A note on weak convergence of singular integrals in metric spaces
aa r X i v : . [ m a t h . C A ] A p r A NOTE ON WEAK CONVERGENCE OF SINGULARINTEGRALS IN METRIC SPACES
VASILIS CHOUSIONIS AND MARIUSZ URBA ´NSKI
Abstract.
We prove that in any metric space (
X, d ) the singular integral oper-ators T kµ,ε ( f )( x ) = Z X \ B ( x,ε ) k ( x, y ) f ( y ) dµ ( y ) . converge weakly in some dense subspaces of L ( µ ) under minimal regularity as-sumptions for the measures and the kernels. Introduction
A Radon measure on a metric space (
X, d ) has s -growth if there exists someconstant c µ such that µ ( B ( x, r )) ≤ c µ r s for all x ∈ X , r > k ( · , · ) : X × X \ { ( x, y ) ∈ X × X : x = y } → R is an s -dimensionalkernel if there exists a constant c > x, y ∈ X , x = y : | k ( x, y ) | ≤ c d ( x, y ) − s . The kernel k is antisymmetric if k ( x, y ) = − k ( y, x ) for all distinct x, y ∈ X .Given a positive Radon measure ν on X and an s -dimensional kernel k , we define T k ν ( x ) := Z k ( x, y ) dν ( y ) , x ∈ X \ spt ν. This integral may not converge when x ∈ spt ν . For this reason, we consider thefollowing ε -truncated operators T kε , ε > T kε ν ( x ) := Z d ( x,y ) >ε k ( x, y ) dν ( y ) , x ∈ X. Given a fixed positive Radon measure µ on X and f ∈ L ( µ ), we write T kµ f ( x ) := T k ( f µ )( x ) , x ∈ X \ spt( f µ ) , and T kµ,ε f ( x ) := T kε ( f µ )( x ) . Concerning the limit properties of the operators T kµ,ε one can ask if the limit, theso called principal value of T , lim ε → T kµ,ε ( f )( x ) , Mathematics Subject Classification.
Primary 32A55, 30L99.
Key words and phrases.
Singular integrals, metric spaces. exists µ almost everywhere. When µ is the Lebesgue measure in R d , and k is a stan-dard Calder´on-Zygmund kernel, due to cancellations and the denseness of smoothfunctions in L , the principal values exist almost everywhere for L -functions. Formore general measures, the question is more complicated. Let n be an integer,0 < n < d , and consider the coordinate Riesz kernels R ni ( x ) = x i | x | n +1 for i = 1 , . . . , d. Tolsa proved in [T] that if E ⊂ R d has finite n -dimensional Hausdorff measure H n the principal values lim ε → Z E \ B ( x,ε ) x i − y i | x − y | m +1 d H n ( y )exist H n almost everywhere in E if and only if the set E is n -rectifiable i.e. if thereexist n -dimensional Lipschitz surfaces M i , i ∈ N , such that H n ( E \ ∪ ∞ i =1 M i ) = 0 . Mattila and Preiss had obtained the same result earlier, in [MP] under some strongerassumptions for the set E . It becomes obvious that the existence of principal valuesis deeply related to the geometry of the set E .Assuming L ( µ )-boundedness for the operators T kµ one could have expected thatmore could be deduced about the structure of µ and the existence of principal val-ues, but this is a hard and, in a large extent, open problem. Dating from 1991 theDavid-Semmes conjecture, see [DS], asks if the L ( µ )-boundedness of the operatorsassociated with the n -dimensional Riesz kernels suffices to imply n -uniform rectifi-abilty, which can be thought as a quantitative version of rectifiability. In the veryrecent deep work [NToV], Nazarov, Tolsa and Volberg resolved the conjecture in thecodimension 1 case, that is for n = d −
1. Mattila, Melnikov and Verdera in [MMV],using a special symmetrization property of the Cauchy kernel, had earlier provedthe conjecture in the case of 1-dimensional Riesz kernels. For all other dimensionsand for other kernels few things are known. In fact, there are several examples ofkernels whose boundedness does not imply rectifiability, see [C], [D] and [H]. Forsome recent positive results involving other kernels see [CMPT].Let µ be a finite Radon measure and let k be an antisymmetric kernel in a completemetric space ( X, d ) where the Vitali covering theorem holds for µ and the family ofclosed balls defined by d . Mattila and Verdera in [MV] showed that in this case the L ( µ )-boundedness of the operators T kµ,ε forces them to converge weakly in L ( µ ).This means that there exists a bounded linear operator T kµ : L ( µ ) → L ( µ ) suchthat for all f, g ∈ L ( µ ),lim ε → Z T kµ,ε ( f )( x ) g ( x ) dµ ( x ) = Z T kµ ( f )( x ) g ( x ) dµ ( x ) . Furthermore notions of weak convergence have been recently used by Nazarov, Tolsaand Volberg in [NToV].
NOTE ON WEAK CONVERGENCE OF SINGULAR INTEGRALS IN METRIC SPACES 3
Motivated by these developments it is natural to ask if limits of this type mightexist if we remove the very strong L -boundedness assumption. We prove that theoperators T kµ,ε converge weakly in dense subspaces of L ( µ ) under minimal assump-tions for the measures and the kernels in general metric spaces. Denote by X B thespace of all finite linear combinations of characteristic functions of balls in X , X B = ( n X i =1 a i χ B ( z i ,r i ) : n ∈ N , a i ∈ R , z i ∈ X, r i > ) . Whenever Vitali’s covering theorem holds for the closed balls in (
X, d ) the space X B is dense in L ( µ ). When X = R d Vitali’s covering theorem holds for any Radonmeasure µ and the closed balls defined by various metrics (including the standard d p metrics for 1 ≤ p ≤ ∞ ) as a consequence of Besicovitch’s covering theorem, see[M, Theorem 2.8]. Furthermore Vitali’s covering theorem holds for any metric space( X, d ) whenever µ is doubling, that is when there exists some constant C such thatfor all balls B , µ (2 B ) ≤ Cµ ( B ), see [F, Section 2.8]. Theorem 1.1.
Let µ be a finite Radon measure with s -growth and k an antisymmet-ric s -dimensional kernel on a metric space ( X, d ) . If the Vitali Covering theoremholds for the closed balls in ( X, d ) then there exists subsets X ′ B ⊂ X B which are densein L ( µ ) and the weak limits lim ε → Z T kµ,ε f ( x ) g ( x ) dµ ( x ) exist for all f, g ∈ X ′ B . Until now Theorem 1.1 was only known for measures with ( d − R d under some smoothness assumptions for the kernels, see [CM]. We thus extend theresult from [CM] to measures with s -growth for arbitrary s in metric spaces whereVitali’s covering theorem holds for the family of closed balls without requiring anysmoothness for the kernels. Our proof follows a completely different strategy usingan “exponential growth” lemma for probability measures on intervals and is selfcontained (unlike the proof from [CM] which depends on several L ( ν ) to L ( µ )boundedness results for separated measures ν and µ ).Recall that if k is the ( d − R d and µ has ( d − d −
1) purely unrectifiable, that is µ ( E ) = 0 for all ( d − E , the principal values diverge µ almost everywhere and the weak convergencein L ( µ ) fails. On the other hand it is of interest that weak convergence in the senseof Theorem 1.1 holds as it holds for any s -dimensional antisymmetric kernel andany finite measure with s -growth.2. Proof of Theorem 1.1
We first prove the following lemma about exponential growth of probability mea-sures on compact intervals. It is motivated by a similar result proved in [SUZ]. Here
VASILIS CHOUSIONIS AND MARIUSZ URBA ´NSKI
Leb stands for the Lebesgue measure on the real line and | I | denotes the length ofan interval I ⊂ R . Lemma 2.1.
For every integer λ > the following holds. Let ν be a probabilityBorel measure on a compact interval ∆ ⊂ R . Then for every interval I ⊂ ∆ thereexists a subset I ′ ( λ ) ⊂ I such that Leb ( I ′ ( λ )) > | I | (1 − λ − + λ − + . . . )) and forevery t ∈ I ′ ( λ ) , ν ([ t − λ n , t + λ n ]) < λ − n for all integers n ≥ .Proof. Let us partition the interval I into λ subintervals J of length | I | λ − . Let B be the family of all intervals J from this partition for which ν ( J ) < λ − . Obviously,there are at most λ intervals in B c . Thus B > λ − λ = λ (cid:18) − λλ (cid:19) and Leb (cid:16)[ { J : J ∈ B } (cid:17) ≥ | I | (cid:18) − λλ (cid:19) = | I | (cid:18) − λ (cid:19) . Next, each interval in B is divided into λ subintervals with disjoint interiors andof length | I | λ − , and we remove those subintervals for which ν ( J ) ≥ λ − . Denotingby B the family of remaining intervals, we see that B ≥ ( λ ) (cid:18) − λλ (cid:19) − λ = ( λ ) (cid:18) − λ − λ (cid:19) and Leb (cid:16)[ { J : J ∈ B } (cid:17) ≥ | I | (cid:18) − λ − λ (cid:19) . Proceeding inductively, we partition the interval I into disjoint intervals of length | I | λ − n . Next, we define in the same way the family B n . It is formed by the intervals J of this partition of n ’th generation, which are contained in some interval of thefamily B n − and for which ν ( J ) < λ − n . ThenLeb (cid:16)[ { J : J ∈ B n } (cid:17) ≥ (cid:18) − λ − λ − · · · − λ n (cid:19) | I | . For any t ∈ I let J n = J n ( t ) be the interval of the n ’th partition such that t ∈ J n .Thus, for every t ∈ T ∞ n =1 S J ∈ B n J , we have that J n ( t ) ∈ B n . Consequently, for all t ∈ T ∞ n =1 S J ∈ B n J , it holds that ν ( J n ( t )) < λ − n for all n ≥
1. Let now C n = { t ∈ I : [ t − | I | λ − n , t + | I | λ − n ] ⊂ J n ( t ) } . It is easy to see that Leb( C cn ) < | I | λ − n , and, therefore,Leb ∞ \ n =1 C n ! > | I | (cid:18) − (cid:18) λ + 1 λ + . . . (cid:19)(cid:19) . NOTE ON WEAK CONVERGENCE OF SINGULAR INTEGRALS IN METRIC SPACES 5
Finally, setting I ′ := ∞ \ n =1 C n ! ∩ ∞ \ i =1 [ J ∈ B i J ! completes the proof. (cid:3) Proof of Theorem 1.1.
We can assume that µ ( X ) ≤
1. We define finite Borel mea-sures on the unit interval for all z ∈ spt µ by µ z ( F ) = µ { x ∈ X : d ( x, z ) ∈ F } , F ⊂ [0 , . Let A z = ∪ λ> I ′ z ( λ ) where I ′ z ( λ ) are the sets we obtain after we apply Lemma2.1 to the measures µ z . Then Lemma 2.1 implies that µ z ( A z ) = µ z ([0 , G z = { r ∈ (0 ,
1] : r ∈ A z } and X ′ B = ( n X i =1 a i χ B ( z i ,r i ) : n ∈ N , a i ∈ R , z i ∈ spt µ, r i ∈ G z i ) . Then X ′ B is dense in L ( µ ).Let f, g ∈ X ′ B such that f = n X i a i χ B i and g = m X j b j χ S j , where a i , b j ∈ R and B i , S j are closed balls. Then for 0 < δ < ε , Z T kµ,ε f ( x ) g ( x ) dµ ( x ) − Z T kµ,δ f ( x ) g ( x ) dµ ( x ) = m X j =1 n X i =1 a i b j Z S j Z B i δ The last inequality follows because by antisymmetry and Fubini’s theorem Z B i ∩ S j Z B i ∩ S j δ Since r ∈ G z there exists some λ ∈ N such that r ∈ I ′ z ( λ ). We write, Z B ( z,r ) o | log d ( x, ∂B ) | dµ ( x ) = Z B ( z,r − λ − ) o | log d ( x, ∂B ) | dµ ( x )+ ∞ X n =1 Z { x : r − λ − n ≤ d ( z,x ) Singular integrals on Sierpinski gaskets , Publ. Mat. 53 (2009), no. 1, 245–256.[CMPT] V. Chousionis, J. Mateu, L. Prat and X. Tolsa, Calder´on-Zygmund kernels and rectifia-bility in the plane , Adv. Math. 231:1 (2012), 535–568.[CM] V. Chousionis and P. Mattila, Singular integrals of general measures separated by Lipschitzgraphs , Bull. London Math. Soc. 42 (2010), no. 1, 109–118.[D] G.David, Des int´egrales singuli`eres born´ees sur un ensemble de Cantor , C. R. Acad. Sci.Paris Sr. I Math. 332 (2001), no. 5, 391–396.[DS] G. David and S. Semmes. Singular Integrals and rectifiable sets in R n : Au-del`a des grapheslipschitziens. Ast´erisque 193, Soci´et´e Math´ematique de France (1991).[F] H. Federer. Geometric Measure Theory Springer-Verlag, 1969.[H] P. Huovinen. A nicely behaved singular integral on a purely unrectifiable set. Proc. Amer.Math. Soc. 129 (2001), no. 11, 3345–3351.[M] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces , Cambridge UniversityPress, (1995).[MMV] P. Mattila, M. Melnikov and J. Verdera, The Cauchy integral, analytic capacity, anduniform rectifiability. Ann. of Math. (2) 144 (1996), no. 1, 127–136.[MV] P. Mattila, J. Verdera, Convergence of singular integrals with general measures , J. Eur.Math. Soc. (JEMS) 11 (2009), no. 2, 257–271.[MP] P. Mattila, D. Preiss, Rectifiable measures in R n and existence of principal values forsingular integrals , J. London Math. Soc., 52 (1995), 482-496.[NToV] F. Nazarov, X. Tolsa and A. Volberg, On the uniform rectifiability of AD-regular measureswith bounded Riesz transform operator: the case of codimension 1. submitted (2012).[T] X.Tolsa, Principal values for Riesz transforms and rectifiability , J. Funct. Anal. 254(2008), no. 7, 1811–1863.[SUZ] M. Szostakiewicz, M. Urba´nski, and A. Zdunik, Fine Inducing and Equilibrium Measuresfor Rational Functions of the Riemann Sphere , Preprint 2011. VASILIS CHOUSIONIS AND MARIUSZ URBA ´NSKI Department of Mathematics, University of Illinois, 1409 West Green St., Ur-bana, IL 61801 E-mail address : [email protected] Department of Mathematics, University of North Texas, General AcademicsBuilding 435, 1155 Union Circle E-mail address ::