J Geom Anal DOI 10.1007/s12220-017-9883-3
Weighted Moser–Onofri–Beckner and Logarithmic Sobolev Inequalities Nguyen Lam1 · Guozhen Lu2
Received: 7 March 2017 © Mathematica Josephina, Inc. 2017
Abstract Motivated by recent results on the best constants and extremal functions for a family of the Caffarelli–Kohn–Nirenberg inequalities in [17,23], we will study weighted Moser–Onofri–Beckner inequalities on the Euclidean space R N . We also set up sharp weighted versions of the logarithmic Sobolev inequalities together with their best constants and optimizers. Keywords Best constants · Extremal functions · Moser–Onofri–Beckner inequalities · Logarithmic Sobolev inequalities Mathematics Subject Classification 26D10 · 46E35
1 Introduction To study the prescribed Gauss curvature problem on two-dimensional sphere S2 , in [25], Moser established the following exponential type inequality on the 2-dimensional sphere S2 with an optimal constant: For every β ≤ 4π , there exists a constant C(β) > 0 such that for all u ∈ H 1 (S2 )
B
Guozhen Lu
[email protected] Nguyen Lam
[email protected]
1
Department of Mathematics, University of British Columbia and The Pacific Institute for the Mathematical Sciences, Vancouver, BC V6T 1Z4, Canada
2
Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA
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N. Lam, G. Lu
β (u − u)2 exp 4π S2 |∇u|2 dω S2
dω ≤ C(β).
Here dω denotes the standard surface measure on S2 , normalized so that S2 dω = 1, and u = S2 udω. Moreover, the constant 4π is sharp in the sense that if β > 4π , then sup u∈H 1 (S2 )
β (u − u)2 exp 4π S2 |∇u|2 dω S2
dω = ∞.
As a consequence of this result, we have that the functional 1 |∇u|2 dω + udω − ln eu dω , J (u) = 4 S2 S2 S2 is bounded from below on H 1 (S2 ). Moser used the following classical Schwarz rearrangement argument: every smooth function u is associated to a function u ∗ such that u ∗ is constant on the parallel circles and such that for any continuous function f : f (u)dω = f (u ∗ )dω. S2
S2
Moreover, the well-known Pólya–Szegö inequality ∗ p ∇u dω ≤ |∇u| p dω S2
(1.1)
S2
plays a crucial role in the approach of Moser and enables him to reduce the consideration to a one-dimensional problem. Onofri, using conformal invariance and results in a paper of Aubin [4], showed in [26] that actually J (u) is bounded from below by 0, and that modulo conformal transformations, u = 0 is the optimizer: For β ≥ 1, inf
u∈H 1 (S2 )
Jβ (u) = 0,
while if 0 ≤ β < 1 inf
u∈H 1 (S2 )
Jβ (u) = −∞.
Here Jβ (u) =
β 4
S2
|∇u|2 dω +
S2
udω − ln
S2
eu dω
Other proofs for this result were provided by Hong in [22] and by Osgood–Phillips– Sarnak in [27]. Also, in 2005, Ghigi made use of the convex analysis, in particular, the well-known Prékopa–Leindler inequality, to give a new proof of the Moser–Onofri
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Moser–Onofri–Beckner and Logarithmic Sobolev Inequalities
inequality in [19]. See Chaps. 16–18 in the book [20] of Ghoussoub and Moradifam for more details. The Moser–Onofri inequality was also extended into higher-dimensional N -spheres S N . For instance, Beckner in [5] proved that for a real-valued function F defined on ∞ the sphere S N with an expansion in spherical harmonics F = Yk , k=0
ln
SN
e F dξ
≤
SN
Fdξ +
1 2N
∞ k=1
(N + k) (N )(k)
SN
|Yk |2 dξ.
Here dξ denotes the normalized surface measure. Moreover, equality happens if and only if e F is given by the Jacobian determinant of a conformal transformation of the N -sphere. Also, in [9], Carlen and Loss used the method of competing symmetries to derive the sharp version of the logarithmic Hardy–Littlewood–Sobolev inequality and deduce Beckner’s result as an application. Sharp Moser inequalities and a weak form of the Moser–Onofri type inequalities were also derived on spheres in complex space C N by Cohn and the second author in [11] using a sharp representation formula for functions on complex spheres in terms of complex tangential gradients (we refer to [10] for more background of complex tangential gradients). In these results, the smaller complex tangential gradient |∇C u| replaces the real tangential gradient |∇t u|, and also the critical exponent is 2N2N−1 rather 2N −1 2N than 2N −2 , which differ from the real Moser inequality on the sphere in R . It is worth noting that if we use the stereographic projection from S2 to R2 , then we could obtain the following Euclidean version of the Moser–Onofri inequality: for 1 all u ∈ L 1 (R2 , dμ) such that ∇u ∈ L 2 (R2 , dx) with dμ = π1 (1+|x| 2 )2 dx, 1 |∇u|2 dx. ln eu dμ − udμ ≤ 2 2 16π R R R2 This Euclidean Moser–Onofri inequality can also be deduced as a limiting procedure based on other functional inequalities [16] or from optimal mass transport [3]. Effort has also been made in [15] in order to get the Euclidean Moser–Onofri– Beckner inequality on R N . More precisely, Del Pino and Dolbeault proved that for any smooth compactly supported function u
u ln e dμ N − udμ N ≤ α N H N (x, ∇u)dx. RN
RN
RN
The best constant α N is given by αN =
N 1−N
N 2
2(N − 1)π
N 2
.
Here we denote for X , Y ∈ R N R N (X, Y ) = |X + Y | N − |X | N − N |X | N −2 X · Y
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and
N −2
H N (X, Y ) = R N −
N |X |− N −1 N
1 + |X | N −1
N −1 Y X, N
.
The Euclidean Moser–Onofri inequality has also been studied in the presence of weights. For example, the following result has been set up recently in [16,17] using the weighted inequalities: Let 0 ≤ s < 2. Then for any
Caffarelli–Kohn–Nirenberg
u : u ∈ L 1 R2 , dμs and |∇u| ∈ L 2 R2 , dx ,
u
ln
R2
e dμs
1 − udμs ≤ 2 8 (2 − s) π R
R2
|∇u|2 dx,
where dμs =
1 dx 2−s
2 |x|s . 2−s 2π 1 + |x|
Motivated by the above discussions, in this article, we will set up the following version of the weighted Euclidean Moser–Onofri–Beckner inequality: Theorem 1.1 Let 0 ≤ s < N and assume that u is any smooth compactly supported function. Then we have
e dμ N ,s −
u
ln
RN
RN
udμ N ,s ≤ α N ,s
RN
H N (x, ∇u) dx.
(1.2)
Here dμ N ,s
N −s −N dx N − s N2 + 1 1 + |x| N −1 = , N |x|s N π2
and α N ,s =
N N −s
N −1
N 2
2(N − 1)N N −1 π
N 2
.
It is easy to check that the equality in (1.2) is attained by constants. However, we do not know whether or not (1.2) can be achieved by nonconstant optimizers. This question is still left open. Corollary 1.1 When N = 2, R2 (X, Y ) = |Y |2 and H2 (X, Y ) = the following weighted Euclidean Moser–Onofri inequality:
ln
123
R2
eu ddμ2,s −
R2
udμ2,s ≤
1 8π (2 − s)
1 4
|Y |2 . So we get
RN
|∇u|2 dx,
Moser–Onofri–Beckner and Logarithmic Sobolev Inequalities
where −2 dx 2−s 1 + |x|2−s . |x|s 2π
dμ2,s = Corollary 1.2 When N = 4,
R4 (X, Y ) = |Y |4 + 2 |X |2 |Y |2 + 4 |Y |2 X · Y + 4(X · Y )2 and H4 (X, Y ) = R N
2
4|X |− 3
3 X, Y − 4 4 1 + |X | 3
4 2 2 3 |X | 3 |Y |2 27 |X |− 3 |Y |2 4 |Y | + 18 = X ·Y + 4 2 4 4 1 + |X | 43 1 + |X | 3 4
|X |− 3 2 + 36 (X · Y ) . 4 2 1 + |X | 3 Hence, we get the following weighted Euclidean Moser–Onofri–Beckner inequality: eu dμ4,s − udμ4,s R4 R4 ⎛ ⎞ 2 1 2 2 |x| 3 |x| 3 |∇u| ⎟ x 1 ⎜ 9 |∇u|2 + 6 ≤ + 18 · ∇u ⎝ ⎠ dx. 4 4 2 |x| 6(4 − s)3 π 2 R4 16 1+|x| 3 1+|x| 3
ln
Here dμ4,s =
4−s −4 dx 4−s 3 |x| 1 + . |x|s 2π 2
As we can see, this inequality contains inhomogeneous Sobolev–Orlicz norms which are quite different (and unexpected) from the results of Onofri [26] and Beckner [5]. From (1.2), for smooth compactly supported function u, we get that 1 ε→0 ε 2 lim
ln
RN
eε(u−u) dμ N ,s
1 ε→0 ε 2
≤ α N ,s lim
RN
H N (x, ε∇ (u − u)) dx,
with u=
RN
udμ N ,s .
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We also note that
ln
RN
ε2 |u − u|2 dμ N ,s + o(ε2 ), eε(u−u) dμ N ,s = 2 RN
and
RN
H N (x, ε∇(u − u)) dx = ε
2 RN
G N (x, ∇ (u − u)) dx + o(ε2 ).
Hence, as a byproduct, we obtain a Poincaré type inequality: Corollary 1.3 For 0 ≤ s < N and u being a smooth compactly supported function, we have 1 |u − u|2 dμ N ,s ≤ α N ,s G N (x, ∇u)dx. 2 RN RN Here N −2 1 1 N −1 N |X |− N −1 Y G N (X, Y ) = lim 2 H N (X, εY ) = lim 2 R N − X, ε N ε→0 ε ε→0 ε N 1 + |X | N −1 N −2 N −1 N |X |− N −1 = LN − Y , X, N N 1 + |X | N −1 and 1 1 d2 |X + tY | N |t=0 R εY = (X, ) N ε→0 ε 2 2 dt 2 1 = N |X | N −4 (N − 2) (X · Y )2 + |X |2 |Y |2 . 2
L N (X, Y ) = lim
As in [17], we will obtain Theorem 1.1 as a result from the limiting process of a particular family of the Caffarelli–Kohn–Nirenberg (CKN) inequalities. More specifically, the proof of Theorem 1.1 will reply on the following weighted CKN inequalities established recently in [17,23]: Theorem A Let 0 ≤ s < N . When r = N Nq−1 −1 and q > N , we have 1 (1− q ) q r N q − N 1− r √ N π ⎞ 1 1− q ⎛ 1 (1− q ) 1 q N −1 N + 1 N ( r ) N r r q−N 2 N q ⎠ ⎝ . × 2 q−N q (N ) N −1 N
CKN(N , s, q, r ) =
N N −s
N
123
q−N
Moser–Onofri–Beckner and Logarithmic Sobolev Inequalities
Moreover, N −1 N −s − q−N V0 (x) = α 1 + β |x| N −1 for some α ∈ R and β > 0, are optimizers of CKN(N , s, q, r ). Here CKN(N , s, q, r ) =
RN
sup N ,q (R N )
u∈D0,s
RN
|∇u| N dx
|u|r
dx |x|s
1/r
N1 (1− qr )
RN
|u|q
dx |x|s
1 r
N R is the completion of the space of smooth compactly supported functions 1/q
1/N dx under the norm R N |∇u| p dx + R N |u|q |x| . s N ,q
and D0,s
Theorem B Assume that 1 < p ≤ p + μ < N, 0 ≤ s = p
Nμ < N, N−p
(C1)
Np N p (q − p) q −1 < ; a= . p−1 N−p ( p − 1) [ pq − N (q − p)] r
p,q Denote by Dμ,s R N the completion of the space of smooth compactly supported 1/ p 1/q dx dx + R N |u|q |x| , and set functions with the norm R N |∇u| p |x| μ s CKN(N , μ, s, p, q, r ) =
sup
u∈Dμ,θ (R N ) p,q
RN
RN
|∇u| p
dx |x|μ
dx |u|r |x| s a
1/r
p
RN
|u|q
dx |x|s
1−a . q
Then CKN(N , μ, s, p, q, r ) 1 + p−1 − 1−a − p−1 (1−a) r p q p N−p = N − p−μ ×
⎛ ⎞ Na p−1 N 1 a q q− p p 2 +1 pq N p − q (N − p) r ⎝ q− p a ⎠ , √ N p−q(N − p) N (q − p) pq p π p−1 N p−1 p q− p p +1
and is achieved when N − p−μ u (x) = A 1 + B |x| N − p
p p−1
− p−1
q− p
for some A ∈ R, B > 0.
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The proof of Theorem A will be provided in Sect. 2 (see Lemma 2.1) for the completeness, while the proof of Theorem B could be found in [23]. The CKN inequalities were introduced by Caffarelli, Kohn, and Nirenberg in their 1984 paper [7]. They play an important role in geometric analysis, partial differential equations, and other branches of modern mathematics. They also generalize many well-known inequalities such as Gagliardo–Nirenberg inequalities, Sobolev inequalities, Hardy–Sobolev inequalities, and Nash’s inequalities. There is another well-known inequality that has a close connection to the CKN inequalities and has been studied extensively in the literature. That is the following sharp L p -logarithmic Sobolev inequality: Theorem C Let p ≥ 1. For any smooth function f such that have RN
| f | p ln | f | p dx ≤
RN
| f | p dx = 1, we
N |∇ f | p dx , ln L p p RN
where
p Lp = N
p−1 e
⎛
p−1 π
− 2p
N
⎝ N
+1
2 p−1 p
⎞p
N
⎠ . +1
The L 1 -logarithmic Sobolev inequalities were studied by Ledoux in [24]. Their optimizers, which are the characteristic functions of the balls, were found by Beckner in [6]. The fact that the sharp L 2 -logarithmic Sobolev inequality is equivalent to the sharp Gross logarithmic Sobolev inequality for the Gaussian measure [21] was pointed out, for example by Carlen in [8]. The optimizers in this case were also
N proved in [8] and are exactly the Gaussians u (x) = (π τ )− 2 exp − 41 τ |x|2 . The optimal L p -logarithmic Sobolev inequalities together with their extremal functions with 1 ≤ p ≤ N were investigated by Del Pino and Dolbeault in [14,15]. Also, Gentil set up in [18] the general sharp L p -logarithmic Sobolev inequalities for all p ≥ 1 and under arbitrary norm on R N , using the Prékopa–Leindler inequality and a particular Hamilton–Jacobi equation. It is also worth noting that one can apply the optimal mass transport to provide other proofs for the sharp L p -logarithmic Sobolev inequalities. See [2,12] for example. The second purpose of this paper is to use Theorem A to derive a version of the sharp weighted L N -logarithmic Sobolev inequalities. More precisely, we will prove the following: N ,N N Theorem 1.2 Let 0 ≤ s < N . For any u ∈ D0,s R such that R N |u| N we have dx |∇u| N dx ≥ |u| N ln |u| N ln SL(N , s) s, N N |x| R R
123
dx |x|s
= 1, (1.3)
Moser–Onofri–Beckner and Logarithmic Sobolev Inequalities
with S L(N , s) =
N N −s
N −1
(N − 1) N −1 1 N2 + 1 . √ N e N −1 (N ) π
The equality happens in (1.3) when N −s u(x) = exp −b |x| N −1 , where N
b > 0 and b
N −1
N − 1 (N − 1) Nπ 2 = N .
2 + 1 N − s N N −1
We will also apply Theorem B to deduce the following sharp weighted L p logarithmic Sobolev inequalities:
p, p dx Theorem 1.3 Assume (C1). For any u ∈ Dμ,s R N such that R N |u| p |x| s = 1, we have dx dx N |∇u| p μ ≥ |u| p ln |u| p s , ln LS(N , p, μ) (1.4) N N |x| |x| p R R with p LS(N , p, μ) = N
N−p N − p−μ
N −1 p N
p−1 e
p−1
⎛
N
⎝ N
+1
⎞p
2 ⎠ p−1 +1 p
N
1 √ π
p .
The equality happens in (1.4) when N − p−μ u (x) = exp −b |x| N − p
p p−1
,
where b > 0 and b
N
p−1 p
N
N π 2 (N − p) ( p − 1) = N
2 + 1 (N − p − μ) p
N p−1 p 1 p−1 . N p p
The rest of the paper is organized as follows: in Sect. 2, we will show several key computations that will be used in the proofs of our main results. Weighted Moser– Onofri–Beckner inequality will then be studied in Sect. 3. Finally in Sects. 4 and 5, we will establish sharp versions of the weighted L N and L p -logarithmic Sobolev inequalities correspondingly.
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2 Some Important Lemmas In this section, we will show some useful estimations and results that will be used in the next sections. Lemma 2.1 Let 0 ≤ s < N . When r = N Nq−1 −1 and q > N , we have 1 (1− q ) q r N q − N 1− r CKN (N , s, q, r ) = √ N π ⎞ 1 1− q ⎛ 1 (1− q ) 1 q N −1 N + 1 N ( r ) N r q−N 2 q N r ⎝ ⎠ . × 2 N −1 N q−N q (N )
N N −s
N
q−N
Moreover, N −1 N −s − q−N V0 (x) = α 1 + β |x| N −1 for some α ∈ R and β > 0, are optimizers of CKN (N , s, q, r ) . Here CKN (N , s, q, r ) =
sup N ,q u∈D0,s (R N )
RN RN
|∇u| N dx
|u|r
dx |x|s
1/r
N1 (1− qr )
RN
|u|q
dx |x|s
1 , r
N R is the completion of the space of smooth compactly supported functions 1/q
1/N dx under the norm R N |∇u| p dx + R N |u|q |x| . s N ,q
and D0,s
Proof Set GN (N , q, r ) =
sup N ,q
u∈D0,0 (R N )
1/r |u|r dx
N1 (1− qr )
RN
RN
|∇u| N dx
RN
|u|q dx
1 . r
Then from the results in [1,13,14], we get that G N (N , q, r ) is equal to
q−N √ N π
123
1− q r
q q−N
1 (1− q ) N
r
N q
1 r
⎛
⎞ N1 (1− qr ) N −1 N2 + 1 q q−N ⎝ ⎠ , N2 (N ) NN−1 q−N
Moser–Onofri–Beckner and Logarithmic Sobolev Inequalities
− N −1 N q−N and can be achieved by U0 (x) = α 1 + β |x| N −1 for some α ∈ R and β > 0. We now set V0 = D −1 N ,s U0 , that is U0 = D N ,s V0 where D N ,s u(x) := D −1 N ,s u(x)
:=
N −s N N N −s
1 N
1 N
u(FN ,s (x)), u(FN−1,s (x)),
and s
FN ,s (x) = |x| N −s x, FN−1,s (x) = |x|− N x. s
We will show that V0 is a maximizer of CKN (N , s, q, r ). Indeed, for any v, we need to show RN
RN
|∇v| N
|v|r
dx |x|s
N1 (1− qr )
dx
1/r
RN
dx |v|q |x| s
RN
1 ≤ r RN
|∇V0
|N
dx
|V0 |r
dx |x|s
1/r
N1 (1− qr )
RN
dx |V0 |q |x| s
1 . r
By Lemma 2.2 in [23], we get
dx N |v| = s |x| N −s
r
RN
RN RN
N N −s
r N
RN
q N
N dx N = |x|s N −s N −s ∇ D N ,s v N dx. |∇v| N dx ≥ |v|q
RN
r D N ,s v (x) dx, D N ,s v (x)
q
dx,
RN
Hence RN
RN
≤
|∇v| N dx
N N −s
|v|r
dx |x|s
1/r
N1 (1− qr )
1 (1− q ) r N
RN
|v|q
dx |x|s
1 r
D N ,s v (x)r dx 1/r 1 (1− q ) q 1 r r ∇ D N ,s v N dx N R N D N ,s v (x) dx
RN
RN
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N. Lam, G. Lu
≤
N N −s
=
|∇V0
1 (1− q ) r N
|N
dx
|V0 |r
|∇U0 | N dx 1/r
|U0 (x)|r dx
N1 (1− qr )
dx |x|s
N1 (1− qr )
dx |V0 |q |x| s
1/r |U0 (x)|q dx
1 r
1 . r
We note that we have the equality in the last row because U0 is radially symmetric. Moreover, we also obtain CKN (N , s, q, r ) =
N N −s
1 (1− q ) r N
GN(N , q, r ).
A direct calculation shows that N −1 N −s − q−N V0 (x) = α 1 + β |x| N −1 for some α ∈ R and β > 0. Lemma 2.2 Let q > N > s ≥ 0 and N −1 N −s − q−N Vq (x) = 1 + |x| N −1 . Then RN
N −N dx Nπ 2 1 =
, |x|s N − s N2 + 1 q dx 1 ω N −1 Vq as q → ∞, → |x|s q RN N (N − s) N −1 N 2N N −2 π 2 1−N ∇Vq N dx = N − s N q N 2 RN 1 as q → ∞. +o N q −1 N −s
1 + |x| N −1
Moreover,
N − s N −2 N − s N − s N Vq = (N − 1) q−N q − N N −1 ⎡ ⎤ − q−1 (N −2)− 2q−N −1 1−s N −s 1−s q−N q−N q−1 N −1 N −1 (N −2)+2 N −1 1 + r r ⎢ q−N ⎥ ×⎣ ⎦. q−1 q−1 N −s − q−N (N −2)− q−N 1−s 2−s−N (N −2)+ N −1 N −1 N −1 − 1+r r
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Moser–Onofri–Beckner and Logarithmic Sobolev Inequalities
Proof If we perform the change of variable: t=
1 1+r
that is r =
N −s N −1
N −1
1−t t
N −s
N −1 and dr = N −s
1−t t
−1+s N −s
·
−1 dt, t2
then
N −s −N dx 1 + |x| N −1 |x|s RN ∞ N −s −N 1 + r N −1 = ω N −1 r N −1−s dr 0
N
N − 1 Nπ 2 =
N − s N2 + 1 N 2
N − 1 Nπ
N − s N2 + 1
=
1
t
N
0 1
1−t t
N −1 N −1−s N −s
1−t t
−1+s N −s
·
1 dt t2
N
(1 − t) N −2 dt =
0
Nπ 2 1 N
. N −s 2 +1
(2.1)
Moreover, RN
N −1 N −s −q q−N dx 1 + |x| N −1 |x|s RN ∞ N −1 N −s −q q−N 1 + r N −1 = ω N −1 r N −1−s dr.
q dx Vq |x|s
=
0
Again, we perform the same change of variables, then
∞
N −s
1 + r N −1
0
N N N = N =
q−N
r N −1−s dr N −1 N −1−s
−1+s 1 − t N −s 1 t · 2 dt t t 0 1 N 2 −q 2 N −N −1 N −1 N −2 q−N B ,N −1 t dt = (1 − t) −s 0 N −s q−N (N − 1) −1 − s N −1 q(N −1) − k k=1 q−N
N −1 = N −s =
−q N −1
N −1 N −s
1
N −1 q q−N
1−t t
N −s
(N − 1) N −1 k=1 [q
(N − 1 − k) + k N ]
(q − N ) N −1 ,
where B is the Euler beta function. Hence, as q → ∞ 1 q
RN
q dx 1 ω N −1 N −1 Vq = . → ω N −1 |x|s N − s N (N − 1) N (N − s)
(2.2)
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N. Lam, G. Lu
Now, noting that Vq is a radial function: Vq (x) = Vq (r ) with r = |x|, we get x ∇Vq = Vq (r ) , r and so N −2 x ∇Vq N −2 ∇Vq = V (r ) Vq (r ) . q r Hence N −2 N Vq = ∇ · ∇Vq ∇Vq N −2 ∂ xi V V = (r ) (r ) q q ∂ xi r N −2 N −2 x r − xi xri xi i Vq (r ) = Vq (r ) + (r ) Vq (r ) V q 2 r r r 2 N −2 −1 N − 2 x x 2 i i 2Vq (r ) Vq (r ) + Vq (r ) Vq (r ) r 2 r N −1 N −2 N −2 + (N − 1) Vq (r ) = Vq (r ) Vq (r ) Vq (r ) r N −2 1 V (r ) + Vq (r ) . = (N − 1) Vq (r ) r q Noting that Vq (r ) = −
q−1 1−s N −s − q−N N −s 1 + r N −1 r N −1 , q−N
and
N −s − q −1 N −s 1 + r N −1 q − N N −1 q−1 2−s−N N −s − q−N 1−s 1 + r N −1 − r N −1 , N −1
N −s Vq (r ) = q−N
2q−N −1 q−N
1−s
r 2 N −1
we obtain N −s q−1 1−s N −2 N −s − q−N N −1 N −1 1+r N Vq = (N − 1) r q − N ⎡ q−1 2q−N −1 1−s ⎤ N −s − q−N 1−s N −s − q−N q−1 N −s N −s N −1 N −1 1 + N −s N −1 1 + r 1 + r r r 2 N −1 ⎥ − ⎢ q−N r q−N q−N N −1 ×⎣ ⎦ q−1 2−s−N N −s − q−N N −s 1−s N −1 N −1 1 + r − q−N r N −1
123
Moser–Onofri–Beckner and Logarithmic Sobolev Inequalities
q−1 N −s − q−N (N −2) 1−s N − s N −2 1 + r N −1 r N −1 (N −2) q−N 2q−N −1 1−s q−1 2−s−N N −s − q−N N −s − q−N N −s N −s q −1 1 + r N −1 × r 2 N −1 − 1 + r N −1 r N −1 q − N N −1 q − N N − s N −2 N − s N − s = (N − 1) q−N q −N N −1 ⎡ ⎤ − q−1 (N −2)− 2q−N −1 1−s N −s 1−s q−N q−N q−1 N −1 N −1 (N −2)+2 N −1 1 + r r ⎢ q−N ⎥ ×⎣ (2.3) ⎦. q−1 q−1 N −s − q−N (N −2)− q−N 1−s 2−s−N −2)+ (N N −1 − 1 + r N −1 r N −1
= (N − 1)
Next, since N −1 N −1 N −s − q−N N −s − q−N −1 N −s N −s x N −1 |x| N −1 −1 1 + |x| N −1 ∇Vq = ∇ 1+|x| N −1 =− , |x| q−N N −1 we get that RN
∇Vq N dx =
=
N −s q−N N −s q−N
N N
RN
q−1 N −s − q−N N 1−s |x| N −1 N dx 1 + |x| N −1
ω N −1
∞
N −s
1 + r N −1
− q−1 N q−N
1−s
r N −1 N r N −1 dr.
0
We perform the same change of variables again: t=
1 1+r
N −s N −1
, that is r =
1−t t
N −1 N −s
N −1 and dr = N −s
1−t t
−1+s N −s
·
−1 dt, t2
and obtain ∇Vq N dx RN
N −1 1−s N N −1 1 q−1 N −s N −1 1 − t N −s (N −1) N −1 N −s N N 1−t q−N = ω N −1 t N −s q − N t t 0 −1+s 1 − t N −s 1 × dt t t2 1 q−1 N −1 N −s N N −N −1 ω N −1 t q−N = (1 − t) N −1 N −s q − N 0 N −1 N −s N q −1 N − N, N = ω N −1 B N −s q − N q−N 2 N N −N N − 1 N − s N Nπ 2 ,N =
B N −s q − N q−N N2 + 1
123
N. Lam, G. Lu
N −1 = N −s = =
N −s N N −s N
2 N −N N N π 2 q−N (N )
(q−1) N2 + 1 Nq−N 2 N −N N N (N ) q−N N N −1 Nπ 2 N
N q−N 2 +1 N (q−1)
N −s q−N
N −1 N −1
N
N −1 N
N q−N
N
q−N
N 2
Nπ N
2 +1
N k=1
(N ) N (q−1) q−N
. − k.
Hence, as q → ∞ q
N −1
RN
∇Vq N dx → (N − 1) (N − s) N −1
N
Nπ 2 N
2 +1
(N ) N −1 k=1 (N −k)
1 N (N − 1)
N
=
(N − s) N −1 N π 2
. N N2 + 1
In other words, RN
N N −1 Nπ 2 1 1−N ∇Vq N dx = (N − s) +o
q N q N −1 N2 + 1 N 1 N − s N −1 2N N −2 π 2 1−N . +o = N q N q N −1 2
(2.4)
Now, we consider the limit when q ↓ N of ⎞ N −1 q q−N q−N q ⎠ N ln + ln ⎝ + ln √ N −1 N 2 q−N N π N q−N ⎤ ⎡ q N −1 N q−N q − N q ⎦ . = ln ⎣ √ q − N N −1 N 2 N π N q−N
⎛
We have N! −1 N −1 N −1 N −1 q = q − (N − 1) −k q q−N q−N q−N =
123
k=1
N −1 N q−N N2
N! −1 k=1
q (N − 1 − k) Nk + q−N q−N
.
Moser–Onofri–Beckner and Logarithmic Sobolev Inequalities
Hence
q−N √ N π
N
q N −1 q−N q q − N N −1 N 2 N q−N
N −1 ! q (N − 1 − k) + N k q = √ N (q − N ) N −1 q−N N π k=1 N −1 ! q = √ N (q (N − 1 − k) + N k) N π k=1 N −1 ! N → √ N (N (N − 1 − k) + N k) N π k=1
N (N − 1) N −1 = √ N [N (N − 1)] N −1 = √ N . N π π So as q ↓ N , N ln
q−N √ N π
+ ln
q q−N
⎞ N −1 N −1 q q−N − 1) (N ⎠ → ln + ln ⎝ . √ N N2 π NN−1 q−N (2.5) ⎛
Lemma 2.3 For 0 ≤ s < N and b > 0, we have dx N −s N −1 1 N −1 exp −N b |x| N −1 = ω (N − 1) N −1 |x|s N − s Nb RN N −s N N −s dx N −1 1 N (N ) exp −b |x| N −1 ln exp −b |x| N −1 s = −bω N −1 |x| N −s N b N −s N N −1 1 N N −s N N −1 |x| exp −b dx = ω (N ). ∇ N −1 N −1 N −s N RN
RN
Proof We first consider the integral RN
dx exp −k |x|n = ω N −1 |x|s
∞
exp −kr n r N −1−s dr.
0
By the change of variable t = kr n , that is r = get
1 n1 k
1
t n and dr =
1 n1 k
1 n1 −1 dt, nt
we
123
N. Lam, G. Lu
dx exp −k |x|n = ω N −1 |x|s
RN
∞
exp (−t)
0
N −1−s 1 n N −1−s 1 1 n 1 1 −1 t n dt t n k k n ∞ N −s exp (−t) t n −1 dt
N −s 1 n k 0 N −s 1 1 n N −s . = ω N −1 n k n 1 n
= ω N −1
(2.6)
Similarly, RN
dx exp −k |x|n |x|n = ω N −1 |x|s
∞
exp −kr n r n+N −1−s dr
0
1 n+N −1−s n n+N −1−s 1 n 1 1 −1 1 t n t n dt k k n 0 N −s ∞ N −s 1 1 1+ n = ω N −1 exp (−t) t n dt n k 0 1+ N −s n 1 1 N −s = ω N −1 (2.7) +1 . n k n = ω N −1
∞
exp (−t)
Hence RN
dx N −s 1 N −1 N −1 exp −N b |x| N −1 = ω (N − 1) , N −1 |x|s N − s Nb
(2.8)
and
N −s N N −s dx exp −b |x| N −1 ln exp −b |x| N −1 s |x| RN N −s dx N −s = −b exp −N b |x| N −1 |x| N −1 |x|s RN N 1 N −1 = −bω N −1 (N ). N − s Nb
N −s N −s N −s −s exp −b |x| N −1 |x| N −1 −1 Also, since ∇ exp −b |x| N −1 = −b NN −1 RN
N −s N N ∇ exp −b |x| N −1 dx = b N N = b N
123
−s −1 −s −1
N N
RN
RN
(2.9)
x |x| ,
we get
N −s 1−s exp −N b |x| N −1 |x| N N −1 dx N −s 1−s dx exp −N b |x| N −1 |x| N N −1 +s |x|s
Moser–Onofri–Beckner and Logarithmic Sobolev Inequalities
N = b N N = b N
N
N −s dx N −s exp −N b |x| N −1 |x| N −1 |x|s RN N N N −1 1 −s ω N −1 (N ). (2.10) −1 N − s Nb −s −1
3 Weighted Moser–Onofri–Beckner Inequality: Proof of Theorem 1.1 Proof of Theorem 1.1 By Lemma 2.1, we have RN
1−θ
1 θ q q dx |∇v| N dx N |v| s N R RN |x| ≥ N −1 N (q−1) dx N (q−1) N −1 R N |v| |x|s
1−θ 1 θ ∇Vq N dx N Vq q dxs q N R |x| , N −1 N (q−1) N (q−1) N −1 dxs R N Vq |x|
where N −1 N −s − q−N . Vq (x) = 1 + |x| N −1 Let u be a smooth compactly supported function such that vq = Vq
RN
udμ N ,s = 0 and set
N −1 1+ u . Nq
We have that q−N q N dx N (N −1) R N vq R N ∇vq q N dx N ∇Vq R N Vq
R
dx |x|s dx |x|s
⎛
⎜ ≥ ⎝
RN RN
N (q−1) vq N −1 N (q−1) Vq N −1
⎞ dx |x|s dx |x|s
⎟ ⎠.
First, we have by (2.1) lim
p→∞ R N
N −1 N (q−1) dx N (q−1) dx N −s − q−N N −1 Vq N −1 N −1 |x| 1 + = lim p→∞ N |x|s |x|s R N −s −N dx 1 ω N −1 , 1 + |x| N −1 = = s |x| N −s RN
and lim
p→∞ R N
N (q−1) N −1 N (q−1) dx N (q−1) N −1 dx N −1 vq N −1 Vq 1+ u = lim s p→∞ R N |x| |x|s Nq N −s −N dx 1 + |x| N −1 = eu s , N |x| R
123
N. Lam, G. Lu
so N α vq N α Vq
RN RN
dx |x|s dx |x|s
→
RN
eu
1 1 N −s ω N −1
N −s −N dx 1 + |x| N −1 = eu dμ N ,s . N |x|s R
Also, by (2.2), when q → ∞
q dx ω N −1 1 Vq , ≈ s |x| N (N − s) q RN q dx vq lim = ∞, q→∞ R N |x|s but lim
q→∞
RN RN
q vq q Vq
dx |x|s dx |x|s
= lim
q→∞
RN
q q Vq 1 + N −1 u Nq q dx s N Vq R
dx |x|s
= 1.
|x|
Now, with N −1 Xq = 1 + u ∇Vq Nq N −1 Vq ∇u, Yq = Nq then N N −2 N −1 N ∇vq N = ∇Vq N 1 + N − 1 u u + Vq ∇Vq ∇Vq · ∇ 1 + Nq Nq
+ R N X q , Yq . We estimate the second term as follows: N −2 N −1 N u Vq ∇Vq ∇Vq · ∇ 1 + Nq RN N N −2 N −1 1+ u =− ∇ Vq ∇Vq ∇Vq Nq RN N N − 1 N N −1 N ∇Vq − 1+ 1+ u u Vq N Vq . =− Nq Nq RN RN
Thus, RN
123
∇vq N = −
RN
N −1 N 1+ u Vq N Vq + R N X q , Yq . Nq RN
Moser–Onofri–Beckner and Logarithmic Sobolev Inequalities
Now, we note that N −1 u q∇Vq q Xq = 1 + Nq N −1 N −s − q−N −1 N − s N −s N −1 x N −1 |x| N −1 −1 u q 1 + |x| N −1 =− 1+ |x| Nq q−N N −1 N −s x 1 → − (N − s) |x| N −1 −1 a.e. as q → ∞ −s |x| 1 + |x| NN −1 and qYq = q
N −1 N −1 Vq ∇u → ∇u. Nq N
As a consequence, both X q and Yq in R N X q , Yq are of the order of q1 . Hence, when q→∞
q N R N X q , Yq → R N − (N − s) |x|
N −s N −1 −1
= H N (x, ∇u) .
x N −1 1 ∇u , −s |x| 1 + |x| NN −1 N
Next, we have from (2.4) that −
RN
Vq N Vq =
RN
∇Vq N dx =
N −s N
N −1
2N N −2 π N2
N 2
q 1−N ,
and from (2.3) that q N −1 Vq N Vq
N − s N −2 N − s q − 1 N − s =q (N − 1) q−N q − N q − N N −1 q−1 2q−N −1 N −1 −2)− − − N −s 1−s 1−s q−N q−N (N q−N × 1 + r N −1 r N −1 (N −2)+2 N −1 N − s N −2 N − s N − s − q N −1 (N − 1) q−N q − N N −1 q−1 q−1 N −1 N −s − q−N − q−N (N −2)− q−N 1−s 2−s−N × 1 + r N −1 r N −1 (N −2)+ N −1
N −s −N 1−s N −s −N +1 1−s (N −2)+2 N1−s (N −2)+ 2−s−N N N −1 N −1 −1 N −1 N −1 N −1 → (N −s) r − 1+r r 1+r N −1
1−s 2−s−N N −s −N 1−s N −s = (N − s) N 1 + r N −1 r N −1 (N −2) r 2 N −1 − 1 + r N −1 r N −1
123
N. Lam, G. Lu
N −s −N 1 = − (N − s) N 1 + r N −1 rs = − (N − s) N −1 ω N −1 μ N ,s as q → ∞.
Hence
−
So RN
1 Vq N Vq udx = N −1 (N − s) N −1 ω N −1 q RN 1 . =o q N −1
RN
udμ N ,s + o
1
q N −1
N −1 1 1 1+ N Vq N Vq + N H N (x, ∇u) u+o Nq q q RN RN 1 ∇Vq N dx + 1 . H ∇u) dx + o (x, N q N RN qN RN
∇vq N = − =
Letting q → ∞, we can deduce that
RN RN
∇vq N dx ∇Vq N dx
Nq−N (N −1)
⎛
⎞
⎜ N H N (x, ∇u) dx ⎜ ≈ ⎜1 + R N −s N −1 2N N −2 π N2 ⎝ q N N
q−N N (N −1)
⎟ ⎟ ⎟ ⎠
2
q−N N (N −1) N (N − 1) α N ,s ≈ 1+ H N (x, ∇u) dx q RN
≈ exp α N ,s H N (x, ∇u) dx . RN
Here α N ,s
1 1 = =
N (N − 1) N −s N −1 2N N−2 π N2 N
N N −s
N −1
RN
eu dμ N ,s ≤ exp α N ,s
RN
N 2
2 (N − 1) N N −1 π
N 2
Finally, putting everything together, we obtain that with
RN
N 2
.
udμ N ,s = 0, then
H N (x, ∇u) dx .
4 Weighted L N -Logarithmic Sobolev Inequality: Proof of Theorem 1.2 Proof of Theorem 1.2 We first recall that from Theorem A, we obtain
123
Moser–Onofri–Beckner and Logarithmic Sobolev Inequalities
|∇u| dx N
RN
1 (1− q ) N
r
RN
|u|q
dx |x|s
1 r
C K N (N , s, q, r ) ≥
RN
|u|r
dx |x|s
1/r .
Hence 1 1 N dx q q q |∇u| N dx |u|q ln 1− + ln + ln (C K N (N , s, q, r )) |x|s r r RN RN 1 dx r |u|r ≥ ln . |x|s RN Equivalently,
N1 N 1 R N |∇u| dx ln ≥ 1 1− q dx q R N |u| |x|s
q r
ln
RN
|u|r
RN
dx |u|q |x| s
Noting that r = N Nq−1 −1 , we have 1 −
q r
=
dx |x|s
1 r
1 1 − 1 − q
q−N N (q−1)
q r
ln (C K N (N , s, q, r )) .
and
1 |∇u| N dx N ln 1 q dx q R N |u| |x|s RN
N (q − 1) ln ≥ q−N
RN
|u|r
dx |x|s
RN
|u|q
dx |x|s
1 r
1 − q
N (q − 1) ln (C K N (N , s, q, r )) . q−N
It is clear that as q ↓ N
1
N1 N |∇u| N dx N R N |∇u| dx ln 1 → ln 1 dx q q N dx N |u| |u| s s N N R R |x| |x|
RN
and by (2.5) N (q − 1) ln (C K N (N , s, q, r )) q−N 1 q−N q N (q − 1) 1 N N 1 = ln + ln + ln + ln √ N N −s N q−N q−N r q N π ⎛ ⎞
N −1 q q−N N2 + 1 1 ⎠ + ln ⎝ 2 N (N ) N −1 N N
q−N
123
N. Lam, G. Lu
1 → ln N
N N −s
N2 + 1 1 1 N −1 (N − 1) N −1 + ln + ln − √ N N N N (N ) π
= S L (N , s) . Now, we consider the limit when q ↓ N of RN
|u|r
dx |x|s
RN
|u|q
dx |x|s
1 ln q − N
1 r
1 =
N −1 N (q−1)
ln
q−1
|u| N N −1
RN
dx |x|s
−
1 q
ln
RN
|u|q
dx |x|s
q−N
q
.
By L’Hospital’s Rule, it is equal to
dx |u| N N −1 ln |u| NN−1 |x| s dx N −1 N Nq−1 −1 |u| ln − N Nq−1 |x|s N (q − 1)2 RN −1 dxs R N |u| |x| dx q dx 1 R N |u| ln |u| |x|s q + 2 ln |u| . q dx |x|s q RN s N |u| q−1
N −1 N (q−1) R N
−
1 q
R
Noting that
RN
|u| N
RN
|u|r
1 ln q − N
RN
=
|x|
|u|q
1 N (N − 1)
dx |x|s
dx |x|s dx |x|s
RN
= 1, we get
1 r
1 q
1 → N −1
|u| N ln |u|
dx 1 |u| ln |u| s − |x| N
N
RN
RN
dx . |x|s
Putting everything together, we obtain
ln S L (N , s)
|∇u| dx
RN
≥
N
RN
|u| N ln |u| N
dx . |x|s
Now, with N −s U (x) = exp −b |x| N −1 , where N
b > 0 and b
123
N −1
N − 1 (N − 1) Nπ 2 = N ,
2 + 1 N − s N N −1
|u| N ln |u|
dx |x|s
Moser–Onofri–Beckner and Logarithmic Sobolev Inequalities
we have by (2.8), (2.9), and (2.10) that 1
N
|∇U | dx N
ln
RN
1 ln = N
N −s N −1
N
N
N −1 Nπ 2 N
2 +1 N −s
1 N
N (N ) N
=
N −1 N −s 1 (N ) π2 1 ln + ln N
+ ln N −1 , N N −1 N 2 +1 N N
and RN
|U | N ln |U |
N −1 1 N dx 1 = − ω (N ) N −1 |x|s b N −1 N −s N
N N −1 1 N N π 2 N2 + 1 (N − s) N N −1 =− (N ) N
N N −s N 2 + 1 N π 2 (N − 1) (N − 1) =−
N −1 . N
Hence ln SL (N , s)
RN
|∇U | N dx
=
RN
|U | N ln |U | N
dx . |x|s
5 Weighted L p -Logarithmic Sobolev Inequality for 1 < p < N Proof of Theorem 1.3 By Theorem B, we have
a 1−a q dx p q dx |∇u| |u| C K N (N , μ, s, p, q, r ) μ s |x| |x| RN RN dx 1/r |u|r ≥ . |x|s RN p
Hence 1 1 q 1−a 1 dx p q dx |∇u| |u| ln ln + + ln C K N (N , μ, s, p, q, r ) μ s |x| |x| a a RN RN 1/r 1 dx |u|r ≥ ln , |x|s a RN
p
123
N. Lam, G. Lu
and RN
|∇u| p
ln
RN
|u|q
dx |x|μ dx |x|s
1
p
+
1 q
RN
1 1 ln C K N (N , μ, s, p, q, r ) ≥ ln a a
|u|r
RN
|u|q
dx |x|s
1/r
dx |x|s
1 . q
(5.1) Now, we will consider the limiting process when q ↓ p of (5.1). In this case, it is easy to see that 1 p p dx 1 |∇u| μ N p R |x| p dx |∇u| → ln . ln 1 μ |x| RN q dx q |u| s RN |x| Now, by L’Hospital’s Rule, we get 1 1 r dx q dx |u| |u| − ln ln s s N N R R r q |x| |x| lim q↓ p q−p ⎡ dx r p 1 p RN |u| ln|u| |x|s r dx |u| − + ln dx r ⎢ ( p−1)r 2 RN r p−1 |x|s |u| |x|s RN ⎢ = lim ⎢ q ln|u| dx |u| N s q↓ p ⎣ |x| dx − q1 R + q12 ln R N |u|q |x| s q dx =
1 p 2 ( p − 1)
RN
RN
|u| p ln |u| p
|u|
⎤ ⎥ ⎥ ⎥ ⎦
|x|s
dx . |x|s
Hence, as q ↓ p
|u|r
RN
1 ln a
RN
|u|q
dx |x|s
1/r
dx |x|s
1 q
1 → N
RN
|u| p ln |u| p
dx . |x|s
Now, we will consider the limit problem lim
q↓ p
1 ln CKN(N , μ, s, p, q, r ). a
First, 1 + p−1 − 1−a − p−1 (1−a) r p q p N−p N − p−μ N−p N−p p−1 1 1 (q − p) (N − p) ln + lim ln = lim q↓ p q↓ p a q − 1 p [ pq − N (q − p)] p N − p−μ N − p−μ N −1 N−p . = ln N N − p−μ
1 lim ln q↓ p a
123
Moser–Onofri–Beckner and Logarithmic Sobolev Inequalities
Next, by L’Hospital’s Rule 1 N p − q (N − p) r pq 1 N p−q (N − p) ( p − 1) ( p−1) [ pq − N (q − p)] r 1 =− . ln = lim q↓ p Np r (q − p) pq p
1 lim ln q↓ p a
Also ⎛
⎛ N N
⎞ Na
⎞ + 1 + 1 1 1 ⎝ ⎠ = 2 ⎠ . lim ln ⎝ 2 ln p−1 q↓ p a N N p +1 N p−1 + 1 p Finally, let us consider the limit ⎡
a
lim
q↓ p
1 ⎢ q−p ln ⎣ √ a p π ⎡
⎢ q−p = lim ln ⎢ ⎣ p √π q↓ p ⎡
⎢ = lim ln ⎢ ⎣ q↓ p
1 √ p π
pq N (q − p)
a
p
pq N (q − p)
pq 1
p
N
⎞a ⎤ N p−1 q q− p ⎥ ⎝ ⎠ ⎦ p−1 δ p q− p ⎛
1
p
p−1 q q− p
⎛
⎥ ⎝ ⎠ ⎥ ⎦ N p−q(N − p) p−1 p q− p ⎛
(q − p)
⎤
⎞ N1
1− 1p
p−1 q q− p
⎞ N1
⎤
⎥ ⎝ ⎠ ⎥ ⎦. N p−q(N − p) p−1 p q− p
p−1 p−1 Noting that with n = q q− p → ∞ as q ↓ p, and z = −N p , we have by the asymptotic approximations of the gamma function and by the Stirling’s formula that
q−p q ( p − 1)
N
1− 1p
p−1 q q− p p−1 N p−q(N − p) p q− p
=
(N )n z → 1 as q ↓ p and n → ∞. (n + z)
Hence ⎡
lim
q↓ p
1 ⎢ q−p ln ⎣ √ a p π
a
pq N (q − p)
a
p
⎞a ⎤ N p−1 q q− p ⎥ ⎝ ⎠ ⎦ p−1 δ p q− p ⎤ ⎛
⎡ 2 1p 1 p 1− 1 ⎣ = ln [ p ( p − 1)] p ⎦ . √ N p π
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N. Lam, G. Lu
Combining all the estimations, we obtain
dx 1 |u| p ln |u| p |x|s N RN RN ⎡ ⎤ ⎞1 ⎛ N N +1 N −1 ( p−1) 2 1p ⎢ p N N−p 1 1 p 1− 1 ⎥ ⎝ 2 ⎠ ≥ ln ⎢ [ p ( p − 1)] p ⎥ √ ⎣ N − p−μ ⎦. p−1 e N p π N +1
1 ln p
|∇u| p
dx |x|μ
−
p
Equivalently, N dx p dx |∇u| |u| p ln |u| p s . ln LS(N , p, μ) μ ≥ N N |x| |x| p R R Moreover, using the identities (2.6) and (2.7), we can check that N dx p dx |∇U | |U | p ln |U | p s , ln LS(N , p, μ) μ = N N |x| |x| p R R where N − p−μ U (x) = exp −b |x| N − p
p p−1
,
with b > 0 and b
N
p−1 p
N
=
N π 2 (N − p) ( p − 1) N
2 + 1 (N − p − μ) p
N p−1 p 1 p−1 . N p p
Acknowledgements Research of this work was partially supported by a US NSF Grant and a Simons Fellowship from the Simons Foundation. Nguyen Lam was also partially supported by a fellowship from the Pacific Institute for the Mathematical Sciences.
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