Around a Sobolev-Orlicz inequality for operators of given spectral density

Abstract

We prove some general Sobolev-Orlicz, Nash and Faber-Krahn inequalities for positive operators of given ultracontractive norms of the spectral projectors on ]0, lambda]. For invariant operators on coverings of finite simplicial complexes this "ultracontractive spectral decay" is equivalent to von-Neumann's spectral density function. This allows in the polynomial decay case to relate the Novikov-Shubin numbers of such coverings to Sobolev inequalities on exact $\ell^2$-cochains, and to the vanishing of the torsion of the $\ell^{p,2}$-cohomology for some $p \geq 2$.

Full text

Page 1

arXiv:0902.2690v1 [math.SP] 16 Feb 2009
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????? ℓ2
? ????????????? ????? ???????????????
??????? ????
??????????? ????? ?????????????? ???????????????? ???? ?? ???????????????????? ???
? ??????
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?
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??
????????
??????? A
??
??????
??? ??
?
????????
???????????
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?
? ??
??????
??????????
??
? F(λ) = ?χA(]0,λ])?1,∞
?? ??
??? ?????? ?? ????????? ??? ???????? ???????????????????????? ???? ???????? ?? ??????????
??? ?? ??????????????? ???????? ???? ???? ????????????????? ????? ?????? ?????????
?? ???? ???? ??? ??? ???????????????? ??
???? ?????? ??? ?????? ????????
????? ????????? ??? ℓp,2
?????????????? ???? p ≥ 2 ?
?????????????? ??????? ???????
??? A
?? ??????????????? (X,µ) ? ???????
?????? ?????? ???????????? e−tA
?? ?????????? ???? ?? L1(X) ? ?????????????? ??? ?????????
???? ?℄??? ????????????? ?????
?e−tA?1,∞≤ Ct−α/2
withα > 2,
??????? ????? ?? ??? ?????????????????
???
?f?p≤ C′?A1/2f?2
for1/p = 1/2 − 1/α.
?????????? ????????? ?????????? ?? ??????? A
?? ?????????????????? ???? ????????????? ???
??????????????????????? ?? ?????? ??? ?????????? ?????????????
??????? ???? ????????? ??????? ?? ??????? ????? ?????? ????????? ??? ???????????????????
????? ???? ???? ???? ??????? ???????????? ?? ??????????????? ??????? ??? ???? ??? ??????????????
?????????????????? ?????? ????? ?????????? ????????????? ???????????????? ???????? ????????
?????? F(λ) = ?Πλ?1,∞
?? ????? ??????????? ??????? Πλ= χA(]0,λ])
?? Eλ
? ???????? ??????
????????? ??? ????? ???????? ??????? ?????? F(λ)
?????? ????????? ????????????????????????
? ?????????? ?? A
????????? ????????? ?? ???????? ???????????????? Γ? ?? ????????????? ??
?????
??
?????
??
????
??? ℓp,q
???? F(λ)
?????????
?
????? ?? ????????? Γ?????????? ?? Eλ
???? ???? F
???????? ?? ????? ?????? ??????? ?????????? A? ?????????????????? ?? ??? ?????????????? ???
???????? ????? F
??????????? ??? ??????? ?????????????????? ??????????? ???????????
???
?P∗P?1,∞= ?P?2
1,2= sup
?f?1,?g?1≤1
|?Pf,Pg?|.
?????? ??????? ????????
???????????????????? ?? ?????????????? ?????? ?????? ?????? ?????? ??????
?????? ?? ??????????? ??? ?????????????????????? ?????????????????????? ????????????????????? ?????? ??
????? ?????
???????????
??????????????? ???????? ???? ????? ??????????????????? ??????
?

Page 2

? ??????? ????
?? ????? ?????? ????????????????????????? ????????? ?? ???????????? ?? ϕ
??? ?????????
????????? ϕ−1
?????????? ???????? ??? ??????? ????????
??????? ????? ?? A
???? ??????????????????? ?? ?????? ?? (X,µ)
?????????? ?????????? ??? ??? ??
????? ?????? Πλ= χA(]0,λ]) ? ???? F(λ) = ?Πλ?1,∞< +∞?
??????????? ???????? ??????????????? ???? G(λ) =
?λ
0
dF(u)
u
? ????? ???????? ??? ???
???? f ∈ L2(X) ∩ (kerA)⊥
?? ?????????
?
X
?? E(f) = ?Af,f?2
?|f(x)|2
4E(f)
????????
???
H
?
dµ ≤ 1,
????? H(y) = y G−1(y) ?
??? ????? ?????? ?????? ?????????? ??????? ????????? ???? ??? ????
???????????? ?? A
???? ??????? ???????????? ???? ?????? (X,µ)
???? ???? L(t) = ?e−tAΠV?1,∞
???????? ???? V = L2(X) ∩ (kerA)⊥
?
??????? ???? ???????? M(t) =
?+∞
t
L(u)du < +∞? ???? ?????? ???? f ∈ V
???????
?????? ????????
???
?
X
N
?|f(x)|2
4E(f)
?
dµ ≤ ln2,
????? N(y) = y/M−1(y)
???? ??????????????????? ?? ??????????????????? ??? ?? ???? ?????????????? ???? ??? F
??
L?
H
?? ????? ????????????? ??????? ????????? ???? F
?? G
?? ??????????? ???????? ????? G
??
????????????? ???????????? F(λ) ≤ Cλα
??? α > 1 ????? G(λ) ≤ C1λα−1
1
2α
1?A1/2f?2
???? C1=
Cα
α−1
?
??? H(y) ≥ C
1
1−α
1
y
α
α−1
? ????? ???????? ?f?2α/(α−1)≤ 2C
?
????????? ????? ???????????? H
??? N
???? ??????????????????? ??? ???????? ???????????
???? ???? ?????????????????????????? ???? ??? ????????????????? ??????????? ????????
?????????????? ??????????? ?????????? ??? ???? ??????? ?? G
?? M
?????? ????????????????
?????????? ?????????????????? ?? ??????? ?????? ???? ???? ????????????? ??? ?????????
??????? ????? ????? ???????????????? ??????????? ?????????? ?? ????????????????? ???
??℄?????? ?????? ??℄???????????? ?????? ??????? ????????????????? ? ??? ?????? ?? ??????
??????? F
??? ????????? ?????? ??????? ???
??????? ????? ?? A
???? ??????? ?????????????? ??????????? ????? F(λ) = ?Πλ?1,∞
? ????
??????? f ∈ V = L2(X) ∩ (kerA)⊥
?
•
???? ??????? ????
???
?
X
|f(x)|2F−1?|f(x)|
2?f?1
?
dµ ≤ 4E(f).
•
?? ϕ
????????????????????? ???? 0 ≤ ϕ(y) ≤ yF−1(y) ?
?f?2
2?f?2
???????????????????????????????
???
1ϕ
??f?2
2
1
?
≤ 2E(f).

Page 3

??????????????????? ??? ????????????????
•
??? ????????? ?? f
??? Af
??? ???? ????? ??? ?????? Ω
??????? ?? ???? ?????? ?? ????? ???
???? ??? ?????????????????
???
Γ?
µ(Ω)ϕ
?
1
2µ(Ω)
?
≤2E(f)
?f?2
2
.
???? ????????????? ???????????? ????? ϕ(y) ≥ CyF−1(y)
??????? ???????? C
? ???????????
ϕ(y) = yF−1(y)
E(f) ≤ λ?f?2
?????? ?? ??????????? ?????? ?? ????? ??????? ????? f ∈ V
????????
2
??????? ??? Ω
??????????? ?????? ??????????? ?????????
???
2µ(Ω)F(4λ/C) ≥ 1.
?? ?????? ??????????? ????????? ?????????????? F(λ)
??? ???????????? ??????? ?? ???????????
?? Eλ
? ??? ?????? ??????????????? ??? ??? ??? ??????? ??????? ?????? ??????????? ??????????
??? ????? ???? ??????? ??????? F(λ) = dimEλ/card(Γ) ? ??????? ????????
2dimE4λ/C≥ card(Γ)/card(Ω).
?????? ??????? ???????????????????? ?? 2
??? 4/C
???????? ??? ??????? ????? ?? ????????
????????????? ???? ?? ?? ???? ???????? Γ
?? ??????? N = card(Γ)/card(Ω)
????????????
??????? Ω ??????????? ??????? ????????????? dimEλ≥ N
??????
??????????????? ????????????
?
?? ?????????? ??? ??? ?????????????????? ?????? ?????????
??? ??????? ???????????? ???????? ???????????? ??? K
???????? ????????????????? ???
X → K = X/Γ
???? ??? ?????????????????? ?? X
??? ????????? ℓ2k
???? ????????????
???????? ??????????
dk: ℓ2Xk→ ℓ2Xk+1
???? ????? ?????? ??????? ∂
?? ???????????????????????℄?
????ℓ2
? ?????????? Hk+1
2
= kerdk+1/Imdk
??????????????????? ???
•
???
kerdk+1∩ kerd∗
??????? ???? Hk+1
2
= kerdk+1/Imdk
? ?????????? ?? ℓ2
????????? ??? ????? Hk+1
2
=
k
•
??? ?????????? Tk+1
2
= Imdk/Imdk
?
???????? ????????????? ???? ???????????? ??????????? ???? ??????????? ?????? ?????
?????? ?? d−1
k
?? Imdk
??? ??????????????????? ???????? ??????
?? ??????? ?? ?????????? ℓp,q
???????????? ??????????? ??? ????? ℓ2Xk
?? ℓpXk
??? p ≥ 2 ?
??? ???????????? ??? p
????? ?????????????
???
dk(ℓ2Xk)
ℓ2
⊂ dk(ℓpXk),
?????? ?????????? ??????? ????????? ??? ???? ???????? ?????
????
∃C such that?α?p≤ C?dkα?2
for all α ∈ (kerdk)⊥⊂ ℓ2.
???????????? ????????????? ??????????? ?????????? ?????? ????????????????? ???????????
X
Hk+1
2
?? ???????? ??????????????????????????????? ?????????? ????????????????????? ????
(X)
???????????????????????????? ?? ?????????????????????????? ??? ℓp,2
???????????
?? X
? ???????????? ???????????
? ????????? ?????????? ????????????????????????????? Γ?????????????????????? Γ?
????????? ???????????? A = d∗
kdk
???????? (kerdk)⊥
????????????????????? FΓ,k(λ) = dimΓEλ

Page 4

? ??????? ????
????? ???????? ?????? Eλ
? ???? ????????? ??????????? ????????? ?????? ???? ????????????
??? ?????????? A?????? ?? ?????????? ?? ???? ?????????? ??? ??????? Tk+1
2
???? ??????????
? ??? ????? ?? FΓ,k(λ)
???? λ ց 0
???? ??????????? ???????? ?????????????????? Γ? ???? ??
???????? ??? ?????? ?? ??????????? ????? K
? ????? ???? ?????? ???????? ??? ?????? ??
?????? ?? ℄?
?????? ??????????????? ??????? ????? ????????? ???? ??????????? ??????????????? ??
??? ??????? ?????????????? ???????????
??????? ????? ?? K
??? ????????????????? ??? ??? X → K = X/Γ
?? ????????? ??
FΓ,k(λ) = dimΓEλ
??????? ?????? ??? ????????? ?????????? A = d∗
kdk
?? (kerdk)⊥
?
?? FΓ,k(λ) ≤ Cλα/2
??????? α > 2 ? ?????????? ???? ?????????? ????? ??? ???????????? ????
??????? 1/p ≤ 1/2 − 1/α?
?? ?????? ?? ??? ??????? ℓ2
??????????? Hk+1
2
(X)
? ??????????????????? ???? ?????????????
ℓp,2
??
???????? ?? X
? ???????? ??????????? ????
????????????????? ?? ?????????????? ????? ??????? ??????????? ??????????? ??????
?? ????????? ???? ?? dk
???? Imdk∩ ℓ2
????????? ?????????????????? ??? ???? H
?
?? ?? ?????? ????????????????
??????? ???? ??? ???? ?????????????? ??? ?? ?? ??????????? ???????? ????????? ?????
????????? ????????? A−1Πλ
??? A−1e−tAΠV
?
???? ??????? ???? •
? ?? A? F
??? G
?? ????? ?? ????? ???? ???????? A−1Πλ
?????? ?????
?????? ????
????
?A−1Πλ?1,∞≤ G(λ) =
?λ
0
dF(u)
u
.
•
? ?? A? L
??? M
?? ????? ???? ????? ?????? ???? A−1e−tAΠV
?+∞
t
?? ?????? ??????????????
????
?A−1e−tAΠV?1,∞≤ M(t) =
L(s)ds.
??? ??? •
????? ?????? ???????? ???
?
]ε,λ]
??
A−1(Πλ− Πε) =
u−1dΠu= λ−1Πλ− ε−1Πε+
?
]ε,λ]
u−2Πudu,
?????????? ?????? ??????????
?A−1(Πλ− Πε)?1,∞≤ λ−1F(λ) + ε−1F(ε) +
= G(λ) − G(ε) + 2ε−1F(ε).
?
]ε,λ]
u−2F(u)du
????? ??????????? G ? ?????? ?Πε/ε?1,∞= F(ε)/ε ≤ G(ε) → 0
?A−1Πλ?1,∞= ?ΠλA−1/2Πλ?2
= lim
???? ε ց 0 ???????????
1,2
ε→0?(Πλ− Πε)A−1/2Πλ?2
= lim
1,2
?????????????
ε→0?A−1(Πλ− Πε)?1,∞≤ G(λ).

Page 5

??????????????????? ??????????? ????????
?? ?????????? ????????
????
G(λ) = λ−1F(λ) +
?λ
0
u−2F(u)du,
????? ????? ??? ?????????????????? ?? ???????????? ???? F
e−sAΠVds
?? G
??? H
2,∞≤ G(λ) ?
?
•
??????????? ?????? ?????????? A−1e−tAΠV =?+∞
t
?? ??? ?? ???????????????
?
??????????? ?????? ??? ?????????? ??????? ??? ???????????? ????????? ?????????????
????????????????? ??????? ?????? ??????????? ?? ????????? ???? ????? ????? L2− Lp
??? ????
?????????? ?? Rn
??? ???? ?????? ??? ???? ??℄? ???????????? ???? ????????????? ??? ?????
{x,|f(x)| > y}
?? ??????? ??????????? ???????? ????????? ?? f = Πλf + Π>λf
??? f ∈ V
?
???? ????? ????????? ???? ???????? ???? ?????? ?A−1/2Πλ?2
2= G(λ)E(f).
?????
????
?Πλf?2
∞≤ G(λ)?A1/2f?2
???????? ??? ???? |f(x)| ≥ y
? ???? y2= 4G(λ)E(f) ??? |Πλf(x)| ≤ y/2
?? ????? ??????
??????????? |Π>λf(x)| ≥ y/2 ≥ |Πλf(x)|
|f(x)|2≤ 4|Π>λf(x)|2
??? ??????
?x ∈ X | |f(x)|2≥ 4G(λ)E(f)?.
????
on
?????????? ??????????? ?? x
?
{x,|f(x)|2≥4E(f)G(λ)}
??? ???
|f(x)|2dµ ≤ 4?Π>λf?2
2,
?????????? ??????????? ?? λ?
4E(f)G−1?|f(x)|2
?
X
|f(x)|2
4E(f)
?
dµ(x) ≤
?+∞
0
?Π>λf?2
E(f)
2
dλ,
????? G−1(y) = sup{λ | G(λ) ≤ y} ??? ????
?+∞
0
?+∞
0
???
?+∞
λ
?? ????????????????? ?????
?+∞
0
?Π>λf?2
2dλ =?dΠµf,f?
=µ?dΠµf,f? = ?Af,f? = E(f),
?????? ???????????
????????? ????????? ?????? ?????? ???????????? ?????? ????????? ??? ??????? ???
?????? f ∈ V
?e−tA/2f?∞≤ M(t)E(f),
?????????
????
|f(x)|2≤ 4|(1 − e−tA/2)f(x)|2
on
?x ∈ X | |f(x)|2≥ 4M(t)E(f)?.
???? ????????????
?
?? x
4E(f)/M−1?|f(x)|2
??? dt/t2
????
X
|f(x)|2
4E(f)
?
dµ(x) ≤
1
E(f)
?+∞
0
?(1 − e−tA/2)f?2
2
dt
t2,