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Register a new userOptimal Hardy-Littlewood type inequalities for polynomials and multilinear operators
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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--Hille type inequalities.
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The Hardy--Littlewood inequalities on $ell _{p}$ spaces provide optimal exponents for some classes of inequalities for bilinear forms on $ell _{p}$ 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 $q_{1},...,q_{m}>0$ and an infinite-dimensional Banach space $Y$ attaining its cotype $cot Y$, if begin{equation*} frac{1}{p_{1}}+...+frac{1}{p_{m}}<frac{1}{cot Y}, end{equation*} then the following assertions are equivalent: (a) There is a constant $C_{p_{1},...,p_{m}}^{Y}geq 1$ such that begin{equation*} left( sum_{j_{1}=1}^{infty }left( sum_{j_{2}=1}^{infty }cdots left( sum_{j_{m}=1}^{infty }leftVert A(e_{j_{1}},...,e_{j_{m}})rightVert ^{q_{m}}right) ^{frac{q_{m-1}}{q_{m}}}cdots right) ^{frac{q_{1}}{q_{2}} }right) ^{frac{1}{q_{1}}}leq C_{p_{1},...,p_{m}}^{Y}leftVert ArightVert end{equation*} for all continuous $m-$linear operators $A:ell _{p_{1}}times cdots times ell _{p_{m}}rightarrow Y.$ (b) The exponents $q_{1},...,q_{m}$ satisfy begin{equation*} q_{1}geq lambda _{m,cot Y}^{p_{1},...,p_{m}},q_{2}geq lambda _{m-1,cot Y}^{p_{2},...,p_{m}},...,q_{m}geq lambda _{1,cot Y}^{p_{m}}, end{equation*} where, for $k=1,...,m,$ begin{equation*} lambda _{m-k+1,cot Y}^{p_{k},...,p_{m}}:=frac{cot Y}{1-left( frac{1}{ p_{k}}+...+frac{1}{p_{m}}right) cot Y}. end{equation*} As an application of the above result we generalize to the $m$-linear setting one of the classical Hardy--Littlewood inequalities for bilinear forms. Our result is sharp in a very strong sense: the constants and exponents are optimal, even if we consider mixed sums.
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.
Let $S_{alpha}$ be the multilinear square function defined on the cone with aperture $alpha geq 1$. In this paper, we investigate several kinds of weighted norm inequalities for $S_{alpha}$. We first obtain a sharp weighted estimate in terms of aperture $alpha$ and $vec{w} in A_{vec{p}}$. By means of some pointwise estimates, we also establish two-weight inequalities including bump and entropy bump estimates, and Fefferman-Stein inequalities with arbitrary weights. Beyond that, we consider the mixed weak type estimates corresponding Sawyers conjecture, for which a Coifman-Fefferman inequality with the precise $A_{infty}$ norm is proved. Finally, we present the local decay estimates using the extrapolation techniques and dyadic analysis respectively. All the conclusions aforementioned hold for the Littlewood-Paley $g^*_{lambda}$ function. Some results are new even in the linear case.
Using elementary techniques, we prove sharp anisotropic Hardy-Littlewood inequalities for positive multilinear forms. In particular, we recover an inequality proved by F. Bayart in 2018.
This paper is devoted to a new family of reverse Hardy-Littlewood-Sobolev inequalities which involve a power law kernel with positive exponent. We investigate the range of the admissible parameters and characterize the optimal functions. A striking open question is the possibility of concentration which is analyzed and related with nonlinear diffusion equations involving mean field drifts.
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