compactness criteria in lp, p>0 -...

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.

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.

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,

Bfdµ =

ˆBfdµ

Doubling condition

µB(x, 2r) 6 cµµB(x, r), x ∈ X, r > 0. (1)

‖f‖Lp = ‖f‖p =

(ˆX|f |pdµ

, p > 0.

Notations

B = B(x, r) = y ∈ X : d(x, y) < r,

Bfdµ =

ˆBfdµ

Doubling condition

µB(x, 2r) 6 cµµB(x, r), x ∈ X, r > 0. (1)

‖f‖Lp = ‖f‖p =

(ˆX|f |pdµ

, p > 0.

Notations

B = B(x, r) = y ∈ X : d(x, y) < r,

Bfdµ =

ˆBfdµ

Doubling condition

µB(x, 2r) 6 cµµB(x, r), x ∈ X, r > 0. (1)

‖f‖Lp = ‖f‖p =

(ˆX|f |pdµ

, p > 0.

Notations

B = B(x, r) = y ∈ X : d(x, y) < r,

Bfdµ =

ˆBfdµ

Doubling condition

µB(x, 2r) 6 cµµB(x, r), x ∈ X, r > 0. (1)

‖f‖Lp = ‖f‖p =

(ˆX|f |pdµ

, p > 0.

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

∣∣∣∣∣f(x)− B(x,r)

∣∣∣∣∣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.

limr→+0

supf∈S

∣∣∣∣∣f(x)− B(x,r)

∣∣∣∣∣p

dµ(x) = 0.

∀ r > 0 infx∈X

µB(x, r) > 0.

limr→+0

supf∈S

∣∣∣∣∣f(x)− B(x,r)

∣∣∣∣∣p

dµ(x) = 0.

∀ r > 0 infx∈X

µB(x, r) > 0.

limr→+0

supf∈S

∣∣∣∣∣f(x)− B(x,r)

∣∣∣∣∣p

dµ(x) = 0.

∀ r > 0 infx∈X

µB(x, r) > 0.

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.

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.

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).

Theorem

infg∈Dη(f)

‖g‖Lp(X) < +∞

Theorem

infg∈Dη(f)

‖g‖Lp(X) < +∞

Theorem

infg∈Dη(f)

‖g‖Lp(X) < +∞

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

η(rB)

( B|f(x)− f(y)|qdµ(y)

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)).

Nqηf(x) = sup

η(rB)

ηf(x)]η(d(x, y)).

Nqηf(x) = sup

η(rB)

ηf(x)]η(d(x, y)).

Nqηf(x) = sup

η(rB)

ηf(x)]η(d(x, y)).

Theorem

Let S ⊂ Lp(X), p > 0, be bounded set. Then1) if 0 < q < p and S is completely bounded, then

∥∥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.

Theorem

< +∞, (3)

Theorem

< +∞, (3)

Theorem

< +∞, (3)

Theorem

Let S ⊂ Lp(X), p > 0, be bounded set. If for some q > 0

limr→+0

supf∈S

B(x,r)

|f(x)− f(y)|q dµ(y)

dµ(x) = 0, (4)

then S is completely bounded.

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

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

B(x,r)

|I(q)B(x,r)f − f(y)|qdµ(y)

dµ(x) = 0 (5)

.The last two theorems looks very similar. But the proofs are quite different.

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

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

B(x,r)

|I(q)B(x,r)f − f(y)|qdµ(y)

dµ(x) = 0 (5)

.The last two theorems looks very similar. But the proofs are quite different.

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.

limr→0

supf∈S

ˆX|I(q)B(x,r)f − f(x)|p dµ(x) = 0. (6)

limr→0

supf∈S

ˆX|I(q)B(x,r)f − f(x)|p dµ(x) = 0. (6)

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,

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,

X is unbounded

Thank you for attention!

compactness criteria in lp, p>0 -...

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