Big-Pieces-of-Lipschitz-Images Implies a Sufficient Carleson Estimate in a Metric Space

Big-Pieces-of-Lipschitz-Images Implies a Sufficien t Carleson Estimate in a Metric Space Raanan Sc h ul Abstract This note is in tended to b e a supplemen t to the bi-Lipsc hitz decomp osition of Lipsc hitz maps sho wn in [Sc h]. W e sho w that in the c ase of 1-Ahlfors- regular se ts, the c ondition of ha ving ‘Big Pieces of bi-Lips c hitz Images’ (BPBI) is equiv alen t to a Carleson cond ition. 1 In tro du ction This note is in tended to b e a supplem en t to the bi-Lipsc hitz decomp osition of Lips- c hitz m aps sho wn in [Sc h]. W e assume familiarity with some metho ds from [Sc har] and some definitions from [DS93] (also in [Sc h]). W e prov e the follo wing theorem. Theorem 1.1. L et E b e a 1- Ahlfors-r e gular set. Supp ose that E has Big Pie c es of Lipschitz Images. Then for any x ∈ E and r < diam( E ) we have X B ∈ b G E B ⊂ Ball ( x, 2 r ) Z Z Z ( B ∩ E ) 3 ∂ ( { x 1 , x 2 , x 3 } )diam( B ) − 3 d H 1 ( x 3 ) d H 1 ( x 2 ) d H 1 ( x 1 ) . r , (1.1) wher e b G E is as define d as in [Schar] and the c ons tant A in the defini tion of b G E is lar ge enough. Remark 1.2. As one may exp e ct, the mai n p oint ab out Lipschitz images, is that we have (1.1) for 1-Ahlfors-r e gular Lipschitz ima ges. It is a p p ar e n t fr om the pr o of that one ma y r eplac e the c ondition BPLI by BP(*), wher e (*) is any c ol le ction of 1-Ahlfors-r e gular sets which satisfies (1.1) . F rom this, together with what app ears in [Sch] and [Hahar], one trivially concludes the fo llowing. Corollary 1.3. L et E b e a 1-A hlfors-r e gular set. Assume that the c onstant A in the definition of b G E is lar ge enough. Then the fol lo w ing c onditions ar e e quiva l e nt 1 2 • F or any x ∈ E an d r < dia m( E ) we have the Carleso n estimate (1.1) . • E has Big Pie c es of Lipschitz Images • E has Big Pie c es of bi-Lipschitz Images This corollary is the motiv atio n for his essa y . In general, getting a similar result to this corollary for k − Ah lfors-D a vid-regular se ts w as a large part of the motiv atio n for [Sc h]. Unfortunately , w e are unable to do this for k > 1. The obstacle is finding a ‘correct’ Carleson condition, and thus far w e hav e encoun tered some tec hnical o b- stacles in our attempts. W e conjecture that one can define a Carleson condition so that the ab ov e corollary will hold for k − Ahlfors-D a vid-regular sets with k > 1, th us o v ercoming t hese tec hnical diffic ulties. In general the author is interes ted in pushing the D a vid-Semmes theory of uniform rectifiability in to the setting of metric spaces. W e see this theory as related t o embedability questions widely studied in the theo- retical computer science comm unit y as well as related to mathematical applications (in pa r t icular, analysis of data s ets). Pr o of of The or em 1.1. W e first note that without loss of generalit y t he w e may replace equation (1 .1 ) b y X Q ∈ ∆ i ( E ) Q ⊂ Q 0 Z Z Z ( E ∩ Q ) × ( E ∩ Q ) × ( E ∩ Q ) ∂ ( { x 1 , x 2 , x 3 } ) diam( Q ) 3 d H 1 ( x 3 ) d H 1 ( x 2 ) d H 1 ( x 1 ) . diam( Q 0 ) , (1.2) where i satisfies 1 ≤ i ≤ P 1 , ∆ i ( E ) is a dyadic filtration constructed fr o m b G E and Q 0 in ∆ i is an arbitrarily c hosen ‘cub e’. This uses Ahlfors-regularit y . (F or suc h constructions see [Chr90 , Dav 91]. See also constructions in section 3.3 of [Sc har ]. Suc h constructions were also used in say [M ¨ ul05]). Note that P 1 dep ends only on the Ahlfors-regularity constant of E . W e will no w follow the outline of the pro o f of Theorem IV.1.3 in [DS9 3]. F irst w e need a John-Niren b erg-Str¨ om brg t yp e Lemma. This lemma is stated and prov ed in section IV.1.2 of [DS93]. (the constant coming out of the pro of is N η 2 ), and so w e only state it: Lemma 1.4. L et ∆ b e a dyadic fil tr ation. L et α : ∆ → [0 , ∞ ) b e given. Supp ose that for some N > 0 a n d η > 0 we have for al l Q 0 ∈ ∆             x ∈ Q 0 : X Q ∋ x Q ⊂ Q 0 , Q ∈ ∆ α ( Q ) ≤ N             ≥ η | Q 0 | . (1.3) Then X Q ⊂ Q 0 Q ∈ ∆ α ( Q ) | Q | . N ,η | Q 0 | (1.4) for al l Q 0 ∈ ∆ . 3 Let i ≤ P 1 and Q 0 ∈ ∆ i b e given. Let ˜ E = ˜ E ( Q 0 ) b e a set s uc h E ∩ ˜ E ∩ Q 0 ≥ θ diam( Q 0 ) , (1.5) and ˜ E is a Lipsc hitz Image as in the definition of Big Pieces of Lipschitz Images (BPLI). W e ma y assume that ˜ E is actually a biLipsch itz image of a subset of the real line b y [Sc h]. Hence, b y extending this map and the results of [Sc har], w e hav e (for an y A ′ ) X Q ∈ ∆ i ( ˜ E ) Q ⊂ Q 0 Z Z Z { x ∈ ˜ E :dis t ( x,Q ) ≤ A ′ diam( Q ) } 3 ∂ ( { x 1 , x 2 , x 3 } ) diam( Q ) 3 d H 1 ( x 3 ) d H 1 ( x 2 ) d H 1 ( x 1 ) . diam( Q 0 ) . (1.6) W e note that (1.6) implies (b y Ahlfors-regularity of E a nd the constan t A ′ b eing large enough) X Q ∈ ∆ i ( E ) Q ⊂ Q 0 Z Z Z { x ∈ ˜ E :dist( x,Q ) ≤ diam( Q ) } 3 ∂ ( { x 1 , x 2 , x 3 } ) diam( Q ) 3 d H 1 ( x 3 ) d H 1 ( x 2 ) d H 1 ( x 1 ) . diam( Q 0 ) . (1.7) W e hav e for a ny Q 1 in ∆ i X Q ⊂ Q 1 , Q ∈ ∆ i ( E ) Q ∩ ˜ E 6 = ∅ Z Q ∩ E dist( · , ˜ E ) diam( Q ) ≤ Z x ∈ Q 1 ∩ E X Q ∋ x, Q ∈ ∆ i ( E ) Q ∈ Q 1 , Q ∩ ˜ E 6 = ∅ dist( · , ˜ E ) diam( Q ) . Z x ∈ Q 1 ∩ E 1 + 1 2 + 1 4 + 1 8 + ... . dia m ( Q 1 ) . (1.8) Similarly to the pro o f of L emma 3.11 in [Sc har ], w e ha v e Z Z Z ( E ∩ Q ) × ( E ∩ Q ) × ( E ∩ Q ) ∂ ( { x 1 , x 2 , x 3 } ) diam( Q ) 3 d H 1 ( x 3 ) d H 1 ( x 2 ) d H 1 ( x 1 ) . Z E ∩ Q dist( · , ˜ E ) diam( Q ) + Z Z Z { x ∈ ˜ E :dist( x,Q ) ≤ diam( Q ) } 3 ∂ ( { x 1 , x 2 , x 3 } ) diam( Q ) 3 d H 1 ( x 3 ) d H 1 ( x 2 ) d H 1 ( x 1 ) . (1.9) Summing eq uation (1.9) we get X Q ⊂ Q 0 , Q ∈ ∆ i ( E ) Q ∩ ˜ E 6 = ∅ Z Z Z ( E ∩ Q ) × ( E ∩ Q ) × ( E ∩ Q ) ∂ ( { x 1 , x 2 , x 3 } ) diam( Q ) 3 d H 1 ( x 3 ) d H 1 ( x 2 ) d H 1 ( x 1 ) ≤ C diam( Q 0 ) (1.10) 4 F rom (1 .5) and taking N = 2 C and η = 1 2 θ and α ( Q ) = Z Z Z { x ∈ E :dis t ( x,Q ) ≤ diam( Q ) } 3 ∂ ( { x 1 , x 2 , x 3 } ) diam( Q ) 4 d H 1 ( x 3 ) d H 1 ( x 2 ) d H 1 ( x 1 ) w e ha ve the conditions for Lemma 1.4, which giv es us Theorerm 1.1. References [Chr90] Mic hael Christ. A T ( b ) theorem with remarks on analytic capacity and the Cauc h y in tegral. Col lo q. Math. , 60/6 1(2):601–628 , 1990. [Da v91] Guy David. Wave lets and singular inte gr als on curves a nd surfac es , v olume 1465 of L e ctur e Notes in Mathematics . Springer- V erlag, Berlin, 1991. [DS93] Guy David a nd Stephen Semmes. A nalysis of and on uniformly r e ctifiable sets , v olume 38 of Mathematic al Surveys and Mono gr aphs . American Math- ematical So ciety , Pro vidence, RI, 1993 . [Hahar] Immo Hahlomaa. Curv ature and Lipsc hitz parametrizations in 1-regular metric spaces. A nnales A c ad emiae Scientiarum F ennic ae , T o app ear. [M ¨ ul05] P aul F. X. M¨ uller. Isomorphism s b etwe en H 1 sp ac es , v olume 66 of Insty- tut Matematyczny Polski e j A kademii Nauk. Mon o gr afi e M atematyczne (New Series) [Mathematics Institute of the Polish A c adem y o f Scienc es. Mathe- matic al Mono gr aphs (New Series)] . Birkh¨ auser V erlag, Basel, 2005. [Sc h] Raanan Sch ul. Bi-Lipsc hitz decomp osition of Lipsc hitz f unctions into a met- ric space. arXiv:math/0702630 . [Sc har] Raanan Sc hul. Ahlfors-regular curv es in metric spaces. A nnales A c ademiae Scientiarum F enn i c ae , T o app ear.