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Register a new userAbstract Hardy-Littlewood Maximal Inequality
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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 deals with the inequalities devoted to the comparison between the norm of a function on a compact hypergroup and the norm of its Fourier coefficients. We prove the classical Paley inequality in the setting of compact hypergroups which further gives the Hardy-Littlewood and Hausdorff-Young-Paley (Pitt) inequalities in the noncommutative context. We establish Hormanders $L^p$-$L^q$ Fourier multiplier theorem on compact hypergroups for $1<p leq 2 leq q<infty$ as an application of Hausdorff-Young-Paley inequality. We examine our results for the hypergroups constructed from the conjugacy classes of compact Lie groups and for a class of countable compact hypergroups.
In this paper, we prove the following reversed Hardy-Littlewood-Sobolev inequality with extended kernel begin{equation*} int_{mathbb{R}_+^n}int_{partialmathbb{R}^n_+} frac{x_n^beta}{|x-y|^{n-alpha}}f(y)g(x) dydxgeq C_{n,alpha,beta,p}|f|_{L^{p}(partialmathbb{R}_+^n)} |g|_{L^{q}(mathbb{R}_+^n)} end{equation*} for any nonnegative functions $fin L^{p}(partialmathbb{R}_+^n)$ and $gin L^{q}(mathbb{R}_+^n)$, where $ngeq2$, $p, qin (0,1)$, $alpha>n$, $0leqbeta<frac{alpha-n}{n-1}$, $p>frac{n-1}{alpha-1-(n-1)beta}$ such that $frac{n-1}{n}frac{1}{p}+frac{1}{q}-frac{alpha+beta-1}{n}=1$. We prove the existence of extremal functions for the above inequality. Moreover, in the conformal invariant case, we classify all the extremal functions and hence derive the best constant via a variant method of moving spheres, which can be carried out emph{without lifting the regularity of Lebesgue measurable solutions}. Finally, we derive the sufficient and necessary conditions for existence of positive solutions to the Euler-Lagrange equations by using Pohozaev identities. Our results are inspired by Hang, Wang and Yan cite{HWY}, Dou, Guo and Zhu cite{DGZ} for $alpha<n$ and $beta=1$, and Gluck cite{Gl} for $alpha<n$ and $betageq0$.
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.
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.
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