Sharp lower bounds for Coulomb energy

Abstract

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

Groundstates and radial solutions to nonlinear Schrödinger–Poisson–Slater equations at the critical frequency

“…In section 6 we study the embeddings of the subspace E α,p rad (R N ) of radially symmetric functions in E α,p (R N ) into Lebesgue spaces, in the spirit of the seminal result of Strauss [52] (see also [53,54]) and its counterpart for some Coulomb-Sobolev spaces [49] (see also [8,14,39]). For α > 1 the radial embedding intervals are wider then the intervals given by Theorem 1: the critical Coulomb-Sobolev exponent 2 2p+α 2+α is replaced by a stronger critical exponent.…”

Sharp Gagliardo–Nirenberg inequalities in fractional Coulomb–Sobolev spaces

“…For d ∈ N, s = 1, α ∈ (0, d) and q ≥ 1 the improved radial inequalities (1.3) were studied in [28,Theorem 4]. The fractional case d = 3, 1/2 < s < 3/2, α = 2, q = 2 was considered in [3].…”

“…It was also shown that no radial improvement occurs when α ≤ 1. In [3], the radial improvement was obtained in E s, 2,2 rad (R 3 ) for 1/2 < s < 3/2. The result however did not include the physically important ultra-relativistic case s = 1/2.…”

Compact embeddings for fractional super and sub harmonic functions with radial symmetry

“…A reasonable idea to prove that the lower endpoint exponent in (1.14) decreases with radial symmetry is to look at a suitable pointwise decay in the spirit of the Strauss lemma [17] (see also [15,16] for Besov and Lizorkin-Triebel classes). In our context where two terms are present, the Sobolev norm and the Riesz potential involving |u|, we have been inspired by [13] where the case s = 1 in (1.14) has been studied (see also [4] and [3]). For our purposes the fact that s is in general not integer makes however the strategy completetly different from the one in [13] and we need to estimate the decay of the high/low frequency part of the function to compute the decay.…”

“…Eventually, using all these tools, we are able to compute a pointwise decay that allows the lower endpoint for (1.14) to be below the threshold p = 2. Computed the pointwise decay we will follow the argument in [4] to estimate the lower endpoint for fractional superharmonic (resp. subharmonic) radially symmetric functions.…”