus see that weighted estimates can be quite powerful. Extrapolation of weighted norm inequalities, rearrangement invariant function spaces, modular inequalities, maximal functions, singular integrals, fractional integrals. Lecture 23 rearrangement inequality holden lee 6411 1 the inequalities we start with an example. A weighted version of a rearrangement inequality, ann.
A weighted l p logarithmic sobolev inequality is derived from these inequalities. More precisely, for suitable values of the parameters we will. Takahashi, on a weighted trudingermoser type inequality on the whole space and its nonexistence of maximizers, differential integral equations, 31 2018, 785806. It can be extended to chebyshevs inequality, and illustrates the practical power of greedy algorithms. Talentia weighted version of a rearrangement inequality. Weighted inequalities and uncertainty principles 3 and 5 are correct, as observed by beckner in section iv. One way to establish this type of control is to start establishing. In this paper, we will study the trudingermoser inequalities with the monomial weight equation in equation with equation.
The ex tremal functions in the weighted sobolev inequalities are fully characterized. Kubiaczyk skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. Vectorvalued inequalities are obtained automatically, without the need of a banachvalued theory. Talenti, a weighted version of a rearrangement inequality.
The rearrangement inequality states that, if is a permutation of a finite set in fact, multiset of real numbers and is a permutation of another finite set of real numbers, the quantity is maximized when and are similarly sorted that is, if is greater than or equal to exactly of the other members of, then is also greater than or equal to exactly of the other members of. We establish the trudingermoser inequality on weighted sobolev spaces in the whole space, and for a class of quasilinear elliptic operators in radial form of the type l u. Finally, it is shown in 8 that weighted cesaro and abel mean summability. Generalization of a class of cyclic inequalities 1. I am looking for a proof of weighted version of the inequality as following. These results are applied to obtain new weighted weaktype and strongtype estimates for. A dual form of the sharp nash inequality and its weighted. One can use the technique of summation by parts for the proof of the rightmost inequality.
Notice that c 1 is the least constant cfor which the above inequality holds for all f. Hence, we dont know whether it is possible to perform the argument using the re. On attainability of mosertrudinger inequality with. You may take 2 bills from one box, 3 bills from another, 4 bills from another, and 5 bills from the remaining box. Our results were motivated by an inequality, involving harmonic means, found in the study. Steinerand intensively used in investigations about isoperimetric inequalities of mathematical physics. Inequalities are often hard to solve, and it is not always possible to nd a nice solution. Aug, 2004 on the isoperimetric nature of a rearrangement inequality and its consequences for some. We first present a sufficient condition, and then obtain the more general, necessary and sufficient, condition. Weighted inequalities and uncertainty principles for the k,a. Moreover, we investigate the trudingermoser inequalities on.
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