Littlewood Inequalities Research Articles

Hardy and Littlewood Inequalities on Time Scales

In this paper, we will prove some new dynamic inequalities of Hardy and Littlewood type on time scales. The results as special cases contain the integral inequalities due to Hardy and the discrete inequalities due to Hardy and Littlewood. The main results will be proved by using some algebraic inequalities, the Holder inequality, and a simple consequence of Keller’s chain rule on time scales.

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.

New formulas for decreasing rearrangements and a class of Orlicz–Lorentz spaces

Using a nonlinear version of the well known Hardy–Littlewood inequalities, we derive new formulas for decreasing rearrangements of functions and sequences in the context of convex functions. We use these formulas for deducing several properties of the modular functionals defining the function and sequence spaces \(M_{\varphi ,w}\) and \(m_{\varphi ,w}\) respectively, introduced earlier in Hudzik et al. (Proc Am Math Soc 130(6): 1645–1654, 2002) for describing the Kothe dual of ordinary Orlicz–Lorentz spaces in a large variety of cases (\(\varphi \) is an Orlicz function and \(w\) a decreasing weight). We study these \(M_{\varphi ,w}\) classes in the most general setting, where they may even not be linear, and identify their Kothe duals with ordinary (Banach) Orlicz–Lorentz spaces. We introduce a new class of rearrangement invariant Banach spaces \(\mathcal M _{\varphi ,w}\) which proves to be the Kothe biduals of the \(M_{\varphi ,w}\) classes. In the case when the class \(M_{\varphi ,w}\) is a separable quasi-Banach space, \(\mathcal M _{\varphi ,w}\) is its Banach envelope.

Expansion by polynomials with respect to Freud-type weights

The weights Wα(x)=exp(−12|x|α),α>1, form an important subclass of Freud weights. For the expansion by orthogonal polynomials with respect to Wα(x), we prove analogues of the Hausdorff–Young and the Hardy–Littlewood inequalities. The special cases of expansion by Hermite functions (α=2), which are new as well, are given explicitly.

Hardy–Littlewood inequalities for norms of positive operators on sequence spaces

We consider estimates of Hardy and Littlewood for norms of operators on sequence spaces, and we apply a factorization result of Maurey to obtain improved estimates and simplified proofs for the special case of a positive operator.

Hardy–Littlewood Inequalities for Fractional Derivatives of Invariant Harmonic Functions

We establish Hardy–Littlewood inequalities for fractional derivatives of Mobius invariant harmonic functions over the unit ball of \({\mathbb R^n}\) in mixed-norm spaces. In doing so we introduce a new criteria for the boundedness of operators in mixed-norm L p -spaces in terms of hyperbolic geometry of the real unit ball.

On Ren-Kähler's Paper “Hardy-Littlewood Inequalities and $Q_p$-Spaces” Z. Anal. Anwendungen 24 (2005), 375 – 388

In this note we prove that a harmonic function u on the unit ball B\subset\mathbb R^n belongs to the harmonic mixed norm space {\cal A}^{p,q}_s(B) , when p,q\in (0,\infty] and s>0 , if and only if all weighted tangential derivatives of order k (with positive orders of derivatives) belong to the related weighted Lebesgue mixed norm space {\cal L}^{p,q}_s(B). Our proof of the result for the case q\in (0,1) and k is odd, corrects the corresponding one in the paper: G. Ren and U. Kähler, Hardy–Littlewood inequalities and Q_p -spaces, Z. Anal. Anwendungen 24 (2005), 375–388.

Cases of equality and strict inequality in the extended Hardy–Littlewood inequalities

Extended Hardy-Littlewood inequalities are where {ui}1≤i≤m are non-negative functions and denote their Schwarz symmetrization.In this paper, we determine appropriate conditions under which equality in (*) occurs if and only if {ui}1≤i≤m are Schwarz symmetric.

Cases of equality and strict inequality in the extended Hardy–Littlewood inequalities

Extended Hardy-Littlewood inequalities are where {ui}1≤i≤m are non-negative functions and denote their Schwarz symmetrization.In this paper, we determine appropriate conditions under which equality in (*) occurs if and only if {ui}1≤i≤m are Schwarz symmetric.

Radial derivative on bounded symmetric domains

We establish weighted Hardy{Littlewood inequalities for radial derivative and fractional radial derivatives on bounded symmetric domains.

An interpolation approach to Hardy–Littlewood inequalities for norms of operators on sequence spaces

We revisit estimates of Hardy and Littlewood for norms of operators on sequence spaces, and exploit recent advances in interpolation theory to provide relatively simple proofs.

Hardy-Landau-Littlewood Inequalities for Fractional Derivatives in Weighted L p Spaces

On etablit l'inegalite de Hardy-Landau-Littlewood dans les espaces L p ponderes pour des poids satisfaisant a la condition (Ap) de Muckenhoupt. On montre que l'integrale indefinie possede un groupe fortement continu de puissances imaginaires dans L w p (0,∞)