The explanation for this can be sought by viewing the r as a kind of weighting of a and b. Weighted norm inequalities for the maximal operator on. Journal of inequalities in pure and applied mathematics. 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. Introduction to inequalities chennai mathematical institute. These results are applied to obtain new weighted weaktype and strongtype estimates for singular integrals. Inequalities are often hard to solve, and it is not always possible to nd a nice solution. Weighted inequalities and uncertainty principles 3 and 5 are correct, as observed by beckner in section iv. Weighted fourier inequalities via rearrangements springerlink. Ams transactions of the american mathematical society.
I and a inequality between arithmetic and geometric mean using the rearrangement inequality. 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. Further, equality holds if and only if every number in the list is the. On a cyclic inequality with exponents and permutations, and its shapirotype analogue. An important role in our analysis is played by a weighted rearrangement inequality, relating any measurable function and its johnstr omberg maximal function. Pdf we prove a weighted version of the wellknown khintchine inequality for rearrangement invariant norms. However, these estimates do not indicate where the support of tfis located, due to the rearrangement invariant nature of the lp norms.
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. Let a 1 a 2 a n and b 1 b 2 b n be two similarly sorted sequences. Sharp weighted trudingermoser inequalities with the l n. Because the lhs of 1 is continuous in k, we can conclude the following. Bilinear weighted hardy inequality for nonincreasing functions 5 if f2m, then f denotes its nonincreasing rearrangement and f the hardylittlewood maximal function of f, i. Using the finite form of jensens inequality for the natural logarithm, we can prove the inequality between the weighted arithmetic mean and the weighted geometric mean stated above. The rearrangement inequality is best proved for two elements, and then.
Generalization of a class of cyclic inequalities 1 introduction hikari. 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. A selfimproving property of sharp function inequalities is established. Among other things, beckners paper records the the optimal constant cp in 4 for p. Jun, 2019 pearson is the worlds learning company, with presence across 70 countries worldwide. Weighted norm inequalities for the maximal operator on variable lebesgue spaces david v. We establish the trudingermoser inequality on weighted sobolev spaces in the whole space. Finally, the paper ends with a weighted rearrangement inequality related to a fourthorder boundary value problem. We thus see that weighted estimates can be quite powerful. Pdf weighted fourier inequalities via rearrangements. We present an extrapolation theory that allows us to obtain, from weighted lp inequalities on pairs of functions for p. In section 3 we obtain the nikolskii and schurtype inequalities in weighted rearrangement invariant spaces for algebraic and trigonometric.
A rearrangement inequality and the permutahedron a. The weighted version of the amgm inequality follows from the original amgm inequality. On a weighted trudingermoser inequality in rn sciencedirect. This inequality was proved earlier by the author in the doubling case. The remeztype inequalities for trigonometric polynomials theorem 2. Applications of a weighted symmetrization inequality to. Some new conditions for weighted fourier inequality. Sharp weighted norm inequalities for littlewoodpaley operators and singular integrals.
Weighted isoperimetric inequalities in cones and applications 3 it was shown in 26 that the isoperimetric set for measures of the type ykdxdy, with k. The amgm, gmhm and amhm inequalities are partic ular cases of a more general kind of inequality called power means inequality. Borell proved that balls centered at the origin are isoperimetric. Such inequalities contain almost all known norm inequalities for the fourier transform. To the best of the authors knowledge, these higher summability results have not been considered before. Extrapolation with weights, rearrangementinvariant. Weighted sobolev norm symmetric decreasing rearrangement.
New results in the case q weighted inequalities for the fractional. Here, we will study the symmetric decreasing rearrangement, which replaces a given nonnegative function f by a radial function f 1. Weighted inequalities and uncertainty principles for the k,a. Let a1, an, b1, bn be two sequences of real numbers. 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. Weighted norm inequalities for integral transforms d. You may take 2 bills from one box, 3 bills from another, 4 bills from another, and 5 bills from the remaining box. Lq inequalities are studied for a variety of integral transforms of fourier type. We also used rearrangement inequality, chebyshevs sum in equality and a weighted holders inequality to generalize our results to the case. Polynomial inequalities on measurable sets and their. Inequalities for averages of convex and superquadratic functions. We will prove it for \ n4 \, and from there it will be clear how one can generalize the method. A weighted l p logarithmic sobolev inequality is derived from these inequalities. Find, read and cite all the research you need on researchgate.
It also follows by the induction that equality holds for a. A selfimproving property of sharp function inequalities. Using the finite form of jensens inequality for the natural logarithm, we can prove the inequality between the weighted arithmetic mean and the weighted. We give inequalities relating the norm of a function and the norm of its average operators p. For all these, besides giving the proof, we present several examples that show how to use them in mathematical olympiad problems. If all w k 1, this reduces to the above inequality of arithmetic and geometric means.
Inequality of arithmetic and geometric means wikipedia. We use pitt inequalities for the fourier transform to prove. The method of using rearrangements to give sufficient conditions for fourier inequalities between weighted lebesgue spaces is revisited, a comparison between two known sufficient conditions is completed, and the method is extended to provide sufficient conditions for a new range of indices. On weighted isoperimetric inequalities with nonradial.
Steinerand intensively used in investigations about isoperimetric inequalities of mathematical physics. Mildorf december 22, 2005 it is the purpose of this document to familiarize the reader with a wide range of theorems and techniques that can be used to solve inequalities of the variety typically appearing on mathematical olympiads or other elementary proof contests. But it is worth approaching an inequality rather than solving it. Pdf weighted rearrangement inequalities for local sharp.
A weighted version of a rearrangement inequality springerlink. A necessary and sufficient condition for the inequality of. In this paper we give some new conditions on weights and for weighted fourier transform norm inequality. Arithmetic mean, geometric mean, harmonic mean inequalities.
Vectorvalued inequalities are obtained automatically, without the need of a banachvalued theory. Vince department of mathematics, university of florida, gainesville, fl 32611 one chapter of the classic book inequalities by hardy, littlewood, and polya 3 is dedicated to inequalities involving sequences with terms rearranged. In this article we obtain improved versions of steinweiss and ca. The lower bound follows by applying the upper bound to. The method of using rearrangements to give sufficient conditions for fourier inequalities between weighted lebesgue spaces is revisited. Higher summability and discrete weighted muckenhoupt and. To prove the lhs, apply the above with y i instead of y i. Oct 16, 2017 the method of using rearrangements to give sufficient conditions for fourier inequalities between weighted lebesgue spaces is revisited, a comparison between two known sufficient conditions is completed, and the method is extended to provide sufficient conditions for a new range of indices. In particular, weighted norm inequalities for the fourier, hankel, and jacobi transforms are derived from calder on type rearrangement estimates.
The rearrangement inequality or the permutation inequality is an elementary inequality and at the same time a powerful inequality. Higher summability and discrete weighted muckenhoupt and gehring type inequalities volume 62 issue 4 s. Our unique insights and worldclass expertise comes from a long history of working closely with renowned teachers, authors and thought leaders, as a result of which, we have emerged as the preferred choice for millions of teachers and learners across the world. Most proofs in the literature of existence of extremals for the trudingermoser inequalities on the whole space rely on finding a radially maximizing sequence through the symmetry and rearrangement technique. Aweighted isoperimetric inequality and applications to symmetrization. Fourier transform maps a given rearrangement invariant banach space into another given space of that type. Introduction weighted rearrangement invariant spaces play an important role in contemporary mathematics. It has an uncanny resemblance to the famous rearrangement inequality, which is about the sum of product of terms, hence its namesake. The arithmetic meangeometric mean amgm inequality states that the arithmetic mean of nonnegative real numbers is greater than or equal to the geometric mean of the same list. These results are applied to obtain new weighted weaktype and strongtype estimates for.
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 rearrangement inequality is a statement about the pairwise products of two sequences. The reverse rearrangement inequality allows us to compare the product of sums of terms in an inequality. A rearrangement inequality and the permutahedron people. 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. Lecture 23 rearrangement inequality holden lee 6411 1 the inequalities we start with an example. In 1,see also 2, the following weighted symmetrization inequality is proved. Introduction this paper continues the study of the rearrangement inequalities in terms of sharp maximal functions bk, bds, l1, l2. We also emphasize how the substitution strategy is used to deduce several inequalities. Cauchys inequality is equivalent to the triangle inequality for the 2norm.
The extremal functions in the weighted sobolev inequalities are fully. Pdf on hardys inequality in weighted rearrangement. Publicationspreprints of andrei lerner title coauthors. Weighted norm inequalities for the maximal operator on variable lebesgue spaces. Several weighted rearrangement inequalities for uncentered and centered local sharp functions are proved. You may think of convexity as meaning that the line segment joining two points of the graph of fis always greater than the graph itself. Weight function, weighted fourier transform norm inequality, weighted lebesgue spaces, hardy inequality, decreasing rearrangement. 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.
Hence, we dont know whether it is possible to perform the argument using the re. A notable difference from the rearrangement inequality is that the variables are now required to be non. Reduction of opialtype inequalities to norm inequalities. This integral inequality was communicated by chebyshev to the french mathemati cian, charles hermite 18221901 who then included it, with the extra assumption that u. The proofs are obtained by employing new discrete weighted hardys type inequalities and their converses for nonincreasing sequences, which, for completeness, we prove in our special setting. Pdf some new conditions for weighted fourier inequality. In this paper, we mainly concern with the sharp weighted trudingermoser inequalities with the l n norm on the whole space see theorem 1. I am wondering if by replacing the lebesgue measure in the definition of symmetricdecreasing function with a weighted lebesgue measure, riesz rearrangement inequality still. The space lp maximal operatorweighted lp application weighted norm inequalities for the maximal operator on variable lebesgue spaces david v. They have some very interesting properties and numerous applications. Thus, when we talk of an inequality, we have the following in mind. They have many applications in various branches of analysis including the theory of function spaces, interpolation theory, mathematical. Since it is reasonable to hold that if china becomes richer, this event should have more impact on the world than if mauritania were to become so, we come to the second type of inequality concept 2.
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