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Register a new userSharp anisotropic Hardy--Littlewood inequality for positive multilinear forms
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Diana Serrano-Rodr\\'iguez M
Publication date
2019
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English
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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.
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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.
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.
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 $betageq0$.
There are at least two directions concerning the extension of classical sharp Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on general manifolds; (2) Extending it for the negative exponent $lambda=n-alpha$ (that is for the case of $alpha>n$). In this paper we confirm the possibility for the extension along the first direction by establishing the sharp Hardy-Littlewood-Sobolev inequality on the upper half space (which is conformally equivalent to a ball). The existences of extremal functions are obtained; And for certain range of the exponent, we classify all extremal functions via the method of moving sphere.
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