Definition and Basic Properties of BMO - Functions of Bounded Mean Oscillation

Exercises

7.1 Functions of Bounded Mean Oscillation

7.1.1 Definition and Basic Properties of BMO

Definition 7.1.1. For f a complex-valued locally integrable function on Rⁿ, set f

BMO=sup

|Q|

f(x)−Avg

f dx,

where the supremum is taken over all cubes Q in Rⁿ. The function f is called of bounded mean oscillation iff

BMO <∞and BMO(Rⁿ)is the set of all locally integrable functions f on Rⁿwithf

BMO<∞.

Several remarks are in order. First it is a simple fact that BMO(Rⁿ)is a linear space, that is, if f,g∈BMO(Rⁿ)andλ∈C, then f+g andλf are also in BMO(Rⁿ) and

f+g

BMO ≤ f

BMO+g

BMO, λf

BMO = |λ|f

BMO. But _BMOis not a norm. The problem is that iff

BMO=0, this does not imply that f =0 but that f is a constant. See Proposition 7.1.2. Moreover, every constant function c satisfiesc

BMO=0. Consequently, functions f and f+c have the same BMO norms whenever c is a constant. In the sequel, we keep in mind that elements of BMO whose difference is a constant are identified. Although _BMO is only a seminorm, we occasionally refer to it as a norm when there is no possibility of confusion.

We begin with a list of basic properties of BMO.

Proposition 7.1.2. The following properties of the space BMO(Rⁿ)are valid:

(1) Iff

BMO=0, then f is a.e. equal to a constant.

(2) L^∞(Rⁿ)is contained in BMO(Rⁿ)andf

BMO≤2f

L^∞.

(3) Suppose that there exists an A>0 such that for all cubes Q in Rⁿthere exists a constant cQsuch that

sup

|Q|

Q|f(x)−cQ|dx≤A. (7.1.1) Then f∈BMO(Rⁿ)andf

BMO≤2A.

(4) For all f locally integrable we have 1

BMO≤sup

|Q|inf

c_Q

Q|f(x)−cQ|dx≤f

BMO.

(5) If f∈BMO(Rⁿ), h∈Rⁿ, andτ^h⁽f)is given byτ^h⁽f)(x) =f(x−h), thenτ^h(f) is also in BMO(Rⁿ)and

τ^h⁽f)

BMO=f

BMO.

(6) If f ∈BMO(Rⁿ)andλ ^>0, then the functionδ^λ⁽f)defined byδ^λ⁽f)(x) = f(λx)is also in BMO(Rⁿ)and

δ^λ⁽f)

BMO=f

BMO.

(7) If f∈BMO then so is|f|. Similarly, if f,g are real-valued BMO functions, then so are max(f,g), and min(f,g). In other words, BMO is a lattice. Moreover,

|f|

BMO ≤ 2f

BMO, max(f,g)

BMO ≤ 3 2

BMO+g

BMO

, min(f,g)

BMO ≤ 3 2

BMO+g

BMO

(8) For locally integrable functions f define f

BMOballs=sup

|B|

f(x)−Avg

f dx, (7.1.2) where the supremum is taken over all balls B in Rⁿ. Then there are positive constants cn,Cnsuch that

cnf

BMO≤f

BMO_balls≤Cnf

BMO.

Proof. To prove (1) note that f has to be a.e. equal to its average cNover every cube [−N,N]ⁿ. Since[−N,N]ⁿis contained in[−N−1,N+1]ⁿ, it follows that cN=cN+1

for all N. This implies the required conclusion. To prove (2) observe that Avg

f−Avg

f ≤2 Avg

Q |f| ≤2f

L^∞.

For part (3) note that

f−Avg

f ≤ |f−cQ|+ Avg

f−cQ ≤ |f−cQ|+ 1

|Q|

Q|f(t)−cQ|dt. Averaging over Q and using (7.1.1), we obtain thatf

BMO≤2A. The lower inequality in (4) follows from (3) while the upper one is trivial. Property (5) is imme- diate. For (6) note that Avg_Qδ^λ⁽^f^{) =}^Avg_λQf and thus

|Q|

f(λx)−Avg

δ^λ⁽f) dx= 1

|λQ|

λQ

f(x)−Avg

λQ f dx.

Property (7) is a consequence of the easy fact that |f| −Avg

|f| ≤ f−Avg

f +Avg

f−Avg

f .

Also, the maximum and the minimum of two functions can be expressed in terms of the absolute value of their difference. We now turn to (8). Given any cube Q in Rⁿ, we let B be the smallest ball that contains it. Then|B|/|Q|=2⁻ⁿvn

√nⁿ, where vnis the volume of the unit ball, and

|Q|

f(x)−Avg

f dx≤|B|

|Q|

|B|

f(x)−Avg

f dx≤vn

√nⁿ 2ⁿ f

BMO_balls.

It follows from (3) that f

BMO≤2¹⁻ⁿvn

√nⁿf

BMO_balls. To obtain the reverse conclusion, given any ball B find the smallest cube Q that contains it and argue similarly using a version of (3) for the space BMOballs. Example 7.1.3. We indicate why L^∞(Rⁿ)is a proper subspace of BMO(Rⁿ). We claim that the function log|x|is in BMO(Rⁿ)but not in L^∞(Rⁿ). To prove that it is in BMO(Rⁿ), for every x0∈Rⁿand R>0, we must find a constant Cx₀,Rsuch that the average of|log|x|−Cx0,R|over the ball{x : |x−x0| ≤R}is uniformly bounded.

Since 1 vnRⁿ

|x−x0|≤R

log|x| −Cx₀,R dx= 1 vn

|z−R⁻¹x₀|≤1

log|z| −Cx₀,R+log R dz, we may take Cx₀,R=C_R−1x₀,1+logR, and things reduce to the case that R=1 and x0is arbitrary. If R=1 and|x0| ≤2, take Cx₀,1=0 and observe that

|x−x0|≤1

log|x| dx≤

|x|≤3

log|x| dx=C.

When R=1 and|x0| ≥2, take Cx0,1=log|x0|. In this case notice that 1

|x−x0|≤1

log|x| −log|x₀| dx= 1 vn

|x−x0|≤1

log |x|

|x0|

dx≤log 2,

since when|x−x₀| ≤1 and|x₀| ≥2 we have that log_|^|x|_x

0|≤log^|x_|⁰_x^|+1

0| ≤log³₂ and log^|x_|x|⁰^|≤log_|x^|x⁰^|

0|−1≤log 2. Thus log|x|is in BMO.

The function log|x|turns out to be a typical element of BMO, but we make this statement a bit more precise later. It is interesting to observe that an abrupt cutoff of a BMO function may not give a function in the same space.

Example 7.1.4. The function h(x) =χx>0log¹_xis not in BMO(R). Indeed, the problem is at the origin. Consider the intervals(−ε^,ε), where 0<ε^<¹₂. We have that

(−ε,ε)Avgh= 1 2ε

₊_ε

−ε h(x)dx= 1 2ε

_ε 0 log1

xdx=1+log¹_ε

2 .

But then 1 2ε

_+ε

−ε h(x)−Avg

(−ε,ε)h dx≥ 1 2ε

₀

−ε Avg

(−ε,ε)h dx=1+log¹_ε

4 ,

and the latter is clearly unbounded asε→0.

Let us now look at some basic properties of BMO functions. Observe that if a cube Q1is contained in a cube Q2, then

Avg

f−Avg

f ≤ 1

|Q1|

Q₁

f−Avg

f dx

≤ 1

|Q1|

Q₂

f−Avg

f dx

≤ |Q2|

|Q1|f

BMO.

(7.1.3)

The same estimate holds if the sets Q1and Q2are balls.

A version of this inequality is the first statement in the following proposition.

For simplicity, we denote byf

BMOthe expression given byf

BMO_ballsin (7.1.2), since these quantities are comparable. For a ball B and a>0, aB denotes the ball that is concentric with B and whose radius is a times the radius of B.

Proposition 7.1.5. (i) Let f be in BMO(Rⁿ). Given a ball B and a positive integer m, we have

Avg

f−Avg

2^mB

f ≤2ⁿmf

BMO. (7.1.4)

(ii) For anyδ ^>0 there is a constant C_n,δ such that for any ball B(x0,R)we have R^δ

Rⁿ

f(x)−Avg_B(x₀_,R)f

(R+|x−x0|)^n+δ dx≤C_n,δf

BMO. (7.1.5)

An analogous estimate holds for cubes with center x0and side length R.

(iii) There exists a constant Cnsuch that for all f∈BMO(Rⁿ)we have

sup

y∈Rⁿ

sup

t>0

Rⁿ|f(x)−(Pt∗f)(y)|Pt(x−y)dx≤Cnf

BMO. (7.1.6)

Here Pt denotes the Poisson kernel introduced in Chapter 2.

(iv) Conversely, there is a constant C_nsuch that for all f∈L¹_loc(Rⁿ)for which

Rⁿ

|f(x)|

(1+|x|)ⁿ⁺¹dx<∞ we have

C_nf

BMO≤sup

y∈Rⁿ

sup

t>0

Rⁿ|f(x)−(Pt∗f)(y)|Pt(x−y)dx. (7.1.7) Proof. (i) We have

Avg

f−Avg

f = 1

|B| _B

f(t)−Avg

f dt

≤ 2ⁿ

|2B|

f(t)−Avg

f dt

≤ 2ⁿf

BMO.

Using this inequality, we derive (7.1.4) by adding and subtracting the terms Avg

f, Avg

2²B

f, . . . , Avg

2^m⁻¹B

(ii) In the proof below we take B(x0,R)to be the ball B=B(0,1)with radius 1 centered at the origin. Once this case is known, given a ball B(x0,R)we replace the function f by the function f(Rx+x0). When B=B(0,1)we have

Rⁿ

f(x)−Avg

f (1+|x|)^n+δ dx

≤

f(x)−Avg

f (1+|x|)ⁿ⁺^δ dx+

∑

∞ k=0

2^k+1B\2^kB

f(x)−Avg

2^k+1B

f + Avg

2^k+1B

f−Avg

f (1+|x|)ⁿ⁺^δ dx

≤

f(x)−Avg

f dx +

∑

∞ k=0

2^−k(n+δ) 2^k+1B

f(x)−Avg

2^k+1B

f + Avg

2^k+1B

f−Avg

f dx

≤vnf

BMO+

∑

∞ k=0

2^−k(n+^δ⁾

1+2ⁿ(k+1)

(2^k+1)ⁿvnf

BMO

=C_n,δf

BMO.

(iii) The proof of (7.1.6) is a reprise of the argument given in (ii). Set Bt=B(y,t).

We first prove a version of (7.1.6) in which the expression(Pt∗f)(y)is replaced by Avg_B_t f . For fixed y,t we have

Rⁿ

t f(x)−Avg

f (t²+|x−y|²)ⁿ⁺¹²

≤

t f(x)−Avg

B_t

f (t²+|x−y|²)ⁿ⁺¹² dx

∑

∞ k=0

2^k+1Bt\2^kBt

t f(x)−Avg

2^k+1Bt

f + Avg

2^k+1Bt

f−Avg

f (t²+|x−y|²)ⁿ⁺¹² dx

≤

f(x)−Avg

B_t

tⁿ dx

∑

∞ k=0

2^−k(n+1) tⁿ

2^k+1Bt

f(x)− Avg

2^k+1Bt

f + Avg

2^k+1Bt

f−Avg

f dx

≤vnf

BMO+

∑

∞ k=0

2^−k(n+1)

1+2ⁿ(k+1)

(2^k+1)ⁿvnf

BMO

=Cnf

BMO.

(7.1.8)

Using the inequality just proved, we also obtain

Rⁿ

(P_t∗f)(y)−Avg

f Pt(x−y)dx = (P_t∗f)(y)−Avg

≤

Rⁿ

Pt(x−y) f(x)−Avg

f dx

≤Cnf

BMO,

which, combined with the inequality in (7.1.8), yields (7.1.6) with constant 2Cn. (iv) Conversely, let A be the expression on the right in (7.1.7). For|x−y| ≤t we have Pt(x−y)≥cnt(2t²)⁻ⁿ⁺¹² =c_nt⁻ⁿ, which gives

A≥

Rⁿ|f(x)−(Pt∗f)(y)|Pt(x−y)dx≥c_n tⁿ

|x−y|≤t|f(x)−(Pt∗f)(y)|dx.

Proposition 7.1.2 (3) now implies that f

BMO≤2A/(vnc_n).

This concludes the proof of the proposition.

Dalam dokumen 250 (Halaman 131-137)