Hardy-Littlewood Inequality Research Articles

Holomorphic extension of certain discrete functions on ℤ n and the related problems

In this paper, we construct an holomorphic extension of certain functions on ℤn. We deduce a discrete Gutzmer’s formula for the discrete Fourier transform (see Theorem 4.4). In addition, we prove the Hardy-Littlewood inequality for the classical Radon transform on ℝn (see Theorem 4.7). In this work, we use the techniques of the diophantine integral geometry and the estimate method in certain weighted L2-spaces.

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.

Necessary conditions for the $L^{p}$-convergence $(0<p<1)$ of single and double trigonometric series

We give necessary conditions in terms of the coefficients for the convergence of a double trigonometric series in the $L^{p}$-metric, where $0<p<1$. The results and their proofs have been motivated by the recent papers of A. S. Belov (2008) and F. Moricz (2010). Our basic tools in the proofs are the Hardy-Littlewood inequality for functions in $H^{p}$ and the Bernstein-Zygmund inequalities for the derivatives of trigonometric polynomials and their conjugates in the $L^{p}$-metric, where $0<p<1$.

Fourier transform of anisotropic Hardy spaces

We show that if f f is in an anisotropic Hardy space H A p H_A^p , 0 &gt; p ≤ 1 0 &gt; p \leq 1 , with respect to a dilation matrix A A , then its Fourier transform f ^ \hat {f} satisfies the pointwise estimate \[ | f ^ ( ξ ) | ≤ C | | f | | H A p ρ ∗ ( ξ ) 1 p − 1 . |\hat f(\xi )| \le C ||f||_{H^p_A} \rho _*(\xi )^{\frac {1}{p}-1}. \] Here, ρ ∗ \rho _* is a quasi-norm associated with the transposed matrix A ∗ A^* . This leads to necessary conditions for functions m m to be multipliers on H A p H_A^p , as well as further pointwise characterizations on f ^ \hat {f} and a generalization of the Hardy-Littlewood inequality on the integrability of f ^ \hat {f} . This last result is strengthened through the use of rearrangement functions.

Estimates of Weighted Hardy-Littlewood Inequality for Differential Forms

Differential forms are interesting and important generalizations of real functions and distributions. Many interesting results and applications of differential forms have recently been found in some fields. As an important tool the Hardy-Littlewood inequality have been playing critical roles in many mathematics, including potential analysis, partial differential equations and the theory of elasticity.in this paper, the local and the global parametric weighted Hardy-Littlewood inequality had been proved on Riemannian manifolds by using the generalized weak reverse Holder inequality for A-harmonic tensors. The required versions can be obtained by selecting the suitable values for the parameters contained in each theorem. These results can be considered as generalizations of the classical Hardy-Littlewood inequality. As applications , the global versions of the Hardy-Littlewood inequality had been extended in L s -averaging domain and L s (w,l) averaging domain. Finally, the global parametric Hardy-Littlewood inequality for the projection operator had been obtained using the global weighted inequality for A -harmonic tensors. These results can also be used to study the integrability of differential forms and estimate the integrals for differential forms.

A new restructured Hardy-Littlewood's inequality

In this paper, we reconstruct the Hardy-Littlewood’s inequality byusing the method of the weight coefficient and the technic of real analysis includinga best constant factor. An open problem is raised.

Hardy-Littlewood and Caccioppoli-Type Inequalities for -Harmonic Tensors

We prove the new versions of the weighted Hardy-Littlewood inequality and Caccioppoli-type inequality for -harmonic tensors. We also explore applications of our results to -quasiregular mappings and -harmonic functions in .

Regularity of the solutions for micro–macro models near equilibrium

In this paper, we study the regularity of solutions to two microscopic–macroscopic models: Hookean spring model and FENE dumbbell model. By making use of the structure of Fokker–Planck operator and establishing the general Hardy–Littlewood's inequality, we show that the classical solutions near the equilibrium obtained in [F. Lin, C. Liu, Ping Zhang, On a micro–macro model for polymeric fluids near equilibrium, Comm. Pure Appl. Math. 60 (6) (2007) 838–866] and [F. Lin, Ping Zhang, The FENE dumbbell model near equilibrium, Acta Math. Sinica (Chin. Ser.) 24 (2008) 529–538] become smooth with respect to the end-to-end vector variable q for any positive time.

A strengthened version of the Hardy-Littlewood inequality

A quantitative version of the Hardy–Littlewood rearrangement inequality, involving a remainder term depending on a distance from the class of functions attaining equality, is established.

On New Extensions of Hilbert's Integral Inequality

It is shown that some new extensions of Hilbert's integral inequality with parameterλ(λ&gt;1/2)can be established by introducing a proper weight function. In particular, whenλ=1, a refinement of Hilbert's integral inequality is obtained. As applications, some new extensions of Widder's inequality and Hardy-Littlewood's inequality are given.

Multiplier Theorems for the Short-Time Fourier Transform

So-called short-time Fourier transform multipliers (also called Anti-Wick operators in the literature) arise by applying a pointwise multiplication operator to the STFT before applying the inverse STFT. Boundedness results are investigated for such operators on modulation spaces and on L p -spaces. Because the proofs apply naturally to Wiener amalgam spaces the results are formulated in this context. Furthermore, a version of the Hardy-Littlewood inequality for the STFT is derived.

Rearrangement inequalities for functionals with monotone integrands

The inequalities of Hardy–Littlewood and Riesz say that certain integrals involving products of two or three functions increase under symmetric decreasing rearrangement. It is known that these inequalities extend to integrands of the form F ( u 1 , … , u m ) where F is supermodular; in particular, they hold when F has nonnegative mixed second derivatives ∂ i ∂ j F for all i ≠ j . This paper concerns the regularity assumptions on F and the equality cases. It is shown here that extended Hardy–Littlewood and Riesz inequalities are valid for supermodular integrands that are just Borel measurable. Under some nondegeneracy conditions, all equality cases are equivalent to radially decreasing functions under transformations that leave the functionals invariant (i.e., measure-preserving maps for the Hardy–Littlewood inequality, translations for the Riesz inequality). The proofs rely on monotone changes of variables in the spirit of Sklar's theorem.

Hardy–Littlewood inequality for quasiregular maps on Carnot groups

We generalize a Hardy–Littlewood inequality for L p -norms of conjugate harmonic functions to horizontal components of quasiregular mappings on Carnot groups.

Conjugate harmonic functions and Clifford algebras

We generalize a Hardy–Littlewood inequality and a Privalov inequality for conjugate harmonic functions in the plane to components of Clifford-valued monogenic functions.

On the generalized Hardy-Hilbert inequality and its applications

A generalized Hardy-Hilbert inequality with weight function of the form B p−2+λ p , q−2+λ q �θr(λ)/(2n+1) λ − 2�λ r (with θr(λ) > 0,r = p, q ,1 � q 1) can be established by means of Euler-Maclaurin summation formula, where B(m,n) is β function. Inparticular, when λ = 1 ,animprovementonHardy-Hilbert'sinequality is obtained. As its applications, Hardy-Littlewood's inequality is extended and refined.

Exponential Growth for a Fractionally Damped Wave Equation

We consider a nonlinear wave equation with an internal damping represented by a fractional time derivative and with a polynomial source. It is proved that the solution is unbounded and grows up exponentially in the Lp-norm for sufficiently large initial data. To this end we use some techniques based on Fourier transforms and some inequalities such as the Hardy-Littlewood inequality.

On the Hilbert Inequality With Weights

In this paper, it is shown that a Hilbert-type inequality with weight \omega(n) = \pi – \frac{\theta}{\sqrt{2n+1}} can be established where \theta = \frac{17}{20} . As application, a quite sharp result of the Hardy-Littlewood inequality is obtained and some further extensions are obtained.

Ar(λ)-Weighted Integral Inequalities for A-Harmonic Tensors

In this paper we prove the Ar(λ)-weighted Caccioppoli-type inequality and weak reverse Hölder inequality for A-harmonic tensors. We also obtain the Ar(λ)-weighted Hardy–Littlewood inequality for conjugate A-harmonic tensors. These inequalities can be considered as extensions of the classical results.

Two-parameter Hardy-Littlewood inequality and its variants

Let s* denote the maximal function associated with the rectangular partial sums $s_{mn}(x,y)$ of a given double function series with coefficients $c_{jk}$. The following generalized Hardy-Littlewood inequality is investigated: $||s*||_{p,μ}≤C_{p,α,β} {Σ_{