Hardy-Littlewood Inequality Research Articles

Extended Hardy-Littlewood inequalities and applications to the calculus of variations

An extension of the Hardy-Littlewood inequality for rearrangements is established. It is used for giving several conditions of existence of a minimum for nonweakly-lower-semicontinuous functionals of the form $J(v) \! = \! \int_0^1f(x,v(x),v'(x)) dx$ with constraints on $v$ and $v'$.

Inégalité de Hardy—Little wood généralisée et applications au calcul des variations

In this Note, we generalize Hardy-Littlewood inequality for integrands of the form ƒ(¦x¦, u), where u is a N-dimensional vector function. We use the result for proving the existence of a minimizer for functionals of the form ∝ 0 1 ƒ(x,v(x),v′(x)) dx under global or pointwise constraints.

On a series inequality for limit–circle and finite problems

Brown and Evans' proposed, a criterion for the validity of their series analogue of Everitt's extension of the Hardy–Littlewood inequality presupposes that the associated difference equation is in the strong–limit point at infinity. In this paper, we investigate the case when the difference equation involved is in the limit–circle case and non–oscillatory and also when the interval is finite.

Ounded Operators on Weak Hardy Spaces and Applications

The atomic decomposition of weak Hardy spaces consisting of Vilenkin martingales is formulated. Some sufficient conditions for a sublinear operator T to be bounded from the weak Hardy space wHp to the weak wLp space are given. As applications a weak version of the Hardy-Littlewood inequality is obtained and it is shown that the maximal operator of the Cesaro means of a Vilenkin-Fourier series is bounded from wHp to wLp and is of weak type (1, 1). This yields that the Cesaro means of a function f ∈ L1 converge a.e. to the function in question, provided that the Vilenkin system is bounded.

On Hilbert's Inequality and Its Applications

In this paper, the weight coefficient of the formπ−θ(n)/2n+1(with θ(n)>0) is introduced. Improvements on Hilbert's inequality and the Hardy–Littlewood inequality are established, and these results are extended.

Generalized Strichartz Inequalities for the Wave Equation

We make a synthetic exposition of the generalized Strichartz inequalities for the wave equation obtained in [6] together with the limiting cases recently obtained in [13] with as simple proofs as possible. The proofs combine stationary phase estimates, dyadic decompositions, the Hardy-Littlewood inequality or the Hölder inequality in time, and abstract duality and interpolation arguments.

Weighted Sobolev inequalities on domains satisfying the chain condition

By similar methods of Iwaniec and Nolder (Hardy-Littlewood inequality for quasiregular mappings in certain domains in R n {\mathbb {R}^n} , Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985)), we obtain weighted Sobolev inequalities on domains satisfying the Boman chain condition.

On an extension of Copson's inequality for infinite series

SynopsisIn 1979 Copson proved the following analogue of the Hardy-Littlewood inequality: if is a sequence of real numbers such that are convergent, where Δan = an+1 – an and Δ2an = Δ(Δan), then is convergent and the constant 4 being best possible. Equality occurs if and only if an = 0 for all n. In this paper we give a result that extends Copson's result to inequalities of the formwhere Mxn =–Δ(pn_l Δxn_l)+qnxn (n = 0, 1, …). The validity of such an inequality and the best possible value of the constant K are determined in terms of the analogue of the Titchmarsh-Weyl m-function for the difference equation Mxn = λwnxn (n = 0, 1, …).

HELP inequalities for limit-circle and regular problems

Everitt’s criterion for the validity of the generalized Hardy-Littlewood inequality presupposes that the associated differential equation is singular at one end-point of the interval of definition and is in the strong-limit-point case at the end-point. In this paper we investigate the cases when the differential equation is in the limit-circle case and non-oscillatory at the singular end-point and when both end-points of the interval are regular.

A weighted version of the Paley–Wiener theorem

A generalization of the classical theorems of Paley and Wiener[5] and Plancherel and Polya[6] concerning entire functions of exponential type is obtained. The proof relies only on the Cauchy theorem and the Hardy–Littlewood inequality for the Fourier transform (see [8, 9]). Since the functions under consideration are supposed to be defined only in two opposite octants in ℂn, a version of the edge of the wedge theorem [7] is derived as a by-product.

Some properties of the relative rearrangement

We develop some new properties of the relative rearrangement. Some of these properties generalize well-known results known as the Hardy-Littlewood inequality.

Supplement to the paper “the averaging operator with respect to a countable partition on a minimal symmetric ideal of the space L1(0, 1)”

Let A be a partition of the segment [0, 1] into a countable number of disjoint subsets of positive measure, let t∈L1(0,1), let Nt be the smallest rearrangement-invariant order ideal vector lattice in L1(0,1), containing t. In the paper one investigates the properties of the image E(Nt¦A) of the averaging operator with respect to A. In particular, one elucidates under what conditions there exists a function g, g∈L1(0,1), such that E(Nt¦A)⊂Ng. One formulates a generalization of the known Hardy-Littlewood inequality, namely Theorem E(t∣A)≺QE(t*∣A*), where ≺ is the Hardy-Littlewood preorder, t* and A* are the decreasing rearrangements of the function ¦t¦ and (in a special sense) of the partition A, while Q is an absolute constant, 1⩽Q⩽25. One formulates the problem of the smallest value of Q.

Hardy-Littlewood inequality for quasiregular mappings in certain domains in R^n

Hardy-Littlewood inequality for quasiregular mappings in certain domains in R^n

A remark on the Hardy-Littlewood inequality

We show that the inequality of the Littlewood conjecture holds precisely for compact abelian groups whose connected component has finite index.

An Extension of the Hardy-Littlewood Inequality

The Hardy-Littlewood inequality is extended from ${L^2}$ to $L_w^2$ where w is any positive nondecreasing function.

An extension of the Hardy-Littlewood inequality

The Hardy-Littlewood inequality is extended from L 2 {L^2} to L w 2 L_w^2 where w is any positive nondecreasing function.

Interpolation und indexbedingungen auf rearrangement—invarianten funktionenräumen. I. Grundlagen und die K-methode

In this paper the K-interpolation method of J. Peetre is built up for rearrangement invariant norms ϱ on (0, ∞). The spaces ( X 1, X 2) θ, ϱ; K (−∞ < θ < ∞), defined by the norm ∥ f∥ θ, ϱ; K = ϱ( t − θ K( t, f)), are shown to be intermediate spaces of the Banach spaces X 1 and X 2 if the condition α < θ ⩽ 1 upon the upper index α of ϱ is assumed. For these spaces an interpolation theorem of M. Riesz-Thorin-type as well as theorems of reiteration and stability are valid, again under certain conditions upon the indices of ϱ. These index-conditions, which turn out to be of central importance in the interpolation theory on rearrangement invariant spaces, are shown to be equivalent to a generalized Hardy-Littlewood inequality, which is established in the first part of the paper.