NON-CONFORMAL LOEWNER TYPE ESTIMATES FOR MODULUS OF CURVE FAMILIES

Transcription

1 Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 35, 200, NON-CONFOMAL LOEWNE TYPE ESTIMATES FO MODULUS OF CUVE FAMILIES Tomasz Adamowicz and Nageswari Shanmugalingam University of Cincinnati, Deartment of Mathematical Sciences PO Box 20025, Cincinnati, OH , USA; University of Cincinnati, Deartment of Mathematical Sciences PO Box 20025, Cincinnati, OH , USA; Abstract We develo various uer and lower estimates for -modulus of curve families on ring domains in the setting of abstract metric measure saces and derive -Loewner tye estimates for continua These estimates are obtained for doubling metric measure saces or Q-Ahlfors regular metric measure saces suorting (, )-Poincaré inequality for the situations of Q and > Q We also study -modulus estimates with resect to iesz otentials Introduction and reliminaries ecently there has been increasing interest in the geometry of the -harmonic world If = n, the relations between n-harmonics and conformal and quasiconformal mas allow us to discover a variety of roerties of n-harmonic functions and maings (see eg Chaters 4, 5 in [HKM] or [MV, MV2] and references therein) The situation changes when n In such a case either new methods and aroaches have to be develoed or when alying the existing methods one has to carefully analyze the stes of reasoning Unfortunately, the use of quasiconformal maings as tools in the study of the -modulus rovides us mainly with qualitative results Thus, the cases where the recise estimates can be obtained are worth contemlation That is the main urose of this note We extend the well known classical results of Gehring [Ge] and Väisälä [Vä] on the n-modulus of the family of curves to the setting of -modulus considered in abstract metric measure saces Additional motivation for our studies comes from the roblem of generalizing the notion of rime ends to the setting of -modulus To accomlish this goal the deeer understanding of -modulus is necessary (see [Nä, Oh] for more on rime ends) The aer is organized as follows In this section we give the needed definitions, state the key technical lemmas and exlain the notation In Section 2 we resent various known results about lower and uer estimates for the -modulus of curves in n from Väisälä s book [Vä] and Caraman s aers [C, C2, C3, C4] to conclude with a Loewner tye theorem for -modulus Section 3 is devoted to studying the uer estimates for -modulus in the abstract setting of metric measure saces In Section 4 we develo the lower estimates in the same setting In Section 5 we discuss the -modulus in relation to iesz otentials The ring domains will be for us the fundamental category of sets to deal with An imortant oint is that the modulus estimates for ring domains even in the Euclidean doi:0586/aasfm Mathematics Subject Classification: Primary 30C65, 28A75, 28A78, 3C5, 46E35 Key words: -modulus of curve family, Loewner tye theorem, metric measure saces, conformal maings, -caacity, -harmonic functions

2 60 Tomasz Adamowicz and Nageswari Shanmugalingam setting are not so widely known for > n Also, our discussion emhasizes the role of the borderline value = n : for n the curves have -caacity zero and therefore the modulus analysis does not result in a viable theory We start by recalling selected definitions and notation commonly aearing in analysis on metric saces Given a metric measure sace (, d, µ), with µ a locally finite Borel measure suorted on, and Γ a family of rectifiable curves in, by F (Γ) we denote the collection of all non-negative Borel measurable functions ρ on such that ρ ds for every Γ The -modulus of the family Γ, denoted M (Γ), is the number M (Γ) := inf ρ dµ ρ F (Γ) Should F (Γ) be emty, we set M (Γ) = One of the fundamental concets of analysis on metric saces is that of the uer gradient (see [He, HK]) Following [He, Section 722] we say that a Borel function ϱ: [0, ] is an uer gradient of a function u: if for all rectifiable curves () u(x) u(y) ϱ ds, where x and y denote the endoints of Observe that ϱ is the uer gradient of every function on and if there are no rectifiable curves in then ϱ 0 is an uer gradient of every function on If condition () holds excet for a fixed family of curves with zero -modulus, then ϱ is called a -weak uer gradient We say that a metric measure sace suorts a (, )-Poincaré inequality if there are constants C > 0 and τ so that for each ball B and each function u : and every uer gradient ϱ of u the following inequality holds: B ( u u B C(diam(B)) τb ϱ ), where u B denotes the mean value of u over B, that is u B = ffl u = u If B µ(b) B a metric sace suorts (, )-Poincaré inequality, then has lenty of rectifiable curves analogous to the Loewner roerty for the -modulus For instance the Euclidean sace n suorts a Poincaré inequality for any <, while the snowflake saces do not suort such inequality for any A doubling sace (see below), suorting Poincaré inequality enables us to establish a first order calculus (see eg [He, HK]) Given a set Ω we will denote its comlement in by Ω c For nonemty sets E, F, K in, Γ = (E, F, K) will denote the family of rectifiable curves from E to F in K More recisely, (E, F, K) consists of comact curves with one endoint in E, the other in F, and \ (E F ) K As in [Vä], the modulus of the curve family M ( (E, F, K)) is the number M ( (E, F, K)) = M ({ K : (E, F, K)}) In what follows we will also aeal to the notion of -caacity and its relation to -modulus Let < < and let the trile (E, F, K) be as above Consider the family of functions u such that u E and u F 0 Following [He] we define the

3 Non-conformal Loewner tye estimates for modulus of curve families 6 -caacity of the trile (E, F, K) as Ca (E, F, K) = inf ϱ ϱ L (K), where the infimum is taken over the set of non-negative Borel-measurable functions ϱ that are uer gradients (or weak uer gradients) of some function u as above One can obtain various caacities by considering different degrees of regularity for functions u, eg by requiring u to be continuous, locally Lishitz With self-exlanatory notation it is immediate that M ( (E, F, K)) Ca (E, F, K) Cont-Ca (E, F, K) locli-ca (E, F, K) As a matter of fact a stronger result is true Namely, the following theorem rovides us with the sufficient conditions for equalities to hold above Theorem (Theorem in [KSh]) If is a roer φ-convex metric measure sace equied with a doubling measure and suorting (, )-Poincaré inequality with < <, and Ω is a domain in, then for all disjoint non-emty subsets E and F of Ω, M ( (E, F, Ω)) = Ca (E, F, Ω) = Cont-Ca (E, F, Ω) = locli-ca (E, F, Ω) (The first equality holds for all comact metric saces, [HK, Proosition 27]) The assumtion on φ-convexity can be easily byassed by the observation that a comlete doubling metric measure sace suorting (, )-Poincaré inequality is quasiconvex (a result due to Semmes, see also [Ko] and references therein) The shere in n, of radius r > 0 and centered at x 0, is denoted S = S(r) = S n (x 0, r), and ω n = m n (S) denotes the Lebesgue measure of S() Suose that Γ is a curve family in S(r) Following Section 0 in [Vä] we denote by M S the - modulus of Γ with resect to the metric sace S equied with the (n )-dimensional Lebesgue measure dm n Let Γ be a family of curves in D n and let f : D n be continuous Then fγ := {f : Γ} Consider f k (x) = kx for k > 0 and f k ( ) = In what follows we will aeal to Proosition 2 (Theorem 82 in [Vä]) With kγ := f k Γ, we have M (kγ) = k n M (Γ) emark 3 Proosition 2 holds also for the sherical modulus M S with n relaced by n, the natural dimension of shere S More secifically, if Γ is a family of curves in S and F k = f k S, then M F k(s) (F k Γ) = k n M S (Γ) In what follows we will need to know the -modulus of the family of non-constant rectifiable curves assing through a given oint The result stating that it is zero is known for = n (see [Vä, Section 79]) Below we discuss the counterart of this result for the general doubling metric measure saces with We also rovide uer and lower estimates for M We say that a metric measure sace is doubling if µ is a Borel regular measure on the metric sace (, d) and there exists C > 0 such that for all x and every r > 0, µ(b(x, 2r)) Cµ(B(x, r))

4 62 Tomasz Adamowicz and Nageswari Shanmugalingam As a consequence of the doubling roerty, if in addition is ath connected, then there exist constants Q Q 2 and C such that for all oints x 0 and 0 < r < diam(), ( ) C ( r ) Q µ(b(x 0, r)) ( r ) µ(b(x 0, )) C Q2 Before we roceed, let us recall the following (Definition 63 in [Vä]) Definition 4 Let Γ and Γ 2 be curve families in We say that Γ 2 is minorized by Γ and denote Γ < Γ 2 if every Γ 2 has a subcurve in Γ It can be seen directly from the definition of -modulus that if Γ < Γ 2, then M (Γ 2 ) M (Γ ) Let us also recall some other concets and results from the metric measure sace theory that will be needed in further discussion Given x and y in we denote by Γ xy the family of comact rectifiable curves in joining x and y To show the measurability of certain functions in Proosition 4 and Theorem 5 we will aeal to the following result Proosition 5 (Corollary 0 in [JJS]) Let be a comlete searable metric sace equied with a σ-finite Borel measure µ, and let ϱ: [0, ] be a Borel function Then for each x 0, the function u: [0, ], defined for all x by { } u(x) = inf ϱ ds: Γ x0 x, is measurable with resect to the σ-algebra generated by analytic sets, and therefore, it is µ-measurable 2 -Modulus estimates in n The urose of this section is to recall uer and lower (-Loewner tye) estimates in n Our goal is also to resent known results in Euclidean setting for the urose of comaring and contrasting with the situation in more general metric measure saces In order to obtain uer bounds for the -modulus it suffices to consider one choice of admissible function ϱ F (Γ) Such estimates in n are nowadays classical (see eg 2 in [HKM]) However, to get a lower bound one needs to consider all admissible functions in F (Γ), and hence lower bounds are more difficult to obtain We recall the following result roved by Caraman Theorem 2 (Theorem 3 in [C]) Let n 2, n < and let E and F be nonemty nonintersecting subsets of the shere S = S n (x 0, r) Consider the curve family Γ = (E, F, S) Then (2) M S (Γ) cn r n+ The equality holds when E = {a} and F = {b} with a and b antiodal oints on S Before stating the next result let us recall the definition of a ring (see Section in [Vä]) Definition 22 A domain A in n is called a ring if A c has exactly two comonents

5 Non-conformal Loewner tye estimates for modulus of curve families 63 In what follows comonents of the comlement of a ring will be denoted by C 0, C and the ring by (C 0, C ) The next lemma is a direct consequence of Theorem 2 and is used to investigate the -modulus on the ring domains Lemma 23 (Theorem 4 in [C]) Suose that 0 < a < b and that E and F are disjoint sets in n such that every shere S n (t) for a < t < b meets both E and F If G contains the sherical ring A = B n (b) \ B n (a) and if Γ = (E, F, G), then for n < < n or > n, (22) M (Γ) cn ( b n a n ), n where c n is the constant in Theorem 2 The equality holds if G = A and E and F are the comonents of L A for L being a line through 0 In order to further analyze the -modulus of ring domains we introduce the following family of rings (comare with Definition 6 in [Vä]) For ositive r and, let (23) Φ n (r, ) = {A = (C 0, C ) n : 0 C 0, a C 0 with a =, C, b C with b = r} For a ring A = (C 0, C ) we set Γ A = (C 0 A, C A, A) We are in a osition to define (24) H (r, ) := inf M (Γ A ) A Φ(r,) Note that by Proosition 2, H n (r, ) = H ( r, ) For this reason it suffices to consider H (r) := H (r, ) and Φ n (r) := Φ n (r, ) Theorem 24 (Theorem 8 in [C2]) The function H (r) has the following roerties () H is a decreasing function, ( ) (2) lim H (r) = 0 for n, while lim H (r) = ω n n r r for < n; moreover H (r) < for all r > 0, (3) lim H (r) = for n, while lim H (r) cn > 0 for n < < n with r 0 r 0 n c n as in Theorem 2, (4) H (r) > 0 for > n and for all r > 0 emark 25 One may exect that in the general setting of metric measure saces the lower bound for H in (4) above follows from a (, )-Poincaré inequality The otimality in cases (2) and (3) for < n is demonstrated by the following examles Examle 26 By Corollary in [C3] we have that ( ) n lim M (Γ AG (r)) ω n, r where for r >, Γ AG (r) = ( B(0, ), {x: r x <, x 2 = = x n = 0}, n) reresents the ath family corresonding to Grötzsch ring The otimality of case (2) for < n now follows immediately from H (r) M (Γ AG (r))

6 64 Tomasz Adamowicz and Nageswari Shanmugalingam A weak otimality in the case (3) for < n, that is lim r 0 H (r) <, is a consequence of the following examle Examle 27 Fix 0 < r <, n = 2, and let E = [, 0], F = [r, ] Then (E, F ) Φ 2 (r), and so H (r) M ( (E, F, 2 )) Let Γ r be the collection of all rectifiable curves connecting E to F in the disc B(0, 2), and Γ be the collection of all rectifiable curves connecting the circle S(0, ) to the circle S(0, 2) Then Γ r (E, F, 2 ), and Γ < (E, F, 2 ) \ Γ r It follows that M ( (E, F, 2 )) M (Γ r ) + M (Γ ) Since the function ρ(z) = (π z ) χ B(0,2) (z) is in F (Γ r ), we have when < 2, M (Γ r ) 2π r 2 2 ( r) 2 Since every curve in Γ has length at least, we also have It follows that M (Γ ) B(0, 2) \ B(0, ) = 3π H (r) 2π r 2 + 3π 2 ( r) 2 Hence when < 2 we have lim H (r) 2π + 3π < r 0 2 In the next observation we exress the lower bound for the -modulus in terms of H (comare to Theorem 9 in [Vä] for the case of n-modulus) Proosition 28 (Theorem 9 in [C2]) Let A = (C 0, C ) be a ring such that a, b C 0 and c, C Then ( ) c a M (Γ A ) H b a n, for > b a The above roosition together with Theorem 3 result in the following characterization of rings with zero -modulus (the roof in the Euclidean setting is similar to the case = n, Theorem 0 in [Vä]) Proosition 29 Let A = (C 0, C ) be a ring Then if n < n, M (Γ A ) = 0 #C 0 = or #C = 3 Uer -modulus estimates in the abstract setting ecall from Section (see ( )) that one of the consequences of the doubling roerty for ath connected saces, is that there exist constants Q Q 2 and C such that for all oints x 0 and 0 < r < diam(), ) Q2 ( ) C ( r ) Q µ(b(x 0, r)) µ(b(x 0, )) C ( r Theorem 3 Let be a ath connected doubling sace with Q 2 > Let x 0 and Γ be a collection of non-constant aths that ass through x 0 If Q 2, then M (Γ) = 0 Let 0 < 2r < and denote by (3) Γ(r, ) = (B(x 0, r), \ B(x 0, ), B(x 0, ))

7 Non-conformal Loewner tye estimates for modulus of curve families 65 the family of curves joining B(x 0, r) and \ B(x 0, ) If < Q 2, we have the uer estimate M (Γ) C( 0, Q ) C, r 4 Q Q 2, C Q 2,r C,2r ( with C,r = ), α = Q 2 2 α r 2 α α and constant 0 > 0 such that r < 0 < If = Q 2, the corresonding estimate reads M Q2 (Γ) C ( 0, Q ) Q Q 2 ( ), where C,r = ln r In the Euclidean setting Q = Q 2 = n and we retrieve the -caacity estimates for the Euclidean annuli (see for examle [HKM, Lemma 2]) Proof We will first aly the telescoing argument to estimate the modulus of Γ(r, ) and then obtain the first art of the lemma by letting r 0 The roof is divided into two cases First, let < Q 2 Consider (32) ϱ(x) = C,2r d(x 0, x), α = Q 2 α >, ( where C,2r = ) Choose k 2 α 2 (2r) α α 0 to be the smallest ositive integer such that 2 k 0 r < 2 k 0 Divide the annulus into dyadic annular regions i = B(x 0, 2 i ) \ B(x 0, 2 i ) with radii 2 i < 2 i, for i = 0,, k 0 Consider any curve Γ(r, ) and denote by i the art of in the i-th subring i (note that i might be disconnected consisting of a family of arcs) Let β i be a subarc of i connecting the inner shere of i with its outer shere Next, we show that ϱ is an admissible function for Γ(r, ) (33) k 0 ϱ ds = ϱ ds i 2 = k 0 k 0 β i ϱ ds k 0 l(β i ) C,2r (2 i ) α k C,2r (2 i ) = 0 α 2C,2r α 2 (α )(k0 ) 2C,2r α 2 α C,2r (2 α 2) ( (2r) α α 2 i(α ) C,2r (2 α 2) α ) =, ( ( ) α ) 2r and hence ϱ F (Γ(r, )) Below the constant C,r may change from line to line, but always has the form C,r = C( ), for C deending on α,, Q r α α 2 and the doubling constant in ( ) The following chain of inequalities leads us to the uer estimate for M (Γ(r, )) M (Γ(r, )) B(x 0,)\B(x 0,r) ϱ dµ = k 0 B(x 0,2 i )\B(x 0,2 i ) ϱ dµ

8 66 Tomasz Adamowicz and Nageswari Shanmugalingam (34) By ( ) again, k 0 2α C,2r µ(b(x 0, 2 i )) C,2r µ(b(x 0, )) α µ(b(x 0, )) C,2r µ(b(x 0, )) C,2r α µ(b(x 0, )) C,2r Q 2 C, r 4 C,2r (2 (i+) ) α k 0 2 i(q 2 α) (by ( )) 2 (α Q2)(k0+) α 2 α Q 2 ( ( ) α Q2 4 α Q 2 ) r ( (4 ) ) α Q2 r α Q 2 µ(b(x 0, )), (as α Q Q 2 = α ) 2 M (Γ(r, )) C, r 4 C,2r µ(b(x 0, 0 )) Q Q 2 Now let = Q 2 and hence α = Comutations similar to these at (33) result in k 0 k0 l(β i ) (35) ϱ ds C,r 2 i = k 0, C,r C,r for a suitable choice of constant C in C,r = C ln( ) By the reasoning analogous to r (34) we arrive at the following estimate: ϱ dµ µ(b(x 0, )) k 0 2 iq 2 B(x 0,)\B(x 0,r) C (by doubling condition),r (2 i ) α (36) µ(b(x 0, )) C,r α µ(b(x 0, )) Q 2 k 0 C,r Q 0 2 i(q 2 α) µ(b(x 0, )) ln ( ) r C,r Q 2 µ(b(x 0, 0 )) 0 Q k 0 Q Q2 C Q 2,r Note that if Q = Q 2 =, then as the exression on the right hand side aroaches 0 On the other hand, if Q = Q 2 >, we can only conclude that lim M (Γ(r, )) C( 0, Q )r Q2 Since our main goal is to estimate the modulus either for r 0 (with r < ) or for large values of, we may assume without loss of generality that > 0 for some ositive constant 0 The above estimates for < Q 2 imly that (37) M (Γ(r, )) C, r µ(b(x 4 0, 0 )) C Q,2r Q Q 2 0

9 Non-conformal Loewner tye estimates for modulus of curve families 67 Denote by Γ() = 0<r< Γ(r, ) Then Γ > n N Γ(/n) (see Definition 4), that is, every curve in Γ has a subcurve that belongs to n N Γ(/n) Hence M (Γ) M ( n N Γ(/n)) Observe that (38) C, r 4 C,2r = C ( 4 r )α α ( ) (2r) α α 0, for r 0 Therefore, if < Q 2 the first art of the lemma follows immediately from the fact that M (Γ()) = lim M (Γ(r, )) = 0, and hence M (Γ) M (Γ (/n)) = 0 r 0 Similar argument allows us to handle the case = Q 2 Namely, by (36) we have that C = C 0, for r 0 (ln ln r),r This comletes the argument in the case = Q 2 emark 32 Another tye of uer estimates have been recently obtained by Garofalo and Marola, see Theorem 34 in [GM] In our setting and notation this result states that for a bounded oen subset Ω and x 0 Ω, if 0 < r < < 0 (Ω), then Ca (B(x 0, r), B(x 0, )) { C(, Ω) µ(b(x 0,r)) C(, Ω) µ(b(x 0,r)) r n=0, < < Q r 2, ( ) ln Q2, = Q r 2 These estimates are not equivalent to ours For instance if Q 2, the above inequalities give the uer bound for M C while we have the uer estimate for M Q Q 2 Similar discreancies aear when = Q 2 For examle, if sace is Q 2 -Ahlfors regular, then for small our estimate is stronger emark 33 If Q = Q 2, that is, if is locally Ahlfors regular, then as a byroduct of the comutations (34), we get the uer bound for the -modulus for large values of : (39) M (Γ(r, )) C, r µ(b(x 4 0, 0 )) C Q,2r C 4α Q2 0 r = C(, Q 2)r Q2, (α Q 2)( ) for emark 34 The above arguments allow us to rove art (2) of Theorem 24 without aealing to Euclidean techniques Namely, using the discussion of (38), emark 33 and estimate (36) with 0 =, = + r and r = we arrive at the following estimates: < Q 2 = Q = n : H (r) M (Γ(, + r)) C +r, 4 C +r,2 C 4α (+r) α ( C(, n), for r, 2 = Q 2 = Q = n : H (r) (+r) α ) C(n) (ln( + r)) Q 2 0, for r

10 68 Tomasz Adamowicz and Nageswari Shanmugalingam However, in this general non-euclidean setting, we do not have exlicit shar exressions for the constants C(, n), C(n) 4 Lower -modulus estimates in the abstract setting The urose of this section is to discuss the counterarts of Theorem 2 and Lemma 23 in the abstract setting The essence of the calculations below reveals that a viable theory of -Loewner saces can be successfully develoed beyond the Euclidean setting The exlicit exressions for the constants involved in the estimates however are lacking in the metric setting Our aroach is very much in the sirit of Heinonen Koskela [HK] Let be a metric measure sace suorting a (, )-Poincaré inequality (see eg Chater 4 in [He] for information on Poincaré inequality) Assume that the following Ahlfors regularity tye condition holds: for every ball B in of radius < diam, (4) C Q µ(b()) C Q, with Q and C Note that (4) imlies the reviously discussed condition ( ) For distinct oints x, y denote by Γ xy the collection of all comact rectifiable curves in connecting x to y For k N such that k > 2009 let d(x,y) Γk xy denote the collection of curves in connecting B(x, /k) to B(y, /k) Proosition 4 With the above notation, if > Q, the sace is Q-Ahlfors regular and suorts a (, )-Poincaré inequality, then (42) M (Γ xy ) Cd(x, y) Q Proof Let k > 2009/d(x, y) and ϱ k F (Γ k xy) Define a function u k : by { } (43) u k (z) = min inf ϱ k ds, connecting B(x,/k) to z The same argument as in [JJS, Corollary 0] gives us that u k is measurable and since ρ k L () is an uer gradient of u k, by the Poincaré inequality u k N, loc () Note that u k on B(y, ), whereas u k k 0 on B(x, ) k Let B 0 = B(x, 2d(x, y)) and for i N consider the family of balls B i = B(x, 2 i d(x, y)), B i = B(y, 2 i d(x, y)) Since x and y are Lebesgue oints of u k, we have by Ahlfors regularity and (, )- Poincaré inequality, (44) = u k (x) u k (y) i Z C i Z C i Z (u k ) Bi (u k ) Bi+ C i Z Cd(x, y) Q ( ) d(x, y)2 i ϱ k dµ B i ( d(x, y) 2 i ) Q ( ( ϱ k dµ ) B i ϱ k dµ ) u k (u k ) Bi dµ

11 Non-conformal Loewner tye estimates for modulus of curve families 69 Therefore ϱ k dµ Cd(x, y)q and therefore M (Γ k xy) Cd(x, y) Q The assumtion that > Q has been used to ensure the convergence of the geometric series in (44) The roof is now comleted by invoking Proosition 42 below Note that Γ xy = k N Γk xy Since Γ k+ xy Γ k xy, the subadditivity of modulus imlies that M (Γ xy) M (Γ 2 xy) M (Γ k xy) M (Γ k+ xy ) This together with Proosition 4 give us that lim k M (Γ k xy) exists and (45) lim k M (Γ k xy) Cd(x, y) Q Proosition 42 Let be a locally comact doubling metric measure sace Then (46) lim k M (Γ k xy) = M (Γ xy ) Proof First suose that µ() < Since for every k we have M (Γ k xy) M (Γ xy ), it suffices to rove that M (Γ xy ) lim k M (Γ k xy) To do so, let us consider ϱ F (Γ xy ) such that (47) ϱ dµ M (Γ xy ) + η Since is locally comact and ϱ L (), the Vitali Carathéodory Theorem imlies that without loss of generality ϱ can be assumed to be lower semicontinuous on (see age 57 in [u]) Fix ɛ > 0 such that with ϱ ɛ = max{ϱ, ɛ}, we have ϱ ɛ dµ ϱ dµ + η We will show that for all δ > 0, there exists k δ N such that for all k N with k > k δ, we have that for all curves Γ k xy, (48) ϱ ɛ ds δ Suose this is not true Then there exists 0 < δ < such that for all k N there is n k > k, so that for some nk Γ n k xy it holds that (49) ϱ ɛ ds < δ nk Since ϱ ɛ ɛ on, ɛl( nk ) ϱ ɛ ds < δ, so l( nk ) < δ, nk ɛ and so for all k, l( nk ) < ( δ)/ɛ Note that nk connects a oint in B(x, n k ) to a oint in B(y, n k ) The Arzela Ascoli theorem imlies that nk uniformly (on a subsequence if necessary) so that l() δ and Γ ɛ xy Comutations similar to these on age 4 of [HK] gives us that by the lower semicontinuity of ϱ ɛ, lim ϱ ɛ ds ϱ ɛ ds k nk Therefore, for the above it holds that ϱ ɛ ds lim ϱ ɛ ds δ <, k nk

12 620 Tomasz Adamowicz and Nageswari Shanmugalingam which contradicts with ϱ F (Γ xy ) Hence condition (48) holds From this we infer that ϱ F δ (Γk xy) for k > k δ Hence by (47) M (Γ k xy) ϱ ( δ) ɛ dµ ( δ) (M (Γ xy ) + 2η), if k > k δ Hence lim M (Γ k xy) k ( δ) (M (Γ xy ) + 2η) The assertion of Proosition 42 follows by letting δ 0 and then η 0 When µ() =, the above argument can be conducted on families of curves restricted to balls in Fixing x 0, for n N let Γ xy,n be the family of curves in Γ xy that lie in the ball B(x 0, n), and Γ k xy,n the family of curves in Γ k xy in B(x 0, n); the above argument yields that M (Γ xy,n ) = lim k M (Γ k xy,n) Now an alication of the fact that whenever (Γ N ) N is a sequence of curve families such that Γ N Γ N+, then lim N M (Γ N ) = M ( N Γ N ) gives the desired conclusion Given a oint x 0 and 0 < r <, we consider the curve family Γ(r, ) = (B(x 0, r), \ B(x 0, ), B(x 0, )) Theorem 43 Let be a Q-Ahlfors regular metric measure sace suorting (, )-Poincaré inequality Assume that > Q If \ B(x 0, 2) is non-emty, then (40) M (Γ(r, )) C µ(b(x 0, )) C Q, with constant C deending quantitatively on sace emark 44 Note that this result is false for Q, since in such a case Theorem 3 gives us that M (Γ(r, )) 0 for r 0 emark 45 Note that Theorem 43 together with Proosition 4 imly that M (Γ) = for > Q, where Γ stands for the family of non-constant curves assing through a given oint in Proof of Theorem 43 Let ϱ F (Γ(r, )) Analogously to Proosition 4 we define u: by { } (4) u(z) = min inf ϱ ds, connecting z to \B(x 0,) The roof slits into two cases Case There exists x B(x 0, r) and y B(x 0, 2) \ B(x 0, ) such that (42) u(x) u B(x,) 4 and u(y) u B(y,) 4 Then and so = u(x) u(y) 4 + u B(x,) u B(y,) + 4, 2 u B(x,) u B(y,)

13 Non-conformal Loewner tye estimates for modulus of curve families 62 Therefore, by the doubling roerty of the measure µ, and by the (, )-Poincaré inequality, ( ) C u u B(x0,0) C ϱ dµ B(x 0,0) B(x 0,0) Hence (43) ϱ dµ ϱ dµ B(x 0,0) C µ(b(x 0, )) By assumtion, \ B(x 0, 2) is non-emty Because of the Poincaré inequality, is ath-connected Therefore there is a oint y B(x 0, 2) \ B(x 0, ) such that d(x 0, y) = 3/2 As u = 0 on \ B(x 0, ), it follows that u(y) u B(y,) = u B(y,) < /4 Hence the only remaining case is the following Case 2 For all x B(x 0, r) we have u(x) u B(x,) > As x is a Lebesgue 4 oint of u we have by the Poincaré inequality with B i = B(x, 2 i ) 4 < u(x) u B(x,) u Bi u Bi+ C u u Bi dµ i N i N B i C ( ) 2 i ϱ ( dµ C 2 i ) ( Q ϱ dµ i N B i i N B i ( ) ( C Q 2 i( Q ) ϱ dµ C Q ϱ dµ i N ) ) To ensure that the above sum is finite we need the assumtion that > Q As observed in emark 44 this assumtion is also necessary The above estimate imlies that ϱ dµ Q C µ(b(x 0, )) C By the above two cases, for all ϱ F (Γ(r, )), ϱ dµ µ(b(x 0, )) C Taking the infimum over all such ϱ, we get M (Γ(r, )) µ(b(x 0, )) C Our discussion of lower bounds for the -modulus will be comlete if we handle the case Q < < Q This will be obtained using the following result of Heinonen Koskela [HK] Theorem 46 (Theorem 59 in [HK]) Suose that (, µ) is a doubling sace where the lower mass bound in (4) holds for some Q Suose further that admits a weak (, )-Poincaré inequality for some Q Let E and F be two comact subsets of a ball B in and assume that, for some Q s Q and λ > 0, we have (44) min{h s (E), H s (F )} λ s Q µ(b )

14 622 Tomasz Adamowicz and Nageswari Shanmugalingam Then there is a constant C, deending only on s and on the data associated with, so that (45) ϱ dµ B C C λµ(b ), whenever u is continuous function in the ball B C with u E 0 and u F, and ϱ is a weak uer gradient of u in B C Here H (E) s denotes the Hausdorff s-content of a set E connected, then H (E) diam E ecall that if E is Proosition 47 Let be a Q-Ahlfors regular metric measure sace that suorts (, )-Poincaré inequality for some > such that Q Q and Q Let E and F be continua contained in a ball B Then (46) M ( (E, F, )) C for C a constant in Theorem 46 Proof Take s = and define λ = min {, C min{diam E, diam F } + Q, } min{h (E), H (F )} Q µ(b ) for given continua E and F If λ =, then by the Ahlfors regularity, C min{h (E), H (F )} Q µ(b ) 2C, and so λ min{h (E), H (F )} C Q µ(b ) An alication of Theorem 46 together with Ahlfors regularity yields (47) ϱ dµ min{diam E, diam F } B C C + Q and the theorem now follows from the fact that M ( (E, F, ))=Cont-Ca (E, F, ), see Theorem in Section The immediate consequence is a -Loewner tye result for Q < Q: Corollary 48 Under the assumtions of Proosition 47 on, let E and F be continua with min{diam E, diam F } > 0 If Q < Q, then (48) M ( (E, F, )) > 0 In order to have comlete analogy with the case > Q we would like to find a lower bound for M (Γ(r, )) when Q Theorem 49 Let be Q-Ahlfors regular metric measure sace suorting (, )-Poincaré inequality Assume also that Q Then we have for 0 < r < such that \ B(x 0, 2), (49) M (Γ(r, )) C r Q Q+ Q

15 Non-conformal Loewner tye estimates for modulus of curve families 623 Indeed, we can relace the condition that \ B(x 0, 2) is non-emty with the condition that there is a oint y with d(x 0, y) = 3/2 Proof Let x 0 and ϱ F (Γ(r, )) be such that ϱ ɛ > 0 on B(x 0, ) \ B(x 0, r) Also assume that ϱ is lsc Let u: be given by (420) u(z) = min inf ϱ ds, connecting \B(x 0,) to z Observe that as ϱ F (Γ(r, )), we have u 0 on \ B(x 0, ), u on B(x 0, r), and 0 u Let B = B(x 0, 2) Note that u B = µ(b) B u dµ µ(b(x 0, r)) µ(b(x 0, 2)) Because \ B(x 0, 2), by the ath-connectedness of (which follows from the Poincaré inequality) there exists y B(x 0, 2) \ B(x 0, ) such that d(x 0, y) = 3 2, B(y, 2 ) B(x 0, 2) \ B(x 0, ) and B C µ(b(y, )) 2 µ(b(x 0, )) C The definition of u, the choice of y, and the Ahlfors regularity imly u u B dµ B µ(b(y, u dµ µ(b) B(y, ))µ(b(x 2 0, r)) 2 ) µ(b(x 0, 2)) + Hence by the (, )-Poincaré inequality, ( µ(b(x 0, r)) (42) C µ(b(x 0, )) C ϱ dµ B ) C µ(b(x 0, )) C ( µ(b(x 0, r)) µ(b(x 0, )) ) ϱ dµ From this we get ϱ dµ µ(b(x 0, r)) C µ(b(x 0, )) Inequality (49) now follows by taking infimum over all such ϱ and by alying the Ahlfors regularity condition emark 40 Following emark 32 we comare the estimates obtained in [GM] to estimates derived in Theorem 43 and in Theorem 49 In our setting and notation Theorem 32 in [GM] states that with constant C = C(, Q, Ω) if 0 < r < < 0 (Ω), then C ( ) r ( ) µ(b(x 0,r)), < < Q, r ( ) r Q(Q ) ( log Ca (B(x 0, r), B(x 0, )) C µ(b(x 0,r)) r C µ(b(x 0,r)) r ( r ) ( ) (2) Q ) Q r, = Q, Q r, > Q However, note that if > Q and r is comarable to then (40) is stronger than the corresonding estimate in [GM], since (40) does not involve r at all Moreover, if Q then inequality (49) for the cases r or r/ 0 gives a more delicate lower estimate than the corresonding art of Theorem 32 in [GM]

16 624 Tomasz Adamowicz and Nageswari Shanmugalingam 5 -Modulus of curves assing through the oint and iesz otentials In Theorem 3 we showed that if is a doubling metric measure sace, then M (Γ) = 0, for Q 2, where Γ stands for the collection of all non-constant curves assing through x 0 If > Q 2, then M (Γ) =, see emark 45 The urose of this section is to show that for the weighted iesz measure, the -modulus of Γ remains ositive for Another motivation for our studies comes from Theorem 2 in Keith s aer [K] Let (, d, µ) be a comlete doubling metric measure sace The roerty of of suorting a (, )-Poincaré inequality with resect to comactly suorted Lischitz functions and their comactly suorted Lischitz uer gradients is equivalent to the roerty of existing a constant C such that M (Γ xy, µ C xy) C d(x, y) for almost every air of distinct oints x, y Here µ C xy denotes the symmetric iesz kernel of µ (dµ C xy = dν x + dν y in the notation below) Let (, d, µ) be a doubling measure sace Consider the iesz otential of a nonnegative function f on (see eg Chater 9 in [He]) f(z) d(x, z) (5) I,B (f)(x) = µ(b(x, d(x, z))) dµ(z) For a given oint x define the measure ν x (z) as follows: (52) dν x (z) = B d(x, z) µ(b(x, d(x, z))) dµ(z) By [He, Theorem 95] we know that if suorts a (, )-Poincaré inequality for some <, then for all balls B in and all bounded continuous functions u on B and for all uer gradients ϱ of u, (53) u(x) u B C(diam B) I,B (ϱ )(x), whenever x 2 B Theorem 5 Let (, d, µ) be a doubling measure sace suorting Poincaré inequality for some < For a given oint x 0 consider measure ν x0 (z) defined as in (52) Then (54) M x 0 (Γ(r, )) := inf ϱ(z) dν x0 (z) > C, ϱ F (Γ) where 0 < r < such that \ B(x 0, 2) is non-emty and the constant C deends only on the constants associated with the doubling roerty and the Poincaré inequality Proof Take 0 < r < and consider the family of curves Γ(r, ) As in Theorem 43 we define u: by { } (55) u(z) = min inf ϱ ds, connecting z to \B(x 0,) From [JJS, Corollary 0], the function u is measurable Observe that as ϱ F (Γ(r, )), we have u 0 on \ B(x 0, ) and u on B(x 0, r) Consider

17 Non-conformal Loewner tye estimates for modulus of curve families 625 y B(x 0, 2) \ B(x 0, ) such that d(x 0, y) = 3 and consider a family of balls 2 defined inductively (56) B 0 = B(x 0, 2d(x 0, y)), B = B(x 0, d(x 0, y)) = B(x 0, 3/2), B i = 2 B i for i >, B = B(y, 3/2), B i = 2 B i+ for i > By construction u B i 0, for i 3 Then by the doubling roerty of the measure and the Poincaré inequality we get the following estimate (57) = u(x 0 ) u(y) u Bi u Bi+ i Z C u(z) u Bi dz + u Bi u Bi+ B i i 3 = C u(z) u Bi dz + u B i u B i+ B i i= ( ) C diam B i ϱ 3 ( ) dz + C d(x 0, y)2 i ϱ dz B i B i Next we aly the Ahlfors regularity and Definition 5 to the first term of (57) ( ) C diam B i µ(b i ) ϱ d(x 0, z) µ(b(x 0, d(x 0, z))) dz B i µ(b(x 0, d(x 0, z))) d(x 0, z) ( ) C diam B i µ(b i ) ϱ d(x 0, z) B i µ(b(x 0, d(x 0, z))) d(x 0, z) Q2 dz (58) ( C (3 2 i+ )(3 2 i+ ) Q 2 (3 2 i+ ) Q 2 C 2 ( ϱ dν x0 ) i= C ( ϱ d(x 0, z) B i µ(b(x 0, d(x 0, z))) dz ϱ dν x0 ) Similar aroach alied to the second term of (57) gives the following: 3 ( ) ( ) (59) d(x 0, y)2 i ϱ dz 3C ϱ dν x0 B i i= Therefore, (58) and (59) result in the inequality: C ϱ dν x0 Take = The roof of Proosition 5 now follows from the fact that the modulus is an outer measure on the sace of all curves in, and so for every 0 < r < < we have: M x 0 (Γ(r, )) M x 0 (Γ(r, )) C )

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