It is shown that the sharp constant in the Hardy-Sobolev-Maz'ya inequality on
the three dimensional upper half space is given by the Sobolev constant. This
is achieved by a duality argument relating the problem to a
Hardy-Littlewood-Sobolev type inequality whose sharp constant is determined as
well.Comment: 9 page

Benguria, Rafael D.

Frank, Rupert L.

Loss, Michael

English

arXiv

It is shown that the sharp constant in the Hardy-Sobolev-Maz’ya inequality on the upper half space H^3 ⊂ R^3 is given by the Sobolev constant. This is achieved by a duality argument relating the problem to a Hardy-Littlewood-Sobolev type inequality whose sharp constant is determined as well

Caltech Authors - Main

The sharp constant in the Hardy-Sobolev-Maz’ya inequality in the three dimensional upper half-space

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The sharp constant in the
Hardy-Sobolev-Maz’ya inequality in the three
dimensional upper half-space
Rafael D. Benguria1, Rupert L. Frank2 and Michael Loss3
1. Department of Physics, P. Universidad Cato´lica de Chile,
Casilla 306, Santiago 22, Chile, email: rbenguri@fis.puc.cl
2. Matematiska institutionen, KTH Stockholm,
100 44 Stockholm, Sweden, email: rupert@math.kth.se
3. School of Mathematics, Georgia Tech, Atlanta, GA 30332
email: loss@math.gatech.edu
May 7, 2007
Abstract
It is shown that the sharp constant in the Hardy-Sobolev-Maz’ya inequality on the
upper half space H3 ⊂ R3 is given by the Sobolev constant. This is achieved by a
duality argument relating the problem to a Hardy-Littlewood-Sobolev type inequality
whose sharp constant is determined as well.
1 Introduction
The present work is concerned with a particular case of the Hardy-Sobolev-Maz’ya inequality
∫
Hn
[
|∇f(x)|2 − 1
4y2
|f(x)|2
]
dx ≥ Cn
(∫
Hn
|f(x)| 2nn−2dx
)n−2
n
(1)
1Work partially supported by Fondecyt (CHILE) projects 106–0651 and 706–0200, and CONICYT/PBCT
Proyecto Anillo de Investigacio´n en Ciencia y Tecnolog´ıa ACT30/2006.
2Work partially supported by the Swedish Foundation for International Cooperation in Research and
Higher Education (STINT).
3Work partially supported by NSF-grant DMS-0600037.
c© 2007 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.
1
BFL/5/7/07 2
where f is a compactly supported function that lives in the half space
H
n := {x = (x, y) : x ∈ Rn−1, y > 0} . (2)
It is quite easy to see that the left side is positive; this is Hardy’s inequality. That (1)
holds for a strictly positive constant Cn was proved by Maz’ya [9] (Section 2.1.6., Corollary
3). In what follows, Cn denotes the sharp constant in the above inequality. It was shown
in recent work by Tertikas and Tintarev [10], that an optimizer for the sharp constant Cn
exists provided the dimension n ≥ 4.
The functional (1) has a number of equivalent formulations. For once it is equivalent to
the inequality
∫
Bn
|∇g(Ω)|2dΩ−
∫
Bn
1
(1− |Ω|2)2 |g(Ω)|
2dΩ ≥ Cn
(∫
Bn
|g(Ω)| 2nn−2dΩ
)n−2
n
(3)
where Bn is the unit ball in Rn. To see this, set
f(x, y) =
(
2
(1 + y)2 + x2
)n−2
2
g(B(x, y)) (4)
where B is the Mo¨bius transformation that maps the upper half space Hn to the unit ball
B
n, i.e.,
Ω = B(x, y) =
(2x, 1− x2 − y2)
(1 + y)2 + x2
. (5)
Inserting (4) into (1) a basically straightforward computation involving some integration by
parts yields (3). Clearly, this functional is invariant under rotation. Note that these two
representations, the one on the half space and the one on the unit ball show the invariance
of the functional under all Mo¨bius transformations that preserve the upper half space. This
indicates that the term containing the expression (1− |Ω|2)−2 has some intrinsic geometric
meaning. A natural way to write the problem (1) is via stereographic projection from the
unit ball to the hyperboloid Pn. Once more, set
g(Ω) =
(
2
1− |Ω|2
)n−2
2
k(P (u)) (6)
where
P (u) =
(2Ω, 1 + |Ω|2)
1− |Ω|2 . (7)
It is easy to check that P maps the unit ball to the upper branch of the hyperboloid u2−v2 =
1, where u = (u, v), u ∈ Rn and v ∈ R. Inserting (6) into (3) yields the equivalent inequality
∫
Pn
|∇k(u)|2dVol− (n− 1)
2
4
∫
Pn
|k(u)|2dVol ≥ Cn
(∫
Pn
|k(u)| 2nn−2dVol
)n−2
n
. (8)
BFL/5/7/07 3
The metric used here on Pn is the one induced by the Euclidean space Rn+1.
As mentioned before the half space problem has been investigated in [10], but in its
formulation on the hyperbolic space it has also been investigated before (see [6] for references)
although under a different point of view. There one asks whether there exists a constant Bn
such that the inequality
∫
Pn
|∇k(u)|2dVol ≥ Sn
(∫
Pn
|k(u)| 2nn−2dVol
)n−2
n
+Bn
∫
Pn
|k(u)|2dVol (9)
holds. Here Sn is the Sobolev constant,
n(n− 2)
4
|Sn| 2n (10)
where |Sn| is the volume of the n-dimensional unit sphere in Rn+1. For n > 3 the sharp
constant Bn =
n(n−2)
4
(see [6]). Note that n(n−2)
4
< (n−1)
2
4
. In this language, the problem
investigated in [10] is different, i.e., replace Bn by the optimal constant and then find the
sharp constant Cn that will replace Sn. Certainly Cn ≤ Sn, in fact Cn < Sn for n > 3. Note
that, in this case the exact value of Cn is not known.
In both formulations the interesting case n = 3 is conspicuously absent and it is this case
we would like to address in this letter. We have
1.1 THEOREM. The inequality
∫
H3
|∇f(x)|2dx ≥
∫
H3
1
4y2
|f(x)|2dx+ S3
(∫
H3
|f(x)|6dx
) 1
3
(11)
holds where S3 is the sharp Sobolev constant in three dimensions, i.e.,
S3 = 3(pi/2)
4/3 . (12)
The inequality is always strict for nonzero f ’s. Using the formulation on hyperbolic space
we have the inequality
∫
P3
|∇k(u)|2dVol ≥ S3
(∫
P3
|k(u)|6dVol
) 1
3
+
∫
P3
|k(u)|2dVol . (13)
In contrast to the case n = 3, for n ≥ 4 the sharp constant is always attained for some
nonzero function (see [10]).
The problem (1) has been generalized to the case where the underlying domain D is a
convex set. In this case one replaces 1
4y2
by 1
4d(x)2
where d(x) is the distance of the point
x ∈ D to the boundary of D. It is conjectured in [10] that the sharp constant for convex
domains is given by the half space problem. This is true for the case where the domain is a
ball. We have
BFL/5/7/07 4
1.2 THEOREM. The inequality∫
Bn
|∇g(Ω)|2dΩ−
∫
Bn
1
4(1− |Ω|)2 |g(Ω)|
2dΩ ≥ Cn
(∫
Bn
|g(Ω)| 2nn−2dΩ
)n−2
n
(14)
holds for all smooth functions compactly supported in the unit ball. For nonzero g’s the
inequality is always strict.
The inequality follows directly from (3) by noting that for |Ω| < 1,
1
(1− |Ω|2)2 >
1
4(1− |Ω|)2 . (15)
That the inequality is sharp and always strict for non-zero functions can be seen by scaling
down a compactly supported ‘almost’ optimizer of the half space problem and use this as
a trial function for the ball problem. Note that this device also works for general convex
domains. The hard part is to establish the analog of (14) for general convex domains.
An amusing consequence of the formulation (3) is that by inversion with respect to the
unit sphere one obtains a sharp inequality on the complement of the unit ball, i.e., we have
1.3 THEOREM. The inequality∫
(Bn)c
|∇g(Ω)|2dΩ−
∫
(Bn)c
1
(1− |Ω|2)2 |g(Ω)|
2dΩ ≥ Cn
(∫
(Bn)c
|g(Ω)| 2nn−2dΩ
)n−2
n
(16)
holds for all functions that are smooth and have compact support on (Bn)c the complement
of the ball Bn in Rn. Moreover, for n > 3 equality can be attained in the sense of [10].
The appropriate formulation of this inequality for general domains, not necessarily con-
vex, is an open problem. Theorem 1.3 suggests that the ‘correct’ inequality is formulated in
terms of either the harmonic radius or the hyperbolic radius of a domain D. For a definition
of these concepts we refer the reader to [1]. Both of these objects are conformally covariant,
i.e., under conformal transformations they scale with the n-th root of the Jacobian. In the
case of a ball, the two concepts coincide and are equal to (1− |Ω|2). Since the ball and the
half space are conformally the same, these two concepts coincide also on the half space and
are given by 2y. Thus, it is natural to ask for which domain D does the inequality∫
D
[
|∇f |2 − 1
R(x)2
|f(x)|2
]
dnx ≥ Cn
(∫
D
|f(x)| 2nn−2dnx
)n−2
n
(17)
hold. Here R(x) is either the harmonic radius or the hyperbolic radius. In this formulation,
due to its conformal invariance, one might be able to show that the Hardy-Sobolev-Maz’ya
inequality for general convex domains holds with the same constant as the one on the half
space.
The plan of the paper is the following. In Section 2 we derive the Green function for
fractional powers of the operator −∆ − 1
4y2
. This yields Hardy-Littlewood-Sobolev type
kernels. In Section 3 we prove Lp estimates for these kernels and recover Theorem 1.1.
BFL/5/7/07 5
2 The Green function
It is convenient to start with the following heat type equation on the upper half space Hn
ut = ∆u+
1
4y2
u , u(x, y; 0) = f(x, y) . (18)
Substituting u =
√
yg one obtains the equation
gt = ∆xg + gyy +
1
y
gy , g(x, y; 0) =
f(x, y)√
y
, (19)
and one see that the right side of the equation is an n+1 dimensional Laplacian. Note that
gyy+
1
y
gy is the two dimensional Laplacian of a radial function. A similar idea has been used
in [2] in a different context. With this in mind one arrives at once at the following formula
for the solution of the heat equation
u(x, y; t) =
∫
Hn
G(x− x′, y, y′; t)f(x′, y′)dx′dy′ (20)
where
G(x− x′, y, y′; t) =
(
1
4pit
)n+1
2 √
yy′e−
(x−x′)2+y2+y′2
4t
∫ 2pi
0
e
yy
′
2t
cos φdφ . (21)
It is not hard to see that this heat kernel is a contraction semigroup on L2(Hn) with Lebesgue
measure. Thus, the generator Q is a selfadjoint operator and it is an extension of −∆− 1
4y2
originally defined on smooth functions with compact support in Hn. Note that the L2-norm
of the gradient of functions in the domain of Q is in general not finite. We shall continue to
use the symbol −∆− 1
4y2
to denote Q.
It is straight forward to see (see e.g., Theorem 7.10 in [8]) that
lim
t→0
1
t
[
‖f‖2L2(Hn) − (f,Gtf)L2(Hn)
]
= 2pi
∫
Hn
(|∇xg|2 + |gy|2) ydydx (22)
where Gtf is the solution of the intial value problem (18) and g =
f√
y
. Note that the right
hand side is manifestly positive and coincides with the interpretation of −∆ − 1
4y2
given in
[10].
Via the heat kernel it is straightforward to find the kernel of the fractional powers
(−∆− 1
4y2
)−
α
2 (x;x′) =
1
Γ(α
2
)
∫ ∞
0
t
α
2G(x− x′, y, y′; t)dt
t
, (23)
for α > 0, and a calculation leads to the expression
(−∆− 1
4y2
)−
α
2 (x;x′) (24)
= 2−αpi−
n+1
2
Γ(n+1−α
2
)
Γ(α
2
)
√
yy′
∫ 2pi
0
[
(x− x′)2 + y2 + y′2 − 2yy′ cosφ]−n+1−α2 dφ (25)
=: Φn,α(x;x
′) . (26)
BFL/5/7/07 6
Similarly, well known expressions hold for (−∆)−α2 on Rn which, for reasons that become
clear later, we write in terms of the variables (x, y) as
(−∆)−α2 (x;x′) (27)
= 2−αpi−
n
2
Γ(n−α
2
)
Γ(α
2
)
[
(x− x′)2 + (y − y′)2]−n−α2 (28)
=: Ψn,α(x;x
′) . (29)
First we state some simple pointwise properties about the kernel Φn,α.
2.1 LEMMA. If n ≤ α ≤ n+ 1, we have that
sup
a
Φn,α(x, y + a; x
′, y′ + a) = lim
a→∞
Φn,α(x, y + a; x
′, y′ + a) ≡ ∞ . (30)
If n− 1 ≤ α < n we have that
sup
a
Φn,α(x, y + a; x
′, y′ + a) = lim
a→∞
Φn,α(x, y + a; x
′, y′ + a) ≡ Ψn,α(x;x′) . (31)
Proof. An elementary calculation shows that
Φn,α(x;x
′) = |x− x′|−n+α2−αpi−n+12 Γ(
n+1−α
2
)
Γ(α
2
)
F (A) , (32)
where
A =
√
yy′
|x− x′| (33)
and
F (A) :=
∫ pi
−pi
A
[1 + 2A2(1− cos(φ))]n+1−α2
dφ . (34)
All the statements are an immediate consequence of Lemma 4.1 with β = n+1−α
2
.
3 Lp-estimates for fractional powers
As a consequence of Lemma 2.1 and Lieb’s sharp constant in the Hardy-Littlewood Sobolev
inequality [7] we have the following corollary.
3.1 COROLLARY. If n ≤ α ≤ n+ 1 then the operator
(−∆− 1
4y2
)−
α
2 (35)
is not bounded on Lp(Hn) for any 1 ≤ p ≤ ∞. If n − 1 ≤ α < n then this operator is a
bounded operator from Lp(Hn) to Lq(Hn) for all 1 < p, q <∞ that satisfy
1
q
=
1
p
− α
n
. (36)
BFL/5/7/07 7
Moreover, for such values of α we have
(f, (−∆− 1
4y2
)−
α
2 f) ≤ 2−αpi−n2 Γ(
n−α
2
)
Γ(α
2
)
C(n, α)‖f‖2p (37)
where p = 2n
n+α
and
C(n, α) = pi
n−α
2
Γ(α
2
)
Γ(n+α
2
)
[
Γ(n
2
)
Γ(n)
]−α
n
(38)
is the sharp constant. This constant is not attained in (37) for nonzero functions.
Proof of Theorem 1.1. We write
|(f, g)| = |(Qα/4f,Q−α/4g)| ≤ (f,Qα/2f)1/2(g,Q−α/2g)1/2 (39)
which by Corollary 3.1 yields the bound
|(f, g)|2 ≤ 2−αpi−n2 Γ(
n−α
2
)
Γ(α
2
)
C(n, α)(f,Qα/2f)‖g‖2p (40)
for n− 1 ≤ α < n and p = 2n
n+α
. Thus,
‖f‖2p′ < 2−αpi−
n
2
Γ(n−α
2
)
Γ(α
2
)
C(n, α)(f,Qα/2f) , (41)
and there is never equality in the above inequality for nonzero functions. Theorem 1.1 follows
by choosing n = 3 and α = 2.
The reader may wonder what happens when 0 < α < n− 1. While we do not succeed in
calculating the sharp constant, it is possible to show that the sharp constant in inequality
(37) is attained. While this constant is strictly bigger than the corresponding constant in
the Hardy-Littlewood-Sobolev inequality, we do not know its exact value.
The procedure for proving this relies on the conformal invariance of the kernel which
allows to transform the problem into one on the unit ball. Then the device of competing
symmetries developed in [4] allows to restrict the maximization problem to radial functions
on the ball. The correction term to Fatou’s lemma ([3], see also [8]) then allows to show the
existence of a maximizer. Thus, we recover some of the results in [10] with a different proof.
Moreover, it is also possible to show that every maximizer is the conformal image of a radial
function. The details will appear elsewhere.
4 Appendix
In this appendix we collect some facts about the function
F (A) :=
∫ pi
−pi
A
(1 + 2A2(1− cos(φ))β dφ , (42)
where β = n+1−α
2
.
BFL/5/7/07 8
4.1 LEMMA. Depending on the value of β, the function F (A) has the following asymptotics
as A→∞.
a) If 0 ≤ β ≤ 1
2
then limA→∞ F (A) =∞.
b) If 1
2
< β ≤ 1, then F (A) is a monotone increasing function and
lim
A→∞
F (A) =
√
pi
Γ(β − 1
2
)
Γ(β)
. (43)
Proof. Since
F (A) =
∫ piA
−piA
1
1 + 2A2(1− cos( φ
A
))β
dφ (44)
the limit as A→∞ is ∫ ∞
−∞
1
(1 + φ2)β
dφ =
√
pi
Γ(β − 1
2
)
Γ(β)
(45)
for β > 1
2
and it is +∞ for β ≤ 1
2
. This proves a). To see that b) holds for β = 1 one easily
performs the φ integration and obtains
F (A) =
2piA√
1 + 4A2
, (46)
which is obviously increasing with A. For 1
2
< β < 1 we use the formula
[
1 + 2A2(1− cosφ)]−β (47)
=
sin(piβ)
pi
∫ ∞
0
[
1 + t+ 2A2(1− cosφ)]−1 t1−β dt
t
. (48)
Integrating with respect to φ yields
F (A) = 2 sin(piβ)
∫ ∞
0
A√
(1 + t)2 + 4(1 + t)A2
t1−β
dt
t
. (49)
Again, this function increases with A.
References
[1] Bandle, C. and Flucher, M., Harmonic radius and concentration of energy; hyperbolic
radius and Liouville’s equations ∆U = eU and ∆U = U
n+2
n−2 , SIAM Review, 38 (1996),
191–238.
[2] Beckner, W., On the Grushin operator and hyperbolic symmetry, Proc. Amer. Math.
Soc. 129 (2000), 1233–1246.
[3] Bre´zis, H. and Lieb, E.H., A relation between pointwise convergence of functions and
convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490.
BFL/5/7/07 9
[4] Carlen, E.A. and Loss, M., On the minimization of symmetric functionals, Rev. Math.
Phys. 6, (1994), 1011–1032.
[5] Filippas, S., Maz’ya, V. G. and Tertikas, A., Sharp Hardy–Sobolev inequalities, C. R.
Math. Acad. Sci. Paris 339 (2004), no. 7, 483–486.
[6] Hebey, E., Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant
Lecture Notes in Mathematics Vol. 5, AMS, Providence, RI, 1999.
[7] Lieb, E.H., Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities,
Ann. Math. 118 (1983), 349–374.
[8] Lieb, E.H. and Loss, M., Analysis, Graduate Studies in Mathematics Vol. 14, AMS,
Providence, 2001.
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[10] Tertikas, A. and Tintarev, K., On existence of minimizers for the Hardy–Sobolev–
Maz’ya inequality, published online in Ann. Mat. Pura Appl., 7 September 2006.


The sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half-space

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