Applications to Differentiation Theory

2.1 Maximal Functions

2.1.3 Applications to Differentiation Theory

1 c_n =

Z

Rⁿ

dx (1+|x|²)ⁿ⁺¹²

=ωn−1 Z _∞

0

rⁿ⁻¹ (1+r²)ⁿ⁺¹²

dr

=ωn−1 Z π/2

0

(sinϕ)ⁿ⁻¹dϕ (r=tanϕ)

= 2πⁿ² Γ(ⁿ₂)

1 2

Γ(ⁿ₂)Γ(¹₂) Γ(ⁿ⁺¹

2 )

= πⁿ⁺¹² Γ(ⁿ⁺¹₂ ),

where we used the formula forωn−1in Appendix A.3 and an identity in Appendix A.4. We conclude that

c_n=Γ(ⁿ⁺¹₂ ) π

n+1 2

and that the Poisson kernel onRⁿis given by P(x) =Γ(ⁿ⁺¹₂ )

πⁿ⁺¹²

1 (1+|x|²)ⁿ⁺¹²

. (2.1.13)

Theorem 2.1.10 implies that the solution of the Dirichlet problem (2.1.12) is point- wise bounded by the Hardy–Littlewood maximal function of f.

We have the following.

Theorem 2.1.14.Let0<p<∞,0<q<∞, and T_ε and T∗as previously. Suppose that for some B>0and all f ∈L^p(X)we have

T∗(f)

_Lq,∞≤B f

_Lp (2.1.15)

and that for all f ∈D,

ε→0limT_ε(f) =T(f) (2.1.16) exists and is finiteν-a.e. (and defines a linear operator on D). Then for all func- tions f in L^p(X,µ)the limit (2.1.16) exists and is finiteν-a.e., and defines a linear operator T on L^p(X)(uniquely extending T defined on D) that satisfies

T(f)

_Lq,∞≤B f

_Lp. (2.1.17)

Proof. Given f inL^p, we define theoscillationof f: O_f(y) =lim sup

ε→0

lim sup

θ→0

|T_ε(f)(y)−T_θ(f)(y)|.

We would like to show that for all f∈L^pandδ >0,

ν({y∈Y: O_f(y)>δ}) =0. (2.1.18) Once (2.1.18) is established, givenf∈L^p(X), we obtain thatO_f(y) =0 forν-almost all y, which implies that T_ε(f)(y)is Cauchy forν-almost ally, and it therefore converges ν-a.e. to someT(f)(y)as ε→0. The operatorT defined this way on L^p(X)is linear and extendsT defined onD.

To approximateO_f we use density. Givenη>0, find a functiong∈Dsuch that

f−g

L^p <η. SinceTε(g)→T(g)ν-a.e, it follows thatO_g=0ν-a.e. Using this fact and the linearity of theTε’s, we conclude that

O_f(y)≤O_g(y) +O_f−g(y) =Of−g(y) ν-a.e.

Now for anyδ>0 we have

ν({y∈Y: O_f(y)>δ}) ≤ν({y∈Y : O_f−g(y)>δ})

≤ν({y∈Y : 2T∗(f−g)(y)>δ})

≤ 2B f−g

L^p/δq

≤(2Bη/δ)^q.

Lettingη→0, we deduce (2.1.18). We conclude thatT_ε(f)is a Cauchy sequence, and hence it convergesν-a.e. to someT(f). Since|T(f)| ≤ |T∗(f)|,the conclusion

(2.1.17) of the theorem follows easily.

We now derive some applications. First we return to the issue of almost every- where convergence of the expressions f∗P_y, wherePis the Poisson kernel.

Example 2.1.15.Fix 1≤p<∞and f ∈L^p(Rⁿ). Let P(x) =Γ(ⁿ⁺¹₂ )

πⁿ⁺¹²

1 (1+|x|²)ⁿ⁺¹²

be the Poisson kernel onRⁿand letPε(x) =ε⁻ⁿP(ε⁻¹x). We deduce from the previ- ous theorem that the family f∗Pεconverges to f a.e. LetDbe the set of all contin- uous functions with compact support onRⁿ. Since the family(P_ε)_ε>0is an approx- imate identity, Theorem 1.2.19 (2) implies that for f inDwe have that f∗Pε→ f uniformly on compact subsets ofRⁿ and hence a.e. In view of Theorem 2.1.10, the supremum of the family of linear operatorsT_ε(f) = f∗P_ε is controlled by the Hardy–Littlewood maximal function, and thus it mapsL^ptoL^p,∞for 1≤p<∞.

Theorem 2.1.14 now gives that f∗P_εconverges to f a.e. for all f ∈L^p.

Here is another application of Theorem 2.1.14. We refer to Exercise 2.1.10 for others.

Corollary 2.1.16.(Lebesgue’s differentiation theorem) For any locally integrable function f onRⁿwe have

limr→0

1

|B(x,r)|

Z

B(x,r)

f(y)dy=f(x) (2.1.19)

for almost all x inRⁿ. Consequently we have|f| ≤M(f)a.e.

Proof. Since Rⁿ is the union of the ballsB(0,N)forN=1,2,3. . ., it suffices to prove the required conclusion for almost allxinside the ballB(0,N). Then we may take f supported in a larger ball, thus working with f integrable over the whole space. LetT_εbe the operator given with convolution withk_ε, wherek=v⁻¹_n χ_B(0,1). We know that the corresponding maximal operatorT∗is controlled by the the cen- tered Hardy–Littlewood maximal functionM, which mapsL¹toL^1,∞. It is straight- forward to verify that (2.1.19) holds for all continuous functions f with compact support. Since the set of these functions is dense in L¹, andT∗mapsL¹ toL^1,∞, Theorem 2.1.14 implies that (2.1.19) holds for a general f inL¹.

The following corollaries were inspired by Example 2.1.15.

Corollary 2.1.17.(Differentiation theorem for approximate identities) Let K be an L¹function onRⁿwith integral1that has a continuous integrable radially decreas- ing majorant. Then f∗K_ε→ f a.e. asε→0for all f ∈L^p(Rⁿ),1≤p<∞.

Proof. It follows from Example 1.2.16 thatKε is an approximate identity. Theorem 1.2.19 now implies thatf∗Kε→funiformly on compact sets when fis continuous.

LetDbe the space of all continuous functions with compact support. Thenf∗Kε→ f a.e. for f ∈D. It follows from Corollary 2.1.12 thatT∗(f) =sup_ε>0|f∗K_ε|maps L^p toL^p,∞ for 1≤p<∞. Using Theorem 2.1.14, we conclude the proof of the

corollary.

Remark 2.1.18.Fix f ∈L^p(Rⁿ)for some 1≤p<∞. Theorem 1.2.19 implies that f∗K_εconverges to f inL^pand hence some subsequencef∗K_εnof f∗K_εconverges tof a.e. asn→∞, (ε_n→0). Compare this result with Corollary 2.1.17, which gives a.e. convergence for the whole family f∗K_ε asε→0.

Corollary 2.1.19.(Differentiation theorem for multiples of approximate identi- ties) Let K be a function onRⁿthat has an integrable radially decreasing majorant.

Let a=^R_RnK(x)dx. Then for all f ∈L^p(Rⁿ)and1≤p<∞,(f∗Kε)(x)→a f(x) for almost all x∈Rⁿasε→0.

Proof. Use Theorem 1.2.21 instead of Theorem 1.2.19 in the proof of Corollary

2.1.17.

The following application of the Lebesgue differentiation theorem uses a simple stopping-time argument. This is the sort of argument in which a selection procedure stops when it is exhausted at a certain scale and is then repeated at the next scale. A certain refinement of the following proposition is of fundamental importance in the study of singular integrals given in Chapter 4.

Proposition 2.1.20.Given a nonnegative integrable function f onRⁿ andα >0, there exist disjoint open cubes Q_j such that for almost all x∈ ^S_jQ_jc

we have f(x)≤αand

α< 1

|Q_j| Z

Qj

f(t)dt≤2ⁿα. (2.1.20)

Proof. The proof provides an excellent paradigm of a stopping-time argument. Start by decomposingRⁿas a union of cubes of equal size, whose interiors are disjoint, and whose diameter is so large that|Q|^−1R_Qf(x)dx≤α for everyQin this mesh.

This is possible since f is integrable and|Q|^−1R_Qf(x)dx→0 as|Q| →∞. Call the union of these cubesE0.

Divide each cube in the mesh into 2ⁿcongruent cubes by bisecting each of the sides. Call the new collection of cubesE1. Select a cubeQinE1if

1

|Q|

Z

Q

f(x)dx>α (2.1.21)

and call the set of all selected cubesS1. Now subdivide each cube inE1\S1into 2ⁿcongruent cubes by bisecting each of the sides as before. Call this new collection of cubesE2. Repeat the same procedure and select a family of cubesS2that satisfy (2.1.21). Continue this way ad infinitum and call the cubes in^S∞_m=1Sm“selected.”

IfQwas selected, then there existsQ1inEm−1containingQthat was not selected at the(m−1)thstep for somem≥1. Therefore,

α< 1

|Q|

Z

Q

f(x)dx≤2ⁿ 1

|Q₁| Z

Q₁

f(x)dx≤2ⁿα.

Now callF the closure of the complement of the union of all selected cubes. If x∈F, then there exists a sequence of cubes containingxwhose diameter shrinks

down to zero such that the average of f over these cubes is less than or equal toα. By Corollary 2.1.16, it follows that f(x)≤α almost everywhere inF. This proves

the proposition.

In the proof of Proposition 2.1.20 it was not crucial to assume that f was defined on allRⁿ, but only on a cube. We now give a local version of this result.

Corollary 2.1.21.Let f ≥0be an integrable function over a cube Q inRⁿand let α≥_|Q|^{1 R}_Qf dx. Then there exist disjoint open subcubes Qjof Q such that for almost all x∈Q\^S_jQ_jwe have f≤α and (2.1.20) holds for all j.

Proof. This easily follows by a simple modification of Proposition 2.1.20 in which

Rⁿis replaced by the fixed cubeQ.

See Exercise 2.1.4 for an application of Proposition 2.1.20 involving maximal functions.

Exercises

2.1.1.A positive Borel measureµ on Rⁿ is calledinner regularif for any open subsetU ofRⁿwe haveµ(U) =sup{µ(K): KjU, Kcompact}andµis called locally finiteifµ(B)<∞for all ballsB.

(a) Letµ be a positive inner regular locally finite measure onRⁿthat satisfies the followingdoubling condition: There exists a constantD(µ)>0 such that for all x∈Rⁿandr>0 we have

µ(3B(x,r))≤D(µ)µ(B(x,r)).

For f ∈L¹_loc(Rⁿ,µ)define the uncentered maximal functionM_µ(f)with respect to µby

M_µ(f)(x) = sup

r>0

sup

z:|z−x|<r µ(B(z,r))6=0

1 µ(B(z,r))

Z

B(z,r)

f(y)dµ(y).

Show that Mµ maps L¹(Rⁿ,µ) to L^1,∞(Rⁿ,µ) with constant at most D(µ) and L^p(Rⁿ,µ)to itself with constant at most 2 _p−1^{p 1}_p

D(µ)^1p.

(b) Obtain as a consequence a differentiation theorem analogous to Corollary 2.1.16.

Hint:Part (a): For f ∈L¹(Rⁿ,µ)show that the set Eα={M_µ(f)>α} is open.

Then use the argument of the proof of Theorem 2.1.6 and the inner regularity ofµ.

2.1.2.OnRconsider the maximal functionM_µof Exercise 2.1.1.

(a) (W. H. Young) Prove the following covering lemma. Given a finite setF of open intervals inR, prove that there exist two subfamilies each consisting of pairwise disjoint intervals such that the union of the intervals in the original family is equal to the union of the intervals of both subfamilies. Use this result to show that the

maximal functionM_µ of Exercise 2.1.1 mapsL¹(µ)→L^1,∞(µ)with constant at most 2.

(b) (Grafakos and Kinnunen [107]) Prove that for anyσ-finite positive measureµ onR,α>0, andf ∈L¹_loc(R,µ)we have

1 α

Z

A

|f|dµ−µ(A)≤ 1 α Z

{|f|>α}

|f|dµ−µ({|f|>α}).

Use this result and part (a) to prove that for allα>0 and all locally integrable f we have

µ({|f|>α}) +µ({M_µ(f)>α})≤ 1 α Z

{|f|>α}|f|dµ+1 α

Z

{M_µ(f)>α}|f|dµ and note that equality is obtained whenα=1 andf(x) =|x|^−1/p.

(c) Conclude thatM_µ mapsL^p(µ)toL^p(µ), 1<p<∞, with bound at most the unique positive solutionA_pof the equation

(p−1)x^p−p x^p−1−1=0.

(d) (Grafakos and Montgomery-Smith [109]) Ifµis the Lebesgue measure show that for 1<p<∞we have

M

L^p→L^p =Ap,

whereApis the unique positive solution of the equation in part (c).

Hint:Part (a): Select a subsetG ofF with minimal cardinality such that^S_J∈GJ= S

I∈FI. Part (d): One direction follows from part (c). Conversely,M(|x|^−1/p)(1) =

p p−1

γ^1/p

0+1

γ+1 , whereγ is the unique positive solution of the equation _p−1^{p γ1/p}

0+1 γ+1 = γ^−1/p.Conclude thatM(|x|^−1/p)(1) =A_pand thatM(|x|^−1/p) =A_p|x|^−1/p. Since this function is not inL^p, consider the family f_ε(x) =|x|^−1/pmin(|x|^−ε,|x|^ε),ε>0, and show thatM(f_ε)(x)≥(1+γ

1 p0+ε

)(1+γ)⁻¹(_p¹0+ε)⁻¹f_ε(x)for 0<ε<p⁰. 2.1.3.Define the centered Hardy–Littlewood maximal functionMc and the uncen- tered Hardy–Littlewood maximal functionM_cusing cubes with sides parallel to the axes instead of balls inRⁿ. Prove that

v_n(n/2)^n/2≤ M(f)

Mc(f)≤2ⁿ/v_n, v_n(n/2)^n/2≤ M(f)

Mc(f)≤2ⁿ/v_n,

wherev_nis the volume of the unit ball inRⁿ. Conclude thatMc andM_care weak type(1,1)and they mapL^p(Rⁿ)to itself for 1<p≤∞.

2.1.4.(a) Prove the estimate:

|{x∈Rⁿ: M(f)(x)>2α}| ≤3ⁿ α

Z

{|f|>α}|f(y)|dy

and conclude that M mapsL^ptoL^p,∞ with norm at most 2·3^n/pfor 1≤p<∞.

Deduce that if flog⁺(2|f|)is integrable over a ballB, thenM(f)is integrable over the same ballB.

(b) (Wiener [291], Stein [255]) Apply Proposition 2.1.20 to |f| andα >0 and Exercise 2.1.3 to show that withc_n= (n/2)^n/2v_nwe have

|{x∈Rⁿ: M(f)(x)>c_nα}| ≥2⁻ⁿ α

Z

{|f|>α}

|f(y)|dy.

(c) Suppose that f is integrable and supported in a ballB(0,ρ). Show that forxin B(0,2ρ)\B(0,ρ)we haveM(f)(x)≤M(ρ²|x|⁻²x).Conclude that

Z

B(0,2ρ)M(f)dx≤(4ⁿ+1) Z

B(0,ρ)M(f)dx and from this deduce a similar inequality forM(f).

(d) Suppose thatf is integrable and supported in a ballBand thatM(f)is integrable overB. Letλ₀=2ⁿ|B|⁻¹

f

_L1. Use part (b) to prove thatflog⁺(λ₀⁻¹c_n|f|)is inte- grable overB.

Hint:Part (a): Write f = fχ_|f|>α+fχ_|f|≤α. Part (c): Letx⁰=ρ²|x|⁻²xfor some ρ<|x|<2ρ. Show that forR>|x| −ρ, we have that

Z

B(x,R)

|f(z)|dz≤ Z

B(x⁰,R)

|f(z)|dz

by showing thatB(x,R)∩B(0,ρ)⊂B(x⁰,R). Part (d): Forx∈/2Bwe haveM(f)(x)≤ λ₀, hence^R_2BM(f)(x)dx≥^R_λ^∞

0|{x∈2B: M(f)(x)>α}|dα.

2.1.5.(A. Kolmogorov) LetSbe a sublinear operator that mapsL¹(Rⁿ)toL^1,∞(Rⁿ) with normB. Suppose that f ∈L¹(Rⁿ). Prove that for any setAof finite Lebesgue measure and for all 0<q<1 we have

Z

A

|S(f)(x)|^qdx≤(1−q)⁻¹B^q|A|^1−q f

q L¹, and in particular, for the Hardy–Littlewood maximal operator,

Z

A

M(f)(x)^qdx≤(1−q)⁻¹3^nq|A|^1−q f

q L¹. Hint:Use the identity

Z

A

|S(f)(x)|^qdx= Z _∞

0

qα^q−1|{x∈A:S(f)(x)>α}|dα and estimate the last measure by min(|A|,^B

α

f

_L1).

2.1.6.LetM_s(f)(x)be the supremum of the averages of|f|over all rectangles with sides parallel to the axes containingx. The operatorM_sis called thestrong maximal function.

(a) Prove thatM_smapsL^p(Rⁿ)to itself.

(b) Show that the operator norm ofM_sisAⁿ_p, whereA_pis as in Exercise 2.1.2(c).

(c) Prove thatM_sis not weak type (1,1).

2.1.7.Prove that if

|ϕ(x₁, . . . ,x_n)| ≤A(1+|x₁|)^−1−ε· · ·(1+|x_n|)^−1−ε

for someA,ε>0, andϕ_t1,...,tn(x) =t₁⁻¹· · ·t_n⁻¹ϕ(t₁⁻¹x₁, . . . ,t_n⁻¹x_n),then the maximal operator

f 7→ sup

t1,...,t_n>0

|f∗ϕ_t1,...,t_n| is pointwise controlled by the strong maximal function.

2.1.8.Prove that for any fixed 1<p<∞, the operator norm ofMonL^p(Rⁿ)tends to infinity asn→∞.

Hint:Let f0be the characteristic function of the unit ball inRⁿ. Consider the aver- ages|B_x|^−1R_B

xf0dy, whereBx=B ¹₂(|x| − |x|⁻¹)_|x|^x,¹₂(|x|+|x|⁻¹)

for|x|>1.

2.1.9.(a) InR²letM₀(f)(x)be the maximal function obtained by taking the supre- mum of the averages of|f|over all rectangles (of arbitrary orientation) containing x. Prove thatM₀is not bounded onL^p(Rⁿ)for p<2 and conclude thatM₀is not weak type(1,1).

(b) LetM₀₀(f)(x)be the maximal function obtained by taking the supremum of the averages of|f|over all rectangles inR²of arbitrary orientation but fixed eccentricity containing x. (The eccentricity of a rectangle is the ratio of its longer side to its shorter side.) Using a covering lemma, show thatM₀₀ is weak type(1,1)with a bound proportional to the square of the eccentricity.

(c) On Rⁿ define a maximal function by taking the supremum of the averages of |f|over all products of intervals I₁× · · · ×I_n containing a point xwith|I₂|= a2|I₁|, . . . ,|I_n|=a_n|I₁|anda2, . . . ,a_n>0 fixed. Show that this maximal function is weak type(1,1)with bound independent of the numbersa2, . . . ,an.

Hint:Part (b): Letbbe the eccentricity. If two rectangles with the same eccentricity intersect, then the smaller one is contained in the bigger one scaled 4btimes. Then use an argument similar to that in Lemma 2.1.5.

2.1.10.(a) Let p,q,X,Y be as in Theorem 2.1.14. Assume thatTε is a family of quasilinear operators defined onL^p(X)[i.e.,|Tε(f+g)| ≤K(|Tε(f)|+|Tε(g)|)for all f,g∈L^p(X)] such that lim_ε→0Tε(f) =0 for all f in some dense subspaceDof L^p(X). Use the argument of Theorem 2.1.14 to prove that lim_ε→0Tε(f) =0 for all

f inL^p(X).

(b) Use the result in part (a) to prove the following improvement of the Lebesgue differentiation theorem: Let f ∈L_loc^p (Rⁿ)for some 1≤p<∞. Then for almost all x∈Rⁿwe have

lim

|B|→0 B3x

1

|B|

Z

B

|f(y)−f(x)|^pdy=0, where the limit is taken over all open ballsBcontainingx.

Hint:Define

T_ε(f)(x) = sup

B(z,ε)3x

1

|B(z,ε)|

Z

B(z,ε)

|f(y)−f(x)|^pdy 1/p

and observe thatT∗(f) =sup_ε>0T_ε(f)≤ |f|+M(|f|^p)^1p. Use Theorem 2.1.14.

2.1.11.OnRdefine the right and left maximal functionsM_RandM_Las follows:

ML(f)(x) =sup

r>0

1 r

Z _x x−r

|f(t)|dt, M_R(f)(x) =sup

r>0

1 r

Z x+r x

|f(t)|dt. (a) (Riesz’s sunrise lemma [218]) Show that

|{x∈R: M_L(f)(x)>α}| = 1 α

Z

{M_L(f)>α}

|f(t)|dt,

|{x∈R: M_R(f)(x)>α}| = 1 α

Z

{M_R(f)>α}|f(t)|dt.

(b) Conclude thatM_LandM_RmapL^ptoL^pwith norm at most p/(p−1)for 1<

p<∞.

(c) Construct examples to show that the operator norms ofM_L andM_R onL^pare exactlyp/(p−1)for 1<p<∞.

(d) (K. L. Phillips) Prove thatM=max(M_R,M_L).

(e) (J. Duoandikoetxea) LetN=min(M_R,M_L). Since M(f)^p+N(f)^p=M_L(f)^p+M_R(f)^p, integrate over the line and use the following consequence of part (a),

Z

R

M_L(f)^p+M_R(f)^pdx= p p−1

Z

R

|f| M(f)^p−1+N(f)^p−1 dx, to prove that

(p−1) M(f)

p L^p−p

f L^p

M(f)

p−1 L^p −

f

p L^p≤0. This provides an alternative proof of the result in Exercise 2.1.2(c).

2.1.12.A cube Q= [a₁2^k,(a₁+1)2^k)× · · · ×[a_n2^k,(a_n+1)2^k) on Rⁿ is called dyadicifk,a1, . . . ,an∈Z. Observe that either two dyadic cubes are disjoint or one

contains the other. Define thedyadic maximal function M_d(f)(x) =sup

Q3x

1

|Q|

Z

Q

f(y)dy,

where the supremum is taken over all dyadic cubesQcontainingx.

(a) Prove thatM_d mapsL¹toL^1,∞with constant at most one, that is, show that for allα>0 andf ∈L¹(Rⁿ)we have

|{x∈Rⁿ: M_d(f)(x)>α}| ≤α⁻¹ Z

{M_d(f)>α}f(t)dt. (b) Conclude thatM_dmapsL^p(Rⁿ)to itself with constant at mostp/(p−1).

2.1.13.Observe that the proof of Theorem 2.1.6 yields the estimate λ|{M(f)>λ}|^1p ≤3ⁿ|{M(f)>λ}|^−1+1p

Z

{M(f)>λ}

|f(y)|dy

forλ>0 and flocally integrable. Use the result of Exercise 1.1.12(a) to prove that the Hardy–Littlewood maximal operatorM maps the space L^p,∞(Rⁿ)to itself for 1<p<∞.

2.1.14.LetK(x) = (1+|x|)^−n−δ be defined onRⁿ. Prove that there exists a constant C_n,δ such that for allε0>0 we have the estimate

sup

ε>ε₀

(|f| ∗K_ε)(x)≤C_n,δ sup

ε>ε₀

1 εⁿ

Z

|y−x|≤ε|f(y)|dy, for all f locally integrable onRⁿ.

Hint:Apply only a minor modification to the proof of Theorem 2.1.10.