The best constant in a weighted Hardy-Littlewood-Sobolev inequality

Best Constant in Sobolev Inequality

The equality sign holds in (1) i] u has the Jorm: (3) u(x) = [a + btxI,~',-'] 1-~1~ , where Ix[ = (x~ @ ...-~x~) 1⁄2 and a, b are positive constants. Sobolev inequalities, also called Sobolev imbedding theorems, are very popular among writers in part ial differential equations or in the calculus of variations, and have been investigated by a great number of authors. Nevertheless there is a ques...

Spherical Reflection Positivity and the Hardy-littlewood-sobolev Inequality

The well-known functions on R , f(x) = c(b + |x − a|2)−p, where a ∈ R , b > 0 and c > 0, appear as the optimizers in some classical functional inequalities, notably the Hardy-Littlewood-Sobolev (HLS) inequality and its dual, the Sobolev inequality. Given their ubiquity, these functions must be endowed with some special property, and this was identified by Y. Y. Li and M. J. Zhu [21]. It is the ...

On the Integral Systems Related to Hardy-littlewood-sobolev Inequality

We prove all the maximizers of the sharp Hardy-Littlewood-Sobolev inequality are smooth. More generally, we show all the nonnegative critical functions are smooth, radial with respect to some points and strictly decreasing in the radial direction. In particular, we resolve all the cases left open by previous works of Chen, Li and Ou on the corresponding integral systems.

Sharp Hardy-littlewood-sobolev Inequality on the Upper Half Space

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 λ = n−α (that is for the case of α > n). In this paper we confirm the possibility for the extension along the first direction by establishing the sharp Hardy-Littlewood-Sobolev inequ...

Determination of the Best Constant in an Inequality of Hardy, Littlewood, and Pólya