Sharp Gagliardo-Nirenberg inequalities in fractional Coulomb ...

SHARP GAGLIARDO–NIRENBERG INEQUALITIES IN FRACTIONAL COULOMB–SOBOLEV SPACES JACOPO BELLAZZINI, MARCO GHIMENTI, CARLO MERCURI, VITALY MOROZ, AND JEAN VAN SCHAFTINGEN

arXiv:1612.00243v1 [math.FA] 1 Dec 2016

Abstract. We prove scaling invariant Gagliardo–Nirenberg type inequalities of the form kϕkLp (Rd ) ≤

CkϕkβH˙ s (Rd )

x Rd ×Rd

|ϕ(x)|q |ϕ(y)|q dx dy |x − y|d−α

!γ

,

involving fractional Sobolev norms with s > 0 and Coulomb type energies with 0 < α < d and q ≥ 1. We establish optimal ranges of parameters for the validity of such inequalities 2d and discuss the existence of the optimisers. In the special case p = d−2s our results include a new refinement of the fractional Sobolev inequality by a Coulomb term. We also prove that if the radial symmetry is taken into account, then the ranges of validity of the inequalities could be extended and such a radial improvement is possible if and only if α > 1.

Contents 1. Introduction and statement of results 2. Completeness of the fractional Coulomb–Sobolev space 3. Gagliardo–Nirenberg inequalities: Proof of Theorems 1.1, 1.2 and 1.3 4. Sharp improvement in the radial case 5. Optimality of the radial embeddings 6. Radial compactness: Proof of Theorem 1.5 Appendix A. Proof of claim (5.8) Acknowledgements References

2 6 7 11 16 21 22 24 24

Date: December 2, 2016. 2010 Mathematics Subject Classification. 46E35 (39B62, 35Q55). Key words and phrases. Interpolation inequalities, fractional Sobolev inequality; Coulomb energy; Riesz potential; radial symmetry. 1

2

J. BELLAZZINI, M. GHIMENTI, C. MERCURI, V. MOROZ, AND J. VAN SCHAFTINGEN

1. Introduction and statement of results 1.1. Introduction. Given d ∈ N, s > 0, α ∈ (0, d) and q ∈ [1, ∞), we define the fractional Coulomb–Sobolev space by ˆ x |ϕ(x)|q |ϕ(y)|q o n s 2 |ξ| ϕ(ξ) dξ < ∞ . b E s,α,q (Rd ) = ϕ : Rd → R : dx dy < ∞ and |x − y|d−α Rd d d R ×R

Since for every measurable function ϕ : Rd → R x 2 ˆ |ϕ|q dx ≤ CRd−α (1.1) BR (0)

Rd ×Rd

|ϕ(x)|q |ϕ(y)|q dx dy, |x − y|d−α

the boundedness of the double integral on the right-hand side of (1.1) ensures that ϕ is a tempered distribution and that its Fourier transform ϕb is a well-defined tempered distribution. In particular |ξ|s ϕb is a well-defined distribution on Rd \{0}. The integrability condition in the definition of E s,α,q (Rd ) means that this distribution can be represented by an L2 –function. s

In the sequel we define the fractional Laplacian (−∆) 2 ϕ by \s2 ϕ)(ξ) = 2π|ξ|2 s2 ϕ(ξ). b ((−∆)

We endow the space E s,α,q (Rd ) with the norm kϕkE s,α,q (Rd ) =

s

(−∆) 2 ϕ 2 2

L (Rd )

+

x

Rd ×Rd


1q ! 12

.

In particular, when s < d2 , a function ϕ is in the space E s,α,q (Rd ) if and only if ϕ ∈ H˙ s (Rd ) and x |ϕ(x)|q |ϕ(y)|q dx dy < ∞. |x − y|d−α d d R ×R

The space E 1,2,2 (R3 ) has been introduced and studied by P.-L. Lions [20, Lemma 4; 21, (55)] and D. Ruiz [27, section 2]. The space E s,α,2 (Rd ) had been studied in [2], while E 1,α,q (Rd ) had been studied in [24]. Following the arguments in [24, Section 2], it is not difficult to see that E s,α,q (Rd ) is a Banach space (see Proposition 2.1 below). The space E s,α,q (Rd ) is the natural domain for the fractional Coulomb–Dirichlet type energy x |ϕ(x)|q |ϕ(y)|q

s

(−∆) 2 ϕ 2 2 d + dx dy, L (R ) |x − y|d−α d d R ×R

which appears in models of mathematical physics related to multi-particle systems, where the Coulomb term with q = 2 typically represents the electrostatic repulsion between the particles. Relevant models include Thomas–Fermi–Dirac–von Weizsäcker (TFDW) models of Density Functional theory [5, 16, 18]; or Schrödinger–Poisson–Slater approximation to Hartree–Fock theory [6]. The fractional case d = 2, s = 1/2 and α = 1 appears in the recent TFDW– type model of charge screening in graphene [22], where relevant powers are q = 2 or q = 1. Interpolation inequalities (1.2) associated with the space E s,2s,2 (Rd ) are in some cases equivalent to the kinetic energy Lieb–Thirring type inequalities [23, Theorem 3], which are fundamental in the study of stability of non-relativistic (s = 1) and ultra-relativistic (s = 1/2) matter [19].

SHARP GAGLIARDO-NIRENBERG INEQUALITIES

3

1.2. Coulomb–Sobolev inequalities. Our first main result in this paper is the continuous embedding E s,α,q (Rd ) ֒→ L

2(2qs+α) 2s+α

(Rd ).

More specifically, we establish a family of scaling–invariant interpolation inequalities for the space E s,α,p (Rd ). Theorem 1.1 (Coulomb–Sobolev inequalities). Let d ∈ N, s > 0, 0 < α < d, q, p ∈ [1, ∞) and q(d − 2s) 6= d + α. There exists a constant C = C(d, s, α, q, p) > 0 such that the scaling invariant inequality (1.2)

p(d+α)−2dq p(d+α−q(d−2s)) H˙ s (Rd )

kϕkp ≤ Ckϕk

x Rd ×Rd


2d−p(d−2s) 2p(d+α−q(d−2s))

holds for every function ϕ ∈ E s,α,q (Rd ) if and only if (1.3) (1.4) (1.5)

d , 2 d if s < 2 d if s < 2

2(2qs + α) 2s + α h 2(2qs + α) 2d i p∈ , 2s + α d − 2s h 2d 2(2qs + α) i p∈ , d − 2s 2s + α

if s ≥

p≥

and and

1 d − 2s > , q d+α 1 d − 2s < . q d+α

Moreover, if p is not an end–point of the intervals (1.3)–(1.5), i.e. p 6= then the best constant for (1.2) is achieved.

2(2qs+α) 2s+α

and p 6=

2d d−2s ,

In the case s = 1 inequality (1.2) was known for d = 3, α = 2 and q = 2 [21, (55); 27, Theorem 1.5]; and for d ∈ N, α ∈ (0, N ) and q ≥ 1 [24, Theorem 1]. The fractional inequality (1.2) first appeared for d = 3, s = 1/2, α = 2 and q = 2 in [4, Proposition 2.1]; and for d ∈ N, s > 0, α ∈ (0, d) and q = 2 in [2, Proposition 2.1]. 1.3. Refined Sobolev inequalities. The special case q(d − 2s) = d + α, which corresponds d+α 2d and q = d−2s , is not covered by the previous theorem and the exponents in to p = d−2s (1.2) are meaningless. In this special case we obtain a refinement of the Sobolev embedding, extending the one observed for s = 1 [24, (1.7)] and for q = 2 [2, Proposition 2.1]. Theorem 1.2 (Endpoint refined Sobolev inequality). Let d ∈ N, 0 < s < Then there exists C = C(d, s, α) > 0 such that the inequality (1.6)

kϕk

2d

L d−2s (Rd )

α(d−2s) d(2s+α) H˙ s (Rd )

≤ Ckϕk

holds for all ϕ ∈ E s,α,q (Rd ).

 

x

Rd ×Rd

d+α d−2s

|ϕ(x)| |ϕ(y)| |x − y|d−α

d+α d−2s

d 2,

0 < α < d.

 s(d−2s)

d(2s+α)

dx dy 

Remark 1.1. It is interesting to compare our refinement for Sobolev embedding with two other improvements. The Gérard–Meyer–Oru improvement [1, Theorem 1.43; 17] states that if 0 < s < d2 and θ ∈ S(Rd ) is such that θb has compact support, has value 1 near the origin

4


and satisfies 0 ≤ θb ≤ 1, then

(1.7)

kϕk

2d L d−2s (Rd )

≤

1− 2s d C(d, s, θ)kϕkH˙ s (R d)

sup λ

d +s 2

λ>0

! 2s d

kθ(λ ·) ⋆ ϕk∞

∀ϕ ∈ H˙ s (Rd ).

The Palatucci–Pisante improvement [25, Theorem 1.1] (see also [30, (4.2)]) states that if 0 < s < d2 , then (1.8)

2s

1− 2s

kϕk

2d L d−2s (Rd )

∀ϕ ∈ H˙ s (Rd ).

d d ≤ C(d, s)kϕkH˙ s (R d ) kϕk 1, d −s

M

2

In the last inequality, the Morrey norm is defined as kϕkMr,γ :=

Rγ

sup R>0, x∈Rd

|u|r BR (x)

1 r

;

one proof of (1.8) relies on (1.7) and on the observation that d

λ 2 +s kθ(λ ·) ⋆ ϕk∞ ≤ Ckϕk

M1,

d−2s 2

.

In our case we have by Hölder’s inequality and monotonicity of the integral

R

d −s 2

d+α d−2s

|ϕ| BR (x)

≤R

d+α 2

|ϕ|

d+α d−2s

BR (x)



≤C

x

d+α

Rd ×Rd

d+α

1

|ϕ(x)| d−2s |ϕ(y)| d−2s dx dy  |x − y|d−α

2

d

so that it is clear that Coulomb norm controls the Morrey norm M1, 2 −s . On the other hand, 2s 1 ˙s the exponent α(d−2s) d(2s+α) = (1 − d ) 1+2s/α for H -norm in our improvement is always less than ˙s the exponent 1 − 2s d for H -norm in (1.7) and (1.8). This suggests that the inequality (1.6) cannot be derived directly from the already known ones. Remark 1.2. The refinement of the Sobolev inequality in Theorem 1.2 is sharp. Indeed, by scaling it can be proved that if a scaling invariant inequality of the following form holds (1.9)

kϕk

2d

L d−2s (Rd )



≤ C(d, s, α)kϕkβH˙ s (Rd ) 

x

Rd ×Rd

d+α

d+α

γ

|ϕ(x)| d−2s |ϕ(y)| d−2s dx dy  , |x − y|d−α

then the exponents γ and β are related by the equation

d − 2s d = − s β + (d + α)γ 2 2

On the other hand, estimates (3.7)–(3.9) in the proof of Theorem 1.1 below imply that d − 2s β ≤ + γ. 2d 2 We conclude that β ≥

α(d−2s) d(2s+α)

is necessary for (1.9) to hold.

Interpolating between the refined and classical Sobolev inequalities, we obtain a new family of interpolation inequalities, for which the best constant is achieved.


5

Theorem 1.3 (Non-endpoint refined Sobolev inequalities). Let d ∈ N, 0 < s < d2 , 0 < α < d s(d−2s) . Then there exists C = C(d, s, α, ε) > 0 such that the inequality and 0 < ε < d(2s+α) (1.10)

||ϕ||

2d d−2s

α(d−2s) 2(α+d) +ε d−2s 2sd+αd H˙ s (Rd )

≤ Ckϕk

 

x

Rd ×Rd

d+α

 s(d−2s) −ε

d+α

|ϕ(x)| d−2s |ϕ(y)| d−2s dx dy  |x − y|d−α

d(2s+α)

holds for all ϕ ∈ E s,α,q (Rd ). Moreover, the best constant for (1.10) is achieved. When ε =

s(d−2s) d(2s+α)

the inequality (1.10) is the classical Sobolev inequality.

The existence of optimizers for the non-endpoint inequality (1.6) provides a partial answer towards the question raised in the case s = 1 in [24, Section 1.5.5]. The existence of optimizers for the endpoint inequality (1.6) remains open. 1.4. Radial improvements. We now consider the question of embeddings for radial functions. Since the symmetric decreasing rearrangement increases the nonlinear nonlocal Coulomb energy term, the situation might be more favorable for radial functions. Our next result shows that for the subspace of radially symmetric functions in the Coulomb–Sobolev space s,α,q Erad (Rd ) the intervals (1.3)–(1.5) of the validity of the Coulomb–Sobolev inequality (1.2) can be extended provided that α > 1. Theorem 1.4 (Sharp Improvement in the radial case for α > 1). Let d ≥ 2, s > 0, 1 < α < d, q, p ∈ [1, ∞), q(d − 2s) 6= d + α and prad

(2s − 1)q + 2 (d − α) := q + . 2s(d + α − 2) + d − α

There exists a constant Crad = Crad (d, s, α, q, p) > 0 such that the scaling invariant inequality (1.11)

p(d+α)−2dq p(d+α−q(d−2s)) H˙ s (Rd )

kϕkLp (Rd ) ≤ Crad kϕk

x Rd ×Rd


2d−p(d−2s) ! 2p(d+α−q(d−2s))

s,α,q (Rd ) if and only if hold for all radially symmetric functions ϕ ∈ Erad

(1.12)

p > prad

(1.13)

p ∈ prad ,

(1.14) (1.15)

2d i d − 2s

2d , prad d − 2s i h 2d p∈ ,q d − 2s

p∈

h

d , 2 1 d − 2s d and > , if s < 2 q d+α d 1 d − 2s 1 1 − 2s if s < and < , 6= , 2 q d+α q 2 1 1 1 − 2s if s < and = . 2 q 2 if s ≥

s,α,q (Rd ) if and only if (1.2) holds on E s,α,q (Rd ). If 0 < α ≤ 1 then inequality (1.11) holds on Erad

In the while for In the was first

important special case s = 1/2 we have the simplified expression prad = q + d−α d−1 , s = 0 we formally obtain prad = 2. special case d = 3, s = 1, α = 2 and q = 2 the improved radial inequality (1.11) established in [27, Theorem 1.2]. For d ∈ N, s = 1, α ∈ (0, d) and q ≥ 1 the

6


improved radial inequalities (1.2) were studied in [24, Theorem 4]. The fractional case d = 3, 1/2 < s < 3/2, α = 2, q = 2 was considered in [3]. We shall emphasise that the radial improvement is possible for any s > 0 but if and only if α > 1. The universality of the threshold α = 1 which does not depend on any other parameter in the problem is quite interesting. 2 Another new and purely fractional phenomenon is the special role of the exponent q = 1−2s in the case s < 1/2. Observe that for s ≥ 1/2 we always have prad > q, while prad < q if d+α 2 2 , the latter requires q > d−2s . If s < 1/2 and q = 1−2s then prad = q s < 1/2 and q > 1−2s s,α,q d d p rad and this is the only case when the endpoint embedding Erad (R ) ֒→ L (R ) is valid. s,α,q (Rd ) ֒→ Lp (Rd ) is compact provided that p is Finally, we prove that the embedding Erad not an endpoint of the embedding intervals.

Theorem 1.5 (Compact embeddings for radial functions). Let d ≥ 2, s > 0, and q ∈ [1, ∞). Moreover we assume that p is away from the endpoints of the intervals in (1.3)–(1.5) when s,α,q (Rd ) ֒→ Lp (Rd ) 0 < α ≤ 1 and in (1.12)–(1.15) when 1 < α < d. Then, the embedding Erad is compact. Compactness of the radial embedding implies in a standard way the existence of radial optimizers associated to the inequalities (1.11), cf. [24, Section 7] where the case s = 1 was considered. 1.5. Outline. The rest of the paper is organised as follows. Section 2 contains a short proof of the completeness of the Coulomb–Sobolev spaces. In Section 3 we discuss the spaces E s,α,q (Rd ) in the nonradial context and show that interpolation inequalities of Theorems 1.1 and 1.2 can be deduced from the standard fractional Gagliardo–Nirenberg inequality (3.3) using a fractional chain rule. We also discuss the existence of the optimisers and prove Theorem 1.3. In Section 4 we derive the radial improvement of Theorem 1.4 as a consequence of Ruiz’s inequality for Coulomb energy (see Theorem 4.1) and de Napoli’s interpolation inequality (see Theorem 4.2), which is a fractional extension of the classical pointwise Strauss type bounds valid only for s > 1/2. In case s ≤ 1/2 we replace de Napoli’s pointwise bounds by Rubin’s inequality (Theorem 4.3), which is a refinement for radial functions of the classical Stein–Weiss inequality. In Section 5 we construct special families of functions which are used to prove the optimality of the radial embeddings, while in Section 6 we prove the compactness of the radial embedding. 1.6. Asymptotic notation. For real valued functions f (t), g(t) ≥ 0, we write: f (t) . g(t) if there exists C > 0 independent of t such that f (t) ≤ Cg(t); f (t) ≃ g(t) if f (t) . g(t) and g(t) . f (t). As usual, C, c, c1 , etc., denote generic positive constants independent of t. 2. Completeness of the fractional Coulomb–Sobolev space As in [24, Section 2], it is not difficult to see that the space E s,α,q (Rd ) is a normed space. Proposition 2.1. For every d ∈ N, s > 0, 0 < α < d and q ∈ [1, ∞), the normed space E s,α,q (Rd ) is complete.


7

s

Proof. If (un )n∈N is a Cauchy sequence in E s,α,q (Rd ), then ((−∆) 2 un )n∈N is a Cauchy sequence s in L2 (Rd ) and there exists thus f ∈ L2 (Rd ) such that ((−∆) 2 un )n∈N converges strongly to f in L2 (Rd ). On the other hand, by (1.1) we have for every R > 0, ˆ |un − um |q = 0. lim m,n→∞ B (0) R

There exists thus a measurable function u : Rd → R such that (un )n∈N converges to u in Lqloc (Rd ). By Fatou’s lemma, we have lim

n→∞

x Rd ×Rd

|un (x) − u(x)|q |un (y) − un (y)|q dx dy |x − y|d−α x

≤ lim lim inf n→∞ m→∞

Rd ×Rd

|un (x) − um (x)|q |un (y) − um (y)|q dx dy. |x − y|d−α

s

It remains now to prove that (−∆) 2 u = f . We observe that by (1.1), ˆ 1 lim sup d−α |un − u|q = 0. n→∞ R>0 R 2 BR (0) bn )n∈N Therefore (un )n∈N converges to u as tempered distributions Rd , and thus the sequence (u cn )n∈N converges b as tempered distributions on Rd . It follows that ((2π)s/2 |ξ|s u converges to u cn )n∈N converges b as distributions on Rd . Since on the other hand, ((2π)s/2 |ξ|s u to 2π s/2 |ξ|s |ξ|s u s to fb it follows that (−∆) 2 u = f .

3. Gagliardo–Nirenberg inequalities: Proof of Theorems 1.1, 1.2 and 1.3

We first establish the endpoint inequality. Theorem 3.1. Let d ∈ N, s > 0, 0 < α < d and q ∈ [1, ∞). Then the following inequality holds s 2qs+α x α

s 2qs+α |ϕ(x)|q |ϕ(y)|q

2 . (−∆) ϕ L2 (Rd ) ∀ϕ ∈ E s,α,q (Rd ). dx dy kϕk 2(2qs+α) d−α d 2s+α |x − y| (R ) L d d R ×R

In particular, E s,α,q (Rd ) ֒→ L

2(2qs+α) 2s+α

(Rd ) continuously.

The above inequality in the particular case q = 1 implies that E s,α,1 (Rd ) embeds continuously into H s (Rd ). Proof of Theorem 3.1. Recall that for all φ ∈ L1loc (Rd ) such that x φ(x)φ(y) dx dy < ∞ (3.1) |x − y|d−α d d R ×R

there holds (3.2)

α

(−∆)− 4 φ 2 2

L (Rd )

=c

x Rd ×Rd

φ(x)φ(y) dx dy. |x − y|d−α

8


Moreover we recall the endpoint Gagliardo–Nirenberg inequality (see for example [1, Theorem 2.44])

α

(−∆) 4 ψ

(3.3) where

2s

Lp (Rd )

s

α

α

+ 2 α+2s

4 ≤ CkψkLα+2s ψ Lr (Rd ) 2 (Rd ) (−∆)

1 2s 1 α 1 = + . p 2 α + 2s r α + 2s

When q = 1 by (3.1) and (3.2) there holds x x ϕ(x)ϕ(y)

α

(−∆)− 4 ϕ 2 2 d = c (3.4) dx dy ≤ c L (R ) |x − y|d−α d d d

R ×Rd

R ×R

|ϕ(x)| |ϕ(y)| dx dy. |x − y|d−α

α

Setting ψ = (−∆)− 4 ϕ and p = r = 2, (3.3) together with (3.4) yields the inequality for q = 1. α

Let q > 1. Setting ψ = (−∆)− 4 |ϕ|q in (3.3), we get

q

|ϕ|

Lp (Rd )

which implies (3.5)

α

2s

s

α

q α+2s

2 ≤ C (−∆)− 4 |ϕ|q Lα+2s 2 (Rd ) (−∆) |ϕ| Lr (Rd )

α

2s

s

α

(q−1)

α

α+2s α+2s

2 kϕkqLqp (Rd ) ≤ C (−∆)− 4 |ϕ|q Lα+2s 2 (Rd ) (−∆) ϕ L2 (Rd ) kϕkL(q−1)l (Rd )

by the fractional chain rule where 1r = such that (q − 1)l = qp, i.e. such that

1 2

+

1 l

[14, Corollary of Theorem 5]. Now choosing l

q−1 1 = l qp we conclude that p = proof.

2α+4qs q(2s+α) .

By (3.5) and setting φ = |ϕ|q in (3.2), this concludes the

Proof of Theorem 1.1 and Theorem 1.2. The exponents for the refined Sobolev inequality given by Theorem 1.2 are derived directly from the endpoint Gagliardo–Nirenberg inequality of Theorem 3.1. The scaling-invariant inequalities of Theorem 1.1 follows from the fact that by interpolation between Theorem 3.1 and the classical fractional Sobolev embedding, E s,α,q (Rd ) ֒→ Lp (Rd ) continuously for

2d 2(2qs + α) , p∈ 2s + α d − 2s 2(2qs + α) 2d , p∈ d − 2s 2s + α

d+α . d − 2s d+α if q > d − 2s

if 1 < q
0

! 2s d

kθ(A ·) ⋆ uk∞

.

Consider a maximizing sequence (ϕn )n∈N for (1.10) such that kϕn kH˙ s (Rd ) = 1 and ||ϕn ||

2d d−2s

s(d−2s) −ε

= (C(d, s, α, ε) + o(1)) D(ϕn )

where for brevity, we denoted D(ϕ) :=

x

,

d+α

d+α

Rd ×Rd

d(2s+α)

|ϕ(x)| d−2s |ϕ(y)| d−2s dx dy. |x − y|d−α

Using the endpoint refined Sobolev inequality we infer that s(d−2s) −ε

D(ϕn ) This implies that

d(2s+α)

. ||ϕn ||

2d d−2s

s(d−2s)

. D(ϕn )

d(2s+α)

.

1 . D(ϕn ) and hence, (3.10)

1 . ||ϕn ||

2d d−2s

.

Let ϕ¯ denotes the weak limit of (ϕn ) in H˙ s (Rd ). Recall that our inequality (1.10) is critical, i.e. it is both scaling and translation invariant. From Theorem 3.2 together with (3.10) there exists sequences (xn )n∈N in Rd of translations and (An )n∈N in R+ of dilations such that ˆ d +s 2 θ(An (xn − y))ϕn (y)dy > 0. inf An n

Rd

This fact implies by the change of variable that x − x s− d n An 2 ϕn ⇀ ϕ¯ 6= 0. An

The fact that ϕ¯ is an optimizer is now standard. By the Brezis–Lieb type splitting properties [7] of the three terms in (1.10) (for the splitting of the nonlocal term D see [24, Proposition 4.7]), we obtain 2d

2d

2d

d−2s

d−2s

¯ d−2s ¯ d−2s C(d, s, α, ε)− d−2s ||ϕ|| 2d + ||ϕn − ϕ|| 2d + o(1)


≥ kϕk ¯ 2H˙ s (Rd ) + kϕn − ϕk ¯ 2H˙ s (Rd ) + o(1) Since

2d(α+d) dα +ε d(2s+α) (d−2s)2

11

D(ϕ) ¯ + D(ϕn − ϕ) ¯

2ds 2d −ε d−2s d(2s+α)

.

2d(α + d) 2ds 2d 2d(α + 2s) dα +ε − ε > 1, + =1+ε 2 d(2s + α) (d − 2s) d(2s + α) d − 2s (d − 2s)2

As a consequence of the discrete Hölder inequality we have 2dα

a d(2s+α)

+ε

4d(α+d) (d−2s)2

2ds

c d(2s+α)

2dα

2d −ε d−2s

+ b d(2s+α)

+ε

4d(α+d) (d−2s)2

2ds

e d(2s+α)

≤ a2 + b2 for all a, b, c, e ≥ 0. Hence

2d

2d

2d −ε d−2s

2d(α+d) dα +ε d(2s+α) (d−2s)2

2ds

(c + e) d(2s+α)

2d −ε d−2s

2d

C(d, s, α, ε)− d−2s ||ϕ|| ¯ d−2s ¯ d−2s 2d + ||ϕn − ϕ|| 2d + o(1) d−2s

d−2s 4d(α+d) 2dα +ε d(2s+α) (d−2s)2

2ds

D(ϕ) ¯ d(2s+α)

≥ kϕk ¯ H˙ s (Rd )

4d(α+d) 2dα +ε d(2s+α) (d−2s)2

+ kϕn − ϕk ¯ H˙ s (Rd )

2d −ε d−2s

2ds

D(ϕn − ϕ) ¯ d(2s+α)

2d −ε d−2s

+ o(1).

Therefore we can conclude that 4d(α+d) 2dα +ε d(2s+α) (d−2s)2

2d

2d

¯ d−2s ≥ kϕk ¯ H˙ s (Rd ) C(d, s, α, ε)− d−2s ||ϕ|| 2d

2ds

D(ϕ) ¯ d(2s+α)

2d −ε d−2s

+ o(1),

d−2s

which implies that ϕ is an optimizer.

4. Sharp improvement in the radial case In order to establish the radial inequality (1.11) we will use a version of the weighted estimate involving the Coulomb term which was originally established by Ruiz [27]. Theorem 4.1 (Ruiz [27, Theorem 1.1], see also [24, Proposition 3.8]). Let d ∈ N, 0 < α < d, q ∈ [1, ∞). Then for every ε > 0 and R > 0 there exists C = C(d, α, q, ε) > 0 such that for 2dq

all ϕ ∈ L d+α (Rd ), ˆ (4.1)

|ϕ(x)|q

Rd \BR (0)

(4.2)

ˆ

BR (0)

|x|

d−α +ε 2

|ϕ(x)|q |x|

d−α −ε 2

C dx ≤ ε R

dx ≤ CR

ε

x Rd ×Rd

x Rd ×Rd

|ϕ(x)|q |ϕ(y)|q dx dy |x − y|d−α |ϕ(x)|q |ϕ(y)|q dx dy |x − y|d−α

21

21

,

.

s (Rd ). In the case s > 1/2 We will also employ two different estimate on the functions in H˙ rad our proof of (1.11) relies on the following interpolation result.

Theorem 4.2 (De Nápoli [9, Theorem 3.1]). Let d ≥ 2, s > 21 , r > 1 and (4.3)

− (d − 1) ≤ a < d(r − 1).

12


Then (4.4)

s

s ∀ϕ ∈ H˙ rad (Rd ) ∩ Lra (Rd ),

|ϕ(x)| ≤ C(d, s, r, a)|x|−σ k(−∆) 2 ϕkθL2 (Rd ) kϕk1−θ Lr (Rd )

where σ = norm

a

2s(d−1)+(2s−1)a , (2s−1)r+2

θ=

2 (2s−1)r+2

kukLra (Rd ) =

and Lra (Rd ) is the weighted Lebesgue space with the ˆ

a

r

|x| |u(x)| dx

Rd

1 r

.

Remark 4.1. The inequality (4.2) has important special cases: i) When r =

2d d−2s

and a = 0 we obtain Cho–Ozawa’s inequality [8]: sup |ϕ(x)| . |x|−

(4.5)

d−2s 2

|x|>0

s ∀ϕ ∈ H˙ rad (Rd ),

kϕkH˙ s (Rd )

ii) When r = 2 and a = 0 we obtain Ni type inequality sup |ϕ(x)| . |x|−

d−1 2

|x|>0

1

1−

1

2s 2s kϕkH ˙ s (Rd ) kϕkL2 (Rd )

s (Rd ). ∀ϕ ∈ H˙ rad

s (Rd ) are no longer available. In the case s ≤ 1/2 pointwise estimates on functions in H˙ rad Instead, our proof of (1.11) relies on the radial version of the classical Stein–Weiss estimate [28].

Theorem 4.3 (Rubin [10; 11, Theorem 1.2; 26]). Let d ≥ 2 and 0 < s < d/2. Then 1 ˆ r s r −βr ∀ϕ ∈ H˙ rad (Rd ), ≤ C(d, s, r, β)kϕkH˙ s (Rd ) |ϕ(x)| |x| dx (4.6) Rd

where r ≥ 2 and

1

(4.7)

− (d − 1)

2

−

1 d ≤β< , r r

1 1 β−s = + . r 2 d Remark 4.2. The difference with the classical (non-radial) Stein–Weiss estimate [28] is only in the extended range (4.7) for β (in the non-radial case we must have 0 ≤ β < dr ). Note special cases of (4.6): (4.8)

i) When β = s and s < d2 we obtain r = 2 which gives the Hardy inequality: 1 ˆ 2 2 −2s s |ϕ(x)| |x| dx ∀ϕ ∈ H˙ rad (Rd ), . kϕkH˙ s (Rd ) Rd

ii) When β = 0 and s < ˆ

Rd

d 2

we obtain r =

|ϕ|

2d d−2s

1−s 2

d

2d d−2s

which gives the Sobolev estimate:

. kϕkH˙ s (Rd )

s ∀ϕ ∈ H˙ rad (Rd ),

2 iii) When β = −(d − 1) 12 − 1r and s < 12 we see from (4.8) that r = 1−2s and hence β = −(d − 1)s, so we obtain a “limiting” inequality ˆ 1 −s 2s(d−1) 2 2 s 1−2s 1−2s |ϕ| |x| dx ∀ϕ ∈ H˙ rad (Rd ). . kϕkH˙ s (Rd ) Rd


13

A corollary of Rubin’s inequality is an integral replacement of the Cho–Ozawa bound (4.5). Lemma 4.1 (Weak Ni’s inequality). Let d ≥ 2, 0 < s ≤ 1/2 and 21 − s ≤ p1 ≤ 12 − ds . Then for R > 0, ˆ d s |ϕ|p ≤ C(d, s, p)Rd−p( 2 −s) kϕkpH˙ s (Rd ) (4.9) ∀ϕ ∈ H˙ rad (Rd ). Rd \BR (0)

rad

Proof. Follows from Rubin’s inequality (4.6) by setting r = p and β =

2d−p(d−2s) . 2p

Using (4.1), (4.4) and (4.6) in the exterior and the classical Sobolev inequality in the interior of a ball we deduce the following. Proposition 4.1. Let d ≥ 2, s > 0, 1 < α < d and s,α,q (Rd ) is continuously embedded into Lp (Rd ) for Erad

(4.10)

p ∈ prad ,

(4.11)

2d i d − 2s

and

p > prad

and

d−2s d+α +

1/2 and d+α s ≤ 1/2. Observe that p > prad > q, since q < d−2s . For a small ε > 0, denote γ :=

d−α + ε. 2

Case s > 1/2. Using successively the inequalities (4.4), (4.1) and (4.5), we estimate ˆ ˆ γ p−q |ϕ(x)|q p p−q |ϕ| ≤ sup |ϕ(x)| |x| (4.12) dx γ |x|>R Rd \BR (0) Rd \BR (0) |x| ˆ ˆ (1−θ)(p−q) q |ϕ(x)|q |ϕ(x)|q θ(p−q) . kϕkH˙ s (Rd ) dx dx γ γ Rd |x| Rd \BR (0) |x| .

+

θ(p−q) kϕkH˙ s (Rd )

kϕkp−q H˙ s (Rd )

21 + (1−θ)(p−q) 2q 1 x |ϕ(x)|q |ϕ(y)|q dx dy 2ε d−α R |x − y| N R

21 ˆ (1−θ)(p−q) q 1 x |ϕ(x)|q |ϕ(y)|q q−γ − d−2s 2 dx dy dx |x| , R2ε N |x − y|d−α BR (0) R

14


where θ =

2 (2s−1)q+2 .

The application of (4.4) requires that γ 2s(d − 1 − γ) + γ ≤σ= , p−q (2s − 1)q + 2

(4.13)

which is fulfilled for a sufficiently small ε > 0 if p > prad . The last integral in (4.12) is finite when (4.14)

−

d − 2s q − γ < −d; 2

this is the case for a sufficiently small ε > 0 when q
p > q and θ ∈ [0, 1] be such that qθ + Hölder inequality together with (4.1) and (4.6), we estimate ˆ (4.15)

p

|ϕ| ≤ Rd \BR (0)

.

ˆ

ˆ

r

γ r−p p−q

|ϕ(x)| |x|

dx

Rd \BR (0) r

−rβ

|ϕ(x)| |x|

dx

Rd \BR (0) r p−q r−q H˙ s (Rd )

. kϕk

p−q ˆ r−q

p−q r−q

1−θ r

= p1 , i.e. θ =

Rd \BR (0)

q r−p p r−q .

By the

r−p r−q

|ϕ(x)|q dx |x|γ

21 r−p r−q 1 x |ϕ(x)|q |ϕ(y)|q dx dy 2ε d−α R |x − y| N R

12 r−p r−q 1 x |ϕ(x)|q |ϕ(y)|q dx dy , 2ε d−α R |x − y| N R

where in view of (4.8) we must express r and β as r=

2(γp − d(p − q)) , 2γ − (d − 2s)(p − q)

β=

Note that β < 0 for sufficiently small ε > 0, since q < β≥−

1 γ(2d − p(d − 2s)) . 2 γp − d(p − q) d+α d−2s

and p
p and r > 2 in (4.6). We conclude that (4.15) holds for p > prad , provided that ε > 0 is sufficiently small.

Proposition 4.2. Let d ≥ 2, 0 < s < d2 , 1 < α < d and s,α,q (Rd ) is continuously embedded into Lp (Rd ) for Erad (4.17) (4.18)

h

2d , prad d − 2s i h 2d p∈ , prad d − 2s

p∈

and and

d+α d−2s

1 1 6 = − s, q 2 1 1 = − s. q 2

< q < ∞. Then the space


15

s,α,q Proof. Note that for 1q 6= 12 − s it is sufficient to establish continuous embedding Erad (Rd ) ֒→ p d L (R ) only for p in a small left neighbourhood of prad , the remaining values of p are then covered by interpolation via Theorem 3.1. s,α,q (Rd ) separately in the Given R > 0, we shall estimate the Lp –norm of a function ϕ ∈ Erad interior and exterior of the ball BR (0). The proof will be splitted into a number of separate cases, which we outline in Table 1.

s s > 1/2 s ≤ 1/2

q q> d+α d−2s

d+α d−2s

1/2. In the exterior of the ball BR (0), for any p > d−2s ˆ d |ϕ|p ≤ CRd−p( 2 −s) kϕkpH˙ s (Rd ) , (4.19) Rd \BR (0)

using the classical Sobolev inequality and Cho–Ozawa’s inequality (4.5). To obtain an estimate in the interior of the ball BR (0), we observe that for s > 1/2 we have q < prad and hence we ´can assume that q < p < prad . For a small ε > 0, set γ := d−α 2 − ε. Then the estimate p on BR (0) |ϕ| is identical to the argument in (4.12), but carried out in the interior of the ball BR (0), which reverses the inequalities in (4.13) and (4.14). d+α 2 Case s ≤ 1/2 and d−2s < q < 1−2s . In the exterior of the ball BR (0) the estimate (4.19) follows directly from the weak Ni’s inequality (4.9). To obtain an estimate in the interior of 2 we have q < prad and hence we can assume that the ball BR (0), observe that for q < 1−2s ´ d−α q < p < prad . For a small ε > 0, set γ := 2 − ε. Then the estimate on BR (0) |ϕ|p is identical to the argument in (4.15), but carried out in the interior of the ball BR (0) with q < p < r. d+α The only difference is that for q > d−2s the inequality in (4.16) reverses and that pε ր prad d−α as ε → 0, since γ < 2 . 2 Note that for 0 < s < 1/2 and q ≥ 1−2s we have q ≥ prad and a Hölder inequality estimate ´ p of type (4.15) on BR (0) |ϕ| is no longer possible.

2 Case s < 1/2 and q = 1−2s . Observe that in this case we have prad = q. In the exterior of the ball BR (0) the estimate ˆ d |ϕ|q ≤ CRd−q( 2 −s) kϕkqH˙ s (Rd ) , (4.20) Rd \BR (0)

16


follows directly from the weak Ni’s inequality (4.9), which is valid for q = ´ q q BR (0) |ϕ| , we can use the L –estimate (1.1), i.e. ˆ

(4.21)

q

|ϕ| ≤ CR

d−α 2

BR (0)

x

Rd ×Rd


21

2 1−2s .

To estimate

.

s,α,q (Rd ) ֒→ Lq (Rd ), the remaining Combining (4.20) and (4.21) together we conclude that Erad range of p follows by interpolation. ´ 2 . Observe that in this case p < prad < q. To estimate BR (0) |ϕ|p , Case s < 1/2 and q > 1−2s we use the Lq –estimate (1.1) to obtain

ˆ

|ϕ|p ≤ CR

BR (0)

1− pq

d−α 2

 

x

Rd ×Rd

|ϕ(x)|q

|ϕ(y)|q

|x − y|d−α

p

2q

dx dy 

.

To obtain an estimate in the exteriour of the ball BR (0), we will use Hölder, Rubin and Ruiz’s inequalities similarly to (4.15), with γ = d−α 2 + ε and r < p < q, which could be carried out 2 for p < prad provided that ε > 0 is sufficiently small, because prad > 1−2s . Proof of Theorem 1.4. The scaling invariant inequalities of Theorem 1.4 follow from Propositions 4.1 and 4.2 by by the same scaling consideration as in the proof of Theorem 1.1. The estimates of Propositions 4.1 and 4.2 improve upon the estimate of Theorem 3.1 only when α > 1. In the next section we show that the intervals of Propositions 4.1 and 4.2 are optimal and that for α ≤ 1 there is no improvement for the radial embedding. 5. Optimality of the radial embeddings The optimality of the intervals in Theorems 1.1 and 1.4 for s ≤ 1 is a consequence of the following. Theorem 5.1. Let d ≥ 2, 1 < α < d, 0 < s < 1/2 and q = is not continuously embedded into Lp (Rd ) for p > q = prad .

2 1−2s .

s,α,q (Rd ) Then the space Erad

Theorem 5.2. Let d ≥ 2, 1 < α < d, 0 < s ≤ 1 and p, q ∈ [1, +∞). Then the space s,α,q Erad (Rd ) is not continuously embedded in Lp (Rd ) for (5.1)

p ≤ prad

and

(5.2)

p ≥ prad

and

1 d − 2s > , q d+α 1 d − 2s 1 1 − 2s < , 6= . q d+α q 2

Theorem 5.3. Let d ≥ 2, 0 < α ≤ 1, 0 < s ≤ 1 and p, q ∈ [1, +∞). Then the space s,α,q Erad (Rd ) is not continuously embedded in Lp (Rd ) for (5.3) (5.4)

2(2qs + α) 2s + α 2(2qs + α) p> 2s + α

p
, q d+α 1 d − 2s < . q d+α


17

The proof of Theorems 5.2 and 5.3 is obtained by constructing counterexamples, i.e a family of functions u such that for a suitable p it holds kuk2H˙ s (Rd ) ≃ 1

x Rd ×Rd

|u(x)|q |u(y)|q dx dy ≃ 1 |x − y|d−α ||u||pLp (Rd ) → +∞.

Given a nonnegative function η ∈ C ∞ (R) \ {0} such that supp η ⊂ [−1, 1], we consider the family of functions (5.5)

uλ,R,S (x) = λ η

|x| − R

, S where R > S > 0 and λ > 0 will be specified in the sequel. By elementary computation we obtain kuλ,R,S kpp ≃ λp Rd−1 S.

(5.6) We also claim that

kuλ,R,S k2H˙ s (Rd ) ≃ λ2 Rd−1 S 1−2s ,

(5.7) and (5.8)

x

(x)|q

Rd ×Rd

|uλ,R,S |uλ,R,S |x − y|d−α

(y)|q

dx dy .

 2q d+α−2 2 S   λ R  

λ2q Rd−1 S 2 log(R/S)

λ2q Rd−1 S 1+α

if 1 < α < d, if α = 1, if 0 < α < 1.

The estimate (5.8) is proved in Appendix A below. To prove (5.7), for any s > 0 choose k ∈ N such that 2k ≥ s. Taking into account that S < R, by the change of variables and scaling we compute 2 ˆ ∞ n 2 ˆ d − 1 ∂ ok d−1 ∂ k 2 2 (5.9) kuλ,R,S kH˙ 2k (Rd ) ≃ dr + u (r) |∆ uλ,R,S | dx ≃ r λ,R,S 2 ∂r r ∂r d 0 R 2 ˆ ∞ n 2k a1 ∂ 2k−1 ak ∂ k o ∂ d−1 = dr + + · · · + u (r) r λ,R,S 2k 2k−1 k ∂r k ∂r r ∂r r 0 ˆ ∞ ˆ ∞ 2 (2k) r−R 2 d−1 (2k−1) r−R d−3 η r dr + |a | dr ≤ λ2 d r η 1 S S 0 0 ˆ ∞ 2 (k) r−R d−1−2k η S r + · · · + |ad | dr 2

.λ S

0

1−4k

R

d−1

+S

1−2(2k−1)

R

d−3

+ · · · + S 1−2k Rd−1−k

. λ2 S 1−4k Rd−1 .

Interpolating between the L2 and H˙ 2k –norm of uλ,R,S (cf. [1, Proposition 1.32]), we conclude from (5.6) and (5.9) that s

2− s

2 d−1 1−2s k k S . kuλ,R,S k2H˙ s (Rd ) ≤ kuλ,R,S kH ˙ 2k (Rd ) kuλ,R,S kL2 (Rd ) . λ R

18


Proof of Theorem 5.1. Let uS := uλ,R,S be the function in (5.5), where we fix R > 0 and for S < R set 1 λ = S− q . Then, since by our assumption 1 < α < d, kuS k2H˙ s (Rd ) . Rd−1 ,

(5.10) x

(5.11)

Rd ×Rd

|uS (x)|q |uS (y)|q dx dy . Rd+α−2 , |x − y|d−α

kuS kpLp (Rd ) ≃ λp SRd−1 ≃ λp−q Rd−1 ≃ S

(5.12)

1− pq

Rd−1 ,

Since R is fixed, we conclude that kuS kLp (Rd ) → ∞ for p > q when S → 0.

Proof of Theorem 5.2. Let uR :=λ,R,S be the function in (5.5), where we set λ = Rβ

and S = λ2 Rd−1

with (5.13)

β=−

1 2s−1

2(d − 1) + (d + α − 2)(2s − 1) , 2q(2s − 1) + 4

γ=

= Rγ , q(d − 1) − (d + α − 2) . q(2s − 1) + 2

Then we compute (5.14) x

(5.15)

kuR k2H˙ s (Rd ) . 1, |uR (x)|q |uR (y)|q dx dy . 1, |x − y|d−α

Rd ×Rd kuR kpLp (Rd )

(5.16)

≃ λp Rd−1 S ≃ Rβ(p−prad ) ,

provided that R > S, that is, either R > 1 and γ < 1 or R < 1 and γ > 1. To complete the proof Theorem 5.2 for p 6= prad we select R according to Table 2. q

1 q

∈

1 q

>

β

d−2s α+d

1−2s , d−2s 2 + α+d

s < 1/2 and

1 q

1.

s,α,q (Rd ) 6⊂ Lprad (Rd ) when Next we prove that Erad we consider the “multibump” sequence

vR,m =

m X

k=1

1 q

uRk ,

6=

1−2s 2 .

Similarly to [24, Lemma 6.4],


19 2β+d−1

where the functions uRk are as in (5.5) with R = Rk , λ = Rkβ , S = Rk 2s−1 and where β is given in (5.13). Note that for R 6= 1 and sufficiently large quotient R/S the functions uRk (k = 1, . . . , m) have mutually disjoint supports. If

1 q

>

d−2s α+d ,

1 q

or s < 1/2 and

j

k=1

If s is an integer the second term vanishes, or if s < 1 then the second term is negative. Otherwise, s = ℓ + σ, with ℓ ∈ N and σ ∈ (0, 1). Thus by the Gagliardo seminorm characterization of H˙ s (Rd ), if uRi and uRj have disjoint supports, x (∇ℓ u i (x) − ∇ℓ u i (y)) · (∇ℓ u j (x) − ∇ℓ u j (y) R R R R (uRi , uRj )H˙ s (Rd ) = dx dy d+2σ |x − y| Rd ×Rd (5.21) x ∇ℓ u i (x) · ∇ℓ u j (y) R R = −2C dx dy. d+2σ |x − y| d d R ×R

Similarly to (5.9), we deduce that kDℓ uλ,R,S kL1 (Rd ) . λRd−1 S 1−ℓ and hence kD ℓ uRk kL1 (Rd ) . Rk(β+d−1+γ(1−ℓ)) .

(5.22)

i j If 1q > d−2s α+d then β < 0 and 0 < γ < 1. For i > j and if R ≫ R we estimate (5.21) as follows,

(uRi , uRj )H˙ s (Rd ) . (5.23)

kD ℓ uRi kL1 (Rd ) kD ℓ uRj kL1 (Rd ) Ri − Rj

d+2σ

. R−i(d+2σ) R(i+j)(β+d−1+γ(1−ℓ)) . R−i(d+2σ) Ri(2(γs−β)+2σγ) . R−i(2σ(1−γ)) ,

since we note that 2(γs − β) < d, provided that q < large R we have kvR,m k2H˙ s (Rd ) . m +

(5.24) The case

1 q

Now, set

∈

1−2s , d−2s 2 + α+d

m X

d+α d−2s .

Then in (5.20) for all sufficiently

R−i(2σ(1−γ)) . m.

i,j=1,i>j

is similar, but letting R → 0 and observing that γ < 0. wR,m (x) = mθ vR,m

x

mσ

.

20


Then by the standard scaling we have (5.25)

kwR,m kpLp (Rd ) ≃ mpθ+σd+1 ,

(5.26)

kwR,m k2H˙ s (Rd ) . m2θ+σ(d−2s)+1 ,

x

(5.27)

Rd ×Rd

|wR,m (x)|q |wR,m (y)|q dx dy . m2qθ+σ(d+α)+1 . |x − y|d−α

If we set

q−1 , d + α − q(d − 2s) then for R → ∞ and m → ∞ we obtain σ=

θ=−

kwR,m k2H˙ s (Rd ) . 1, x |w (x)|q |w

(5.28)

R,m

(5.29)

|x

Rd ×Rd

q R,m (y)| − y|d−α

2s + α , 2(d + α − q(d − 2s))

dx dy . 1, 2s(α−1)

kwR,m kpLp (Rd ) ≃ mpθ+σd+1 ≃ m 2s(d+α−2)+d−α → ∞,

(5.30)

since α > 1 and d ≥ 2. The case

1 q

∈

1−2s , d−2s 2 + α+d

is similar, by letting R → 0.

Proof of Theorem 5.3. The strategy in the case 0 < α < 1 and 1q 6= 1−2s 1+α is the same as in the first part of the proof of Theorem 5.2. Let uR := uλ,R,S be the function in (5.5) and we choose λ = Rβ ,

S = λ2 Rd−1

where β=−

(d − 1)(2s + α) , 2(q(2s − 1) + 1 + α)

1 2s−1

γ=

= Rγ ,

(d − 1)(q − 1) . q(2s − 1) + 1 + α

Then (5.14) and (5.15) hold, and kuR kpLp (Rd ) ≃ λp Rd−1 S ≃ Rβ(p−

2(2qs+α) ) 2s+α

,

provided that R > S. Then to construct the required counterexamples, we select R according to Table 3. q

1 q

∈

1 q

>

β

d−2s α+d

1−2s d−2s 1+α + , α+d

s < 1/2 and

1 q

0 and for S < R we set

λ=S

α+2s − 2(q−1)

=S

2s−1 2

.

Then kuS k2H˙ s (Rd ) ≃ Rd−1 ,

(5.31) x

(5.32)

Rd ×Rd

|uS (x)|q |uS (y)|q dx dy . Rd−1 , |x − y|d−α 2

kuS kpLp (Rd ) ≃ λp SRd−1 ≃ λp− 2s−1 Rd−1 ≃ S 1−

(5.33)

Since R is fixed, we conclude that kuS kLp (Rd ) → ∞ for p >

p(1−2s) 2

2(2qs+α) 2s+α

Rd−1 .

=

2 1−2s

when S → 0.

The case α = 1 is similar, but takes into account the logarithmic correction in (5.8). We omit the details. 6. Radial compactness: Proof of Theorem 1.5 We need the following preliminary local compactness result. Lemma 6.1 (Local compactness). Let d ∈ N, s > 0, α ∈ (0, d) and q ∈ [1, ∞). Then the embedding E s,α,q (Rd ) ֒→ L1loc (Rd ) is compact. Proof. Multiplication by θ ∈ S(Rd ) is a continuous mapping E s,α,q (Rd ) → H˙ s (Rd ). Indeed by the fractional Leibniz rule, see e.g. [15, Theorem 1.4], we obtain s

s

s

k(−∆) 2 θukL2 (Rd ) . k(−∆) 2 ukL2 (Rd ) kθkL∞ (Rd ) + k(−∆) 2 θkLr (Rd ) kuk

L

with r such that

2s+α 2(2qs+α)

+

1 r

2(2qs+α) 2s+α

(Rd )

,

= 21 . For q = 1, we set r = ∞. Hence by Theorem 3.1, kθukH˙ s (Rd ) ≤ C(θ)kukE s,α,q (Rd ) .

For every ρ > 0, we choose θ ∈ C ∞ (Rd ) such that θ = 1 on Bρ and θ = 0 in Rd \ B2ρ . Let (un )n∈N be a bounded sequence in E s,α,q (Rd ). Setting vn = θun , theorem 3.1 implies that (vn )n∈N is also bounded in H s (Rd ). We can assume that vn converges weakly to some v in L2 (Rd ). By testing against suitable test functions, it follows that v is also supported in B2ρ and thus vˆ ∈ L∞ (Rd ). By Plancharel’s identity we have ˆ ˆ 2 2 b b |vbn (ξ) − vb(ξ)|2 dξ. |vn (ξ) − v (ξ)| dξ + kvn − vkL2 (Rd ) = |ξ|≤R

|ξ|>R

By showing that the right hand side goes to zero we will infer by Hölder’s inequality that kun − vkL1 (Bρ ) → 0. We have ˆ ˆ C 1 2 s 1 + |ξ| |vbn (ξ) − vb(ξ)|2 dξ ≤ . |vbn (ξ) − vb(ξ)|2 dξ ≤ 2 )s (1 + R (1 + R2 )s d R |ξ|>R

22


Since eix·ξ ∈ L2x (B2ρ ), by weak convergence in L2 (B2ρ ) we have vbn (ξ) → vb(ξ) almost everywhere. To conclude it suffices to show that ˆ |vbn (ξ) − vb(ξ)|2 dξ = o(1). (6.1) |ξ|≤R

1

1

Notice that kvbn k∞ ≤ kvn kL1 (B2ρ ) ≤ µ(B2ρ ) 2 kvn kL2 (B2ρ ) ≤ µ(B2ρ ) 2 kvn kH s (Rd ) and hence |vbn (ξ)−vb(ξ)|2 is estimated by a uniform constant so that by Lebesgue’s dominated convergence theorem (6.1) holds. This concludes the proof.

Proof of theorem 1.5 . We sketch the proof only in the most interesting case α > 1, s < 1/2, 2 , namely when prad ≤ q. Notice that for all R > 0, by (1.1) and Lemma 6.1, and q ≥ 1−2s s,α,q (Rd ) ֒→ Lploc (Rd ) for interpolation between q and p′ = 1 yields the compact embedding Erad s,α,q all 1 ≤ p < q. Thus it suffices to show that for any bounded sequence (un )n∈N in Erad (Rd ) there holds ˆ sup |un |p → 0, R → ∞. n∈N Rd \BR (0)

When p ≤

2 1−2s ,

we use Lemma 4.1 which yields ˆ |un |p ≤ o(1)kun kpE s,α,q (Rd ) ,

R → ∞.

Rd \BR (0)

2 When p > 1−2s the same conclusion holds by arguing as in the proof of (4.15) and using the strict inequality p < prad . This is enough to prove the theorem for α > 1, s < 1/2, and 2 . q ≥ 1−2s The other cases are similar, estimating the various integrals as in Proposition 4.1 for q < d+α d+α d−2s and according to Table 1 for q > d−2s . This concludes the proof.

Appendix A. Proof of claim (5.8) Proof of (5.8). We use an estimate for radial functions from [24]. Similar estimates were previously obtained in [12, 26, 29]. Lemma A.1 ([24, Lemma 6.3]). Let d ≥ 2 and α ∈ (0, d), then for every measurable function f : [0, ∞) → [0, ∞) ˆ ∞ˆ ∞ x f (|x|)f (|y|) R f (r)Kα,d (r, s)f (s)r d−1 sd−1 dr ds dx dy = d−α |x − y| 0 0 d d R ×R

R : [0, ∞) × [0, ∞) → ∞ is defined for r, s ∈ [0, ∞) × [0, ∞) by where the kernel Kα,d d−3 d−3 ˆ 1 z 2 (1 − z) 2 R Kα,d (r, s) = Cd d−α dz. 0 ((s + r)2 − 4srz) 2

Moreover, there exists M > 0 such that

(A.1)

R Kα,d (r, s) ≤ M

 1 d−1  ) 2 |r−s|1 1−α ( rs    d−1

1 ( rs ) 2 ln     1 d−α 2

( rs )

2|r+s| |r−s|

if α < 1, if α = 1, if α > 1.


23

Case α > 1. From (A.1) we obtain for radially symmetric functions that ˆ ∞ˆ ∞ x |ϕ(x)|q |ϕ(y)|q |ϕ(r)|q |ϕ(s) |q r d−1 sd−1 dx dy ≤ C dr ds, d−α d−α |x − y| 2 0 0 (rs) d d R ×R

and hence that x Rd ×Rd

|ϕ(x)|q |ϕ(y)|q dx dy ≤ C |x − y|d−α

ˆ

∞

q

|ϕ(r)| r

d +α −1 2 2

dr

0

2

.

Let u = uλ,R,S be defined in (5.5). Then x Rd ×Rd

|u(x)|q |u(y)|q dx dy ≤ Cλ2q |x − y|d−α

ˆ

!q

S − r − R S

R+S R−S

r

d +α −1 2 2

dr

!2

.

S− r−R S

Using the trivial estimate < 1 it follows that x |u(x)|q |u(y)|q 2 d d 2q +α +α 2 2 − (R − S) 2 2 dx dy ≤ Cλ (R + S) |x − y|d−α d d R ×R

and we get the desired estimate. Case α = 1. From (A.1) we obtain for radially symmetric functions that ˆ ∞ˆ ∞ x |ϕ(x)|q |ϕ(y)|q |ϕ(r)|q |ϕ(s)|q r d−1 sd−1 2|r + s| dx dy ≤ C dr ds, ln d−1 |x − y|d−α |r − s| 2 0 0 (rs) d d R ×R and hence that ˆ ∞ˆ ∞ x |ϕ(x)|q |ϕ(y)|q d 1 d 1 2|r + s| dr ds. dx dy ≤ C |ϕ(r)|q |ϕ(s)|q r 2 − 2 s 2 − 2 ln d−α |x − y| |r − s| 0 0 d d R ×R

S− r−R S

Let u = uλ,R,S be defined in (5.5). Using the estimates < 1 and r ≤ R + S, s ≤ R + S we have ˆ R+S ˆ R+S x |u(x)|q |u(y)|q 2|r + s| 2q d−1 dx dy ≤ Cλ (R + S) dr ds ln d−α |x − y| |r − s| R−S R−S d d R ×R

and we can conclude that x |u(x)|q |u(y)|q Rd ×Rd

i.e.

x Rd ×Rd

|x − y|d−α

dx dy ≤ Cλ2q Rd−1

ˆ

R+S

R−S

ˆ

R+S

ln R−S

2|r + s| dr ds |r − s|

|u(x)|q |u(y)|q dx dy ≤ Cλ2q Rd−1 S 2 (ln R − ln S + 1) = O(λ2q Rd−1+β S 2 ). |x − y|d−α

Case 0 < α < 1. This case is similar to α = 1, we omit the details.

24


Acknowledgements J. Bellazzini and M. Ghimenti were supported by GNAMPA 2016 project “Equazioni non lineari dispersive”. M. Ghimenti was partially supported by P.R.A. 2016, University of Pisa. J. Van Schaftingen was supported by the Projet de Recherche (Fonds de la Recherche Scientifique–FNRS) T.1110.14 “Existence and asymptotic behavior of solutions to systems of semilinear elliptic partial differential equations”. References [1] H. Bahouri, J.-Y. Chemin, and R. Danchin, Fourier analysis and nonlinear partial differential equations, Grundlehren der Mathematischen Wissenschaften, vol. 343, Springer, Heidelberg, 2011. ↑3, 8, 9, 17 [2] J. Bellazzini, R. L. Frank, and N. Visciglia, Maximizers for Gagliardo–Nirenberg inequalities and related non-local problems, Math. Ann. 360 (2014), no. 3–4, 653–673. ↑2, 3, 9, 10 [3] J. Bellazzini, M. Ghimenti, and T. Ozawa, Sharp lower bounds for Coulomb energy, Math. Res. Lett. 23 (2016), no. 3, 621–632. ↑6, 9 [4] J. Bellazzini, T. Ozawa, and N. Visciglia, Ground states for semi-relativistic Schrödinger-Poisson-Slater energies, To appear in: Funkcial. Ekvac., available at arXiv:1103.2649v2. ↑3 [5] R. Benguria, H. Brézis, and E. H. Lieb, The Thomas-Fermi-von Weizsäcker theory of atoms and molecules, Comm. Math. Phys. 79 (1981), no. 2, 167–180. ↑2 [6] I. Catto, J. Dolbeault, O. Sánchez, and J. Soler, Existence of steady states for the Maxwell-Schrödinger– Poisson system: exploring the applicability of the concentration-compactness principle, Math. Models Methods Appl. Sci. 23 (2013), no. 10, 1915–1938. ↑2 [7] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486–490. ↑10 [8] Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Commun. Contemp. Math. 11 (2009), no. 3, 355–365. ↑12 [9] P. L. De Napoli, Symmetry breaking for an elliptic equation involving the fractional Laplacian, available at arXiv:1409.7421. ↑11 [10] P. L. De Napoli and I. Drelichman, Elementary proofs of embedding theorems for potential spaces of radial functions, Methods of Fourier Analysis and Approximation Theory (M. Ruzhansky and S. Tikhonov, eds.), Birkhäuser, Basel, 2016, pp. 115–138. ↑12 [11] P. L. De Nápoli, I. Drelichman, and R. G. Durán, On weighted inequalities for fractional integrals of radial functions, Illinois J. Math. 55 (2011), no. 2, 575–587 (2012). ↑12 [12] J. Duoandikoetxea, Fractional integrals on radial functions with applications to weighted inequalities, Ann. Mat. Pura Appl. (4) 192 (2013), no. 4, 553–568. ↑22 [13] J. Fröhlich, E. H. Lieb, and M. Loss, Stability of Coulomb systems with magnetic fields. I. The one-electron atom., Comm. Math. Phys. 104 (1986), no. 2, 251–270. ↑9 [14] A. E. Gatto, Product rule and chain rule estimates for fractional derivatives on spaces that satisfy the doubling condition, J. Funct. Anal. 188 (2002), no. 1, 27–37. ↑8 [15] A. Gulisashvili and M. A. Kon, Exact smoothing properties of Schrödinger semigroups., Amer. J. Math. 118 (1996), no. 6, 1215–1248. ↑21 [16] C. Le Bris and P.-L. Lions, From atoms to crystals: a mathematical journey, Bull. Amer. Math. Soc. (N.S.) 42 (2005), no. 3, 291–363. ↑2 [17] M. Ledoux, On improved Sobolev embedding theorems, Math. Res. Lett. 10 (2003), no. 5-6, 659–669. ↑3 [18] E. H. Lieb, Thomas-Fermi and related theories of atoms and molecules, Rev. Modern Phys. 53 (1981), no. 4, 603–641. ↑2 [19] E. H. Lieb and R. Seiringer, The stability of matter in quantum mechanics, Cambridge University Press, Cambridge, 2010. ↑2 [20] P.-L. Lions, Some remarks on Hartree equation, Nonlinear Anal. 5 (1981), no. 11, 1245–1256. ↑2 [21] , Solutions of Hartree–Fock equations for Coulomb systems, Comm. Math. Phys. 109 (1987), no. 1, 33–97. ↑2, 3 [22] J. Lu, V. Moroz, and C. B. Muratov, Orbital-free density functional theory of out-of-plane charge screening in graphene, J. Nonlinear Sci. 25 (2015), no. 6, 1391–1430. ↑2


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