Averages on annuli of Eulidean space
aa r X i v : . [ m a t h . D S ] F e b AVERAGES ON ANNULI OF EULIDEAN SPACE
FRANC¸ OIS HAVARD AND EMMANUEL LESIGNE
Abstract.
We study the range of validity of differentiation theorems andergodic theorems for R d actions, for averages on “thick spheres” of Eu-clidean space. October 22, 2018
Contents
Introduction 11. Averages on spheres 32. Averages on annuli. Basic facts 53. Averages on annuli. Maximal inequality 74. Averages on annuli. Ergodic and differentiation theorems. 125. Dimension 1 13References 14
Introduction
The classical Lebesgue differentiation Theorem states that, for any locallyintegrable function f on the d -dimensional Euclidean space R d ,lim r → | B r | Z B r f ( x + t ) d t = f ( x ) for almost all x. Here B r denotes the ball centered at the origin and of radius r >
0, and | A | denotes the Euclidean volume (i.e. the Lebesgue measure) of the set A . Thereference measure on R d is Lebesgue measure.The main fact which ensures this differentiation theorem is the Hardy-Littlewood maximal inequality, which can be written in the following form:for any non-negative integrable function f on R d , (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) x ∈ R d | sup r> | B r | Z B r f ( x + t ) d t > (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( d ) Z R d f ( x ) d x, where C ( d ) is a constant depending only on the dimension d .It is known since the works of Wiener that the maximal inequality is alsothe cornerstone of the following ergodic theorem.Let (Ω , T , µ ) be a probability space and ( T t ) t ∈ R d be a measurable and measure preserving action of the group R d on this space. Then, for all integrablefunction φ on (Ω , T , µ ),lim r → + ∞ | B r | Z B r φ ( T t ω ) d t exists for µ -almost all ω. Moreover if the action is ergodic, this limit is Z Ω φ ( ω ) d µ ( ω ) . (Basic definitions are recalled at the end of Introduction)These classical results have been extended in many different directions, andthe literature on differentiation and ergodic theorems is extremely wide (amonggeneral reviews see [St2], [K], [TEAG] ...). Variations can be made on thetype of averaging region, the type of acting group, the type of averages andthe function spaces under consideration.In the present article, we study averages on domains of the Euclidean spacewith spherical symmetry, more precisely annuli C r,e := { x ∈ R d | r − e ≤ k x k ≤ r } , where k · k denotes the standard Euclidean norm and r , e are positive realnumbers with e ≤ r .We will consider annuli for which the thickness e is a function of the radius r . Thus we assume, we are given a function r e ( r ) from (0 , + ∞ ) into itself,with e ( r ) ≤ r . And we define the following averaging operators.If f is a locally integrable function on R d , M r f ( x ) := 1 (cid:12)(cid:12) C r,e ( r ) (cid:12)(cid:12) Z C r,e ( r ) f ( x + t ) d t. If (Ω , T , µ, ( T t ) t ∈ R d ) is a measurable and measure preserving action of thegroup R d on a measure space, and if φ ∈ L ( µ ), A r φ ( ω ) := 1 (cid:12)(cid:12) C r,e ( r ) (cid:12)(cid:12) Z C r,e ( r ) φ ( T t ω ) d t. It is not difficult to verify that the family ( A r φ ) r> is µ -almost everywherewell defined (see Proposition 1). Considering R d acting on itself by translation,we have in particular that the family ( M r f ) r> is almost everywhere welldefined.We are interested in the validity of a differentiation theorem, describing thebehaviour of M r f when r goes to zero, and of an ergodic theorem, describingthe behaviour of A r φ when r goes to infinity. We know that a maximal inequal-ity is the cornerstone of each of these results, when one looks for pointwiseconvergence.Not surprisingly, the case of dimension d = 1 differs significantly from thecase of higher dimensions. In the present article we will be mostly interestedin the case of dimension d ≥
2. In this case, our study is related to the studyof averages on spheres. Although less classical and more difficult to studythan averages on balls, the averages on spheres have been thoroughly studiedby Stein ([St1], differentiation theorem, d ≥ VERAGES ON ANNULI OF EULIDEAN SPACE 3 theorem, d = 2), Jones ([J], ergodic theorem, d ≥
3) and Lacey ([L], ergodictheorem, d = 2). (A survey and numerous references can be found in [TEAG].)These results are described in Section 1. In particular, we recall that themaximal inequality for spherical averages is valid under the L p -integrabilitycondition, with p > dd − .In Section 2, we describe the first basic results concerning averages on an-nuli: almost everywhere existence of averages, and their convergence in themean.Section 3 contains the most original part of our study. We describe therange of L p spaces in which a maximal inequality is satisfied by the familiesof averages ( M r f ) and ( A r φ ). The reader could expect that this range varies“continuously” with the choice of the thickness function e , from the condi-tion p ≥ p > dd − corresponding to the case of spheres. This is not true, and we establish adichotomy theorem: under some mild natural condition of regularity on thefunction e , either the averages M r and A r behave like averages on balls, orthey behave like averages on spheres. As soon as the thickness of the annuliis asymptotically negligible with respect to the radius, there is no L d/ ( d − maximal inequality.In Section 5, we present quickly what happens in dimension 1.We thank Anthony Quas for helpful discussions, and the suggestion of arescaling argument which plays a crucial role in the negation of the weak- L d/ ( d − maximal inequality.For the reader who is not familiar with ergodic theory, we recall some basicdefinitions.A measurable and measure preserving action of R d on a measure space (Ω , T , µ )is given by a family ( T t ) t ∈ R d of maps from Ω into itself such that • The map R d × Ω → Ω , ( t, ω ) T t ( ω ) is measurable; • For all t ∈ R d and all A ∈ T , µ (cid:0) T − ( A ) (cid:1) = µ ( A ); • T = Id Ω and, for all t, s ∈ R d , T t + s = T t ◦ T s .Moreover this action is ergodic if for A ∈ T , h ∀ t ∈ R d , T t ( A ) = A i = ⇒ µ ( A ) = 0 or µ (Ω \ A ) = 0 . Averages on spheres
This section does not contain any original results, but the facts that werecall are necessary for the understanding of the remainder.Let us denote by σ r the uniform probability on the sphere S r centered at theorigin and of radius r , in the standard Euclidean space R d . If f is a continuousfunction from R d into C , we denote by M Sr f ( x ) its mean value on the spherecentered at x and of radius r . M Sr f ( x ) = Z S r f ( x + t ) d σ r ( t ) = Z S f ( x + rt ) d σ ( t ) . FRANC¸ OIS HAVARD AND EMMANUEL LESIGNE
The following strong maximal inequality is due to Stein for d ≥ d = 2. Theorem 1.
Let d be an integer ≥ and p be a real number > dd − . Thereexists a positive constant C ( d, p ) such that, for all continuous function f on R d , with compact support, (1) Z R d sup r> (cid:12)(cid:12) M Sr f ( x ) (cid:12)(cid:12) p d x ≤ C ( d, p ) Z R d | f ( x ) | p d x. For this theorem, we refer to original articles [St1], [Bo] or textbooks [St2],[TEAG]. It is not very difficult to see that the lower bound d/ ( d −
1) is optimalin this statement.Thanks to this maximal inequality, it makes sense to consider the maximalfunction sup r> (cid:12)(cid:12) M Sr f (cid:12)(cid:12) for any f ∈ L p ( R d ) with p > dd − , and the maximalinequality (1) remains valid for all f ∈ L p ( R d ).As a direct consequence, we have the following differentiation theorem(which is evident for continuous functions, and which extends by densitythanks to the maximal inequality). Corollary 1.
Let d ≥ and p > dd − . For all f ∈ L p loc ( R d ) , for almost all x ∈ R d , lim r → M Sr f ( x ) = f ( x ) . The Calder`on transference principle (see for example [TEAG], Section 2.3)allows us to transfer the maximal inequality (1) to the general context of ameasure preserving R d -action.Let (Ω , T , µ ) be a finite or σ -finite measure space and ( T t ) t ∈ R d be a mea-surable and measure preserving action of the group R d on this space. If φ isan integrable function on Ω, we denote A Sr φ ( ω ) := Z S r φ ( T t ω ) d σ r ( t ) . After transfer, the maximal inequality (1) takes the following form: let d ≥ p > dd − ; if φ ∈ L p ( µ ), the family (cid:0) A Sr φ (cid:1) r> is µ -almost everywhere welldefined and Z Ω sup r> (cid:12)(cid:12) A Sr φ ( ω ) (cid:12)(cid:12) p d µ ( ω ) ≤ C ( d, p ) Z Ω | φ ( ω ) | p d µ ( ω ) . The pointwise ergodic theorem for spherical averages is the following result.Here we suppose that µ is a probability measure. Theorem 2.
Let d ≥ , p > dd − , and φ ∈ L p ( µ ) . For µ -almost every ω , lim r → + ∞ A Sr φ ( ω ) exists . Moreover, if the R d -action on (Ω , T , µ ) is ergodic, then this limit is Z Ω φ d µ . VERAGES ON ANNULI OF EULIDEAN SPACE 5
The preceding theorem is due to Jones in the case d ≥ d = 2. Of course, the maximal inequality plays a crucial role in the proof,but we notice that the pointwise ergodic theorem is not a direct consequenceof the maximal inequality since the spheres in R d do not form a Følner family.The mean ergodic theorem is much easier to prove. Via the spectral the-orem, it is a direct consequence of the fact that the Fourier transform of themeasure σ r tends to zero at any non-zero point when r tends to infinity. Inorder to set out the mean ergodic theorem, notice first that, by Fubini The-orem, for all p ≥ φ ∈ L p ( µ ), for each r >
0, the function A Sr φ is welldefined as an element of L p ( µ ). The mean ergodic theorem then states thatthere exists ˜ φ ∈ L p ( µ ), invariant under the transformations T t , such thatlim r → + ∞ Z Ω (cid:12)(cid:12)(cid:12) A Sr φ − ˜ φ (cid:12)(cid:12)(cid:12) p d µ = 0 . Averages on annuli. Basic facts
A function e , from (0 , + ∞ ) into itself, is given such that e ( r ) ≤ r . Weconsider averaging operators M r and A r defined in Introduction. The firstthing to make sure is that the objects we want to study are well defined.Let (Ω , T , µ ) be a measure space and ( T t ) t ∈ R d be a measurable and measurepreserving action of the group R d on this space. Proposition 1. If φ is an integrable function on Ω , then, for µ -almost all ω ,the function t φ ( T t ω ) is locally integrable on R d . If the function φ is equalto zero µ -almost everywhere, then, for µ -almost all ω , φ ( T t ω ) = 0 for almostall t .Proof. Let φ ∈ L (Ω). By Fubini Theorem, for all R >
0, we have, Z Ω (cid:18)Z B R | φ | ( T t ω ) d t (cid:19) d µ ( ω ) = Z B R (cid:18)Z Ω | φ | ( T t ω ) d µ ( ω ) (cid:19) d t = | B R | Z Ω | φ | ( ω ) d µ ( ω ) . Hence, for all
R >
0, for µ -almost all ω , the function t φ ( T t ω ) is integrableon B R . This implies that for µ -almost all ω , for all R ∈ N , the function t φ ( T t ω ) is integrable on B R , which means that this function is locallyintegrable.If φ = 0 µ -almost everywhere, the preceding calculus gives:for all R >
0, for µ -almost all ω , φ ( T t ω ) = 0 for almost all t in B R .Once more we exchange the quantifiers and we obtain that for µ -almost all ω ,for all R > φ ( T t x ) = 0 for almost all t in B R . (cid:3) Corollary 2.
For all φ ∈ L (Ω) , the family ( A r φ ) r> is well defined µ -almosteverywhere. In particular, if f is a locally integrable function on R d , then the family( M r f ) r> is almost everywhere well defined. FRANC¸ OIS HAVARD AND EMMANUEL LESIGNE
Now we will see that the mean ergodic theorem for these averages on annulidoes not present any difficulty. Suppose that µ is a probability measure. If φ belongs to the Hilbert space L ( µ ), we denote by ˜ φ its orthogonal projectionon the subspace of functions which are invariant under the transformations T t , t ∈ R d . This operation extends to L ( µ ): ˜ φ is the conditionnal expectation of φ with respect to the sub- σ -algebra of T t -invariant events. Theorem 3 (Mean ergodic theorem for averages on annuli) . Let d ≥ . Forall p ≥ and for all φ ∈ L p ( µ ) , lim r → + ∞ Z Ω (cid:12)(cid:12)(cid:12) A r φ − ˜ φ (cid:12)(cid:12)(cid:12) p d µ = 0 . In the case when the thickness of the annuli tends to infinity with the radius(i.e. when e ( r ) → + ∞ if r → + ∞ ), one can show that the family ( C r,e ( r ) ) r> has the Følner property. In this case, the mean ergodic theorem follows directlyfrom general classical arguments (see for example [TEAG], Section 2.2). Butin the case when the thickness does not tend to infinity, the Følner property isnot satisfied anymore and another argument is required. We will use a Fouriertransform argument; since it is not original, we only give its main lines. Sketch of the proof of Theorem 3.
Using spherical coordinates, the Fourier trans-form b c r ( z ) = 1 (cid:12)(cid:12) C r,e ( r ) (cid:12)(cid:12) Z C r,e ( r ) exp(2 πiz · x ) d x ( z ∈ R d )of the uniform measure on the annulus C r,e ( r ) can be expressed b c r ( z ) := k d (cid:12)(cid:12) C r,e ( r ) (cid:12)(cid:12) Z rr − e ( r ) ρ d − c σ ( ρz ) d ρ, where c σ is the Fourier transform of the uniform probability on the unit sphereand k d is a constant. We know that, for all z ∈ R d \ { } , lim ρ → + ∞ c σ ( ρz ) = 0(see for example ([St2], page 347 ) or ([TEAG], Section 5.1). From this, weeasily deduce that, for all non zero z , lim r → + ∞ b c r ( z ) = 0.Consider now φ ∈ L ( µ ). The spectral theorem associates to φ a positivefinite measure ν on R d such that, for all t ∈ R d Z Ω φ ( T t ω ) φ ( ω ) d µ ( ω ) = Z R d exp(2 πiz · t ) d ν ( z ) . Moreover the function φ is orthogonal to the subspace of ( T t ) t ∈ R d -invariantfunctions if and only if ν ( { } ) = 0.We have Z Ω | A r φ ( ω ) | d µ ( ω ) = Z R d | b c r ( z ) | d ν ( z ) , and this last quantity tends to ν ( { } ) as r goes to infinity, by dominatedconvergence.This proves the mean ergodic theorem for p = 2. VERAGES ON ANNULI OF EULIDEAN SPACE 7
For any p ≥
1, we conclude along the following lines:for all φ ∈ L ∞ ( µ ), ˜ φ = lim r → + ∞ A r φ in L ( µ );by dominated convergence, this convergence takes also place in L p ( µ );by density, this convergence extends to any φ ∈ L p ( µ ). (cid:3) Averages on annuli. Maximal inequality
Introduction.
As in the preceding section the family of annuli C r := C r,e ( r ) is given. Let p ∈ [1 , + ∞ ). We say that the family of operators ( A r ) r> satisfies the strong- L p maximal inequality if there exists a constant C ( d, p )such that, for any measure preserving system (Ω , T , µ, ( T t ) t ∈ R d ) and for any φ ∈ L p ( µ ), Z Ω sup r> | A r φ ( ω ) | p d µ ( ω ) ≤ C ( d, p ) Z Ω | φ ( ω ) | p d µ ( ω ) . We say that the family of operators ( A r ) r> satisfies the weak- L p maximal in-equality if there exists a constant C ( d, p ) such that, for any measure preservingsystem (Ω , T , µ, ( T t ) t ∈ R d ) and for any φ ∈ L p ( µ ), µ (cid:26) ω ∈ Ω | sup r> | A r φ ( ω ) | > (cid:27) ≤ C ( d, p ) Z Ω | φ ( ω ) | p d µ ( ω ) . In the study of maximal inequalities for averages on annuli, we will keep inmind the following two classical facts. • Thanks to Calder`on transference principle, the validity of a maxi-mal inequality in the general context of a measure preserving system(Ω , T , µ, ( T t ) t ∈ R d ) will be guaranteed by the validity of this maximalinequality in the particular context of the action of R d on itself bytranslation. • Thanks to the general construction of Rokhlin towers, the negation ofa maximal inequality in this particular context implies the negation ofthis maximal inequality in any aperiodic system.3.2.
Direct consequences of known results on spheres and balls.
Theaverage operator on an annulus can always be written as an average of sphericalaverages: M r f ( x ) = c d | C r | Z rr − e ( r ) ρ d − (cid:18)Z S f ( x + ρθ ) d σ ( θ ) (cid:19) d ρ , where c d is a constant. As a consequence, any maximal inequality for sphericalaverages gives a maximal inequality for averages on annuli. Thus, the followingtheorem is a corollary of Theorem 1. Theorem 4.
Let d ≥ . If p > dd − , then the family of operators ( A r ) r> satisfies the strong- L p maximal inequality. FRANC¸ OIS HAVARD AND EMMANUEL LESIGNE
For averages on balls, the strong- L p maximal inequality ( p >
1) and theweak- L maximal inequality are known ([W]). Just writing down the trivialinequality that the integral of a non-negative function on C r is bounded by itsintegral on the ball B r , we obtain maximal inequalities for averages on annuli,as soon as these annuli have a volume comparable to the volume of the ball. Theorem 5.
If there exists γ > such that, for all r > , e ( r ) ≥ γr , then, forall p > , the family ( A r ) r> satisfies the strong- L p and the weak- L maximalinequalities. In the remainder of Section 3, we suppose that d ≥ L p maximalinequality as soon as p ≤ dd − . We could have expected that the range ofvalidity of a maximal inequality would evolve “continuously” with the choiceof the thickness function e . Next theorem shows that it is not the case. It saysessentially that, as soon as the function e is reasonably regular and does notsatisfy the hypothesis of Theorem 5, averages on annuli behave like averageson spheres.3.3. The dichotomy theorem.Theorem 6.
If one of the following three hypothesis is satisfied (h1) the function e is non-decreasing and inf r> e ( r ) r = 0 , (h2) lim r → + ∞ e ( r ) r = 0 , (h3) lim r → e ( r ) r = 0 ,then, for all p ≤ dd − , the family of operators ( A r ) r> does not satisfy theweak- L p maximal inequality. In fact, we will consider the following hypothesis on the thickness function e :the function e is defined on a sub-interval I of (0 , + ∞ ) and(2) ∃ a > , ∀ δ > , ∃ h > , such that [ h, ah ] ⊂ I and ∀ r ∈ [ h, ah ] , e ( r ) ≤ δr. This is not the optimal hypothesis, but it keeps a useful form. It is notdifficult to verify that each of the Hypothesis (h i ) of the Theorem impliesHypothesis (2).We will prove that, under Hypothesis (2), there is no weak- L d/ ( d − maximalinequality. Following the same construction or using a classical interpolationargument, the result extends to any L p , with p ≤ dd − .Let us begin by a geometrical lemma. Lemma 1.
Let C be an annulus of thickness ǫ ∈ ]0 , with center at x andexternal radius k x k > . If ρ ∈ [ ǫ, then σ ρ ( S ρ ∩ C ) ≥ c d ǫ/ρ . In dimension 2, this lemma says that the length of an arc defined as theintersection of the annulus C k x k ,ǫ with a circle of radius ρ centered on the VERAGES ON ANNULI OF EULIDEAN SPACE 9 exterior boundary of the annulus is greater than ǫ . This can be considered asa geometrical evidence. However, we propose a proof, given here with detailsin dimension 3. Proof.
We choose a Cartesian system of coordinates such that point x hascoordinates (0 , , k x k ), and we also use spherical coordinates ( ρ, φ, α ), ρ > < φ < π , 0 < α < π . We haveArea( S ρ ∩ C ) = Z { θ ∈ S (0 , |k x k− ǫ ≤k ρθ − x k≤k x k} ρ d σ ( θ )= Z π Z π I ( ρ, φ, α ) ρ sin α d α d φ, where I is the integration domain.We see that I = (cid:26) ( ρ, φ, α ) | ρ k x k ≤ ρ cos α ≤ ǫ k x k + ρ − ǫ k x k (cid:27) . With the change of variable u = ρ cos α , we can writeArea( S ρ ∩ C ) = 2 πρ Z ρ − ρ ρ k x k ≤ u ≤ ǫ k x k + ρ − ǫ k x k ( u ) d u = 2 πρ λ (cid:18) [ − ρ, ρ ] ∩ (cid:20) ρ k x k , ǫ k x k + ρ − ǫ k x k (cid:21)(cid:19) = 2 πρ min (cid:18) ρ − ρ k x k , ǫ k x k + ρ − ǫ k x k − ρ k x k (cid:19) = 2 πρ (cid:18) ǫ k x k + ρ − ǫ k x k − ρ k x k (cid:19) , since k x k > ≥ ρ ≥ ǫ, ≥ c ρǫ. (cid:3) Lemma 2.
There exists a non-negative function f on R d such that (3) Z R d f ( x ) d/ ( d − d x < ∞ , and, if k x k > and e ( k x k ) ≤ , then M k x k f ( x ) ≥ ln | ln e ( k x k ) |k x k d − . Proof.
We consider a non-negative function f such that f ( x ) = − k x k d − ln( k x k ) si x ∈ B / . Outside the ball B / the function f can be chosen to be zero. It is alsopossible to choose f to be smooth (except at zero) and with bounded support.In any case the integrability assumption (3) is satisfied. For all x ∈ R d , we have M k x k f ( x ) ≥ − c d k x k d − e ( k x k ) Z k x k− e ( k x k ) ≤k t k≤k x kk x + t k≤ / d t k x + t k d − ln( k x + t k ) ≥ − c d k x k d − e ( k x k ) Z k x k− e ( k x k ) ≤k t ||≤k x k e ( k x k ) ≤k x + t k≤ / d t k x + t k d − ln( k x + t k ) ≥ − c d k x k d − e ( k x k ) Z k x k− e ( k x k ) ≤k t − x k≤k x k e ( k x k ) ≤k t k≤ / d t k t k d − ln( k t k ) . Using spherical coordinates, we can write Z k x k− e ( k x k ) ≤k t − x k≤k x k e ( k x k ) ≤k t k≤ / d t k t k d − ln( k t k )= c d Z / e ( k x k ) ρ d − ln( ρ ) Z { θ ∈ S |k x k− e ( k x k ) ≤k ρθ − x k≤k x k} ρ d − d σ ( θ ) ! d ρ ≥ c d Z / e ( k x k ) ρ ln( ρ ) e ( k x k ) d ρ , the last identity beeing a consequence of Lemma 1 (since we suppose that k x k > M k x k f ( x ) ≥ − c d k x k d − e ( k x k ) e ( k x k ) Z e ( k x k ) ≤ ρ ≤ / dρρ ln( ρ ) ≥ c d k x k d − (ln | ln e ( k x k ) | − ln | ln(1 / | ) ≥ c d k x k d − ln | ln e ( k x k ) | , since e ( k x k ) ≤ / . In order to obtain the function announced in Lemma 2 we just have to multiplythis function f by a well chosen real constant. (cid:3) We know fix a function f satisfying the properties presented in Lemma 2. Lemma 3.
Let a > and < δ < be given such that, for all r ∈ [1 , a ] , e ( r ) ≤ δ .There exists λ > such that λ d/ ( d − (cid:12)(cid:12)(cid:12)n x ∈ R d | M k x k f ( x ) ≥ λ o(cid:12)(cid:12)(cid:12) ≥ c d (1 − a − d )(ln | ln δ | ) d/ ( d − . Proof.
From Lemma 2, we deduce that, for all λ >
0, the set of points x such that M k x k f ( x ) ≥ λ contains the annulus defined by k x k ∈ (1 , a ] andln | ln e ( k x k ) |k x k d − ≥ λ . Hence this set contains the annulus defined by k x k ∈ (1 , a ]and ln | ln δ |k x k d − ≥ λ . VERAGES ON ANNULI OF EULIDEAN SPACE 11
We choose λ = ln | ln δ | a d − , so that the set of points x such that M k x k f ( x ) ≥ λ contains the annulus defined by k x k ∈ (1 , a ]. We have (cid:12)(cid:12)(cid:12)n x ∈ R d | M k x k f ( x ) ≥ λ o(cid:12)(cid:12)(cid:12) ≥ c d ( a d − , and multiplying each term by λ d/ ( d − , we obtain the inequality stated inLemma 3. (cid:3) End of the proof of Theorem 6.
Now suppose that Hypothesis (2) is satisfied.Let δ ∈ (0 , ). We fix a number h > e ( r ) ≤ δr/a for all r ∈ [ h, ah ].We consider the function g defined by g ( x ) = f ( h x ). The mean value of g onthe annulus of center x , radius k x k and thickness e ( k x k ) is the mean value of f on the annulus of center h x , radius k x k /h and thickness e ( k x k ) /h . We have e ( r ) /h ≤ δ when r/h ∈ [1 , a ]. From Lemma 3, we deduce that there exists λ > λ d/ ( d − (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) h x | M k x k g ( x ) ≥ λ (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) ≥ c d (1 − a − d )(ln | ln δ | ) d/ ( d − . Note also that R R d g ( x ) d/ ( d − d x = h d R R d f ( x ) d/ ( d − d x . This last integral isa given constant denoted by C . We conclude that λ d/ ( d − (cid:12)(cid:12)(cid:8) x | M k x k g ( x ) ≥ λ (cid:9)(cid:12)(cid:12) ≥ c d C (1 − a − d )(ln | ln δ | ) d/ ( d − Z R d g ( x ) d/ ( d − d x. Since for arbitrarily small δ we can find such a function g , we conclude thatthe weak- L d/ ( d − is not satisfied, and the theorem is proved. (cid:3) How to get rid of the dichotomy.
The conclusion of the precedingstudy is that if the thickness function e is regular enough, the critical exponentfor the maximal inequality is either d/ ( d −
1) or 1. It is however possible toget rid of this dichotomy, by using results concerning spherical averages A Sr when the radius r is only considered in a sparse subset of the real line. Wewill refer here to the article by Seeger, Wainger and Wright ([SeWaWr]).Let p be any fixed number in the interval (1 , dd − ). Following [SeWaWr],there exists a sequence of positive numbers ( r n ) n ∈ N such that the family ofaverage operators (cid:0) A Sr n (cid:1) n ∈ N satisfies the strong- L p maximal inequality for all p > p and does not satisfy the strong- L p maximal inequality for any p < p .Coming back to our setting, this suggests the following construction. Let usconsider a sequence of positive reals ( ε n ) and define the thickness function e by e ( r n ) = ε n , and e ( r ) = r for the values of r that do not appear in the sequence( r n ). If the sequence ( ε n ) tends to zero rapidly enough, we will obtain that:for all p > p , the family ( A r ) r> satisfies the strong- L p maximal inequality;for all p < p , the family ( A r ) r> does not satisfy the strong- L p maximalinequality. Averages on annuli. Ergodic and differentiation theorems.
It is well known that a maximal inequality is necessary in order to have adifferentiation theorem or a pointwise ergodic theorem for a family of averages.This fact is a consequence of the
Banach principle , described, for example in[G] and [TEAG].On the other hand, the differentiation propertylim r → M r f ( x ) = f ( x ) for a.e. x is valid (for any f ∈ L p ( R d )) as soon as a L p maximal inequality is satisfied.And writing averages on annuli as averages of averages on spheres, we seeeasily that the ergodic theoremlim r → + ∞ A r φ ( ω ) exists for µ -almost all ω (and equals the integral of φ when the action is ergodic) is valid as soon as itis valid for spherical averages.Thus we have enough information in the preceding sections to conclude thatin the case when e ( r ) /r is bounded from below the differentiation theoremand the ergodic theorem for averages on annuli are satisfied under the soleintegrability assumption, and in the case when d ≥ e ( r ) /r tends tozero at zero (resp. at infinity) the differentiation theorem (resp. the pointwiseergodic theorem) is valid under the integrability assumption p > d/ ( d −
1) andinvalid under the integrability assumption p ≤ d/ ( d − p ≤ d/ ( d −
1) meansthat there exists a function f on R d , which is locally of power d/ ( d − r → M r f does not exist in the almost everywheresense (of course, it exists and equal f in the L d/ ( d − loc sense). Saying thatthe pointwise ergodic theorem is invalid for p ≤ d/ ( d −
1) means that, for anyaperiodic probability measure preserving dynamical system (Ω , T , µ, ( T t ) t ∈ R d ),there exists φ ∈ L d/ ( d − ( µ ) such that the ergodic averages A r φ are not almosteverywhere convergent when r tends to infinity. (Recall that they converge inthe mean, by Theorem 3.)4.1. Remark on non-spherical annuli.
The facts that are presented inSections 3 and 4 concern spherical annuli. They can certainly be extended toother geometrical domains; indeed, it is known that the maximal inequalityand differentiation theorem for spherical averages can be extended to averageson homotetic images of very general hypersurfaces, and there is no difficultyto define an associated notion of annuli. However, these hypersurfaces haveto satisfy some curvature conditions in order, roughly speaking, to avoid any“flat part”. These facts are very well described in Stein’s book [St2], SectionXI.3.
VERAGES ON ANNULI OF EULIDEAN SPACE 13
We will not enter in this level of generality in the present article, but wepropose to state, as a typical example, that “cubic annuli” are in general badfor differentiation and ergodic theorems.Mimicing the notation we used for spherical annuli, we define D r,e := { x ∈ R d | r − e ≤ k x k ∞ ≤ r } , where k ( x , x , . . . , x d ) k ∞ = max ≤ i ≤ d | x i | .Of course Hardy-Littlewood maximal inequality and Wiener ergodic theo-rem are valid for k · k ∞ -“balls” as well as they are valid for Euclidean balls.Thus, if we consider a thickness function r e ( r ) such that e ( r ) /r staysbounded by below, we have an analogue of Theorem 5 and differentiation the-orem, as well are pointwise ergodic theorem, are true for averages on D r,e ( r ) for the class of integrable functions.If in contrast inf r> e ( r ) /r = 0 the situation is radically different. Us-ing negative results on the maximal inequality that are known in dimen-sion 1, it is possible to prove that if e is non-decreasing and inf r> e ( r ) /r = 0then the averages on D r,e ( r ) do not satisfy any L p maximal inequality. Iflim r → e ( r ) /r = 0, there is no hope to obtain a differentiation theorem foraverages on cubic annuli, even in the class of bounded measurable functions f . If lim r → + ∞ e ( r ) /r = 0, there is no hope to obtain a pointwise ergodic the-orem for averages on cubic annuli, even in the class of bounded measurablefunctions φ . 5. Dimension 1
In dimension 1, our averages take the following form : M r f ( x ) = 12 e ( r ) Z − r + e ( r ) − r f ( x + t ) d t + Z rr − e ( r ) f ( x + t ) d t ! and A r φ ( ω ) = 12 e ( r ) Z − r + e ( r ) − r φ ( T t ω ) d t + Z rr − e ( r ) φ ( T t ω ) d t ! . They are similar to “moving ergodic averages” studied in the discrete timecase in [BeJR].We have the following facts.(i) The mean ergodic theorem (cf Theorem 3) is satisfied by the averages( A r ) r> if and only if lim r →∞ e ( r ) = + ∞ .(ii) If inf r> e ( r ) /r >
0, then a weak- L maximal inequality and a strong- L p maximal inequality are satisfied by these averages, for all p ∈ (1 , + ∞ ).(iii) If one of Hypothesis (h1), (h2) or (h3) of Theorem 6 is satisfied, noneof the preceding maximal inequality is satisfied.The “if” part of fact (i) is a direct consequence of the computation of theFourier transform of the kernel of M r ; it can also be verified easily that the Følner property is satisfied. The “only if” part of fact (i) can be verified byan explicit construction, using Rokhlin towers.Fact (ii) follows directly from the classical Hardy-Littlewood maximal in-equality in dimension 1.Fact (iii) can be verified by using a cone condition similar to the conditiondescribed in [BeJR]. We do not give here the details of this argument, whichis described in the doctorat thesis of the first author ([H]).
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