2016
DOI: 10.4310/mrl.2016.v23.n3.a2
|View full text |Cite
|
Sign up to set email alerts
|

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 … Show more

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
2
1
1

Citation Types

0
8
0

Year Published

2016
2016
2024
2024

Publication Types

Select...
6
1

Relationship

2
5

Authors

Journals

citations
Cited by 14 publications
(10 citation statements)
references
References 11 publications
0
8
0
Order By: Relevance
“…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.…”
Section: Radial Estimates and Existence Of Radial Groundstatesmentioning
confidence: 99%
“…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.…”
Section: Radial Estimates and Existence Of Radial Groundstatesmentioning
confidence: 99%
“…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].…”
Section: Theorem 14 (Sharp Improvement In the Radial Case Formentioning
confidence: 99%
“…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.…”
mentioning
confidence: 91%
“…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.…”
Section: Introductionmentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%