Hardy Littlewood Inequalities Research Articles

Marcinkiewicz's interpolation theorem for Hardy-type spaces and its applications

A series of results similar to Marcinkiewicz's theorem on the interpolation of operators is put forward. The difference from the classical forms of this theorem is that spaces of integrable functions are replaced by some function classes that are extensions of various Hardy spaces. Some applications of these results to the extension of Carleson's embedding theorem and the Hardy-Littlewood inequalities for analytic functions in Hardy classes are presented. Bibliography: 41 titles.

Super-critical Hardy-Littlewood inequalities for multilinear forms

The multilinear Hardy-Littlewood inequalities provide estimates for the sum of the coefficients of multilinear forms T ∶ ℓ p 1 n × ⋯ × ℓ p m n → R ( or C ) when 1 / p 1 × ⋯ × 1 / p m < 1 . In this paper we investigate the critical and super-critical cases; i.e., when 1 / p 1 × ⋯ × 1 / p m ≥ 1.

Fourier transform of variable anisotropic Hardy spaces with applications to Hardy-Littlewood inequalities

Fourier transform of variable anisotropic Hardy spaces with applications to Hardy-Littlewood inequalities

Extensions of the vector-valued Hausdorff–Young inequalities

In this paper we study the vector-valued analogues of several inequalities for the Fourier transform. In particular, we consider the inequalities of Hausdorff–Young, Hardy–Littlewood, Paley, Pitt, Bochkarev and Zygmund. The Pitt inequalities include the Hausdorff–Young and Hardy–Littlewood inequalities and state that the Fourier transform is bounded from Lp(Rd, | · | βp) into Lq(Rd, | · | -γq) under certain condition on p, q, β and γ. Vector-valued analogues are derived under geometric conditions on the underlying Banach space such as Fourier type and related geometric properties. Similar results are derived for Td and Zd by a transference argument. We prove sharpness of our results by providing elementary examples on lp-spaces. Moreover, connections with Rademacher (co)type are discussed as well.

On the Sobolev embedding properties for compact matrix quantum groups of Kac type

We study the optimal order of natural analogues of Sobolev embedding properties within the framework of compact matrix quantum groups of Kac type. One of the main results of this paper is that the optimal order is given by the polynomial growth order of dual discrete quantum groups in a broad class, which covers all connected compact Lie groups, duals of polynomially growing discrete groups, $ O_2^+ $ and $ S_4^+ $. Outside the realm of co-amenable compact quantum groups, we prove that the optimal order is $ 3 $ for duals of free groups and free quantum groups $ O_N^+ $ and $ S_N^+ $, and that Sobolev embedding properties can be generalized for all compact matrix quantum groups of Kac type whose duals have the rapid decay property. In addition, we generalize sharpened Hausdorff-Young inequalities, compute degrees of the rapid decay property for duals of $ O_N^+,S_N^+ $ and prove sharpness of Hardy-Littlewood inequalities on duals of free groups.

Sharp Anisotropic Hardy–Littlewood Inequality for Positive Multilinear Forms

Using elementary techniques, we obtain the optimal anisotropic Hardy–Littlewood inequalities for positive multilinear forms. In particular, we recover an inequality proved by Bayart (J Funct Anal 274(4):1129–1154, 2018).

Hardy-Littlewood, Hausdorff-Young-Paley inequalities, and Lp-Lq Fourier multipliers on compact homogeneous manifolds

In this paper we prove new inequalities describing the relationship between the “size” of a function on a compact homogeneous manifold and the “size” of its Fourier coefficients. These inequalities can be viewed as noncommutative versions of the Hardy-Littlewood inequalities obtained by Hardy and Littlewood [17] on the circle. For the example case of the group SU(2) we show that the obtained Hardy-Littlewood inequalities are sharp, yielding a criterion for a function to be in Lp(SU(2)) in terms of its Fourier coefficients. We also establish Paley and Hausdorff-Young-Paley inequalities on general compact homogeneous manifolds. The latter is applied to obtain conditions for the Lp-Lq boundedness of Fourier multipliers for 1<p≤2≤q<∞ on compact homogeneous manifolds as well as the Lp-Lq boundedness of general (non-invariant) operators on compact Lie groups. We also record an abstract version of the Marcinkiewicz interpolation theorem on totally ordered discrete sets, to be used in the proofs with different Plancherel measures on the unitary duals.

Strong Factorizations of Operators with Applications to Fourier and Cesàro Transforms

Consider two continuous linear operators T:X1(μ)→Y1(ν) and S:X2(μ)→Y2(ν) between Banach function spaces related to different σ-finite measures μ and ν. By means of weighted norm inequalities we characterize when T can be strongly factored through S, that is, when there exist functions g and h such that T(f)=gS(hf) for all f∈X1(μ). For the case of spaces with Schauder basis, our characterization can be improved, as we show when S is, for instance, the Fourier or Cesàro operator. Our aim is to study the case where the map T is besides injective. Then we say that it is a representing operator—in the sense that it allows us to represent each element of the Banach function space X(μ) by a sequence of generalized Fourier coefficients—providing a complete characterization of these maps in terms of weighted norm inequalities. We also provide some examples and applications involving recent results on the Hausdorff–Young and the Hardy–Littlewood inequalities for operators on weighted Banach function spaces.

Hardy-Littlewood inequalities for multipolynomials

‎The notion of multipolynomials was recently introduced and explored by T‎. ‎Velanga [Linear Multilinear Algebra‎. ‎66 (2018)‎, ‎no‎. ‎11‎, ‎2328-2348] as an attempt to encompass the theories of polynomials and multilinear operators‎. ‎In the present paper‎, ‎we push this subject further‎, ‎by proving Hardy-Littlewood inequalities for multipolynomials and‎, ‎en passant‎, ‎a variant of the Kahane-Salem-Zygmund inequality in this framework‎.

Critical Hardy–Littlewood inequality for multilinear forms

The Hardy–Littlewood inequalities for m-linear forms on $$\ell _{p}$$ spaces are known just for $$p>m$$. The critical case $$p=m$$ was overlooked for obvious technical reasons and, up to now, the only known estimate is the trivial one. In this paper we deal with this critical case of the Hardy–Littlewood inequality. More precisely, for all positive integers $$m\ge 2$$ we have $$\begin{aligned} \sup _{j_{1}}\left( \sum _{j_{2}=1}^{n}\left( \ldots \left( \sum _{j_{m}=1} ^{n}\left| T\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right| ^{s_{m} }\right) ^{\frac{1}{s_{m}}\cdot s_{m-1}}\ldots \right) ^{\frac{1}{s_{3}}s_{2} }\right) ^{\frac{1}{s_{2}}}\le 2^{\frac{m-2}{2}}\left\| T\right\| \end{aligned}$$for all m-linear forms $$T{:}\,\ell _{m}^{n}\times \cdots \times \ell _{m} ^{n}\rightarrow \mathbb {K}=\mathbb {R}$$ or $$\mathbb {C}$$ with $$s_{k} =\frac{2m(m-1)}{mk-2k+2}$$ for all $$k=2,\ldots ,m$$ and for all positive integers n. As a corollary, for the classical case of bilinear forms investigated by Hardy and Littlewood in 1934 our result is sharp in a strong sense (both exponents and constants are optimal for real and complex scalars).

Anisotropic regularity principle in sequence spaces and applications

We refine a recent technique introduced by Pellegrino, Santos, Serrano and Teixeira and prove a quite general anisotropic regularity principle in sequence spaces. As applications, we generalize previous results of several authors regarding Hardy–Littlewood inequalities for multilinear forms.

Universal Bounds for the Hardy–Littlewood Inequalities on Multilinear Forms

The Hardy–Littlewood inequalities for multilinear forms on sequence spaces state that for all positive integers $$m,n\ge 2$$ and all m-linear forms $$T:\ell _{p_{1}}^{n}\times \cdots \times \ell _{p_{m}}^{n}\rightarrow {\mathbb {K}}$$ ( $${\mathbb {K}}={\mathbb {R}}$$ or $${\mathbb {C}}$$ ) there are constants $$C_{m}\ge 1$$ (not depending on n) such that $$\begin{aligned} \left( \sum _{j_{1},\ldots ,j_{m}=1}^{n}\left| T(e_{j_{1}},\ldots ,e_{j_{m}})\right| ^{\rho }\right) ^{\frac{1}{\rho }}\le C_{m}\sup _{\left\| x_{1}\right\| ,\ldots ,\left\| x_{m}\right\| \le 1}\left| T(x_{1},\ldots ,x_{m})\right| , \end{aligned}$$ where $$\rho =\frac{2m}{m+1-2\left( \frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}}\right) }$$ if $$0\le \frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}}\le \frac{1}{2}$$ or $$\rho =\frac{1}{1-\left( \frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}}\right) }$$ if $$\frac{1}{2}\le \frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}}<1$$ . Good estimates for the Hardy–Littlewood constants are, in general, associated to applications in Mathematics and even in Physics, but the exact behavior of these constants is still unknown. In this note we give some new contributions to the behavior of the constants in the case $$\frac{1}{2}\le \frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}}<1$$ . As a consequence of our main result, we present a generalization and a simplified proof of a result due to Aron et al. on certain Hardy–Littlewood type inequalities.

Optimal Hardy–Littlewood inequalities uniformly bounded by a universal constant

The Hardy--Littlewood inequality for $m$-linear forms on $\ell _{p}$ spaces and $m<p\leq 2m$ asserts that \begin{equation*} \left( \sum_{j_{1},...,j_{m}=1}^{\infty }\left\vert T\left( e_{j_{1}},\ldots ,e_{j_{m}}\right) \right\vert ^{\frac{p}{p-m}}\right) ^{\frac{p-m}{p}}\leq 2^{\frac{m-1}{2}}\left\Vert T\right\Vert \end{equation*} for all continuous $m$-linear forms $T:\ell _{p}\times \cdots \times \ell _{p}\rightarrow \mathbb{R}$ or $\mathbb{C}.$ The case $m=2$ recovers a classical inequality proved by Hardy and Littlewood in 1934. As a consequence of the results of the present paper we show that the same inequality is valid with $2^{\frac{m-1}{2}}$ replaced by $2^{\frac{\left( m-1\right) \left( p-m\right) }{p}}$. In particular, for $m<p\leq m+1$ the optimal constants of the above inequality are uniformly bounded by $2.$

Hardy–Littlewood inequalities on compact quantum groups of Kac type

The Hardy-Littlewood inequality on $\mathbb{T}$ compares the $L^p$-norm of a function with a weighted $\ell^p$-norm of its Fourier coefficients. The approach has recently been studied for compact homogeneous spaces and we study a natural analogue in the framework of compact quantum groups. Especially, in the case of the reduced group $C^*$-algebras and free quantum groups, we establish explicit $L^p-\ell^p$ inequalities through inherent information of underlying quantum group, such as growth rate and rapid decay property. Moreover, we show sharpness of the inequalities in a large class, including $C(G)$ with compact Lie group, $C_r^*(G)$ with polynomially growing discrete group and free quantum groups $O_N^+$, $S_N^+$.

Polynomial and multilinear Hardy-Littlewood inequalities: analytical and numerical approaches

We investigate the growth of the polynomial and multilinear Hardy--Littlewood inequalities. Analytical and numerical approaches are performed and, in particular, among other results, we show that a simple application of the best known constants of the Clarkson inequality improves a recent result of Araujo et al. We also obtain the optimal constants of the generalized Hardy--Littlewood inequality in some special cases.

A note on the Hardy-Littlewood inequalities for multilinear forms

A note on the Hardy-Littlewood inequalities for multilinear forms

A regularity principle in sequence spaces and applications

We prove a nonlinear regularity principle in sequence spaces which produces universal estimates for special series defined therein. Some consequences are obtained and, in particular, we establish new inclusion theorems for multiple summing operators. Of independent interest, we settle all Grothendieck's type (ℓ1,ℓ2) theorems for multiple summing multilinear operators. We further employ the new regularity principle to solve the classification problem concerning all pairs of admissible exponents in the anisotropic Hardy–Littlewood inequalities.

Some applications of the regularity principle in sequence spaces

The Hardy–Littlewood inequalities for m-linear forms have their origin with the seminal paper of Hardy and Littlewood (Q J Math 5:241–254, 1934). Nowadays it has been extensively investigated and many authors are looking for the optimal estimates of the constants involved. For \(m<p\le 2m\) it asserts that there is a constant \(D_{m,p}^{\mathbb {K}}\ge 1\) such that $$\begin{aligned} \left( \sum _{j_{1},\ldots ,j_{m}=1}^{n}\left| T(e_{j_{1}},\ldots ,e_{j_{m} })\right| ^{\frac{p}{p-m}}\right) ^{\frac{p-m}{p}}\le D_{m,p} ^{\mathbb {K}}\left\| T\right\| , \end{aligned}$$ for all m-linear forms \(T:\ell _{p}^{n}\times \cdots \times \ell _{p} ^{n}\rightarrow \mathbb {K}=\mathbb {R}\) or \(\mathbb {C}\) and all positive integers n. Using a regularity principle recently proved by Pellegrino, Santos, Serrano and Teixeira, we present a straightforward proof of the Hardy–Littlewood inequality and show that: (1) If \(m<p_{1}\le p_{2}\le 2m\) then \(D_{m,p_{1}}^{\mathbb {K}}\le D_{m,p_{2}}^{\mathbb {K}}\); (2) \(D_{m,p}^{\mathbb {K}}\le D_{m-1,p}^{\mathbb {K}}\) whenever \(m<p\le 2\left( m-1\right) \) for all \(m\ge 3\).

Optimal exponents for Hardy–Littlewood inequalities for m-linear operators

The Hardy–Littlewood inequalities on ℓp spaces provide optimal exponents for some classes of inequalities for bilinear forms on ℓp spaces. In this paper we investigate in detail the exponents involved in Hardy–Littlewood type inequalities and provide several optimal results that were not achieved by the previous approaches. Our first main result asserts that for q1,...,qm>0 and an infinite-dimensional Banach space Y attaining its cotype cot⁡Y, if1p1+...+1pm<1cot⁡Y, then the following assertions are equivalent:(a) There is a constant Cp1,...,pmY≥1 such that(∑j1=1∞(∑j2=1∞⋯(∑jm=1∞‖A(ej1,...,ejm)‖qm)qm−1qm⋯)q1q2)1q1≤Cp1,...,pmY‖A‖ for all continuous m-linear operators A:ℓp1×⋯×ℓpm→Y.(b) The exponents q1,...,qm satisfyq1≥λm,cot⁡Yp1,...,pm,q2≥λm−1,cot⁡Yp2,...,pm,...,qm≥λ1,cot⁡Ypm, where, for k=1,...,m,λm−k+1,cot⁡Ypk,...,pm:=cot⁡Y1−(1pk+...+1pm)cot⁡Y. As an application of the above result we generalize one of the classical Hardy–Littlewood inequalities for bilinear forms to the m-linear setting. Our result is sharp in a very strong sense: the constants and exponents are optimal, even if we consider mixed sums.

Remarks on an inequality of Hardy and Littlewood

An inequality of Hardy and Littlewood for m-homogeneous polynomials on ℓp spaces is valid for p > m. In this note, among other results, we present an optimal version of this inequality for the case p = m and obtain the optimal constant, when restricted to the case of 2-homogeneous polynomials on ℓ2(ℝ2). In an Appendix we justify why, curiously, the optimal exponents of some Hardy-Littlewood type inequalities do not behave smoothly.