Littlewood Inequality Research Articles

Paley inequality for the Weyl transform and its applications

Abstract In this paper, we prove several versions of the classical Paley inequality for the Weyl transform. As for some applications, we prove a version of the Hörmander’s multiplier theorem to discuss L p {L^{p}} - L q {L^{q}} boundedness of the Weyl multipliers and prove the Hardy–Littlewood inequality. We also consider the vector-valued version of the inequalities of Paley, Hausdorff–Young, and Hardy–Littlewood and their relations. Finally, we also prove Pitt’s inequality for the Weyl transform.

Weighted norm inequalities for the Opdam–Cherednik transform

In this paper, we study several weighted norm inequalities for the Opdam–Cherednik transform. We establish different versions of the Heisenberg–Pauli–Weyl inequality for this transform. In particular, we give an extension of this inequality using weights with different exponents and present a variation of the inequality that incorporates [Formula: see text]-norms for the Opdam–Cherednik transform. Also, we prove a version of the Hardy–Littlewood inequality for this transform. Finally, we give other variations of the Heisenberg–Pauli–Weyl inequality such as the Nash-type and Clarkson-type inequalities for the Opdam–Cherednik transform.

$$L^{p}$$–$$L^{q}$$ Boundedness of Fourier Multipliers on Fundamental Domains of Lattices in $$\mathbb {R}^d$$

In this paper we study the \(L^{p}\)–\(L^{q}\) boundedness of Fourier multipliers on the fundamental domain of a lattice in \(\mathbb {R}^{d}\) for \(1< p,q < \infty \) under the classical Hörmander condition. First, we introduce Fourier analysis on lattices and have a look at possible generalisations. We then prove the Hausdorff–Young inequality, Paley’s inequality and the Hausdorff–Young–Paley inequality in the context of lattices. This amounts to a quantitative version of the \(L^{p}\)–\(L^{q}\) boundedness of Fourier multipliers. We will show that this delivers also some \(L^{p}\)-estimates by using some standard Lebesgue space embeddings, which come from the finite measure of the fundamental domain. Moreover, the Paley inequality allows us to prove the Hardy–Littlewood inequality. As an application we treat some Sobolev embedding results, which also indicate the sharpness of our main inequalities.

A contractive Hardy–Littlewood inequality

We prove a contractive Hardy–Littlewood type inequality for functions from H p ( T ) , 0 < p ⩽ 2 which is sharp in the first two Taylor coefficients and asymptotically at infinity.

Reduction principle for a certain class of kernel‐type operators

AbstractThe classical Hardy–Littlewood inequality asserts that the integral of a product of two functions is always majorized by that of their non‐increasing rearrangements. One of the pivotal applications of this result is the fact that the boundedness of an integral operator acting near zero is equivalent to the boundedness of the same operator restricted to the cone of positive non‐increasing functions. It is well known that an analogous inequality for integration away from zero is not true. We will show in this paper that, nevertheless, the equivalence of the two inequalities is still preserved for certain rather general class of kernel‐type operators under a mild restriction and regardless of the measure of the underlying integration domain.

Asymptotic Estimates for Unimodular Multilinear Forms with Small Norms on Sequence Spaces

The existence of unimodular forms with small norms on sequence spaces is crucial in a variety of problems in modern analysis. We prove that the infimum of $\left\Vert A\right\Vert $ over all unimodular $d$-linear (complex or real) forms $A$ on $\ell_{p_1}^{n_{1}} \times \cdots \times \ell_{p_d}^{n_{d}}$, for all $p_1,\dots,p_d \in [2, \infty]$ and all positive integers $n_1,\dots,n_d$, behaves (asymptotically) as $\left(n_{1}^{1/2} + \cdots + n_{d}^{1/2}\right) \prod_{j=1}^{d}n_{j}^{\frac{1}{2} - \frac{1}{p_j}}$. Applications to the theory of the multilinear Hardy--Littlewood inequality are also presented.

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).

On minimal Hölder gaps and Shannon entropy balance

When estimating a bilinear form in x and y by a product of two terms depending solely on x or y , the well known Hölder inequality which uses the product of a p -norm and its dual comes easily into play. However, if one can choose p freely, one could reduce this Hölder gap accordingly. This note addresses this elementary but apparently not too popular issue by using strict log-convexity of the p -norm in \frac 1p (sometimes called Littlewood's inequality). The optimal p is characterized by a balance condition on the Shannon entropies of distributions related to x and y .

Fractional smoothness of distributions of polynomials and a fractional analog of the Hardy–Landau–Littlewood inequality

We prove that the distribution density of any non-constant polynomial $f(\xi_1,\xi_2,\ldots)$ of degree $d$ in independent standard Gaussian random variables $\xi$ (possibly, in infinitely many variables) always belongs to the Nikol'skii--Besov space $B^{1/d}(\mathbb{R}^1)$ of fractional order $1/d$ (and this order is best possible), and an analogous result holds for polynomial mappings with values in $\mathbb{R}^k$. Our second main result is an upper bound on the total variation distance between two probability measures on $\mathbb{R}^k$ via the Kantorovich distance between them and a suitable Nikol'skii--Besov norm of their difference. As an application we consider the total variation distance between the distributions of two random $k$-dimensional vectors composed of polynomials of degree $d$ in Gaussian random variables and show that this distance is estimated by a fractional power of the Kantorovich distance with an exponent depending only on $d$ and $k$, but not on the number of variables of the considered polynomials.

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.

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 $2.$

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 \(m\ge 3\).

Some applications of the Hölder inequality for mixed sums

We use the Holder inequality for mixed exponents to prove some optimal variants of the generalized Hardy–Littlewood inequality for m-linear forms on \(\ell _{p}\) spaces with mixed exponents. Our results extend recent results of Araujo et al.

Optimal constants for a mixed Littlewood type inequality

For $$p\in [2,\infty ]$$ a mixed Littlewood-type inequality asserts that there is a constant $$C_{(m),p}\ge 1$$ such that $$\begin{aligned} \left( \sum _{i_{1}=1}^{\infty }\left( \sum _{i_{2},\ldots ,i_{m}=1}^{\infty }|T(e_{i_{1}},\ldots ,e_{i_{m}})|^{2}\right) ^{\frac{1}{2}\frac{p}{p-1}}\right) ^{\frac{p-1}{p}}\le C_{(m),p}\Vert T\Vert \end{aligned}$$ for all continuous real-valued m-linear forms on $$\ell _{p}\times c_{0} \times \dots \times c_{0}$$ (when $$p=\infty $$ , $$\ell _{p}$$ is replaced by $$c_{0})$$ . We prove that for $$p>2.18006$$ the optimal constants $$C_{(m),p}$$ are $$\left( 2^{\frac{1}{2}-\frac{1}{p}}\right) ^{m-1}$$ . When $$p=\infty ,$$ we recover the best constants of the mixed $$\left( \ell _{1},\ell _{2}\right) $$ -Littlewood inequality.

On the Constants of the Bohnenblust–Hille and Hardy–Littlewood Inequalities

For \(\mathbb {K}=\mathbb {R}\) or \(\mathbb {C}\), the Hardy–Littlewood inequality for m-linear forms asserts that for \(4\le 2m\le p\le \infty \) there exists a constant \(C_{m,p}^{\mathbb {K}}\ge 1\) such that, for all m-linear forms \(T:\ell _{p}^{n}\times \cdots \times \ell _{p}^{n}\rightarrow \mathbb {K}\), and all positive integers n, Open image in new window This result was proved by Hardy and Littlewood (QJ Math 5:241–254, 1934) for bilinear forms and extended to m-linear forms by Praciano-Pereira (J Math Anal Appl 81:561–568, 1981). The case \(p=\infty \) recovers the Bohnenblust–Hille inequality (Ann Math 32:600–622, 1931). In this paper, among other results, we show that for \(p>2m(m-1)^2\) the optimal constants satisfying the Hardy–Littlewood inequality for m-linear forms are dominated by the best known constants of the corresponding Bohnenblust–Hille inequality. For instance, we show that if \(p>2m(m-1)^2\), then $$\begin{aligned} \textstyle C_{m,p}^{\mathbb {C}}\le \prod \limits _{j=2}^{m}\Gamma \left( 2-\frac{1}{j}\right) ^{\frac{j}{2-2j}}<m^{\frac{1-\gamma }{2}}, \end{aligned}$$ where \(\gamma \) is the Euler–Mascheroni constant.

New Lower Bounds for the Constants in the Real Polynomial Hardy–Littlewood Inequality

ABSTRACTIn this article, we obtain new lower bounds for the constants of the real scalar-valued Hardy–Littlewood inequality form-homogeneous polynomials on spaces when p = 2m and for certain values of m.

Lower bounds for the complex polynomial Hardy–Littlewood inequality

The Hardy–Littlewood inequality for complex homogeneous polynomials asserts that given positive integers m≥2 and n≥1, if P is a complex homogeneous polynomial of degree m on ℓpn with m<p≤∞ given by P(x1,…,xn)=∑|α|=maαxα, then there exists a constant CC,m,ppol≥1 (which does not depend on n) such that(∑|α|=m|aα|ρ)1ρ≤CC,m,ppol⋅supz∈Bℓpn⁡|P(z)|, with ρ=pp−m if m<p<2m and ρ=2mpmp+p−2m if 2m≤p≤∞. In this short note we provide nontrivial lower bounds for the constants CC,m,ppol. For instance we prove that, for m≥2 and m<p<∞,CC,m,ppol≥2mp for m even, andCC,m,ppol≥2m−1p for m odd. Estimates for the case p=∞ (this is the particular case of the complex polynomial Bohnenblust–Hille inequality) were recently obtained by D. Nuñez-Alarcón in 2013.

On the polynomial Hardy–Littlewood inequality

We investigate the behavior of the constants of the polynomial Hardy–Littlewood inequality.

Lower bounds for the constants of the Hardy–Littlewood inequalities

Given an integer m≥2, the Hardy–Littlewood inequality (for real scalars) says that for all 2m≤p≤∞, there exists a constant Cm,pR≥1 such that, for all continuous m-linear forms A:ℓpN×⋯×ℓpN→R and all positive integers N,(∑j1,...,jm=1N|A(ej1,...,ejm)|2mpmp+p−2m)mp+p−2m2mp≤Cm,pR‖A‖. The limiting case p=∞ is the well-known Bohnenblust–Hille inequality; the behavior of the constants Cm,pR is an open problem. In this note we provide nontrivial lower bounds for these constants.

On the upper bounds for the constants of the Hardy–Littlewood inequality

The best known upper estimates for the constants of the Hardy–Littlewood inequality for m-linear forms on ℓp spaces are of the form (2)m−1. We present better estimates which depend on p and m. An interesting consequence is that if p≥m2 then the constants have a subpolynomial growth as m tends to infinity.