compactness criteria in lp, p>0 -...
TRANSCRIPT
Compactness criteria in Lp, p > 0
V.G.Krotov
Belarusian State University
Bedlewo, 19.09.2017
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 1 / 17
Introduction
Compactness in C(X)
Historically first compactness criterion in function space was a followingstatement, giving condition of completely boundedness in the space ofcontinuous function.
Theorem (C.Arzela–G.Ascoli criterion)
Let X be compact metric space. The set S ⊂ C(X) is completely boundedif and only if S is uniformly bounded and equicontinuous, that is
∃M > 0 ∀ f ∈ S, x ∈ X |f(x)| 6M
∀ ε > 0 ∃ δ > 0 ∀ f ∈ S ∀x1, x2 ∈ X
d(x1, x2) < δ =⇒ |f(x1)− f(x2)| < ε.
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 2 / 17
Introduction
Compactness in C(X)
Historically first compactness criterion in function space was a followingstatement, giving condition of completely boundedness in the space ofcontinuous function.
Theorem (C.Arzela–G.Ascoli criterion)
Let X be compact metric space. The set S ⊂ C(X) is completely boundedif and only if S is uniformly bounded and equicontinuous, that is
∃M > 0 ∀ f ∈ S, x ∈ X |f(x)| 6M
∀ ε > 0 ∃ δ > 0 ∀ f ∈ S ∀x1, x2 ∈ X
d(x1, x2) < δ =⇒ |f(x1)− f(x2)| < ε.
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 2 / 17
Compactness in Lp(X)
Compactness in Lp(X)
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 3 / 17
Compactness in Lp(X)
Notations
Let (X, d, µ) be bounded metric space with metric d and Borel measure µ,
B = B(x, r) = y ∈ X : d(x, y) < r,
rB is the radius of B,
fB =
Bfdµ =
1
µB
ˆBfdµ
Doubling condition
µB(x, 2r) 6 cµµB(x, r), x ∈ X, r > 0. (1)
‖f‖Lp = ‖f‖p =
(ˆX|f |pdµ
)1/p
, p > 0.
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 4 / 17
Compactness in Lp(X)
Notations
Let (X, d, µ) be bounded metric space with metric d and Borel measure µ,
B = B(x, r) = y ∈ X : d(x, y) < r,
rB is the radius of B,
fB =
Bfdµ =
1
µB
ˆBfdµ
Doubling condition
µB(x, 2r) 6 cµµB(x, r), x ∈ X, r > 0. (1)
‖f‖Lp = ‖f‖p =
(ˆX|f |pdµ
)1/p
, p > 0.
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 4 / 17
Compactness in Lp(X)
Notations
Let (X, d, µ) be bounded metric space with metric d and Borel measure µ,
B = B(x, r) = y ∈ X : d(x, y) < r,
rB is the radius of B,
fB =
Bfdµ =
1
µB
ˆBfdµ
Doubling condition
µB(x, 2r) 6 cµµB(x, r), x ∈ X, r > 0. (1)
‖f‖Lp = ‖f‖p =
(ˆX|f |pdµ
)1/p
, p > 0.
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 4 / 17
Compactness in Lp(X)
Notations
Let (X, d, µ) be bounded metric space with metric d and Borel measure µ,
B = B(x, r) = y ∈ X : d(x, y) < r,
rB is the radius of B,
fB =
Bfdµ =
1
µB
ˆBfdµ
Doubling condition
µB(x, 2r) 6 cµµB(x, r), x ∈ X, r > 0. (1)
‖f‖Lp = ‖f‖p =
(ˆX|f |pdµ
)1/p
, p > 0.
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 4 / 17
Compactness in Lp(X)
Theorem (A.N.Kolmogorov criterion)
S ⊂ Lp(X), p > 1, is completely bounded if and only if S is bounded and
limr→+0
supf∈S
ˆ
X
∣∣∣∣∣f(x)− B(x,r)
fdµ
∣∣∣∣∣p
dµ(x) = 0.
A.N.Kolmogorov (1931), X ⊂ Rn is bounded and measurable (all functionsare zero outside of X).A.Ka lamajska (1999), on metric measure spaces with the property
∀ r > 0 infx∈X
µB(x, r) > 0.
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 5 / 17
Compactness in Lp(X)
Theorem (A.N.Kolmogorov criterion)
S ⊂ Lp(X), p > 1, is completely bounded if and only if S is bounded and
limr→+0
supf∈S
ˆ
X
∣∣∣∣∣f(x)− B(x,r)
fdµ
∣∣∣∣∣p
dµ(x) = 0.
A.N.Kolmogorov (1931), X ⊂ Rn is bounded and measurable (all functionsare zero outside of X).A.Ka lamajska (1999), on metric measure spaces with the property
∀ r > 0 infx∈X
µB(x, r) > 0.
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 5 / 17
Compactness in Lp(X)
Theorem (A.N.Kolmogorov criterion)
S ⊂ Lp(X), p > 1, is completely bounded if and only if S is bounded and
limr→+0
supf∈S
ˆ
X
∣∣∣∣∣f(x)− B(x,r)
fdµ
∣∣∣∣∣p
dµ(x) = 0.
A.N.Kolmogorov (1931), X ⊂ Rn is bounded and measurable (all functionsare zero outside of X).A.Ka lamajska (1999), on metric measure spaces with the property
∀ r > 0 infx∈X
µB(x, r) > 0.
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 5 / 17
Compactness in Lp(X)
Theorem (A.N.Kolmogorov criterion)
S ⊂ Lp(X), p > 1, is completely bounded if and only if S is bounded and
limr→+0
supf∈S
ˆ
X
∣∣∣∣∣f(x)− B(x,r)
fdµ
∣∣∣∣∣p
dµ(x) = 0.
A.N.Kolmogorov (1931), X ⊂ Rn is bounded and measurable (all functionsare zero outside of X).A.Ka lamajska (1999), on metric measure spaces with the property
∀ r > 0 infx∈X
µB(x, r) > 0.
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 5 / 17
Luzin theorem and compactness
Luzin theorem and compactness
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 6 / 17
Luzin theorem and compactness
Notations
Let Ω be the class of functions η : (0, 1]→ R+,
η(+0) = 0, η(t) ↑ .
Let Dη(f) be the set of functions 0 6 g ∈ L0(X) with
∃E ⊂ X µE = 0
|f(x)− f(y)| 6 [g(x) + g(y)]η(d(x, y)), x, y ∈ X \ E. (2)
This inequality (2) is called local smoothness inequality.
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 7 / 17
Luzin theorem and compactness
Notations
Let Ω be the class of functions η : (0, 1]→ R+,
η(+0) = 0, η(t) ↑ .
Let Dη(f) be the set of functions 0 6 g ∈ L0(X) with
∃E ⊂ X µE = 0
|f(x)− f(y)| 6 [g(x) + g(y)]η(d(x, y)), x, y ∈ X \ E. (2)
This inequality (2) is called local smoothness inequality.
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 7 / 17
Luzin theorem and compactness
Main point of new criterion of compactness is the following: the functionη ∈ Ω in local smoothness inequality is the same for whole compact, andfunctions g from this inequality satisfy some uniform estimate.
Theorem
The set S ⊂ Lp(X), p > 0, is completely bounded if and only if S isbounded and
∃ η ∈ Ω supf∈S
infg∈Dη(f)
‖g‖Lp(X) < +∞
If we replace here Lp by C, then we obtain perfectly Arzela–Ascoli criterionfor C(X).
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 8 / 17
Luzin theorem and compactness
Main point of new criterion of compactness is the following: the functionη ∈ Ω in local smoothness inequality is the same for whole compact, andfunctions g from this inequality satisfy some uniform estimate.
Theorem
The set S ⊂ Lp(X), p > 0, is completely bounded if and only if S isbounded and
∃ η ∈ Ω supf∈S
infg∈Dη(f)
‖g‖Lp(X) < +∞
If we replace here Lp by C, then we obtain perfectly Arzela–Ascoli criterionfor C(X).
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 8 / 17
Luzin theorem and compactness
Main point of new criterion of compactness is the following: the functionη ∈ Ω in local smoothness inequality is the same for whole compact, andfunctions g from this inequality satisfy some uniform estimate.
Theorem
The set S ⊂ Lp(X), p > 0, is completely bounded if and only if S isbounded and
∃ η ∈ Ω supf∈S
infg∈Dη(f)
‖g‖Lp(X) < +∞
If we replace here Lp by C, then we obtain perfectly Arzela–Ascoli criterionfor C(X).
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 8 / 17
Luzin theorem and compactness
Main point of new criterion of compactness is the following: the functionη ∈ Ω in local smoothness inequality is the same for whole compact, andfunctions g from this inequality satisfy some uniform estimate.
Theorem
The set S ⊂ Lp(X), p > 0, is completely bounded if and only if S isbounded and
∃ η ∈ Ω supf∈S
infg∈Dη(f)
‖g‖Lp(X) < +∞
If we replace here Lp by C, then we obtain perfectly Arzela–Ascoli criterionfor C(X).
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 8 / 17
Maximal functions and compactness
Maximal functions and compactness
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 9 / 17
Maximal functions and compactness
The construction of functions g from local smoothness inequality goes backto works of A.Calderon, K.I.Oskolkov and V.I.Kolyada.For q > 0 and η ∈ Ω denote by
Nqηf(x) = sup
B3x
1
η(rB)
( B|f(x)− f(y)|qdµ(y)
)1/q
.
X = Rn, η(t) = tα A.Calderon (1972), and later A.Calderon–R.Scott(1978),X = [0, 1], η(t)t−1 ↓ K.I.Oskolkov (1977, in implicit form),X = [0, 1]n, η(t)t−1 ↓ V.I.Kolyada (1987), extended Oskolkov results onfunctions of several variables, in general case I.A.Ivanishko (2004).The main property now is the local smoothness type inequality
|f(x)− f(y)| 6 cq[Nqηf(x) + Nq
ηf(x)]η(d(x, y)).
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 10 / 17
Maximal functions and compactness
The construction of functions g from local smoothness inequality goes backto works of A.Calderon, K.I.Oskolkov and V.I.Kolyada.For q > 0 and η ∈ Ω denote by
Nqηf(x) = sup
B3x
1
η(rB)
( B|f(x)− f(y)|qdµ(y)
)1/q
.
X = Rn, η(t) = tα A.Calderon (1972), and later A.Calderon–R.Scott(1978),X = [0, 1], η(t)t−1 ↓ K.I.Oskolkov (1977, in implicit form),X = [0, 1]n, η(t)t−1 ↓ V.I.Kolyada (1987), extended Oskolkov results onfunctions of several variables, in general case I.A.Ivanishko (2004).The main property now is the local smoothness type inequality
|f(x)− f(y)| 6 cq[Nqηf(x) + Nq
ηf(x)]η(d(x, y)).
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 10 / 17
Maximal functions and compactness
The construction of functions g from local smoothness inequality goes backto works of A.Calderon, K.I.Oskolkov and V.I.Kolyada.For q > 0 and η ∈ Ω denote by
Nqηf(x) = sup
B3x
1
η(rB)
( B|f(x)− f(y)|qdµ(y)
)1/q
.
X = Rn, η(t) = tα A.Calderon (1972), and later A.Calderon–R.Scott(1978),X = [0, 1], η(t)t−1 ↓ K.I.Oskolkov (1977, in implicit form),X = [0, 1]n, η(t)t−1 ↓ V.I.Kolyada (1987), extended Oskolkov results onfunctions of several variables, in general case I.A.Ivanishko (2004).The main property now is the local smoothness type inequality
|f(x)− f(y)| 6 cq[Nqηf(x) + Nq
ηf(x)]η(d(x, y)).
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 10 / 17
Maximal functions and compactness
The construction of functions g from local smoothness inequality goes backto works of A.Calderon, K.I.Oskolkov and V.I.Kolyada.For q > 0 and η ∈ Ω denote by
Nqηf(x) = sup
B3x
1
η(rB)
( B|f(x)− f(y)|qdµ(y)
)1/q
.
X = Rn, η(t) = tα A.Calderon (1972), and later A.Calderon–R.Scott(1978),X = [0, 1], η(t)t−1 ↓ K.I.Oskolkov (1977, in implicit form),X = [0, 1]n, η(t)t−1 ↓ V.I.Kolyada (1987), extended Oskolkov results onfunctions of several variables, in general case I.A.Ivanishko (2004).The main property now is the local smoothness type inequality
|f(x)− f(y)| 6 cq[Nqηf(x) + Nq
ηf(x)]η(d(x, y)).
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 10 / 17
Maximal functions and compactness
Theorem
Let S ⊂ Lp(X), p > 0, be bounded set. Then1) if 0 < q < p and S is completely bounded, then
∃ η ∈ Ω supf∈S
∥∥Nqηf∥∥Lp(X)
< +∞, (3)
2) if for some q > 0 the condition (3) is fulfield then S is completelybounded.
The first statement of this theorem is not true for q = p.The second part this theorem can be strengthened, and the condition (3)can be weakened.
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 11 / 17
Maximal functions and compactness
Theorem
Let S ⊂ Lp(X), p > 0, be bounded set. Then1) if 0 < q < p and S is completely bounded, then
∃ η ∈ Ω supf∈S
∥∥Nqηf∥∥Lp(X)
< +∞, (3)
2) if for some q > 0 the condition (3) is fulfield then S is completelybounded.
The first statement of this theorem is not true for q = p.The second part this theorem can be strengthened, and the condition (3)can be weakened.
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 11 / 17
Maximal functions and compactness
Theorem
Let S ⊂ Lp(X), p > 0, be bounded set. Then1) if 0 < q < p and S is completely bounded, then
∃ η ∈ Ω supf∈S
∥∥Nqηf∥∥Lp(X)
< +∞, (3)
2) if for some q > 0 the condition (3) is fulfield then S is completelybounded.
The first statement of this theorem is not true for q = p.The second part this theorem can be strengthened, and the condition (3)can be weakened.
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 11 / 17
Maximal functions and compactness
Theorem
Let S ⊂ Lp(X), p > 0, be bounded set. Then1) if 0 < q < p and S is completely bounded, then
∃ η ∈ Ω supf∈S
∥∥Nqηf∥∥Lp(X)
< +∞, (3)
2) if for some q > 0 the condition (3) is fulfield then S is completelybounded.
The first statement of this theorem is not true for q = p.The second part this theorem can be strengthened, and the condition (3)can be weakened.
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 11 / 17
Maximal functions and compactness
Theorem
Let S ⊂ Lp(X), p > 0, be bounded set. If for some q > 0
limr→+0
supf∈S
ˆ
X
B(x,r)
|f(x)− f(y)|q dµ(y)
p/q
dµ(x) = 0, (4)
then S is completely bounded.
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 12 / 17
New conditions for compactness
New conditions for compactness
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 13 / 17
New conditions for compactness
Let B ⊂ X be a ball. Then there exists the number I(p)B f , such thatˆB|f − I(p)B f |p dµ = inf
c∈R
ˆB|f − c|p dµ.
Theorem
Let S ⊂ Lp(X), p > 0, be bounded set. If for some q > 0 the followingcondition holds
limr→+0
supf∈S
ˆ
X
B(x,r)
|I(q)B(x,r)f − f(y)|qdµ(y)
p/q
dµ(x) = 0 (5)
then S is completely bounded.
.The last two theorems looks very similar. But the proofs are quite different.
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 14 / 17
New conditions for compactness
Let B ⊂ X be a ball. Then there exists the number I(p)B f , such thatˆB|f − I(p)B f |p dµ = inf
c∈R
ˆB|f − c|p dµ.
Theorem
Let S ⊂ Lp(X), p > 0, be bounded set. If for some q > 0 the followingcondition holds
limr→+0
supf∈S
ˆ
X
B(x,r)
|I(q)B(x,r)f − f(y)|qdµ(y)
p/q
dµ(x) = 0 (5)
then S is completely bounded.
.The last two theorems looks very similar. But the proofs are quite different.
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 14 / 17
New conditions for compactness
It is natural to consider the following condition
limr→0
supf∈S
ˆX|I(q)B(x,r)f − f(x)|p dµ(x) = 0. (6)
It is similar to Kolmogorov condition.But I don’t know if this condition is sufficient for compactness of S.
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 15 / 17
New conditions for compactness
It is natural to consider the following condition
limr→0
supf∈S
ˆX|I(q)B(x,r)f − f(x)|p dµ(x) = 0. (6)
It is similar to Kolmogorov condition.But I don’t know if this condition is sufficient for compactness of S.
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 15 / 17
New conditions for compactness
It is natural to consider the following condition
limr→0
supf∈S
ˆX|I(q)B(x,r)f − f(x)|p dµ(x) = 0. (6)
It is similar to Kolmogorov condition.But I don’t know if this condition is sufficient for compactness of S.
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 15 / 17
X is unbounded
All Lp-criterion don’t work for unbounded X. This was observed firstly byJ.Tamarkin (1932). We need the following extra condition for compactness
limR→+∞
supf∈S
ˆX\B(x0,R)
|f |p dµ = 0,
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 16 / 17
X is unbounded
All Lp-criterion don’t work for unbounded X. This was observed firstly byJ.Tamarkin (1932). We need the following extra condition for compactness
limR→+∞
supf∈S
ˆX\B(x0,R)
|f |p dµ = 0,
V.G.Krotov (BSU) Compactness criteria Bedlewo, 19.09.2017 16 / 17