Some inequalities of Hardy-Littlewood type

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This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant #234.     

Optimal Hardy-Littlewood type inequalities for polynomials and multilinear operators

In this paper we obtain quite general and definitive forms for Hardy-Littlewood type inequalities. Moreover, when restricted to the original particular cases, our approach provides much simpler and straightforward proofs and we are able to show that in most cases the exponents involved are optimal. The technique we used is a combination of probabilistic tools and of an interpolative approach; this former technique is also employed in this paper to improve the constants for vector-valued Bohnenblust-Hilletype inequalities.     

Hardy-Littlewood inequalities for Ciesielski-Fourier series

Summary The inequality (^∞_n=2Σn^p-2│^{^}f(n) │^p)≤C_p││f││_H (0 is proved for the one- and two-dimensional Ciesielski-Fourier coefficients of functions in Hardy spaces.     

On a class of integral inequalities of Hardy-Littlewood type

Journal d'Analyse Mathématique -     

Extended Hardy-Littlewood inequalities and some applications

We establish conditions under which the extended Hardy-Little- wood inequality

where each is non-negative and denotes its Schwarz symmetrization, holds. We also determine appropriate monotonicity assumptions on such that equality occurs in the above inequality if and only if each is Schwarz symmetric. We end this paper with some applications of our results in the calculus of variations and partial differential equations.

     

The Hardy-Littlewood theorem for trigonometric series with generalized monotone coefficients

Earlier we introduced a continuous scale of monotony for sequences (classes M _α, α ≥ 0), where, for example, M ₀ is the set of all nonnegative vanishing sequences, M ₁ is the class of all nonincreasing sequences, tending to zero, etc. In addition, we extended several results obtained for trigonometric series with monotone convex coefficients onto more general classes. The main result of this paper is a generalization of the well-known Hardy—Littlewood theorem for trigonometric series, whose coefficients belong to classes M _α, where α ∈ ( $ \tfrac{1} {2} Earlier we introduced a continuous scale of monotony for sequences (classes M _α, α ≥ 0), where, for example, M ₀ is the set of all nonnegative vanishing sequences, M ₁ is the class of all nonincreasing sequences, tending to zero, etc. In addition, we extended several results obtained for trigonometric series with monotone convex coefficients onto more general classes. The main result of this paper is a generalization of the well-known Hardy—Littlewood theorem for trigonometric series, whose coefficients belong to classes M _α, where α ∈ (, 1). Namely, the following assertion is true. Let α ∈ (, 1), < p < 2, a sequence a ∈ M _α, and . Then the series cos nx converges on (0,2π) to a finite function f(x) and f(x) ∈ L _p (0,2π). Original Russian Text ? M.I. D’yachenko, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii, Matematika, 2008, No. 5, pp. 38–47.     

Markov-Bernstein type inequalities for constrained polynomials with real versus complex coefficients

LetP _n,k ^c denote the set of all polynomials of degree at mostn withcomplex coefficients and with at mostk(0≤k≤n) zeros in the open unit disk. Let denote the set denote the set of all polynomials of degree at mostn withreal coefficients and with at mostk(0≤k≤n) zeros in the open unit disk. Associated with0≤k≤n andx∈[?1, 1], let $B_{n,k,x}^* : = \max \{ \sqrt {\frac{{n(k + 1)}}{{1 - x^2 }}} ,n\log (\frac{e}{{1 - x^2 }}\} ,B_{n,k,x}^* : = \sqrt {\frac{{n(k + 1)}}{{1 - x^2 }}} ,$ , andM _n,k ^* ?max{n(k+1),nlogn},M _n,k ?n(k+1). It is shown that $M_{n,k}^* : = \max \{ n(k + 1),n\log n\} ,M_{n,k}^* :n(k + 1)$ for everyx∈[?1, 1], wherec ₁>0 andc ₂>0 are absolute constants. Here ‖·‖_[?1,1] denotes the supremum norm on [?1,1]. This result should be compared with the inequalities $c3\min \{ B_{n,k,x,} B_{n,,k,} \} \leqslant _{p \in P_{n,k} }^{\sup } \frac{{|p'(x)|}}{{||p||[1,1]}} \leqslant \{ B_{n,k,x,} B_{n,,k,} \} ,$ , for everyx∈[?1,1], wherec ₃>0 andc ₄>0 are absolute constants. The upper bound of this second result is also fairly recent; and it may be surprising that there is a significant difference between the real and complex cases as far as Markov-Bernstein type inequalities are concerned. The lower bound of the second result is proved in this paper. It is the final piece in a long series of papers on this topic by a number of authors starting with Erdös in 1940.     

Inequalities of Hardy-Littlewood type

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Weighted norm inequalities for the vectorvalued Hardy-Littlewood maximal functions of one parameter rectangles

Let N denote the Hardy-Littlewood maximal operator for the familyR of one parameter rectangles. In this paper, we obtain that for 1 _w ^p (l^r) to L _W ^P (l^r) if and only if w ∈ A_P(R); for 1≤p<∞, N is bounded from L _W ^P (l^r) to weak L _W ^P (l^r) if and only if W ∈ A_P(R). Here we say W∈A_p (1), if $$\begin{gathered} \mathop {sup}\limits_{R \in R} \left( {\tfrac{1}{{|R|}}\smallint _r wdx} \right)\left( {\tfrac{1}{{|R|}}\smallint _R w^{ - 1/(p - 1)} dx} \right)^{p - 1}< \infty ,1< p< \infty , \hfill \\ (Nw)(x) \leqslant Cw(x)a.e.,p = 1 \hfill \\ \end{gathered} $$ ,     

Markov-Bernstein type inequalities of multivariate polynomials with positive coefficients and applications

The Cauchy’s formula of entire functions f:C^k→C is used to establish Markov-Bernstein type inequalities of multivariate polynomials with positive coeffeicients on the k-dimensional simplex T_k⊂R^k and on the cube [0,1]^k. The main results generalize and improve those of G.G. Lorentz, etc. Some applications of these inequalities are also considered in polynomial constrained approximation.     

Many-dimensional analogues of Hardy-Littlewood estimates for Taylor coefficients of functions of class HP

Moscow. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 4, pp. 216–218, July–August, 1988.     

Generalized Grunsky coefficients and inequalities

The generalized Grunsky coefficients are defined in this paper for all locally univalent meromorphic functions in any domain in the complete complex plane. Various explicit formulas for these coefficients are established. Necessary conditions for univalence are obtained in arbitrary domains and in the unit disc in particular. The first one generalizes Grunsky inequalities and the second one is an extension of the Nehari-Schwarzian derivative condition.     

Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems

This paper is devoted to inequalities of Lieb-Thirring type. Let V be a nonnegative potential such that the corresponding Schrödinger operator has an unbounded sequence of eigenvalues (λi(V))_i∈N^∗. We prove that there exists a positive constant C(γ), such that, if γ>d/2, then

(∗)     

Chebyshev type inequalities for pseudo-integrals

Chebyshev type inequalities for two classes of pseudo-integrals are shown. One of them concerning the pseudo-integrals based on a function reduces on the g-integral where pseudo-operations are defined by a monotone and continuous function g. Another one concerns the pseudo-integrals based on a semiring ([a,b],max,⊙), where ⊙ is generated. Moreover, a strengthened version of Chebyshev’s inequality for pseudo-integrals is proved.     

Some inequalities for fourier coefficients in inner product spaces

Some inequalities for Fourier coefficients in inner product spaces and related results are given. These inequalities complement, in a sense, some of the results in a recent book due to Mitrinovi, Peari and Fink.This research was supported by a grant from La Trobe University     

Strong Haagerup inequalities with operator coefficients

We prove a Strong Haagerup inequality with operator coefficients. If for an integer d, denotes the subspace of the von Neumann algebra of a free group F_I spanned by the words of length d in the generators (but not their inverses), then we provide in this paper an explicit upper bound on the norm on , which improves and generalizes previous results by Kemp–Speicher (in the scalar case) and Buchholz and Parcet–Pisier (in the non-holomorphic setting). Namely the norm of an element of the form ∑_{i=(i1,…,id)}a_iλ(g_i1g_id) is less than , where M₀,…,M_d are d+1 different block-matrices naturally constructed from the family (a_i)_iI^d for each decomposition of I^dI^l×I^d−l with l=0,…,d. It is also proved that the same inequality holds for the norms in the associated non-commutative L_p spaces when p is an even integer, pd and when the generators of the free group are more generally replaced by *-free -diagonal operators. In particular it applies to the case of free circular operators. We also get inequalities for the non-holomorphic case, with a rate of growth of order d+1 as for the classical Haagerup inequality. The proof is of combinatorial nature and is based on the definition and study of a symmetrization process for partitions.