The local sharp maximal function and BMO on locally homogeneous spaces
aa r X i v : . [ m a t h . F A ] N ov The local sharp maximal function and BMOon locally homogeneous spaces
Marco Bramanti, Maria Stella Fanciullo ∗ June 27, 2018
Abstract
We prove a local version of Fefferman-Stein inequality for the localsharp maximal function, and a local version of John-Nirenberg inequal-ity for locally BMO functions, in the framework of locally homogeneousspaces, in the sense of Bramanti-Zhu [3].
Real analysis and the theory of singular integrals have been developed firstin the Euclidean setting and then in more general contexts, in view of theirapplications to harmonic analysis, partial differential equations, and complexanalysis. Around 1970 the theory of spaces of homogeneous type , that is quasi-metric doubling measure spaces, started to be systematically developed in themonograph by Coifman-Weiss [10] and was successfully applied to several fields.Much more recently, some problems arising from the quest of a-priori estimatesfor PDEs suggested that real analysis would be a more flexible and useful toolif its concepts and results were stated also in a local version. The meaningof this localization is, roughly speaking, the following: we want an abstracttheory which, when applied to the concrete setting of a bounded domain Ω ⊂ R n endowed with a local quasidistance ρ and a locally doubling measure dµ (typically, the Lebesgue measure), brings to integrals over metric balls B r ( x ) properly contained in Ω, and never requires to compute integrals over sets ofthe kind B r ( x ) ∩ Ω , as happens when we apply the standard theory of spacesof homogeneous type to a bounded domain Ω. These versions, however, arenot easily obtained a posteriori from the well established theory; instead, theyrequire a careful analysis which often poses nontrivial new problems.For instance, global L p estimates for certain operators of Ornstein-Uhlenbecktype were proved in [2] using results from a theory of nondoubling spaces, de-veloped in [1], which in particular applies to certain locally doubling spaces. ∗ L p and Schauder estimates for nonvariational operators structured on H¨ormander’svector fields.The aim of this paper is to continue the theory of locally doubling spaces, asstarted in [3], with two main results: a local version of Fefferman-Stein’s theoremregarding the sharp maximal function, and a local version of John-Nirenberginequality about BM O functions. The first of these results, in its Euclideanversion, is a key ingredient of a novel approach to the proof of L p estimatesfor nonvariational elliptic and parabolic operators with possibly discontinuouscoefficients, first devised by Krylov in [16]; the present extension can open theway to the application of these techniques to more general differential operators,as shown in [5]. Comparison with the existent literature and main results.
The “sharpmaximal function” was introduced in the Euclidean context by Fefferman-Steinin [13], where the related L p inequality was proved. In some spaces of general-ized homogeneous type, this operator has been introduced and studied by Laiin [17]. Local sharp maximal functions have been studied by several authors(with different definitions). Here we follow the approach of Iwaniec [14] whoproves, in the Euclidean context, a version of local sharp maximal inequality.His proof relies on a clever adaptation of Calder´on-Zygmund decomposition,and the striking fact is that this construction can be adapted quite naturallyto the abstract context of locally homogeneous spaces, exploiting the proper-ties of the “dyadic cubes” abstractly constructed in this framework in [3]. Ourfirst main result is the sharp maximal inequality stated in Theorem 3.4. Thisstatement involves dyadic cubes and the dyadic local sharp maximal function(see Definition 3.3); since, however, these “cubes” are abstract objects which inthe concrete application of the theory are not easily visualized, it is convenientto derive from Theorem 3.4 some consequences formulated in the language ofmetric balls and the local sharp maximal function (defined by means of balls, in-stead of dyadic cubes, see Definition 3.5). These results, more easily applicable,are Corollaries 3.7, 3.8, 3.9.The space
BM O of functions with bounded mean oscillation was introducedin [15], where the famous “John-Nirenberg inequality” is proved. Versions ofthis space and this inequality in spaces of homogeneous type have been givenby several authors, starting with Buckley [6] (see also Caruso-Fanciullo [7] andDafni-Yue [11]). To adapt this result to our context, we follow the approachcontained in Mateu, Mattila, Nicolau, Orobitg [18, Appendix], see also Castillo,Ramos Fern´andez, Trousselot [8]. Our main result is Theorem 4.2, with its usefulconsequence, Theorem 4.5, stating that we can equivalently compute the meanoscillation of a function or its L p version for any p ∈ (1 , ∞ ), always computingaverages over small balls. Plan of the paper.
Section 2 contains some basic facts about locally doublingspaces; in Section 3 the local sharp maximal function is studied and several L p We start recalling the abstract context of locally homogeneous spaces, as intro-duced in [3].(H1) Let Ω be a set, endowed with a function ρ : Ω × Ω → [0 , ∞ ) such thatfor any x, y ∈ Ω:(a) ρ ( x, y ) = 0 ⇔ x = y ;(b) ρ ( x, y ) = ρ ( y, x ) . For any x ∈ Ω , r > , let us define the ball B ( x, r ) = { y ∈ Ω : ρ ( x, y ) < r } . These balls can be used to define a topology in Ω, saying that A ⊂ Ω is open iffor any x ∈ A there exists r > B ( x, r ) ⊂ A . Also, we will say that E ⊂ Ω is bounded if E is contained in some ball.Let us assume that:(H2) (a) the balls are open with respect to this topology;(H2) (b) for any x ∈ Ω and r > B ( x, r ) is contained in { y ∈ Ω : ρ ( x, y ) ≤ r } . It can be proved (see [3, Prop. 2.4]) that the validity of conditions (H2) (a)and (b) is equivalent to the following:(H2’) ρ ( x, y ) is a continuous function of x for any fixed y ∈ Ω . (H3) Let µ be a positive regular Borel measure in Ω . (H4) Assume there exists an increasing sequence { Ω n } ∞ n =1 of bounded mea-surable subsets of Ω , such that: ∞ [ n =1 Ω n = Ω (2.1)and such for, any n = 1 , , , ... :(i) the closure of Ω n in Ω is compact;(ii) there exists ε n > { x ∈ Ω : ρ ( x, y ) < ε n for some y ∈ Ω n } ⊂ Ω n +1 ; (2.2)We also assume that:(H5) there exists B n ≥ x, y, z ∈ Ω n ρ ( x, y ) ≤ B n ( ρ ( x, z ) + ρ ( z, y )) ; (2.3)(H6) there exists C n > x ∈ Ω n , < r ≤ ε n we have0 < µ ( B ( x, r )) ≤ C n µ ( B ( x, r )) < ∞ . (2.4)(Note that for x ∈ Ω n and r ≤ ε n we also have B ( x, r ) ⊂ Ω n +1 ).3 efinition 2.1 We will say that (Ω , { Ω n } ∞ n =1 , ρ, µ ) is a locally homogeneousspace if assumptions (H1) to (H6) hold. Dependence on the constants.
The numbers ε n , B n , C n will be called “theconstants of Ω n ”. It is not restrictive to assume that B n , C n are nondecreasingsequences, and ε n is a nonincreasing sequence. Throughout the paper our esti-mates, for a fixed Ω n , will often depend not only on the constants of Ω n , butalso (possibly) on the constants of Ω n +1 , Ω n +2 , Ω n +3 . We will briefly say that“a constant depends on n ” to mean this type of dependence.In the language of [10], ρ is a quasidistance in each set Ω n ; we can also saythat it is a local quasidistance in Ω . We stress that the two conditions appearingin (H2) are logically independent each from the other, and they do not followfrom (2.3), even when ρ is a quasidistance in Ω , that is when B n = B > n . If, however, ρ is a distance in Ω, that is B n = 1 for all n , then (H2) isautomatically fulfilled. The continuity of ρ also implies that (2.3) still holds for x, y, z ∈ Ω n . Also, note that µ (Ω n ) < ∞ for every n , since Ω n is compact.The basic concepts about Vitali covering lemma, the local maximal functionand its L p bound can be easily adapted to this context: Lemma 2.2 (Vitali covering Lemma)
Let E be a measurable subset of Ω n and let { B ( x λ , r λ ) } λ ∈ Λ be a family of balls with centers x λ ∈ Ω n and radii < r λ ≤ r n ≡ ε n / (cid:0) B n +1 + 3 B n +1 (cid:1) , such that E ⊂ [ λ ∈ Λ B ( x λ , r λ ) . Then onecan select a countable subcollection (cid:8) B (cid:0) x λ j , r λ j (cid:1)(cid:9) ∞ j =1 of mutually disjoint ballsso that E ⊂ ∞ [ j =1 B (cid:0) x λ j , K n r λ j (cid:1) (2.5) with K n = (cid:0) B n +1 + 3 B n +1 (cid:1) and, for some constant c depending on Ω , ∞ X j =1 µ (cid:0) B (cid:0) x λ j , r λ j (cid:1)(cid:1) ≥ cµ ( E ) . (2.6)We can then give the following Definition 2.3
Fix Ω n , Ω n +1 and, for any f ∈ L (Ω n +1 ) define the local max-imal function M Ω n , Ω n +1 f ( x ) = sup B ( x,r ) ∋ xr ≤ r n µ ( B ( x, r )) Z B ( x,r ) | f ( y ) | dµ ( y ) for x ∈ Ω n where r n = 2 ε n / (cid:0) B n +1 + 3 B n +1 (cid:1) is the same number appearing in VitaliLemma. (Actually, the following theorem still holds if this number r n is re-placed by any smaller number). Theorem 2.4
Let f be a measurable function defined on Ω n +1 . The followinghold:(a) If f ∈ L p (Ω n +1 ) for some p ∈ [1 , ∞ ] , then M Ω n , Ω n +1 f is finite almosteverywhere in Ω n ;(b) if f ∈ L (Ω n +1 ) , then for every t > , µ (cid:0)(cid:8) x ∈ Ω n : (cid:0) M Ω n , Ω n +1 f (cid:1) ( x ) > t (cid:9)(cid:1) ≤ c n t Z Ω n +1 | f ( y ) | dµ ( y ) ; (c) if f ∈ L p (Ω n +1 ) , < p ≤ ∞ , then M Ω n , Ω n +1 f ∈ L p (Ω n ) and (cid:13)(cid:13) M Ω n , Ω n +1 f (cid:13)(cid:13) L p (Ω n ) ≤ c n,p k f k L p (Ω n +1 ) . By standard techniques, from the above theorem one can also prove thefollowing:
Theorem 2.5 (Lebesgue differentiation theorem)
For every f ∈ L loc (Ω n +1 ) and a.e. x ∈ Ω n there exists lim r → + µ ( B ( x, r )) Z B ( x,r ) f ( y ) dµ ( y ) = f ( x ) . In particular, for every f ∈ L loc (Ω n +1 ) and a.e. x ∈ Ω n , | f ( x ) | ≤ M Ω n , Ω n +1 f ( x ) . A deep construction which is carried out in [3, Thm. 8.3], adapting toour local context an analogous construction developed in doubling spaces byChrist [9] is that of dyadic cubes . Their relevant properties are collected in thefollowing:
Theorem 2.6 (Dyadic cubes) (See [3, Thm. 3.1]). Let (Ω , { Ω n } ∞ n =1 , ρ, µ ) be a locally homogeneous space. For any n = 1 , , , ... there exists a collection ∆ n = (cid:8) Q kα ⊂ Ω , k = 1 , , ..., α ∈ I k (cid:9) (where I k is a set of indices) of open sets called “dyadic cubes subordinatedto Ω n ” , positive constants a , c , c , c , δ ∈ (0 , and a set E ⊂ Ω n of zeromeasure, such that for any k = 1 , , ... we have:(a) ∀ α ∈ I k , each Q kα contains a ball B (cid:0) z kα , a δ k (cid:1) ; (b) [ α ∈ I k Q kα ⊂ Ω n +1 ; (c) ∀ α ∈ I k , ≤ l ≤ k, there exists Q lβ ⊇ Q kα ; (d) ∀ α ∈ I k , diam (cid:0) Q kα (cid:1) < c δ k and Q kα ⊂ B (cid:0) z kα , c δ k (cid:1) ⊂ Ω n +2 ; (e) ℓ ≥ k = ⇒ ∀ α ∈ I k , β ∈ I l , Q ℓβ ⊂ Q kα or Q ℓβ ∩ Q kα = ∅ ;5 f ) Ω n \ [ α ∈ I k Q kα ⊂ E ; (g) ∀ α ∈ I k , x ∈ Q kα \ E, j ≥ there exists Q jβ ∋ x ; (h) µ (cid:0) B ( x, r ) ∩ Q kα (cid:1) ≤ c µ (cid:0) B ( x, r ) ∩ Q kα (cid:1) (2.7) for any x ∈ Q kα \ E, r > . More precisely, for these x and r we have: µ (cid:0) B ( x, r ) ∩ Q kα (cid:1) ≥ (cid:26) c µ ( B ( x, r )) for r ≤ δ k c µ (cid:0) Q kα (cid:1) for r > δ k (2.8)The sets Q kα can be thought as dyadic cubes of side length δ k . Note that k is a positive integer, so we are only considering small dyadic cubes. Thecubes (cid:8) Q kα (cid:9) are subordinated to a particular Ω n , meaning that they essentiallycover Ω n (that is, their union covers Ω n up to a set of zero measure) and arecontained in Ω n +1 . Note that the cubes Q kα and all the constants depend on n ,so we should write, more precisely n Q ( n ) ,kα o α ∈ I ( n ) k ; δ ( n ) ; a , ( n ) , c , ( n ) , c , ( n ) , c , ( n ) ,but we will usually avoid this heavy notation.In the proof of the above theorem, δ is chosen small enough , so it is notrestrictive to assume c δ < ε n +1 , which implies that the ball B (cid:0) z kα , c δ k (cid:1) appearing in point (d) is ⊂ Ω n +2 . (We remark this fact because in [3] theinclusion B (cid:0) z kα , c δ k (cid:1) ⊂ Ω n +2 is not stated).Point (h) contains a crucial information: the triple (cid:0) Q kα , ρ, dµ (cid:1) is a spaceof homogeneous type in the sense of Coifman-Weiss, that is the measure µ of ρ -balls restricted to Q kα is doubling . Note that in our context this property couldfail to be true, instead, for the measure µ of ρ -balls restricted to a fixed ρ -ball .We will also need the following: Lemma 2.7 (Covering Lemma)
For every n and every positive integer k large enough, the set Ω n can be essentially covered by a finite union of dyadiccubes Q kα (subordinated to Ω n +1 ) with the following properties:(i) Q kα ⊂ B (cid:0) z kα , c δ k (cid:1) ⊂ Ω n +1 (ii) B (cid:0) z kα , c δ k (cid:1) ⊂ F kα (essentially), F kα ⊂ B (cid:0) z kα , c ′ δ k (cid:1) ⊂ Ω n +1 , where theset F kα is a finite union of dyadic cubes Q kβ α , hence F kα is a space of homogeneoustype, that is satisfies (h) of the previous theorem. Proof.
Since the whole Ω n +1 can be essentially covered by the union of thedyadic cubes Q kα subordinated to Ω n +1 , Ω n is essentially covered by a subfamilyof these. By (d) in the previous theorem, for each Q kα of these cubes, there existsa ball B (cid:0) z kα , c δ k (cid:1) such that Q kα ⊂ B (cid:0) z kα , c δ k (cid:1) ⊂ Ω n +2 . However, since Q kα , and then B (cid:0) z kα , c δ k (cid:1) , contains a point of Ω n , for k large enough B (cid:0) z kα , c δ k (cid:1) ⊂ Ω n +1 , that is (i) holds.Let F kα the union of all the dyadic cubes Q kβ intersecting B (cid:0) z kα , c δ k (cid:1) . Since B (cid:0) z kα , c δ k (cid:1) ⊂ Ω n +1 which is essentially covered by the union of all the dyadic6ubes Q kβ , then B (cid:0) z kα , c δ k (cid:1) is essentially covered by F kα . Since, by (d) in theprevious theorem, diam (cid:16) Q kβ (cid:17) < c δ k , and each Q kβ contained in F kα intersects B (cid:0) z kα , c δ k (cid:1) , diam F kα is comparable to δ k , so F kα ⊂ B (cid:0) z kα , c ′ δ k (cid:1) , which is againcontained in Ω n +1 , for k large enough, since the ball contains a point of Ω n .Finally, any finite union of dyadic cubes satisfies the doubling condition, by(2.7) and [3, Corollary 3.9]. As we have explained in the Introduction, the proof of the sharp maximal in-equality will be achieved following the approach in [14], which exploits a suitableversion of Calder´on-Zygmund decomposition. We start proving in the contextof locally homogeneous spaces the following decomposition lemma.For any measurable set E and function f ∈ L ( E ), let f E = 1 | E | Z E f. Lemma 3.1
For fixed Ω n , Ω n +1 we consider the family ∆ n of dyadic cubesbuilt in Theorem 2.6. Let Q α be a fixed dyadic cube (“of first generation”)and let f ∈ L (cid:0) Q α (cid:1) . For any λ ≥ a ≡ | f | Q α there exists a countable family C λ = { Q λ,j } j =1 , ,... of pairwise disjoint dyadic subcubes of Q α such that:(i) λ < | f | Q λ,j ≤ c n λ for j = 1 , , ... ;(ii) if λ ≥ µ ≥ a then each cube Q λ,j is a subcube of one from the family C µ ;(iii) | f ( x ) | ≤ λ for a.e. x ∈ Q α \ S j Q λ,j ;(iv) P j | Q λ,j | ≤ (cid:12)(cid:12)(cid:12)n x ∈ Q α : M f ( x ) > λc ′ n o(cid:12)(cid:12)(cid:12) ;(v) (cid:12)(cid:12)(cid:8) x ∈ Q α : M f ( x ) > c ′′ n λ (cid:9)(cid:12)(cid:12) ≤ c ′′′ n P j | Q λ,j | where c n , c ′ n , c ′′ n , c ′′′ n are constants > only depending on n and we let for sim-plicity M f = M Ω n +1 , Ω n +2 (cid:16) f χ Q α (cid:17) (i.e., the local maximal function is computed after extending f to zero outside Q α ). Remark 3.2
As will be apparent from the proof, this lemma still holds if insteadof a fixed dyadic cube Q α of the first generation we fix a cube Q k α k of some fixedgeneration k > . Throughout this section we will stick to the convention ofconsidering Q α a cube of first generation, just to simplify notation, however wemust keep in mind that the results still hold under the more general assumptionon Q k α k . Or, saying this with other words, we can think that the cube Q α appearing in Theorem 3.4 and Corollaries 3.7 and 3.8 has diameter as small aswe want. roof. By point (c) in Thm. 2.6 for every dyadic cube Q ⊂ Q α there existsan increasing chain of dyadic cubes Q = Q kα k ⊂ Q k − α k − ⊂ ... ⊂ Q α . (3.1)For a fixed λ ≥ a = | f | Q α , in order to define the family C λ we say that Q ∈ C λ if, with the notation (3.1), λ < | f | Q and | f | Q sαs ≤ λ for s = 1 , , ..., k − . Note that any two cubes in C λ are disjoint, otherwise by point (e) in Thm. 2.6one should be contained in the other, so they would be two different steps inthe same chain (3.1), which contradicts our rule of choice. Let us show that C λ satisfies properties (i)-(iv).(i). For Q = Q kα k ∈ C λ , by construction, λ < | f | Q and | f | Q k − αk − ≤ λ hence | f | Q ≤ (cid:12)(cid:12) Q kα k (cid:12)(cid:12) Z Q k − αk − | f | ≤ (cid:12)(cid:12)(cid:12) Q k − α k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q kα k (cid:12)(cid:12) | f | Q k − αk − ≤ c n λ since by points (a) and (d) in Thm. 2.6, B (cid:0) z kα , a δ k (cid:1) ⊂ Q kα k ⊂ Q k − α k − ⊂ B (cid:16) z k − α k − , c δ k − (cid:17) , hence by the locally doubling condition (cid:12)(cid:12)(cid:12) Q k − α k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q kα k (cid:12)(cid:12) ≤ c n for some c n only depending on n (in particular, independent of k ). Hence (i) isproved.(ii). For Q = Q kα k ∈ C λ , λ ≥ µ ≥ a we have | f | Q kαk > λ ≥ µ hence in the chain (3.1) there is an l such that | f | Q lαl > µ, | f | Q l − αl − ≤ µ . Thismeans that in the chain (3.1) there is a cube Q ′ = Q lα l ∈ C µ , and Q ′ ⊃ Q .(iii). Let x ∈ Q α \ S j Q λ,j and let Q be any dyadic cube such that x ∈ Q ⊂ Q α . Consider again the chain (3.1) starting with Q . By our choice of x , none ofthe cubes Q lα l in this chain belongs to C λ , and this means that | f | Q lαl ≤ λ . Thenby point (g) in Thm. 2.6, for a.e. x ∈ Q α \ S j Q λ,j there exists a decreasingsequence of dyadic cubes (cid:8) Q lα l (cid:9) such that | f | Q lαl ≤ λ and ∩ Q lα l = { x } . ByLebesgue’s differentiation theorem, (iii) follows.8iv). Let f ∗ ( x ) = sup x ∈ Q ∈ ∆ n | f | Q . In the previous point we have proved that for a.e. x ∈ Q α \ S j Q λ,j and dyadiccube Q such that x ∈ Q ⊂ Q α , we have | f | Q ≤ λ . Hence f ∗ ( x ) ≤ λ for a.e. x ∈ Q α \ [ j Q λ,j . Conversely, if x ∈ S j Q λ,j then | f | Q λ,j > λ hence f ∗ ( x ) > λ. These two factsmean that, up to a set of zero measure, [ j Q λ,j = (cid:8) x ∈ Q α : f ∗ ( x ) > λ (cid:9) hence, since the { Q λ,j } j are pairwise disjoint, X j | Q λ,j | = (cid:12)(cid:12)(cid:8) x ∈ Q α : f ∗ ( x ) > λ (cid:9)(cid:12)(cid:12) . However, again by points (a) and (d) in Thm. 2.6 and the locally doublingcondition, f ∗ ( x ) ≤ c n M Ω n +1 , Ω n +2 (cid:16) f χ Q α (cid:17) ( x ) ≡ c n M f ( x ) , hence (iv) follows.(v). Let Q λ,j ∈ C λ . For some k = 2 , , ..., we will have Q λ,j = Q kα k and bypoints (a) and (d) in Thm. 2.6, B (cid:0) z kα , a δ k (cid:1) ⊂ Q kα k ⊂ B (cid:0) z kα k , c δ k (cid:1) for some z kα . For a K > KQ λ,j = B (cid:0) z kα k , Kc δ k (cid:1) . Forany x / ∈ S j KQ λ,j and any ball B = B r ( x ) such that x ∈ B r ( x ) and r ≤ r wehave, extending f to zero outside Q α if B Q α , Z B | f | = Z B \ S j Q λ,j | f | + X j Z B ∩ Q λ,j | f | by point (iii) ≤ λ | B | + X j : B ∩ Q λ,j = ∅ Z Q λ,j | f | by point (i) ≤ λ | B | + c n λ X j : B ∩ Q λ,j = ∅ | Q λ,j | . (3.2)9ext, we need the following Claim.
There exists
K, H > n ) such that if x / ∈ S i KQ λ,i and B r ( x ) ∩ Q λ,j = ∅ then Q λ,j ⊂ B Hr ( x ). Proof of the Claim.
Recall that B (cid:0) z kα , a δ k (cid:1) ⊂ Q λ,j ⊂ B (cid:0) z kα k , c δ k (cid:1) KQ λ,j = B (cid:0) z kα k , Kc δ k (cid:1) . Since B r ( x ) ∩ Q λ,j = ∅ , in particular B r ( x ) ∩ B (cid:0) z kα , a δ k (cid:1) = ∅ hence ρ (cid:0) x, z kα (cid:1) ≤ B n +1 (cid:0) r + a δ k (cid:1) . Since x ∈ B r ( x ), ρ (cid:0) x, z kα (cid:1) ≤ B n +2 (cid:0) r + ρ (cid:0) x, z kα (cid:1)(cid:1) ≤ B n +2 (cid:0) r + B n +1 (cid:0) r + a δ k (cid:1)(cid:1) and since x / ∈ KQ λ,j ,B n +2 (cid:0) r + B n +1 (cid:0) r + a δ k (cid:1)(cid:1) > Kc δ k which, picking K = B n +1 B n +2 a c , gives δ k < (1 + B n +1 ) B n +1 a r. Then Q λ,j ⊂ B (cid:0) z kα k , c δ k (cid:1) ⊂ B Hr ( x ) for a suitable H depending on n , namelyfor z ∈ Q λ,j ,ρ ( z, x ) ≤ B n +1 (cid:0) c δ k + ρ (cid:0) x, z kα (cid:1)(cid:1) ≤ B n +1 (cid:0) c δ k + B n +1 (cid:0) r + a δ k (cid:1)(cid:1) ≤ δ k (cid:0) B n +1 c + B n +1 a (cid:1) + B n +1 r ≤ (1 + B n +1 ) B n +1 a r (cid:0) B n +1 c + B n +1 a (cid:1) + B n +1 r ≡ Hr, which proves the Claim.Let us come back to the proof of (v). By (3.2) and the Claim we have (sincethe dyadic cubes in the sum are disjoint) Z B r ( x ) | f | ≤ λ | B r ( x ) | + c n λ | B Hr ( x ) | and, by the locally doubling condition, for r ≤ r n small enough, | f | B ≤ c ′ n λ for every B ∋ x ∈ Q α \ S j KQ λ,j , that is M f ( x ) ≤ c ′ n λ x , so that (cid:8) x ∈ Q α : M f ( x ) > c ′ n λ (cid:9) ⊂ [ j KQ λ,j and (cid:12)(cid:12)(cid:8) x ∈ Q α : M f ( x ) > c ′ n λ (cid:9)(cid:12)(cid:12) ≤ X j | KQ λ,j | ≤ c ′′ n X j | Q λ,j | . We can now prove the following local analog of Fefferman-Stein inequality.Let us first define a version of dyadic sharp maximal function : Definition 3.3
For f ∈ L (cid:0) Q α (cid:1) , x ∈ Q α , let f ( x ) = sup x ∋ Q ∈ ∆ n Q ⊂ Q α | Q | Z Q | f − f Q | . Note that this definition involves only the values of f in Q α (there is noneed of extending f outside that cube). Theorem 3.4 (Local Fefferman-Stein inequality)
Let f ∈ L (cid:0) Q α (cid:1) andassume f ∈ L p (cid:0) Q α (cid:1) for some p ∈ [1 , + ∞ ) . Then
M f ∈ L p (cid:0) Q α (cid:1) and (cid:12)(cid:12) Q α (cid:12)(cid:12) Z Q α ( M f ) p ! /p ≤ c n,p (cid:12)(cid:12) Q α (cid:12)(cid:12) Z Q α (cid:16) f (cid:17) p ! /p + (cid:12)(cid:12) Q α (cid:12)(cid:12) Z Q α | f | ! (3.3) for some constant c n,p only depending on n, p, where, as above, M f = M Ω n +1 , Ω n +2 (cid:16) f χ Q α (cid:17) . Proof.
Let a = | f | Q α . We start proving the following estimate: for every λ ≥ c n a and every A > X j | Q λ,j | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) x ∈ Q α : f ( x ) > λA (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) + 2 A X Q ∈C λ/ cn | Q | . (3.4)Fix λ ≥ c n a . By point (i) in Lemma 3.1, for any Q λ,k ∈ C λ ,λ < | f | Q λ,j ≤ c n λ ;also, for any Q ∈ C λ/ c n , | f | Q ≤ c n λ c n = λ Q λ,j and Q ,1 | Q λ,j | Z Q λ,j | f − f Q | ≥ | f | Q λ,j − | f | Q > λ − λ λ | Q λ,j | < λ Z Q λ,j | f − f Q | . (3.5)By point (ii) in Lemma 3.1, since λ > λ/ c n , any Q λ,j ∈ C λ is contained in some Q ∈ C λ/ c n ; also, the cubes Q ∈ C λ/ c n , like the cubes Q λ,j ∈ C λ are pairwisedisjoint, hence we can write X j | Q λ,j | = X Q ∈C λ/ cn X Q λ,j ∈C λ Q λ,j ⊂ Q | Q λ,j | . (3.6)For any Q ∈ C λ/ c n , by (3.5) and since the Q λ,j are disjoint X Q λ,j ∈C λ Q λ,j ⊂ Q | Q λ,j | ≤ X Q λ,j ∈C λ Q λ,j ⊂ Q λ Z Q λ,j | f − f Q | ≤ λ Z Q | f − f Q | . Let now fix a number
A > | Q | R Q | f − f Q | ≤ λA , then X Q λ,j ∈C λ Q λ,j ⊂ Q | Q λ,j | ≤ λ λA | Q | = 2 A | Q | . b. If | Q | R Q | f − f Q | > λA , then for every x ∈ Qf ( x ) > λA ,that is Q ⊂ (cid:26) x ∈ Q α : f ( x ) > λA (cid:27) and X Q λ,j ∈C λ Q λ,j ⊂ Q | Q λ,j | ≤ (cid:12)(cid:12)(cid:12)(cid:12) Q ∩ (cid:26) x ∈ Q α : f ( x ) > λA (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) . In any case we can write X Q λ,j ∈C λ Q λ,j ⊂ Q | Q λ,j | ≤ (cid:12)(cid:12)(cid:12)(cid:12) Q ∩ (cid:26) x ∈ Q α : f ( x ) > λA (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) + 2 A | Q | . Adding up these inequalities for Q ∈ C λ/ c n , recalling (3.6) and the fact thatthe cubes Q are pairwise disjoint, we get (3.4).12y (3.4) and points (v), (iv) in Lemma 3.1, we have (cid:12)(cid:12)(cid:8) x ∈ Q α : M f ( x ) > c ′′ n λ (cid:9)(cid:12)(cid:12) ≤ c ′′′ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) x ∈ Q α : f ( x ) > λA (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) + 2 A X Q ∈C λ/ cn | Q | ≤ c ′′′ n (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)(cid:26) x ∈ Q α : f ( x ) > λA (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) + 2 A (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) x ∈ Q α : M f ( x ) > λ c n c ′ n (cid:27)(cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (3.7)for any A > , λ ≥ c n a .We now want to compute integrals using the identity (for any F ∈ L p (cid:0) Q α (cid:1) , ≤ p < ∞ ) Z Q α | F ( y ) | p dy = Z + ∞ pt p − (cid:12)(cid:12)(cid:8) x ∈ Q α : | F ( x ) | > t (cid:9)(cid:12)(cid:12) dt. Letting µ ( t ) = (cid:12)(cid:12)(cid:8) x ∈ Q α : | M f ( x ) | > t (cid:9)(cid:12)(cid:12) and integrating (3.7) for λ ∈ (2 c n a, N ) and any fixed N > c n | f | Q α aftermultiplying by pλ p − we have Z N c n a pλ p − µ ( c ′′ n λ ) dλ ≤ c ′′′ n Z N c n a pλ p − (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) x ∈ Q α : f ( x ) > λA (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) dλ + 2 A Z N c n a pλ p − µ (cid:18) λ c n c ′ n (cid:19) dλ ! (3.8)Changing variable in each of the three integrals in (3.8) we get: Z c ′′ n N c n c ′′ n a pt p − µ ( t ) dt ≤ ( c ′′ n ) p c ′′′ n (cid:18) A p Z + ∞ pt p − (cid:12)(cid:12)(cid:12)n x ∈ Q α : f ( x ) > t o(cid:12)(cid:12)(cid:12) dt + (3.9)2 A (2 c n c ′ n ) p Z N cnc ′ n pt p − µ ( t ) dt ! . Using also the elementary inequality Z c n c ′′ n a pt p − µ ( t ) dt ≤ Z c n c ′′ n a pt p − (cid:12)(cid:12) Q α (cid:12)(cid:12) dt = (cid:12)(cid:12) Q α (cid:12)(cid:12) (2 c n c ′′ n a ) p , N c n c ′ n < N < c ′′ n N we get Z c ′′ n N pt p − µ ( t ) dt ≤ ( c ′′ n ) p c ′′′ n (cid:18) A p Z + ∞ pt p − (cid:12)(cid:12)(cid:12)n x ∈ Q α : f ( x ) > t o(cid:12)(cid:12)(cid:12) dt + 2 A (2 c n c ′ n ) p Z c ′′ n N pt p − µ ( t ) dt ! + (cid:12)(cid:12) Q α (cid:12)(cid:12) (2 c n c ′′ n a ) p . Letting finally A = 4 (2 c n c ′ n c ′′ n ) p c ′′′ n we conclude Z c ′′ n N pt p − µ ( t ) dt ≤ c n,p (cid:18)Z + ∞ pt p − (cid:12)(cid:12)(cid:12)n x ∈ Q α : f ( x ) > t o(cid:12)(cid:12)(cid:12) dt + (cid:12)(cid:12) Q α (cid:12)(cid:12) | f | pQ α (cid:19) which implies that M f ∈ L p (cid:0) Q α (cid:1) and k M f k pL p ( Q α ) ≤ c n,p (cid:18)(cid:13)(cid:13)(cid:13) f (cid:13)(cid:13)(cid:13) pL p ( Q α ) + (cid:12)(cid:12) Q α (cid:12)(cid:12) | f | pQ α (cid:19) that is (3.3).We now want to reformulate the above theorem in terms of balls, insteadof dyadic cubes, to make it more easily applicable to concrete situations. Thisreformulation can be done in several ways. First of all, we introduce the localsharp maximal functions defined by balls instead of cubes. Definition 3.5
For f ∈ L loc (Ω n +1 ) , x ∈ Ω n , let f n , Ω n +1 ( x ) = sup B ( x,r ) ∋ xx ∈ Ω n ,r ≤ ε n | B ( x, r ) | Z B ( x,r ) (cid:12)(cid:12) f − f B ( x,r ) (cid:12)(cid:12) . Let us compare this function with its dyadic version f : Lemma 3.6
With the above notation, for any x ∈ Q α ,f ( x ) ≤ c n f n +1 , Ω n +2 ( x ) for some constant c n only depending on n . Here the function f can be assumedeither in L loc (Ω n +2 ) or in L (cid:0) Q α (cid:1) and extended to zero outside Q α . Proof.
For any dyadic cube Q = Q kα k ⊂ Q α ⊂ Ω n +1 we have (see points (a),(d) in Theorem 2.6) B (cid:0) z kα , a δ k (cid:1) ⊂ Q ⊂ B (cid:0) z kα k , c δ k (cid:1) ⊂ Ω n +2 . B , B in place of B (cid:0) z kα , a δ k (cid:1) , B (cid:0) z kα k , c δ k (cid:1) . Then by thelocally doubling condition | B || Q | ≤ | B || B | ≤ c n and we can write1 | Q | Z Q | f − f Q | ≤ | Q | Z B | f − f Q |≤ c n | B | Z B | f − f Q |≤ c n (cid:18) | B | Z B | f − f B | + | f Q − f B | (cid:19) . Also, | f Q − f B | = (cid:12)(cid:12)(cid:12)(cid:12) | Q | Z Q ( f − f B ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | Q | Z Q | f − f B |≤ c n | B | Z B | f − f B | hence 1 | Q | Z Q | f − f Q | ≤ c n (1 + c n ) 1 | B | Z B | f − f B | and the assertion follows.Exploiting the previous Lemma and Theorem 2.5 we can now rewrite thestatement of Theorem 3.4 as follows: Corollary 3.7
Let f ∈ L (cid:0) Q α (cid:1) and assume f n +1 , Ω n +2 ∈ L p (cid:0) Q α (cid:1) for some p ∈ [1 , + ∞ ) . Then f ∈ L p (cid:0) Q α (cid:1) and (cid:12)(cid:12) Q α (cid:12)(cid:12) Z Q α | M f | p ! /p ≤ c n,p (cid:12)(cid:12) Q α (cid:12)(cid:12) Z Q α (cid:16) f n +1 , Ω n +2 (cid:17) p ! /p + (cid:12)(cid:12) Q α (cid:12)(cid:12) Z Q α | f | ! for some constant c n,p only depending on n, p . Here M is defined as in Theo-rem 3.4 and, again, the function f can be assumed either in L loc (Ω n +2 ) or in L (cid:0) Q α (cid:1) and extended to zero outside Q α . The following is also useful:
Corollary 3.8
Let B ⊂ Q α ⊂ B with B , B concentric balls of compara-ble radii (like in the proof of Lemma 3.6) and assume that f ∈ L ( B ) with n +1 , Ω n +2 ∈ L p (cid:0) Q α (cid:1) for some p ∈ [1 , + ∞ ) and f B = 0 (where the function f can be assumed either in L loc (Ω n +2 ) or in L ( B ) and extended to zero outside B ). Then f ∈ L p (cid:0) Q α (cid:1) and we have (cid:12)(cid:12) Q α (cid:12)(cid:12) Z Q α | f | p ! /p ≤ c n,p (cid:12)(cid:12) Q α (cid:12)(cid:12) Z Q α (cid:16) f n +1 , Ω n +2 (cid:17) p ! /p (cid:18) | B | Z B | f | p (cid:19) /p ≤ c ′ n,p (cid:18) | B | Z B (cid:16) f n +1 , Ω n +2 (cid:17) p (cid:19) /p for some constants c n,p , c ′ n,p only depending on n, p . Also, removing the assump-tion f B = 0 we can write (cid:18) | B | Z B | f − f B | p (cid:19) /p ≤ c ′ n,p (cid:18) | B | Z B (cid:16) f n +1 , Ω n +2 (cid:17) p (cid:19) /p . Proof.
We can write1 (cid:12)(cid:12) Q α (cid:12)(cid:12) Z Q α | f | = 1 (cid:12)(cid:12) Q α (cid:12)(cid:12) Z Q α | f − f B | ≤ c n | B | Z B | f − f B | ≤ c n f n +1 , Ω n +2 ( x )for every x ∈ B . Averaging this inequality on Q α we get1 (cid:12)(cid:12) Q α (cid:12)(cid:12) Z Q α | f | ≤ c n (cid:12)(cid:12) Q α (cid:12)(cid:12) Z Q α f n +1 , Ω n +2 ≤ c n (cid:12)(cid:12) Q α (cid:12)(cid:12) Z Q α (cid:16) f n +1 , Ω n +2 (cid:17) p ! /p so that, by recalling Corollary 3.7 (cid:12)(cid:12) Q α (cid:12)(cid:12) Z Q α | f | p ! /p ≤ c n,p (cid:12)(cid:12) Q α (cid:12)(cid:12) Z Q α (cid:16) f n +1 , Ω n +2 (cid:17) p ! /p , which also implies the second inequality, by the locally doubling condition andthe comparability of the radii of B , B .Although, in the previous Corollary, the second inequality has the pleasantfeature of involving balls instead of dyadic cubes (however, note the two differentballs appearing at the left hand side of the last inequality), remember that wecannot choose these balls as we like, since they are related to dyadic cubes.In concrete applications of this theory, we could use this result to bound k f k L p (Ω n ) . To this aim, recall that the domain Ω n can be covered (up to a zeromeasure set) by a finite union of dyadic cubes of the kind Q α , but Ω n is notcovered by the union of the corresponding smaller balls B . We then need toimprove the previous corollary, replacing the dyadic cube Q on the left handside with a larger ball: Corollary 3.9
For any n and every k large enough, the set Ω n can be coveredby a finite union of balls B R ( x i ) of radii comparable to δ k such that for any uch ball B R and every f supported in B R such that f ∈ L ( B R ) , R B R f = 0 ,and f n +2 , Ω n +3 ∈ L ploc (Ω n +1 ) for some p ∈ [1 , ∞ ) , one has k f k L p ( B R ) ≤ c n,p (cid:13)(cid:13)(cid:13) f n +2 , Ω n +3 (cid:13)(cid:13)(cid:13) L p ( B γR ) with γ > absolute constant. Proof.
Applying Lemma 2.7, let us (essentially) cover Ω n with a finite unionof dyadic cubes Q kα ⊂ B (cid:0) z kα , c δ k (cid:1) ⊂ F kα ⊂ B (cid:0) z kα , c ′ δ k (cid:1) (where the inclusion B (cid:0) z kα , c δ k (cid:1) ⊂ F kα is only essential). We claim that theballs B (cid:0) z kα , c δ k (cid:1) are the required covering of Ω n . To see this, let f be supportedin B (cid:0) z kα , c δ k (cid:1) and with vanishing integral. Then the same is true for f withrespect to the larger ball B (cid:0) z kα , c ′ δ k (cid:1) . We can then apply Corollary 3.8 to eachdyadic cube Q kβ which constitutes F kα , writing: (cid:12)(cid:12)(cid:12) Q kβ (cid:12)(cid:12)(cid:12) Z Q kβ | f | p /p ≤ c n,p (cid:12)(cid:12)(cid:12) Q kβ (cid:12)(cid:12)(cid:12) Z Q kβ (cid:16) f n +2 , Ω n +3 (cid:17) p /p (note that the local sharp function is f n +2 , Ω n +3 because we are using dyadicballs related to Ω n +1 ), that is Z Q kβ | f | p ≤ c pn,p Z Q kβ (cid:16) f n +2 , Ω n +3 (cid:17) p . Adding these inequalities for all the cubes Q kβ in F kα we get Z B ( z kα ,c δ k ) | f | p ! /p ≤ Z F kα | f | p ! /p ≤ c n,p Z F kα (cid:16) f n +2 , Ω n +3 (cid:17) p ! /p ≤ c n,p Z B ( z kα ,c ′ δ k ) (cid:16) f n +2 , Ω n +3 (cid:17) p ! /p which is our assertion, with R = c δ k , γ = c ′ /c . BM O and John-Nirenberg inequality
We start defining the space of functions with locally bounded mean oscillationin a locally homogeneous space:
Definition 4.1
Let f ∈ L (Ω n +1 ) . We say that f belongs to BM O (Ω n , Ω n +1 ) if [ f ] n ≡ sup x ∈ Ω n ,r ≤ ǫ n µ ( B ( x, r ) Z B ( x,r ) | f ( y ) − f B ( x,r ) | dµ ( y ) < ∞ . Theorem 4.2 (Local John-Nirenberg inequality)
There exist positive con-stants b n , R n such that ∀ f ∈ BM O (Ω n , Ω n +1 ) and for any ball B ( a, R ) , with a ∈ Ω n and R ≤ R n , the following inequality holds true µ (cid:0) { x ∈ B ( a, R ) : (cid:12)(cid:12) f ( x ) − f B ( a,R ) (cid:12)(cid:12) > λ } (cid:1) ≤ e − bnλ [ f ] n µ ( B ( a, R )) ∀ λ > . (4.1) Remark 4.3
As will appear from the proof, the constant R n is strictly smallerthan the number ε n appearing in the definition of BM O (Ω n , Ω n +1 ) . Explicitly,we will see that R n = 2 ε n B n +1 (cid:0) B n +1 + 3 B n +1 + 1 (cid:1) . (4.2) Proof.
We can assume [ f ] n = 1, since (4.1) does not change dividing both f and λ for a constant.Let K n = 2 B n +1 + 3 B n +1 be the constant appearing in Vitali covering Lemma 2.2, α n = B n +1 (cid:18) K n + 1 (cid:19) R n = 2 ε n α n . Let a ∈ Ω n , R ≤ R n and let S = B ( a, R ) (since R < ε n , S ⊆ Ω n +1 ). Theproof consists in an iterative construction. Step 1.
We will prove that there exists a family of balls { S j } ∞ j =1 ⊂ S andconstants c, λ ≥ n such that:i) { x ∈ S : | f ( x ) − f S | > λ } ⊂ ∞ [ j =1 S j ⊂ S ;ii) ∞ X j =1 µ ( S j ) ≤ µ ( S ) ;iii) (cid:12)(cid:12) f S − f S j (cid:12)(cid:12) ≤ cλ To prove this we start defining the maximal operator associated to S letting,for any x ∈ S,M S f ( x ) = sup (cid:26) µ ( B ) Z B | f ( y ) − f S | dµ ( y ) : B ball , x ∈ B, B ⊆ α n S (cid:27) where α n S = B ( a, α n R ) ⊆ Ω n +1 since α n R ≤ ε n .18e claim that there exists A = A ( n ) > t > µ ( { x ∈ S : M S f ( x ) > t } ) ≤ At µ ( S ) . (4.3)To show this, let t > U t = { x ∈ S : M S f ( x ) > t } . For every x ∈ U t there exists a ball B x such that x ∈ B x ⊆ α n S and µ ( B x ) < t Z B x | f ( y ) − f S | dµ . Now, by Vitali Lemma 2.7 there exists a countable subcollection of disjoint balls { B ( x i , r i ) } such that U t ⊆ ∞ [ i =1 B ( x i , K n r i ) . Then, since by definition of S and M S f , ∪ ∞ i =1 B ( x i , r i ) ⊆ α n S ⊂ Ω n +1 , for someconstant A = A ( n ) which can vary from line to line we have µ ( U t ) ≤ µ ∞ [ i =1 B ( x i , K n r i ) ! ≤ ∞ X i =1 µ ( B ( x i , K n r i )) ≤ A ∞ X i =1 µ ( B ( x i , r i )) ≤ ∞ X i =1 At Z B ( x i ,r i ) | f − f S | dµ = At Z ∪ ∞ i =1 B ( x i ,r i ) | f − f S | dµ ≤ At Z α n S | f − f S | dµ ≤ At (cid:26)Z α n S | f − f α n S | dµ + Z α n S | f α n S − f S | dµ (cid:27) ≤ At { µ ( α n S )[ f ] n + µ ( α n S ) | f S − f α n S |}≤ At µ ( α n S )[ f ] n ≤ At µ ( S )where we exploited the assumption [ f ] n = 1. Hence (4.3) is proved.Let now λ > A , we consider the following open set U = { x ∈ S : M S f ( x ) > λ } . We have, by (4.3), µ ( U ∩ S ) = µ ( U ) ≤ Aλ µ ( S ) < µ ( S ) , (4.4)from which S ∩ U c = ∅ . x ∈ S we set r ( x ) = 12 K n ρ ( x, U c ) ∀ x ∈ U. If x, y ∈ S we have ρ ( x, y ) ≤ B n +1 R .
Then ∀ x ∈ U (taking a point y ∈ U c ∩ S in the following inequality) r ( x ) ≤ K n ρ ( x, y ) ≤ B n +1 (2 + 3 B n +1 ) 2 B n +1 R ≤ R y ∈ B ( x, K n r ( x )) for some x ∈ U , we have ρ ( y, x ) < K n r ( x ) = K n K n ρ ( x, U c ) < ρ ( x, U c )then y ∈ U , from which B ( x, K n r ( x )) ⊆ U . (4.5)On the other hand U ⊂ [ x ∈ U B ( x, r ( x )) (4.6)and from the Vitali Lemma there exists a countable sequence of disjoint balls { B ( x j , r j ) } ( r j = r ( x j )) such that U ⊂ ∞ [ j =1 B ( x j , K n r j )which by the inclusion (4.5) means that U = ∞ [ j =1 B ( x j , K n r j ) . (4.7)Moreover B ( x j , K n r j ) ∩ U c = ∅ ∀ j ∈ N and B ( x j , K n r j ) ⊆ α n S since α n = B n +1 (cid:0) K n + 1 (cid:1) .If y ∈ B ( x j , K n r j ) ∩ U c , then M s f ( y ) ≤ λ and1 µ ( B ( x j , K n r j )) Z B ( x j , K n r j ) | f − f S | dµ ≤ λ . (4.8)We now set S j = B ( x j , K n r j )and we have, by (4.8) (cid:12)(cid:12) f S − f S j (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ ( S j ) Z S j f dµ − f S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ µ ( S j ) Z S j | f − f S | dµ (4.9) ≤ cµ ( B ( x j , K n r j )) Z B ( x j , K n r j ) | f − f S | dµ ≤ cλ x ∈ S \ ∪ j S j (that by(4.7) implies that x ∈ U c so that M S f ( x ) ≤ λ ) | f ( x ) − f S | ≤ λ . This means that { x ∈ S : | f ( x ) − f S | > λ } ⊂ ∞ [ j =1 S j ⊂ S which is point i).Moreover, by the doubling property (H7) and (4.4), ∞ X j =1 µ ( S j ) ≤ c ∞ X j =1 µ ( B ( x j , r j )) = cµ ∞ [ j =1 B ( x j , r j ) ≤≤ cµ ( U ) ≤ c Aλ µ ( S ) = 12 µ ( S ) , having finally chosen λ = 2 cA, so that also point ii) is proved and step 1 iscompleted. Step 2 consists in doing the same construction on each ball S j constructedin Step 1, which allows to conclude that, for every j = 1 , , ... there exists asequence of balls { S j j } ∞ j =1 ⊂ S j such that (for the same constants c, λ ofStep 1)i) { x ∈ S j : (cid:12)(cid:12) f ( x ) − f S j (cid:12)(cid:12) > λ } ⊂ ∞ [ j =1 S j j ⊂ S j ;ii) ∞ X j =1 µ ( S j j ) ≤ µ ( S j ) ;iii) (cid:12)(cid:12) f S j − f S j j (cid:12)(cid:12) ≤ cλ . Point ii) of Step 2 and Step 1 imply ∞ X j ,j =1 µ ( S j j ) ≤ ∞ X j =1 µ ( S j ) ≤ µ ( S ) . Also, point i) of Step 2 and point iii) of Step 1, imply that for a.e. x ∈ S j \ ∞ S j =1 S j j , | f ( x ) − f S | ≤ (cid:12)(cid:12) f ( x ) − f S j (cid:12)(cid:12) + (cid:12)(cid:12) f S j − f S (cid:12)(cid:12) ≤ λ + cλ < cλ { x ∈ S j : | f ( x ) − f S | > cλ } ⊂ ∞ [ j =1 S j j ⊂ S j . (4.10)However, point i) of Step 1 implies { x ∈ S : | f ( x ) − f S | > cλ } ⊂ { x ∈ S : | f ( x ) − f S | > λ } ⊂ ∞ [ j =1 S j hence (4.10) rewrites as { x ∈ S : | f ( x ) − f S | > cλ } ⊂ ∞ [ j ,j =1 S j j and, letting λ = cλ ,µ ( { x ∈ S : | f ( x ) − f S | > λ } ) ≤ µ ( S ) . (4.11)Relation (4.11) summarizes the joint consequences of Steps 1 and 2.Proceeding this way the iterative construction, at Step N we will have that: µ ( { x ∈ S : | f ( x ) − f S | > N λ } ) ≤ N µ ( S ) . (4.12)Now, let λ >
0. If λ ≥ λ , let N be the positive integer such that N λ < λ ≤ ( N + 1) λ , then µ ( { x ∈ S : | f ( x ) − f S | > λ } ) ≤ µ ( { x ∈ S : | f ( x ) − f S | > N λ } ) ≤ N µ ( S ) = e − N log 2 µ ( S ) ≤ e − (cid:16) log 2 λ (cid:17) λ µ ( S )Finally, if 0 < λ ≤ λ ,µ ( { x ∈ S : | f ( x ) − f S | > λ } ) ≤ µ ( S ) ≤ e − (cid:16) log 2 λ (cid:17) λ µ ( S )and the assertion follows (recall we are assuming [ f ] n = 1), with b n = log 2 λ . Definition 4.4
Let p ∈ (1 , + ∞ ) . We say that f belongs to BM O p (Ω n , Ω n +1 ) if f ∈ L p (Ω n +1 ) and [ f ] p,n ≡ sup x ∈ Ω n ,r ≤ R n µ ( B ( x, r )) Z B ( x,r ) | f ( y ) − f B ( x,r ) | p dµ ( y ) ! /p < ∞ where R n is the constant appearing in (4.2), strictly smaller than ε n . BM O (Ω n , Ω n +1 ) and BM O p (Ω n , Ω n +1 ). Theorem 4.5
For any p ∈ (1 , ∞ ) and n we have BM O p (Ω n , Ω n +1 ) = BM O (Ω n , Ω n +1 ) . Moreover, there exists a positive constant c n,p such that for any f ∈ BM O (Ω n , Ω n +1 ) , [ f ] p,n ≤ c n,p [ f ] n . (4.13) In particular,
BM O (Ω n , Ω n +1 ) ⊆ L p (Ω n ) for every p ∈ (1 , ∞ ) . Remark 4.6
Comparing this result with those about the local Fefferman-Steinfunction proved in the previous section (for instance, Corollary 3.9), we see thatthe present theorem is a “local” result in a different sense. Here, in the upperbound (4.13), there is not an enlargement of the domain, passing from the leftto the right hand side; instead, the local seminorms [ f ] p,n are computed takingthe supremum over balls of radii r ≤ R n , which is a stricter condition than thebound r ≤ ε n defining the seminorm [ f ] n . Proof.
Let f ∈ BM O p (Ω n , Ω n +1 ). By H¨older’s inequality we can write, forevery x ∈ Ω n , r ≤ R n ,1 µ ( B ( x, r )) Z B ( x,r ) | f ( y ) − f B ( x,r ) | dµ ( y ) ≤ µ ( B ( x, r )) Z B ( x,r ) | f ( y ) − f B ( x,r ) | p dµ ( y ) ! /p ≤ [ f ] p,n < ∞ . On the other hand, if R n < r < ε n we have1 µ ( B ( x, r )) Z B ( x,r ) | f ( y ) − f B ( x,r ) | dµ ( y ) ≤ µ ( B ( x, r )) Z B ( x,r ) | f ( y ) | dµ ( y ) ≤ µ ( B ( x, R n )) Z Ω n +1 | f ( y ) | dµ ( y ) ≤ c n k f k L p (Ω n +1 ) because inf x ∈ Ω n µ ( B ( x, R n )) ≥ c n > , as can be easily proved as a consequence of the local doubling condition. There-fore [ f ] n < ∞ and f ∈ BM O (Ω n , Ω n +1 ).Conversely, to prove (4.13), let B be a ball centered in x ∈ Ω n with radius r ≤ R n . Then by Theorem 4.2 we have Z B | f ( y ) − f B | p dµ ( y ) = Z + ∞ pλ p − µ ( { z ∈ B : | f ( z ) − f B | > λ } ) dλ ≤ Z + ∞ pλ p − e − b n λ/ [ f ] n µ ( B ) dλ = µ ( B )[ f ] pn p Z + ∞ t p − e − b n t dt, (cid:18) µ ( B ) Z B | f ( y ) − f B | p dµ ( y ) (cid:19) /p ≤ (cid:18) p Z + ∞ t p − e − b n t dt (cid:19) /p [ f ] n = c n,p [ f ] n which gives (4.13) and the inclusion BM O (Ω n , Ω n +1 ) ⊆ BM O p (Ω n , Ω n +1 ).Finally, to show that f ∈ L p (Ω n ) we can cover Ω n with a finite collection ofballs B ( x, R n ) with x ∈ Ω n , writing (cid:18) µ ( B ) Z B | f ( y ) | p dµ ( y ) (cid:19) /p ≤ (cid:18) µ ( B ) Z B | f ( y ) − f B | p dµ ( y ) (cid:19) /p + | f B |≤ [ f ] p,n + 1 µ ( B ) k f k L (Ω n +1 ) < ∞ which implies the finiteness of k f k L p (Ω n ) . References [1] M. Bramanti: Singular integrals in nonhomogeneous spaces: L and L p continuity from H¨older estimates. Rev. Mat. Iberoam. 26 (2010), no. 1,347–366.[2] M. Bramanti, G. Cupini, E. Lanconelli, E. Priola: Global L p estimatesfor degenerate Ornstein-Uhlenbeck operators. Math. Z. 266 (2010), no. 4,789–816.[3] M. Bramanti, M. Zhu: Local real analysis in locally homogeneous spaces.Manuscripta Math. 138 (2012), no. 3-4, 477–528.[4] M. Bramanti, M. Zhu: L p and Schauder estimates for nonvariational opera-tors structured on H¨ormander vector fields with drift. Anal. PDE 6 (2013),no. 8, 1793–1855.[5] M. Bramanti, M. Toschi: The sharp maximal function approach to L p estimates for operators structured on H¨ormander’s vector fields. Preprint(2015).[6] S. M. Buckley: Inequalities of John-Nirenberg type in doubling spaces. J.Anal. Math. 79 (1999), 215–240.[7] A. O. Caruso, M. S. Fanciullo: BM O on spaces of homogeneous type: adensity result on C-C spaces. Annales Academiae Scientiarum Fennicae 32(2007), 13-26.[8] R. E. Castillo, J. C. Ramos Fern´andez, E. Trousselot: Functions of bounded( ϕ, p ) mean oscillation. Proyecciones 27 (2008), no. 2, 163–177.249] M. Christ: A T ( b ) theorem with remarks on analytic capacity and theCauchy integral. Colloq. Math. 60/61 (1990), no. 2, 601–628.[10] R. R. Coifman, G. Weiss: Analyse harmonique non-commutative sur cer-tains espaces homog`enes. Lecture Notes in Mathematics, Vol. 242. Springer-Verlag, Berlin-New York, 1971.[11] G. Dafni, H. Yue: Some characterizations of local bmo and h on metricmeasure spaces. Anal. Math. Phys. 2 (2012), no. 3, 285–318.[12] L. Diening, M. R˚uˇziˇcka, K. Schumacher: A decomposition technique forJohn domains. Ann. Acad. Sci. Fenn. Math. 35 (2010), no. 1, 87–114.[13] C. Fefferman, E. M. Stein: H p spaces of several variables. Acta Math. 129(1972), no. 3-4, 137–193.[14] T. Iwaniec: On L p -integrability in PDEs and quasiregular mappings forlarge exponents. Ann. Acad. Sci. Fenn. Ser. A I Math. 7 (1982), no. 2,301–322.[15] F. John, L. Nirenberg: On functions of bounded mean oscillation. Comm.Pure Appl. Math. 14 1961 415–426.[16] N. V. Krylov: Parabolic and elliptic equations with VMO coefficients.Comm. Partial Differential Equations 32 (2007), no. 1-3, 453–475.[17] Q. Lai: The sharp maximal function on spaces of generalized homogeneoustype. J. Funct. Anal. 150 (1997), no. 1, 75–100.[18] J. Mateu, P. Mattila, A. Nicolau, J. Orobitg: BMO for nondoubling mea-sures. Duke Math. J. 102 (2000), no. 3, 533–565. Marco BramantiDipartimento di MatematicaPolitecnico di MilanoVia Bonardi 920133 Milano, ITALY [email protected]
Maria Stella FanciulloDipartimento di Matematica e InformaticaUniversit`a di CataniaViale Andrea Doria 695125 Catania, ITALY [email protected]@dmi.unict.it