Rearrangement inequalities for functionals with monotone. We prove that c is the largest root of the quadratic equation 12c2. Mean values with an arbitrary function and the theory of convex functions. Symmetrisations and other rearrangement inequalities. Thus the averages arf are uniformly bounded in size as r varies. Hardy and john edensor littlewood, states that if f and g are nonnegative measurable real functions vanishing at infinity that are defined on ndimensional euclidean space r n then. One chapter of the classic book inequalities by hardy, littlewood, and polya.
A note on hardylittlewood maximal operators journal of. In this paper we prove a rearrangement inequality that generalizes in equalities given in the book by hardy, littlewood and polyai and by luttinger and friedberg. For the classical hardylittlewood maximal function mf, a well known and important estimate due to herz and stein, gives the equivalence mf t. A general rearrangement inequality for multiple integrals, j. From schurs test or youngs inequality we know that these ar are contractions on every lprd, 1. The inequality for an integral of a product of functions of one variable is further extended to the case of. On an inequality of hardy, littlewood and polya by hoffman, a. Sharp constants in the hardylittlewoodsobolev and related. Weighted hardylittlewoodsobolev inequalities on the. Sharp hardylittlewoodsobolev inequalities on quaternionic. For the classical hardylittlewood maximal operator m. It follows that it is enough to prove the theorem for simple functions u and u. The best constant for the centered hardylittlewood maximal inequality by antonios d.
It seems to be 2n instead of 6n, but im not sure and at least hard to prove. As far as the rightmost inequality is concerned, there is a vast literature on the subject. Abstract the inequalities of hardylittlewood and riesz say that certain integrals involving. On a generalization of an inequality of hardy, littlewood, and polya1 milton sobel two sets of n real numbers each are given. For certain exponent, we classify all extremal functions via the method of moving sphere, and compute the best constants for the sharp inequality. There is a direct and selfcontained proof of hls inequality in analysis by lieb and loss, theorem 4. A sharp rearrangement inequality for the fractional maximal operator. Welcome,you are looking at books for reading, the inequalities, you will able to read or download in pdf or epub books and notice some of author may have lock the live reading for some of country. A hardylittlewood inequality 283 tions, and its rearrangement ux is the limit of the rearrangements u,x, a consequence of the fact that ulx s uzx implies urx 5 urx. Sharp hardylittlewoodsobolev inequality on the upper. A rearrangement inequality and the permutahedron people. Polya in the last chapter of their book inequalities. Its main implication is that symmetric decreasing rearrangement decreases l2distances 3.
A general, albeit simple, approach to obtaining rearrangement inequalities of. Its main implication is that rearrangement decreases l2distances 25. First published in 1934, it presents both the statement and proof of all the standard inequalities of analysis. Congming li, john villavert department of applied mathematics university of colorado at boulder, boulder, co, usa 80309. Hardylittlewood maximal function denote the average of f on a by h a f. Inequalities 1 and 2 contain the basic information to study m, and the operators it controls, in rearrangement invariant function spaces. It uses nothing but layer cake representation, holders inequality, and clever manipulation of integrals. The hardylittlewood inequality is a very basic inequality that holds, with suitably defined rearrangements, on arbitrary measure spaces 4. Jan 20, 2016 the basic realvariable construct was introduced by hardy and littlewood for \n1\, and by wiener for \n\ge2\. Inequalities arithmetic mean geometric mean inequal. Extended hardylittlewood inequalities and applications to the calculus of variations cardaliaguet, pierre and tahraoui, rabah, advances in differential equations, 2000. The existence of extremal functions, called extremizers, was first proved by lieb in, combining the riesz rearrangement inequality, refined fatou lemma and. On the other hand, the hardy littlewoodpolya hlp inequality1, inequality 381, p. Mar 18, 2018 on an inequality of hardy, littlewood and polya.
A short course on rearrangement inequalities almut burchard june 2009 these notes grew out of introductory courses for graduate students that i gave at the first. A strengthened version of the hardylittlewood inequality cianchi 2008 journal of the london mathematical society wiley online library. Rupert frank california institute of technology, pasadena, usa diogo oliveira e silva university of bonn, germany christoph thiele university of bonn, germany supported by hausdor center for mathematics, bonn. Hardylittlewoodsobolev inequalities via fast diffusion. To obtain the best constant in the weighted hardylittlewoodsobolev whls inequality 2, one can maximize the functional jf. The theory of rearrangement inequalities is welldeveloped in rn. In contrast, the riesz rearrangement inequal ity is speci. It is well known that the hardy littlewood maximal function plays an important role in many parts of analysis. Note that some of the problems can be solved by di. These numbers will be regarded as distinct although two or more may be equal in value.
Hardys inequality on hardy spaces ho, kwokpun, proceedings of the japan academy, series a, mathematical sciences, 2016. Sharp inequalities in harmonic analysis summer school, kopp august 30th september 4th, 2015 organizers. A strengthened version of the hardylittlewood inequality. Kolm 1969 was the very rst one, followed by dasgupta, sen and starrett 1973, to point out the relevance of this result in establishing the foundations of inequality measurement. The hardy, littlewood and polyas theorem is the key mathematical result in the area of inequality measurement. 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. Increasing rearrangement and hardylittlewood inequality. For the classical hardylittlewood maximal function mf,awell known and important estimate due to herz and stein gives the equivalence mf t. If it available for your country it will shown as book reader and user fully subscribe will benefit by having full. This is a study of the inequalities used throughout mathematics. Rearrangement inequalities for functionals with monotone integrands almut burchard.
General rearrangement inequality rims, kyoto university. For certain exponent, we classify all extremal functions via the method of moving sphere, and compute the. Recently, frank and lieb see 16 have given a new and rearrangementfree proof of this inequality. Let a 1 a 2 a n and b 1 b 2 b n be two similarly sorted sequences. Sharp constants in the hardy littlewood sobolev and related inequalities. Rearrangement inequalities were studied by hardy, littlewood and. See also 6, 15 for the other rearrangement free proofs for some special cases of the sharp hardylittlewoodsobolev inequality. A consequence of the hardylittlewood inequality is that the lp distance between two functions is larger than the lp distance between their symmetric decreasing rearrangements, as seen in the following proposition. In this paper we prove a rearrangement inequality that generalizes inequalities given in the book by hardy, littlewood and polya1 and by luttinger and friedberg. In mathematical analysis, the hardylittlewood inequality, named after g. Some generalizations of this inequality include the power mean inequality and the jensens inequality see below.
It s main implication is that rearrangement decreases l2distances 5. Dedicated to albert baernstein, ii on the occasion of his 65th birthday. To prove the lhs, apply the above with y i instead of y i. Sharp hardylittlewoodsobolev inequality on the upper half. A quantitative version of the hardylittlewood rearrangement inequality, involving a remainder term depending on a distance from the class of functions attaining equality, is established. One way to prove youngs inequality is using the rieszthorin interpolation theorem and similarly one can use marcinkiewicz interpolation theorem to prove the weak young inequality. Mar 29, 2011 we prove that supermodularity is a necessary condition for the generalized hardylittlewood and riesz rearrangement inequalities. The sharp hardylittlewoodsobolev inequality on the upper half space is proved. The lower bound follows by applying the upper bound to. We prove that supermodularity is a necessary condition for the generalized hardylittlewood and riesz rearrangement inequalities. Such inequalities are the continuous versions of the classical rearrangement inequalities for discrete sets of numbers. The fundamental hardylittlewood maximal inequality asserts that they are also uniformly bounded in shape. 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.
Generalized hardylittlewoodsobolev inequality mathoverflow. Therefore it need a free signup process to obtain the book. The inequality for an integral of a product of functions of one variable is further extended to the case of functions of several variables. Here are several problems from the putnam exam, which can be solved using the amgm inequality. A consequence of the hardy littlewood inequality is that the lp distance between two functions is larger than the lp distance between their symmetric decreasing rearrangements, as seen in the following proposition. In this paper, the hardylittlewoodpolya rearrangement inequality is extended to hermitian matrices with respect to determinant, trace, kronecker product, and hadamard product. Decreasing rearrangements of nonnegative c0 sequences.
It was found by hardy, littlewood and sobolev almost a century ago. Generalization of a hardylittlewoodpolya inequality. Their method was also used to prove the sharp hardylittlewoodsobolev inequality in the heisenberg group see 17. Among the classical rearrangement inequalities, the hardylittlewood inequality has received the least attention. Oct 16, 20 the sharp hardylittlewoodsobolev inequality on the upper half space is proved. The best constant for the centered hardylittlewood. In this paper, we establish a weighted hardylittlewoodsobolev hls inequality on the upper half space using a weighted hardy type inequality on the upper half space with boundary term, and discuss the existence of extremal functions based on symmetrization argument. In contrast, the riesz rearrangement inequality is. We also show the necessity of the monotonicity of the kernels involved in the riesztype integral. Their method was also used to prove the sharp hardy littlewood sobolev inequality in the heisenberg group see 17. A rearrangement inequality and the permutahedron a. The computation based on the ow as was done in 22 can. Rearrangementfree proof of the sharp hardylittlewoodsobolev inequality 28. The best constant for the centered hardylittlewood maximal.