uncorrected proof
DOI 10.1007/s00526-016-1065-9
Calculus of Variations
Metric measure spaces supporting Gagliardo–Nirenberg inequalities: volume non-collapsing and rigidities
Alexandru Kristály1,2
Received: 22 February 2016 / Accepted: 23 August 2016
© Springer-Verlag Berlin Heidelberg 2016
Abstract Let(M,d,m)be a metric measure space which satisfies the Lott–Sturm–Villani
1
curvature-dimension condition CD(K,n) for some K ≥ 0 and n ≥ 2, and a lower
2
n-density assumption at some point of M. We prove that if (M,d,m) supports the
3
Gagliardo–Nirenberg inequality or any of its limit cases (Lp-logarithmic Sobolev inequality
4
or Faber–Krahn-type inequality), then aglobal non-collapsing n-dimensional volume growth
5
holds, i.e., there exists a universal constantC0>0 such thatm(Bx(ρ))≥C0ρnfor allx ∈M
6
andρ ≥0,whereBx(ρ)= {y ∈ M :d(x,y) < ρ}. Due to the quantitative character of
7
the volume growth estimate, we establish several rigidity results on Riemannian manifolds
8
with non-negative Ricci curvature supporting Gagliardo–Nirenberg inequalities by explor-
9
ing a quantitative Perelman-type homotopy construction developed by Munn (J Geom Anal
10
20(3):723–750,2010). Further rigidity results are also presented on some reversible Finsler
11
manifolds.
12
Mathematics Subject Classification Primary 53C23; Secondary 35R06·53C60
13
Contents
14
1 Introduction . . . .
15
1.1 Recalling optimal Gagliardo–Nirenberg inequalities on normed spaces . . . .
16
1.2 Statement of main results . . . .
17
1.2.1 Volume non-collapsing on metric measure spaces . . . .
18
1.2.2 Applications: rigidity results in smooth settings . . . .
19
2 Volume non-collapsing via Gagliardo–Nirenberg inequalities . . . .
20
2.1 Casesα >1 & 0< α <1: usual Gagliardo–Nirenberg inequalities. . . .
21
Communicated by L. Ambrosio.
B
Alexandru Kristályalex.kristaly@econ.ubbcluj.ro; alexandrukristaly@yahoo.com
1 Department of Economics, Babe¸s-Bolyai University, 400591 Cluj-Napoca, Romania 2 Romania and Institute of Applied Mathematics, Óbuda University, Budapest 1034, Hungary
Author Proof
uncorrected proof
2.2 Limit case I(α→1):Lp-logarithmic Sobolev inequality . . . .
22
2.3 Limit case II(α→0): Faber–Krahn-type inequality. . . .
23
3 Rigidity results in smooth settings. . . .
24
3.1 Gagliardo–Nirenberg inequalities on Riemannian manifolds with Ricci≥0. . . .
25
3.2 Gagliardo–Nirenberg inequalities on Finsler manifolds withn-Ricci≥0 . . . .
26
References. . . .
27
1 Introduction
28
An important role in the theory of geometric functional inequalities is played by the
29
Gagliardo–Nirenberg interpolation inequality and its limit cases. The present paper is devoted
30
to the study of Gagliardo–Nirenberg inequalities on metric measure spaces; to be more pre-
31
cise, we shall
32
(a) establishquantitative volume non-collapsing propertiesof metric measure spaces satis-
33
fying the Lott–Sturm–Villani curvature-dimension conditionCD(K,n)for someK ≥0
34
andn≥2,in the presence of a Gagliardo–Nirenberg inequality or one of its limit cases
35
(Lp-logarithmic Sobolev inequality or Faber–Krahn-type inequality);
36
(b) provide rigidityresults in the framework of Riemannian and Finsler manifolds with
37
non-negative Ricci curvature which support (almost)opti mal Gagliardo–Nirenberg
38
inequalities by using the volume non-collapsing property from (a) and a quantitative
39
homotopy construction due to Munn [17] and Perelman [22].
40
In Sect.1.1, we recall the optimal Gagliardo–Nirenberg inequalities on normed spaces which
41
play a comparison role in our investigations; in Sect.1.2, we present the main results of the
42
paper.
43
1.1 Recalling optimal Gagliardo–Nirenberg inequalities on normed spaces
44
The optimal Gagliardo–Nirenberg inequality in the Euclidean case has been obtained by Del
45
Pino and Dolbeault [7] for a certain range of parameters by using symmetrization arguments.
46
By using mass transportation argument, Cordero-Erausquin et al. [6] extended the results
47
from [7] to prove optimal Gagliardo–Nirenberg inequalities on arbitrary normed spaces. In
48
the sequel, we recall the main theorems from [6] and some related results.
49
Let · be an arbitrary norm onRn;without loss of generality, we may assume that the
50
Lebesgue measure of the unit ball in(Rn, · )is the volume of then-dimensional Euclidean
51
unit ballωn =πn2Ŵ(n2+1)−1. The dual norm · ∗of · is given byx∗=supy≤1x·y
52
where′·′is the Euclidean inner product. Let p∈ [1,n)andLp(Rn)be the Lebesgue space
53
of orderp. As usual, we consider the Sobolev spaces
54
W˙1,p(Rn)= {u∈Lp⋆(Rn): ∇u∈Lp(Rn)}
55
and
56
W1,p(Rn)= {u∈Lp(Rn): ∇u∈Lp(Rn)},
57
wherep⋆ = n−pnp and∇is the gradient operator. On account of the Finslerian duality (see
58
also Sect.3.2), ifu∈ ˙W1,p(Rn),the norm of∇uis defined by
59
∇uLp =
Rn
∇u(x)∗pd x 1/p
,
60
Author Proof
uncorrected proof
whered xis the Lebesgue measure onRn.1
61
Fixn≥2,p∈(1,n)andα∈(0,n−pn ]\{1}; for everyλ >0,let
62
hλα,p(x)=(λ+(α−1)xp′)
1 1−α
+ , x ∈Rn,1
63
wherep′= p−1p is the conjugate top, andr+=max{0,r}forr∈R.The followingoptimal
64
Gagliardo–Nirenberg inequalitiesare known on normed spaces:
65
Theorem A. (see [6, Theorem 4])Let n≥2, p∈(1,n)and · be an arbitrary norm on
66
Rn.
67
• If1< α≤ n−np, then
68
uLαp ≤Gα,p,n∇uθLpu1−θ
Lα(p−1)+1, ∀u∈ ˙W1,p(Rn), (1.1)
69
where
70
θ= p⋆(α−1)
αp(p⋆−αp+α−1), (1.2)
71
and the best constant
72
Gα,p,n = α−1
p′ θ
p′ n
θp+θn
α(p−1)+1 α−1 − pn′
αp1
α(p−1)+1 α−1
θp−αp1
ωnB
α(p−1)+1
α−1 − pn′,pn′
θn
73
is achieved by the family of functions hλα,p,λ >0;
74
• If0< α <1, then
75
uLα(p−1)+1 ≤Nα,p,n∇uγLpu1−γLαp, ∀u∈ ˙W1,p(Rn), (1.3)
76
where
77
γ = p⋆(1−α)
(p⋆−αp)(αp+1−α), (1.4)
78
and the best constant
79
Nα,p,n= 1−α
p′ γ
p′ n
γp+γn α(p−1)+1
1−α + pn′
γp−α(p−1)+11 α(p−1)+1
1−α
α(p−1)+11
ωnBα(p−1)+1
1−α ,np′
γn
80
is achieved by the family of functions hλα,p,λ >0.
81
Hereafter,B(·,·)is the Euler beta-function.
82
The borderline caseα= n−pn (thusθ=1) reduces to theoptimal Sobolev inequality, see
83
Aubin [3] and Talenti [26] in the Euclidean case, and Alvino et al. [1] for normed spaces.
84
Furthermore, inequalities (1.1) and (1.3) degenerate to theoptimal Lp-logarithmic Sobolev
85
inequalitywheneverα → 1 (called also as the entropy-energy inequality involving the
86
Shannon entropy), while (1.3) reduces to aFaber–Krahn-type inequalitywheneverα→0,
87
respectively. More precisely, one has
88
1The functionhλα,pis positive everywhere forα >1 whilehλα,phas always a compact support forα <1.
Author Proof
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Theorem B. Let n≥2, p∈(1,n)and · be an arbitrary norm onRn.
89
• Limit case I(α→1)(see [9, Theorem 1.1]2): One has
90
Entd x(|u|p)=
Rn
|u|plog|u|pd x ≤ n plog
Lp,n∇upLp
,
91
∀u∈W1,p(Rn), uLp =1, (1.5)
92
where the best constant
93
Lp,n= p n
p−1 e
p−1 ωnŴ
n p′+1
−pn
94
is achieved by the family of functions
95
lλp(x)=λ
n pp′ω−
1 n pŴ
n p′+1
−1p
e−λpxp
′
, λ >0;
96
• Limit case II(α→0)(see [6, p. 320]):One has
97
uL1 ≤Fp,n∇uLp|supp(u)|1−p1⋆, ∀u∈ ˙W1,p(Rn) (1.6)
98
and the best constant
99
Fp,n = lim
α→0Nα,p,n=n−1pω−
1
nn(p′+n)−
1 p′ 100
is achieved by the family of functions
101
fpλ(x)= lim
α→0hλα,p(x)=(λ− xp′)+, x∈Rn,
102
wheresupp(u)stands for the support of u and|supp(u)|is its Lebesgue measure.
103
1.2 Statement of main results
104
As we already pointed out, the primordial purpose of the present paper is to establish fine
105
topological properties of metric measure spaces curved in the sense of Lott–Sturm–Villani
106
which support Gagliardo–Nirenberg-type inequalities. In fact, the metric spaces we are work-
107
ing on are supposed to satisfy the curvature-dimension conditionCD(K,n)for someK ≥0
108
andn≥2,introduced by Lott and Villani [15] and Sturm [24,25]; see Sect.2for its formal
109
definition.
110
1.2.1 Volume non-collapsing on metric measure spaces
111
Let(M,d,m)be a metric measure space (with a strictly positive Borel measurem) and
112
Lip0(M)be the space of Lipschitz functions with compact support onM. Foru∈Lip0(M),
113
let
114
|∇u|d(x):=lim sup
y→x
|u(y)−u(x)|
d(x,y) , x∈M. (1.7)
115
Note thatx → |∇u|d(x)is Borel measurable onMforu∈Lip0(M).
116
2Gentil [9] proved an optimalLp-logarithmic Sobolev inequality for even,q-homogeneous(q>1), strictly convex functionsC:Rn→ [0,∞). In our case,C(x)=xp
′ p′ .
Author Proof
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As before, letn ≥2 be an integer, p ∈ (1,n)andα ∈ (0,n−np]\{1}. Throughout this
117
section we assume that thelower n-density of the measuremat a pointx0 ∈M is unitary,
118
i.e.,
119
(D)nx
0 :lim inf
ρ→0
m(Bx0(ρ)) ωnρn =1,
120
whereBx(r)= {y∈M:d(x,y) <r}.
121
Throughout the whole paper, we shall keep the notations from Theorems A and B [i.e.,
122
the four best constants from the Gagliardo–Nirenberg inequalities on normed spaces and the
123
numbersθ andγ from (1.2) and (1.4), respectively]; the Lebesgue spaces Lp are defined
124
on the measure space(M,m). We now are the position to state our quantitative, globally
125
non-collapsing volume growth results:
126
Theorem 1.1 (Gagliardo–Nirenberg inequalities)Let(M,d,m)be a proper metric measure
127
space which satisfies the curvature-dimension conditionCD(K,n) for some K ≥ 0and
128
n ≥2. Let p ∈(1,n)and assume that(D)nx
0 holds for some x0 ∈ M. Then the following
129
statements hold:
130
(i) If1< α≤ n−np and the inequality
131
uLαp ≤C|∇u|dθLpu1−θ
Lα(p−1)+1, ∀u∈Lip0(M) (GN1)α,pC
132
holds for someC≥Gα,p,n, then K =0and
133
m(Bx(ρ))≥ G
α,p,n
C nθ
ωnρn f or all x ∈M andρ≥0.
134
(ii) If0< α <1and the inequality
135
uLα(p−1)+1 ≤C|∇u|dγLpu1−γLαp, ∀u∈Lip0(M) (GN2)α,pC
136
holds for someC≥Nα,p,n, then K =0and
137
m(Bx(ρ))≥ N
α,p,n
C γn
ωnρn f or all x∈M andρ≥0.
138
In the limit caseα→1, we can state
139
Theorem 1.2 (Lp-logarithmic Sobolev inequality) Under the same assumptions as in
140
Theorem1.1, if
141
Entdm(|u|p)=
M
|u|plog|u|pdm≤ n plog
C|∇u|dpLp
, ∀u∈Lip0(M),
142
uLp =1 (LS)Cp
143
holds for someC≥Lp,n,then K =0and
144
m(Bx(ρ))≥ L
p,n
C np
ωnρn f or all x∈M andρ≥0.
145
In the remaining limit caseα→0, one can prove
146
Author Proof
uncorrected proof
Theorem 1.3 (Faber–Krahn-type inequality)Under the same assumptions as in Theorem
147
1.1, if
148
uL1 ≤C|∇u|dLpm(supp(u))1−p1⋆, ∀u∈Lip0(M) (FK)Cp
149
holds for someC≥Fp,n, then K =0and
150
m(Bx(ρ))≥ F
p,n
C n
ωnρn f or all x∈M andρ≥0.
151
Some remarks are in order.
152
Remark 1.1 (a) The proofs of Theorems1.1–1.3aresyntheticwhere we shall exploit some
153
basic features of metric measure spaces satisfying the CD(K,n) condition (such as
154
generalized Bonnet–Myers and Bishop–Gromov comparison inequalities) and direct
155
constructions. Although the lines of the proofs of these results are similar, our arguments
156
require different technics, deeply depending on theshapeof certain test functions whose
157
profiles come from the family of extremals in normed spaces (cf. Theorems A & B).
158
Note that instead of theCD(K,n)condition it is enough to consider the slightly weaker
159
measure contraction propertyMCP(K,n), see Ohta [20].
160
(b) The casep = 2 andα = n−2n (n ≥3) is contained in Kristály and Ohta [12], where
161
the authors studied Caffarelli–Kohn–Nirenberg inequalities on metric measure spaces.
162
We notice that the roots of Theorem1.1(i) on Riemannian manifolds with non-negative
163
Ricci curvature can be found in do Carmo and Xia [8], Ledoux [13] and Xia [28].
164
(c) The generalized Bishop–Gromov inequality and density assumption (D)nx
0 imply
165
m(Bx0(ρ)) ≤ ωnρn for allρ ≥ 0. In particular, the latter inequality and the con-
166
clusions of Theorems1.1–1.3imply the Ahlforsn-regularity at the pointx0; therefore,
167
the Hausdorff dimension of(M,d)is preciselyn.
168
(d) (D)nx
0clearly holds for every pointx0onn-dimensional Riemannian and Finsler mani-
169
folds endowed with the canonical Busemann–Hausdorff measure.
170
1.2.2 Applications: rigidity results in smooth settings
171
Having fine volume growth estimates in Theorems1.1–1.3, importantrigidityresults can
172
be deduced in the context of Riemannian and Finsler manifolds supporting Gagliardo–
173
Nirenberg-type inequalities.
174
In order to state such results, let(M,g)be ann-dimensional complete Riemannian mani-
175
fold with non-negative Ricci curvature(n≥2)endowed with its canonical volume formdvg.
176
LetαM P(k,n)∈(0,1]be the so-calledMunn–Perelman constantfor everyk =1, . . . ,n,
177
see Munn [17]. In fact, based on the double induction argument of Perelman [22], Munn
178
determined explicit lower bounds for the volume growth in terms of the constantαM P(k,n)
179
which guarantee the triviality of thek-th homotopy groupπk(M)of(M,g);see details in
180
Sect.3.
181
For sake of simplicity, we restrict here our attention to the Lp-logarithmic Sobolev
182
inequality(LS)Cp on(M,g) by proving that onceC > 0 is closer and closer to the opti-
183
mal Euclidean constantLp,n, the manifold(M,g)approaches topologically more and more
184
to the Euclidean spaceRn.
185
Theorem 1.4 Let(M,g) be an n-dimensional complete Riemannian manifold with non-
186
negative Ricci curvature(n≥2)and assume the Lp-logarithmic Sobolev inequality(LS)Cp
187
holds on(M,g)for some p∈(1,n)andC>0. Then the following assertions hold:
188
Author Proof
uncorrected proof
(i) C≥Lp,n;
189
(ii) The order of the fundamental groupπ1(M)is bounded above by C
Lp,n
np
;
190
(iii) IfC< αM P(k0,n)−pnLp,nfor some k0 ∈ {1, . . . ,n}thenπ1(M)= · · · =πk0(M)=0;
191
(iv) IfC< αM P(n,n)−npLp,nthen M is contractible;
192
(v) C=Lp,nif and only if(M,g)is isometric to the Euclidean spaceRn.
193
Remark 1.2 (a) Theorem 1.4(v) answers an open question of Xia [29] for generic p ∈
194
(1,n). Forp=2 the latter equivalence is well known by using sharp analytic estimates
195
for the heat kernel on complete Riemannian manifolds with non-negative Ricci curva-
196
ture; see Bakry et al. [4], Ni [18], and Li [14]. Details are presented in Sect.3.1(see
197
Remark3.1).
198
(b) The conclusionC≥Lp,nin Theorem1.4(i) is in a perfect concordance with the assump-
199
tion of Theorem1.2. Analogous statements hold for the other Gagliardo–Nirenberg
200
inequalities.
201
(c) Similar results to Theorem1.4can be stated also for Gagliardo–Nirenberg inequalities
202
(GN1)Cand(GN2)C, and Faber–Krahn inequality(FK)Cwith trivial modifications. In
203
particular, we have:
204
Corollary 1.1 (Optimality vs. flatness) Let (M,g) be an n(≥2)-dimensional complete
205
Riemannian manifold with non-negative Ricci curvature. The following statements are equiv-
206
alent:
207
(i) (GN1)α,pG
α,p,n holds on(M,g)for some p∈(1,n)andα∈(1,n−np];
208
(ii) (GN2)α,pN
α,p,nholds on(M,g)for some p∈(1,n)andα∈(0,1);
209
(iii) (LS)Lp
p,n holds on(M,g)for some p∈(1,n);
210
(iv) (FK)Fp
p,n holds on(M,g)for some p∈(1,n);
211
(v) (M,g)is isometric to the Euclidean spaceRn.
212
Remark 1.3 (a) The equivalence (i)⇔(v) in Corollary 1.1is precisely the main result of
213
Xia [28].
214
(b) A similar rigidity result to Corollary1.1can be stated on reversible Finsler manifolds
215
endowed with the natural Busemann–Hausdoff measured VFof(M,F); roughly speak-
216
ing, we can replace the notions ‘Riemannian’ and ‘Euclidean’ in Corollary1.1by the
217
notions ‘Berwald’ and ‘Minkowski’, respectively (see Theorem3.2). The latter notions
218
will be introduced in Sect.3.2.
219
Notations.When no confusion arises, · Lp abbreviates: (a) · Lp(M,dm)on the metric
220
measure space(M,d,m); (b) · Lp(M,dvg)on the Riemannian manifold(M,g)wheredvg
221
stands for the canonical Riemannian measure on(M,g); (c) · Lp(M,d VF)on the Finsler
222
manifold(M,F)whered VFdenotes the Busemann-Hausdoff measure on(M,F); and (d)
223
· Lp(Rn,d x)on the Euclidean/normed spaceRn whered xis the usual Lebesgue measure,
224
respectively. WhenAis not the whole space we are working on, we shall use the notation
225
uLp(A)for theLp-norm of the functionu: A→R.
226
2 Volume non-collapsing via Gagliardo–Nirenberg inequalities
227
Before the presentation of the proofs of Theorems1.1–1.3, we recall for completeness some
228
notions and results from Lott and Villani [15] and Sturm [24,25], which are indispensable in
229
our arguments.
230
Author Proof
uncorrected proof
Let(M,d,m)be a metric measure space, i.e.,(M,d)is a complete separable metric space
231
andmis a locally finite measure onMendowed with its Borelσ-algebra. In the sequel, we
232
assume that the measuremonMis strictly positive, i.e., supp[m] =M.As usual,P2(M,d)
233
is theL2-Wasserstein space of probability measures onM, whileP2(M,d,m)will denote
234
the subspace ofm-absolutely continuous measures.(M,d,m)is said to be proper if every
235
bounded and closed subset ofMis compact.
236
For a given numberN ≥1,theRényi entropy functional SN(·|m):P2(M,d)→Rwith
237
respect to the measuremis defined bySN(µ|m) = − Mρ−N1dµ, ρbeing the density of
238
µcinµ=µc+µs =ρm+µs, whereµcandµsrepresent the absolutely continuous and
239
singular parts ofµ∈P2(M,d),respectively.
240
LetK,N ∈Rbe two numbers with K ≥0 andN ≥1. For everyt ∈ [0,1]ands≥0,
241
let
242
τK,N(t) (s)=
⎧
⎪⎪
⎨
⎪⎪
⎩
+∞, if K s2 ≥(N −1)π2;
tN1 sin
K N−1t s
sin
K
N−1s1−N1
, if 0<K s2< (N−1)π2;
t, if K s2 =0.
243
We say that(M,d,m)satisfies thecurvature-dimension conditionCD(K,N)if for each
244
µ0, µ1 ∈ P2(M,d,m) there exists an optimal couplingγ ofµ0, µ1 and a geodesicŴ :
245
[0,1] →P2(M,d,m)joiningµ0andµ1such that
246
SN′(Ŵ(t)|m)≤ −
M×M
τK(1−t),N′(d(x0,x1))ρ−
1 N′
0 (x0)+τK(t),N′(d(x0,x1))ρ−
1 N′ 1 (x1)
dγ (x0,x1)
247
for everyt∈ [0,1]andN′≥N, whereρ0andρ1are the densities ofµ0andµ1with respect
248
tom. Clearly, whenK =0, the above inequality reduces to the the geodesic convexity of
249
SN′(·|m)on theL2-Wasserstein spaceP2(M,d,m).
250
It is well known that CD(K,n) holds on a complete Riemannian manifold (M,g)
251
endowed with the Riemannian volume elementdvg if and only if its Ricci curvature≥K
252
and dim(M)≤n.
253
LetBx(r)= {y ∈M:d(x,y) <r}. In the sequel we shall exploit properties which are
254
resumed in the following results.
255
Theorem 2.1 (see [25]) Let (M,d,m) be a metric measure space with strictly positive
256
measuremsatisfying the curvature-dimension conditionCD(K,N)for some K ≥ 0and
257
N >1. Then every bounded set S⊂M has finitem-measure and the metric spheres∂Bx(r)
258
have zerom-measures. Moreover, one has:
259
(i) [Generalized Bonnet–Myers theorem]If K > 0,then M =supp[m]is compact and
260
has diameter less than or equal to N−1
K π.
261
(ii) [Generalized Bishop–Gromov inequality]If K = 0,then for every R > r > 0and
262
x ∈M,
263
m(Bx(r))
rN ≥m(Bx(R)) RN .
264
Lemma 2.1 Let (M,d,m) be a metric measure space which satisfies the curvature-
265
dimension conditionCD(0,n)for some n≥2. If
266
ℓx∞0 :=lim sup
ρ→∞
m(Bx0(ρ))
ωnρn ≥a (2.1)
267
Author Proof
uncorrected proof
for some x0∈M and a>0, then
268
m(Bx(ρ))≥aωnρn, ∀x∈M, ρ≥0.
269
Proof Let us fixx ∈Mandρ >0; then we have
270
m(Bx(ρ))
ωnρn ≥lim sup
r→∞
m(Bx(r))
ωnrn [Bishop−Gromov inequality]
271
≥lim sup
r→∞
m(Bx0(r−d(x0,x)))
ωnrn [Bx(r)⊃Bx0(r−d(x0,x))]
272
=lim sup
r→∞
m(Bx0(r−d(x0,x)))
ωn(r−d(x0,x))n ·(r−d(x0,x))n rn
273
=ℓ∞x0
274
≥a, [cf. (2.1)]
275
which concludes the proof. ⊓⊔
276
We are now in the position to prove our volume non-collapsing results.
277
2.1 Casesα >1 & 0< α <1: usual Gagliardo–Nirenberg inequalities
278
In this subsection we present the proof of Theorem1.1by distinguishing two cases:
279
Proof of Theorem1.1(i): the case 1< α≤ n−np. In this part, we follow the line of [12];
280
the proof is divided into several steps. We clearly may assume thatC>Gα,p,nin(GN1)α,pC ;
281
indeed, ifC =Gα,p,n we can consider the subsequent arguments forC :=Gα,p,n +εwith
282
smallε >0 and then takeε→0+.
283
Step 1(K =0). If we assume thatK >0 then the generalized Bonnet-Myers theorem
284
(see Theorem2.1(i)) implies thatMis compact andm(M)is finite. Taking the constant map
285
u(x)=m(M)in(GN1)α,pC as a test function, one gets a contradiction. Therefore,K =0.
286
Step 2(ODE from the optimal Euclidean Gagliardo–Nirenberg inequality I). We consider
287
the optimal Gagliardo–Nirenberg inequality (1.1) in the particular case when the norm is
288
precisely the Euclidean norm| · |.After a simple rescaling, one can see that the function
289
x → (λ+ |x|p′)1−α1 , λ > 0,is a family of extremals in (1.1); therefore, we have the
290
following first order ODE
291
1−α
α(p−1)+1h′G(λ) αp1
=Gα,p,n p′
α−1 θ
hG(λ)+ α−1
α(p−1)+1λh′G(λ) θp
292
hG(λ)α(p−1)+11−θ , (2.2)
293
wherehG:(0,∞)→Ris given by
294
hG(λ)=
Rn
λ+ |x|p′α(p−1)+11−α
d x, λ >0.
295
For further use, we shall represent the functionhGin two different ways, namely
296
hG(λ)=ωn n p′B
α(p−1)+1
α−1 − n
p′, n p′
λ
α(p−1)+1
1−α +n
p′ 297
= ∞
0
ωnρnfG(λ, ρ)dρ, (2.3)
298
Author Proof
uncorrected proof
where
299
fG(λ, ρ)= p′α(p−1)+1 α−1
λ+ρp′1−ααp
ρp′−1. (2.4)
300
Step 3 (Differential inequality from (GN1)α,pC ). By the generalized Bishop-Gromov
301
inequality (see Theorem2.1(ii)) and hypothesis (D)nx
0 one has that
302
m(Bx0(ρ))
ωnρn ≤lim inf
r→0
m(Bx0(r))
ωnrn =1, ρ >0. (2.5)
303
Inspired by the form ofhG, we consider the functionwG:(0,∞)→Rdefined by
304
wG(λ)=
M
λ+d(x0,x)p′α(p−1)+11−α
dm(x), λ >0.
305
By using the layer cake representation, it follows thatwGis well-defined and of classC1;
306
indeed,
307
wG(λ)= ∞
0
m
x ∈M:
λ+d(x0,x)p′α(p−1)+11−α
>t
dt
308
= ∞
0
m(Bx0(ρ))fG(λ, ρ)dρ [changet=
λ+ρp′α(p−1)+11−α
and see(2.5)]
309
≤ ∞
0
ωnρnfG(λ, ρ)dρ [see(2.5)]
310
=hG(λ),
311
thus
312
0< wG(λ)≤hG(λ) <∞, λ >0. (2.6)
313
For everyλ >0 andk∈N, we consider the functionuλ,k:M→Rdefined by
314
uλ,k(x)=(min{0,k−d(x0,x)} +1)+
λ+maxd(x0,x),k−1p′
1 1−α
.
315
Note that since(M,d,m)is proper, the set supp(uλ,k) = Bx0(k+1)is compact. Conse-
316
quently,uλ,k ∈Lip0(M)for everyλ >0 andk ∈N; thus we can apply these functions in
317
(GN1)α,pC , i.e.,
318
uλ,kLαp ≤C|∇uλ,k|dθLpuλ,k1−θ
Lα(p−1)+1.
319
Moreover,
320
k→∞lim uλ,k(x)=
λ+d(x0,x)p′1−α1
=:uλ(x).
321
By using the dominated convergence theorem, it turns out from the above inequality thatuλ
322
also verifies(GN1)α,pC , i.e.,
323
uλLαp ≤C|∇uλ|dθLpuλ1−θ
Lα(p−1)+1. (2.7)
324
The non-smooth chain rule gives that
325
|∇uλ|d(x)= p′ α−1
λ+d(x0,x)p′1−αα
d(x0,x)p′−1|∇d(x0,·)|d(x), x ∈M. (2.8)
326
Author Proof
uncorrected proof
Sinced(x0,·)is 1-Lipschitz (therefore,|∇d(x0,·)|d(x) ≤1 for allx ∈ M), due to (2.7),
327
(2.8) and the form of the functionwG, we obtain the differential inequality
328
1−α
α(p−1)+1w′G(λ) αp1
329
≤C p′
α−1 θ
wG(λ)+ α−1
α(p−1)+1λw′G(λ) θp
wG(λ)α(p−1)+11−θ . (2.9)
330
Step 4(Compar i sono fwGandhGnear t heor igi n).We claim that
331
lim
λ→0+
wG(λ)
hG(λ) =1. (2.10)
332
By hypothesis(D)nx
0, for everyε >0 there existsρε>0 such that
333
m(Bx0(ρ))≥(1−ε)ωnρnfor allρ∈ [0, ρε]. (2.11)
334
By (2.11), one has that
335
wG(λ)= ∞
0
m(Bx0(ρ))fG(λ, ρ)dρ
336
≥(1−ε) ρε
0
ωnρnfG(λ, ρ)dρ=(1−ε)λ
α(p−1)+1
1−α +n
p′
ρελ− 1 p′
0
ωnρnfG(1, ρ)dρ.
337
Thus, by the representation (2.3) ofhGand a change of variables, it turns out that
338
lim inf
λ→0+
wG(λ)
hG(λ) ≥(1−ε)lim inf
λ→0+
ρελ− 1 p′
0
ωnρnfG(1, ρ)dρ
∞ 0
ωnρnfG(1, ρ)dρ
=1−ε.
339
The above inequality (withε > 0 arbitrary small) combined with (2.6) proves the claim
340
(2.10).
341
Step 5(Globalcompar i sono fwGandhG).We now claim that
342
wG(λ)≥ G
α,p,n
C nθ
hG(λ)= ˜hG(λ), λ >0. (2.12)
343
Since we assumed thatC>Gα,p,n,by (2.10) one has
344
lim
λ→0+
wG(λ) h˜G(λ) =
C Gα,p,n
nθ
>1.
345
Therefore, there existsλ0>0 such that for everyλ∈(0, λ0), one haswG(λ) >h˜G(λ).
346
By contradiction to (2.12), we assume that there existsλ#>0 such thatwG(λ#) <h˜G(λ#).
347
Ifλ∗=sup{0< λ < λ#:wG(λ)= ˜hG(λ)}, then 0< λ0≤λ∗< λ#.In particular,
348
wG(λ)≤ ˜hG(λ), ∀λ∈ [λ∗, λ#].
349
Author Proof
uncorrected proof
The latter relation and the differential inequality (2.9) imply that for everyλ∈ [λ∗, λ#],
350
1−α
α(p−1)+1w′G(λ) αθ1
351
≤Cθp p′
α−1 p
h˜G(λ)+ α−1
α(p−1)+1λw′G(λ)
h˜G(λ)θ (α(p−1)+1)(1−θ )p . (2.13)
352
Moreover, sinceh˜G(λ)=G
α,p,b C
nθ
hG(λ), the ODE in (2.2) can be equivalently transformed
353
for everyλ >0 into the equation
354
1−α
α(p−1)+1h˜′G(λ) αθ1
355
=Cθp p′
α−1 p
h˜G(λ)+ α−1
α(p−1)+1λh˜′G(λ)
h˜G(λ)
(1−θ )p
θ (α(p−1)+1). (2.14)
356
Forλ >0 fixed we introduce the increasing functionjGλ:(0,∞)→Rdefined by
357
jGλ(t)=
α−1 α(p−1)+1t
αθ1 +Cpθ
p′ α−1
p
α−1
α(p−1)+1λh˜G(λ)
(1−θ )p θ (α(p−1)+1)t.
358
Relations (2.13) and (2.14) can be rewritten into
359
jGλ(−w′G(λ))≤Cθp
p′ α−1
p
h˜G(λ)1+
(1−θ )p
θ (α(p−1)+1) = jGλ(− ˜h′G(λ)), ∀λ∈ [λ∗, λ#],
360
which implies that
361
−w′G(λ)≤ − ˜h′G(λ), ∀λ∈ [λ∗, λ#],
362
i.e., the functionh˜G−wGis non-increasing in[λ∗, λ#]. In particular, 0< (h˜G−wG)(λ#)≤
363
(h˜G−wG)(λ∗)=0, a contradiction. This concludes the proof of (2.12).
364
Step 6(Asympt oti cvolumegr owt hesti mat ew.r.t.x0). We claim that
365
ℓ∞x0 :=lim sup
ρ→∞
m(Bx0(ρ)) ωnρn ≥
G
α,p,n
C nθ
. (2.15)
366
By assuming the contrary, there existsε0 >0 such that for someρ0>0,
367
m(Bx0(ρ)) ωnρn ≤
G
α,p,n
C nθ
−ε0, ∀ρ≥ρ0.
368
By (2.12) and from the latter relation, we have for everyλ >0 that
369
0≤wG(λ)− G
α,p,n
C nθ
hG(λ)
370
= ∞
0
m(Bx0(ρ)) ωnρn −
G
α,p,n
C nθ
ωnρnfG(λ, ρ)dρ
371
≤
1+ε0− G
α,p,n
C
nθ ρ0 0
ωnρnfG(λ, ρ)dρ−ε0
∞ 0
ωnρnfG(λ, ρ)dρ
372
