Abstract. We prove L p lower bounds for Coulomb energy for radially symmetric functions inḢ s (R 3 ) with 1 2 < s < 3 2 . In case 1 2 < s ≤ 1 we show that the lower bounds are sharp.In this paper we prove lower bounds for the Coulomb energy In the general case, without restricting to radial functions, the upper bound for the Coulomb energy is given by the well known Hardy-Littlewood-Sobolev inequality while lower bounds have been proved only very recently. In particular if one can control suitable homogeneous Sobolev spaceḢ s (R 3 ) the L p lower bound for the Coulomb energy is given by the following inequalitieswith θ = 6−5p 3−2ps−2p. Here the parameters s > 0 and 1 < p ≤ ∞ satisfy .We shall underline that in many physical applications involving Sobolev norms and Coulomb energy the radially symmetric assumption of ϕ is natural due to the rotational invariance of energy functionals (see e.g [8] in the context of stability of matter). Our purpose is to see if it is possible to control lower L p norms if one assumes radial symmetry of ϕ. In the sequel we use two theorems that are crucial for our improvement in case of radial symmetry. The first is the following pointwise decay for radial functions inḢis the weighted Lebesgue space with the normwhere θ = where Jd−2 2 is the Bessel function of order. The argument is similar to the one developed in [5] for the pointwise decay of radial function inḢ s (R d ), i.e to split ϕ into low and high frequency parts. The pointwise decay of high frequency part of ϕ will be controlled by the boundess of Sobolev norm while the decay of low frequency part by the boundness of the weighted Lebesgue norm.The second theorem is the following lower bound for the Coulomb energy by Ruiz, see [9]. , there exists c = c(α) > 0 such that for any measurable ϕ : R d → R we have
SHARP LOWER BOUNDS FOR COULOMB ENERGY 3Let us defineFollowing the argument of Ruiz [9] it is easy to show that || · || E s is a norm andRuiz proved that for E 1 the following continuous embeddingThe result by Ruiz follows from two steps: first, Theorem 0.2 proves that , second, a weighted Sobolev embedding for radial function proved by Su, Wang and Wilem [10] gives the inclusionḢThe aim of our paper is to find continuous embeddings and hence better lower bounds for the Coulomb energy assuming radial symmetry when 1 2