A Hölder-type inequality on a regular rooted tree

Transcription

1 A Hölder-type nequalty on a regular rooted tree K.J. Falconer Mathematcal Insttute, Unversty of St Andrews, North Haugh, St Andrews, Ffe, KY6 9SS, Scotland arxv:40.635v [math.ca] 23 Oct 204 Abstract We establsh an nequalty whch nvolves a non-negatve functon defned on the vertces of a fnte m-ary regular rooted tree. The nequalty may be thought of as relatng an nteracton energy defned on the free vertces of the tree summed over automorphsms of the tree, to a product of sums of powers of the functon over vertces at certan levels of the tree. Conjugate powers arse naturally n the nequalty, ndeed, Hölder s nequalty s a key tool n the proof whch uses nducton on subgroups of the automorphsm group of the tree. Introducton The energy of an nteractng partcle system wth a fnte number of stes s typcally gven by a sum (, 2 µ( µ( 2 F (, 2 over pars of partcles (, 2 where F (, 2 represents the force or nteracton between the partcles located at and 2 whch have masses or weghts µ( and µ( 2 respectvely. A more complex system may nvolve nteractons between three or more partcles smultaneously rather than just two. Thus we are led to consder energes of the form (,..., n µ( µ( n F (,..., n resultng from nteractons F (,..., n whch depend on n partcles and ther confguraton. Typcally each partcle may be affected most by those other partcles that are closest, so the nteracton should take nto account the nearest neghbor structure of the partcle confguraton. One convenent way of ncorporatng such a structure s by representng the stes as the free vertces (.e. vertces of valence of a regular m-ary tree, so that the dstance between a par of stes s an ultrametrc determned by the level of ther frst common ancestor. The arrangement of common ancestors or jons of a collecton of n partcles determnes ther nearest neghbor confguraton. Our man results and detaled notaton are set out n Secton 2, but to fx deas we gve here a bref overvew and an example. We work on an m-ary regular rooted tree T of k levels, where the level or generaton of a vertex s ts edge dstance from the root. In the usual parlance, each vertex has m chldren, except for the free vertces at level k whch have no chldren; we let T 0 denote the set of free vertces. We wrte for the jon of, T 0, that s the vertex j T of maxmum level that les on both of the paths from the root to and from to. The jon set of n partcles located at stes,..., n T 0, s the set of vertces of T gven by j for all < j n (ths s made a lttle more precse n the next secton where we allow for multple jon ponts.

2 For f a non-negatve functon defned on the vertces of T, we wll consder nteractons that can be expressed as the product of the values of f over the jon ponts, that s, F (,..., n = f(j f(j 2 f(j n where j, j 2,..., j n are the jon ponts of,..., n. Then the energy s of the form µ( µ( n F (,..., n. (. (,..., n Typcally, f(j wll be large f the jon pont j s a hgh level vertex correspondng to a sgnfcant nteracton component resultng from nearby partcles. For many problems one needs to bound the energy of a system, perhaps n the lmt as the number of generatons becomes large. Such estmates are requred, for example, n estmatng hgh moments of certan measures, see for [2, 3]. Whlst one may wsh to estmate the sum (. over all arrangements of n partcles, the sum breaks down naturally nto sub-sums over confguratons of partcles whch have somorphc jon structures. Thus we consder the set of confguratons of n partcles on T 0 that may be obtaned from each other under some automorphsm of the rooted tree, n other words the equvalence classes of confguratons defned by the automorphsms. These equvalence classes are the orbts of the automorphsm group of T actng on the ordered n-tuples from T 0. Wrtng [I] for such an equvalence class or orbt, we are led to consder the sums µ( µ( n F (,..., n. (,..., n [I] We wll obtan upper bounds for these sums over the equvalence class [I] n terms of pth powers of f summed across certan levels of the tree T. Fgure : Partcles and jon ponts n the example To llustrate ths, consder a specfc case of the bnary rooted tree of k = 3 levels wth n = 4 partcles. Startng wth the confguraton (, 2, 2, 22 of ponts of T 0 (wth the usual codng of vertces of the bnary tree the jon ponts are at, and 2, see Fgure. Let [I] be the class of 64 dfferent equvalent ordered 4-tuples 2

3 (g(, g(2, g(2, g(22 obtanable under an automorphsm g of the rooted tree T. Assgn each free vertex T 0 a postve weght µ(. For each vertex j T wrte µ(j for the total weght of the free vertces below j, so, for example, µ(2 = µ(2 + µ(22. In ths specal case, our man nequalty becomes, for any non-negatve functon f on the vertces of T and for all p, p 2, p 3 > 2 satsfyng p + p 2 + p 3 =, (, 2, 3, 4 [I] µ( µ( 2 µ( 3 µ( 4 F (, 2, 3, 4 ( /p 4 /p2 2 /p 3 ( f( µ( +p /p ( f( p 2 µ( +p 2 + f(2 p 2 µ(2 +p 2 /p 2 ( f( p 3 µ( +p 3 + f(2 p 3 µ(2 +p 3 + f(2 p 3 µ(2 +p 3 + f(22 p 3 µ(22 +p 3 /p 3. (.3 The sum (.2 has 64 terms, each a product over 3 jon ponts, one at each level 0, and 2. The bound (.3 s a product of weghted p th power sums of f across each of the 3 levels of the tree at whch the jon ponts occur. A check shows that equalty holds f the weghts µ( are equal for all T 0 and f s constant across each level, that s f f( = f(2 and f( = f(2 = f(2 = f(22. We wll prove results of ths type n a general context by employng nducton wth respect to automorphsms fxng the jon ponts and ncorporatng Hölder s nequalty n a natural way. 2 Notaton and statement of results To state the man results we set out some further notaton relatng to the tree T. For ntegers m 2 and k, we ndex the vertces of T, the m-ary regular rooted tree of k + levels (ncludng the level of the root, by the symbolc space of words formed from the symbols {, 2,..., m}. Thus the vertces of T are gven by {(, 2,..., l : 0 l k, j m}, wth the root of the tree as the empty word. We often abbrevate a word by = (, 2,..., l and wrte = l for ts level or generaton. The set of free vertces, that s those wth = k, s denoted by T 0. We wrte j to mean that s a curtalment of j, that s s an ntal subword of j. If, T then the jon s the maxmal word such that both and. Wth each j T we assocate the cylnder C j = { T 0 : j } comprsng those ponts of T 0 below j. We next consder automorphsms of the rooted tree T, regarded as a graph, whch nduce permutatons of the vertces at each level of the tree. For each vertex v T and n m k we wrte S v (n = { (,..., n : j T 0, j v ( j n, j h (j h } (2. for the set of all ordered n-tuples of dstnct elements of T 0 that are descendents of v. Let Aut v be the group of automorphsms of the rooted tree T that fx v. Defne an equvalence relaton on S v (n by (,..., n (,..., n f there exsts g Aut v such that g( r = r for all r n; (2.2 3

4 thus the equvalence classes are the orbts of S v (n under Aut v. We wrte [I] v for the equvalence class contanng I = (,..., n. For notatonal smplcty, we often omt the subscrpt when v =, so that S(n = S (n and [I] = [I]. We requre some termnology relatng to the jons of subsets of T 0. Let I = (,..., n S v (n. The jon set of I, denoted by (I = (,..., n, s the set of vertces {j,..., j n } T consstng of the jon ponts j for all < j n, wth w (I occurrng wth multplcty r f there are (r + dstnct ndces <... < r+ n such that s t = w for all s t. Note that the jon set of n ponts always conssts of n ponts countng by multplcty. Moreover, f T s a bnary tree, so m = 2, all jon ponts have multplcty. Fgure 2: A 3-ary tree wth 8 partcles and 7 jon ponts, 2 of whch have multplcty 2 The set of jon levels L(I of I S v (n s { j,..., j n : j (I} wth levels repeated accordng to multplcty. Notce that f I I then L(I = L(I =: L([I] v,.e. the set of levels s constant across each equvalence class [I] v of S v (n. For a level L of T we wll, by slght abuse of notaton, wrte j L to mean that j = L, so we thnk of a vertex j belongng to a level of the tree. We now assgn a weght µ( 0 to each T 0. Then the weght of each cylnder s defned to be the sum of the weghts of the ponts n the cylnder, that s µ(c j = T 0,j µ( for each j T. Next we assgn a postve value f(j to each vertex j of T, that s f : T R +. Then for each I S(n we take the product of these values over the jon ponts of I. Thus f n 2 we defne F : S(n R + by F (,..., n = f(j f(j 2 f(j n, (2.3 where I = (,..., n S(n has jon set (,..., n = {j,..., j n }. For I = ( S( we make the trval assgnment F ( =. Note that f j s a jon pont of multplcty r then the factor f(j wll occur n (2.3 r tmes We can now state the man theorem. 4

5 Theorem 2. Let T be the rooted m-ary tree of k levels. Let 2 n m k. Let I S(n have jon ponts at levels L,..., L n and let p,..., p n > 0 satsfy n =. Then (,..., n [I] n ( /p, µ( µ( n F (,..., n K f(j p µ(c j +p (2.4 j L where K does not depend on f or µ. The value of the constant K wll be dscussed n Secton 4. We obtan the followng estmate, though ths s not n general optmal. j (m! Corollary 2.2 We may take K = (m n n (2.4, where the (m r! product s over the dstnct jon ponts whch have multplctes r,..., r j. In partcular, for a bnary tree, where m = 2, we can take K =. For a bnary tree we can, under certan condtons, mprove on ths to obtan the best possble value of K. Corollary 2.3 Let T be the rooted bnary tree of k levels. Let 2 n 2 k. Let I S(n have jon ponts {j,..., j n } at levels L,..., L n and let p,..., p n > 0 satsfy n p =. Suppose also that the followng condton s satsfed: If the jon ponts of I are ordered so that j s the top jon pont (.e. j j for all and {j 2,..., j m } and {j m+,..., j n } are the jon ponts below the left and rght edges abuttng j respectvely, then p m =2 p 2 and n =m+ p 2. (2.5 Then (,..., n [I] µ( µ( n F (,..., n n ( /p, f(j p µ(c 2 n j +p (2.6 j L wth equalty f µ( s constant for all T 0 and, for each level L, f(j s constant for all j L. 3 Proof of Theorem 2. We wll establsh the key Lemma 3. by nducton on the jon ponts, usng repeated applcatons of Hölder s nequalty and at each step carryng forward some of the weght to the next jon pont. Note that the proof s smpler when T s a bnary tree or when all the jon ponts have multplcty, n whch case d = 2 throughout the nducton. Theorem 2.3 follows mmedately from the lemma on takng v = to be the root of the tree T. 5

6 In the proof of Lemma 3. we wll refer to the numbers K(m; a,..., a m, defned as follows for m and a 0. Wth S m as the symmetrc group of all permutatons σ of (, 2,..., m, K(m; a,..., a m s the least number such that x a σ( xa 2 σ(2 xam σ(m K(m; a,..., a m (x + + x m a + +a m (3. σ S m for all x 0, where we make the consstent conventon that 0 0 =. It s easy to see that K(m; a,..., a m m! ; we wll dscuss the value of K n more detal n Secton 4. Lemma 3. Let T be the rooted m-ary tree of k levels. Let v T and n m k v. Let I S v (n have jon ponts at levels L,..., L n and let p,..., p n > 0 satsfy =. Then n p n ( µ( µ( n F (,..., n K (,..., n [I] v where K does not depend on f or µ. j L,j v f(j p µ(c j +p /p, (3.2 Proof. We proceed by strong nducton. We take the nductve hypothess P(n for n N to be as follows. P(n (n : For all v T, for all p,..., p n > 0 and α satsfyng n p =, f I S α v(n has jon levels L,..., L n, there s a constant K such that n ( µ( µ( n F (,..., n Kµ(C v /α (,..., n [I] v j L,j v f(j p µ(c j +p /p, for all µ and f. When n = then α = and we take the empty product to equal n (3.3. The proof falls nto several parts. (a Verfcaton of P(. If v T and I = ( S v ( then µ( F ( = µ( = µ(c v, ( [I] v v whch s P( wth K =, notng that α =. (3.3 (b The nductve step. Assume, for some nteger n 2, that P(l holds for all l n. We wll show that P(n holds. Let v T and I = (,..., n S v (n where, wthout loss of generalty, (,..., n are ndexed n lexcographcal order. We consder two cases. Case (b. Suppose that v T s a jon pont of I of multplcty d (2 d m. Let v,..., v d be the vertces of T that are chldren of v and are above at least one of the j, that s for each j, v j = v + and there s an k such that v v j k. Then I splts nto d non-empty parts, I j = ( j,..., j n j for j d, so that j v j for all, j, wth n j and n + + n d = n. Note that the jon ponts of I comprse the jon ponts of each of the I j together wth v. 6

7 Let p,..., p n > 0 be the exponents assocated wth the jon ponts of I where n p = wth < α. Re-ndex these exponents so that α pj,..., p j n j are those assocated wth the jon ponts of I j for each j and the exponents p 0,..., p 0 d are those assocated wth v. Let the correspondng jon levels of I j be Λ(I j = {L j,..., L j n j } for j d. Defne α,..., α d by n j p j = α j < (f n j 2 or α j = (f n j =. (3.4 Summng over j gves + + = d α α d p 0 p 0 d α = d + β (3.5 where, for convenence, we wrte /β = /p /p 0 d + /α. Now let (v,..., v d,..., v m be the set of all vertces of T mmedately below v. Let G be a subgroup of Aut v such that {(g(v,..., g(v m : g G} ncludes every permutaton of (v,..., v m exactly once, so that G has order m!. The orbts of I may be decomposed as [I] v = {([g(i ] g(v,..., [g(i d ] g(vd } g G, but f m d + 2 then (m d! elements g of G wll yeld each such decomposton of [I] v. Then, by defnton of F, (m d! µ( µ( n F (,..., n (,..., n [I] v = ( f(v d µ( µ( n F (,..., n g G (,..., n [g(i ] g(v ( µ( d µ( d n d F ( d,..., d n d f(v d g G ( K µ(c g(v /α ( d,...,d n d [g(i d ] g(vd ( K d µ(c g(vd /α d ( j L,j g(v d ( j L d,j g(v d f(j p µ(cj +p /p f(j pd µ(cj +pd /p d (on applyng (3.3 to each I j n the decomposton of I, wth K j as the correspondng constant K = K K d f(v ( ( d µ(c g(v /α µ(c g(vd /α d g G ( j L,j g(v K K d f(v d ( g G f(j p µ(cj +p /p d /β µ(c g(v β/α µ(c g(vd β/α d ( j L d,j g(v d f(j pd µ(cj +pd /p d 7

8 ( g G j L,j g(v f(j p µ(cj +p /p (usng Hölder s nequalty, notng that β + d j= nj d p j = ( g G j L d,j g(v d f(j pd µ(cj +pd K K d f(v d K(m; β/α,..., β/α d, 0,..., 0 /β( µ(c v + + µ(c vm /α + +/α d ((m! j L,j v f(j p (where K(m; s gven by (3. µ(cj +p /p = K 0 f(v d µ(c v d +/p0 + +/p0 d +/α j d, n j d ((m! j L d,j v f(j pd µ(cj +pd ( f(j p j /p j µ(cj +pj j L j,j v (where K 0 = K K d K(m; β/α,..., β/α d, 0,..., 0 /β (m! /β and usng (3.5 (3.6 = K 0 µ(c v /α( f(v p0 µ(cv /p +p0 0 (f(v p0 d µ(cv d /p +p0 0 d ( /p j f(j pj µ(cj +pj j d, n j = K 0 µ(c v /α j L,j v 0 j d, n j ( j L,j v f(j pj µ(cj +pj /p j (on ncorporatng the terms nvolvng p j 0 as sngle terms n the product n ( /p. = K 0 µ(c v /α f(j p µ(c j +p j L,j v Thus P(n holds wth K = K 0 /(m d!. Case (b2. Now suppose that v s not a jon pont of I. As before let p,..., p n > 0 be the exponents assocated wth the jon ponts of I where n p = wth < α. α Let w be the frst jon pont of I below v, so that,..., n w v, wth n 2. Let v,..., v r be the vertces of T such that v j v and v j = w, and let g j Aut v be such that g j (w = v j. We may decompose the orbts of I as [I] v = {[g j (I] vj } r j=. Applyng case (b to [g j (I] vj for j r, and lettng K be the correspondng constant gven by (3.3, whch wll be the same for each j, we obtan (,..., n [I] v µ( µ( n F (,..., n = r j= µ( µ( n F (,..., n (,..., n [g j (I] vj r n ( Kµ(C vj /α f(j p µ(c j +p j L,j v j j= 8 /p /p d /p d

9 ( r K µ(c vj j= /α n ( r n ( = Kµ(C v /α j L,j v j= j L,j v j f(j p µ(c j +p f(j p µ(c j +p usng Hölder s nequalty, to get (3.3 n ths case. Thus by nducton P(n holds for all n, and the Lemma follows f n > takng α =. /p Proof of Theorem 2. The theorem s mmedate on settng v = n ( Value of the constant K The constant K that occurs n (2.4 arses from a product of terms K(m; β/α,..., β/α d, 0,..., 0 that are ncorporated at (3.6 at each step of the nducton. Thus to estmate K we frst need to bound K(m; a,..., a m. Lemma 4. Let m and let a 0 for =, 2,..., m, wth 0 < a + a a m = s. Let S m be the symmetrc group of all permutatons σ of (, 2,..., m. Recall that K(m; a,..., a m s the least number such that x a σ( xa 2 σ(2 xam σ(m K(m; a,..., a m (x + x x m s (4. σ S m for all x 0, wth the conventon that 0 0 =. ( If 0 < s then K(m; a,..., a m = m! m s. ( If s then m! m s K(m; a,..., a m (m!. ( If s and a (s /m for all =,..., m then K(m; a,..., a m = m! m s. (v For m = 2, f (a a 2 2 a + a 2 then K(2; a, a 2 = 2 s = 2 a a 2. Wth the values gven n (, ( and (v there s equalty n (4. f and only f the x are all equal. Proof. It s enough to prove (4. wth the x > 0 and take the lmt of the nequalty for any x = 0. Modfyng the exponents to sum to and then usng Murhead s nequalty (see [4, 5] gves m! σ S m x a σ( xa 2 σ(2 xam σ(n = m! σ S m (x s σ( a/s (x s σ(2 a2/s (x s σ(m am/s 9 /p

10 m (xs + x s x s m. Then ( follows drectly on applyng the generalzed mean nequalty and ( follows usng Mnkowsk s nequalty. Note that n case (, settng K = m!m s n (4. we get equalty when the x are all equal but the nequalty may fal wth ths K for other x. For ( we have m! σ S m x a σ( xa 2 σ(2 xam σ(n = m! x a (s /m σ( σ S m x am (s /m ( (s /m σ(n xσ( x σ(2 x σ(m m (x + x x m ( (x + x x m /m s usng Murhead s nequalty and the geometrc-arthmetc mean nequalty. For (v we frst clam that for all r, q 0 such that (r q 2 r + q cosh(r qθ (cosh θ r+q < for all 0 θ R. (4.2 To see ths, assume, wthout loss of generalty, that 0 q < r. Dfferentatng and usng the addton formula for hyperbolc functons gves [ ] d cosh(r qθ = r snh ( (r q θ q snh ( (r q + θ. (4.3 dθ (cosh θ r+q (cosh θ r+q cosh θ Note that r(r q q(r q + = (r q 2 (r + q 0 from our assumpton. Snce (r q 2 < (r q + 2 and q(r q + 0 t follows nductvely that r(r q 2k+ q(r q + 2k+ < 0 (4.4 for all ntegers k. But the left-hand expresson n (4.4 s just the coeffcent of θ 2k+ /(2k +! n the power seres expanson of the numerator of the quotent n (4.3. Thus the dervatve (4.3 s strctly postve for θ < 0 and strctly negatve for θ > 0, from whch (4.2 follows. If x = λx 2 where λ > 0 and settng λ /2 = e θ, x r x q 2 + x r 2x q (x + x 2 = λ(r q/2 + λ (q r/2 r+q (λ /2 + λ /2 = e(r qθ + e (r qθ r q cosh(r qθ = 2 r+q (e θ + e θ r+q (cosh θ r+q 2 r q by (4.2, wth equalty f and only f x = x 2. Takng r = a and q = a 2 gves (v. Note that case (v of nequalty (4. when m = 2 s related to Murhead means and Schur convexty, whch has a substantal lterature, see [, 4]. Some related nequaltes are obtaned n [, 6] but we were unable to fnd the partcular nequalty that we requred. Note also n relaton to case (v that f (a a 2 2 > a + a 2 then the maxmum of σ S 2 x a σ( xa 2 σ(2/ (x + x 2 a +a 2 does not necessarly occur when x = x 2, n whch case K(2; a, a 2 > 2 a a2. Proof of Corollary 2.2 Each tme the nducton step s appled at a jon pont wth multplcty d the constant gets multpled at (3.6 by K(m; β/α,..., β/α d, 0,..., 0 /β (m! /β (m d! 0

11 (m! /β (m! /β (m d! = (m!(m d! usng Lemma 4.(. The nducton step s appled once at each jon pont, so the estmate for K follows. Notng that (m!/(m r! (m r for each and j r = n gves the stated nequalty. In general the value of K stated n Corollary 2.2 wll not be optmal whch n the case of a bnary tree gves K =. However, provded that the exponents p are reasonably well dstrbuted n the manner stated precsely n Corollary 2.3, we can reduce ths to K = 2 (n. Proof of Corollary 2.3 For a bnary tree, m = 2, all the jon ponts are of multplcty, and the nductve step s appled n tmes. Thus the corollary wll follow f we can show that the multpler ncorporated at (3.6 each tme the nductve step s used satsfes K(2; β/α, β/α 2 /β 2. Note that (2.5 mples that at each stage of the nducton /α, /α 2 each tme 2 (3.4 s used. In partcular (/α /4(/α 2 /4 /6 whch rearranges to Then + 4. (4.5 α α 2 α α 2 ( 2 = + 2 α α 2 α 2 α2 2 α α ( ( + α 2 α2 2 α α 2 α α ( 2 = + ( + α α 2 α α ( 2 = + α α 2 β usng (3.5, gvng Thus Lemma 4. (v gves ( β α β α 2 2 β α + β α 2. as requred. K(2; β/α, β/α 2 /β 2 ( β/α β/α 2 /β = 2 Fnally note that condtons for equalty are not n general easy to specfy. Ths requres equalty at each step of the nducton when Hölder s nequalty and (4. are appled. There wll be equalty n Theorem 2. f µ( s constant for all T 0 and f(j s constant for all j L for each level L provded that we have the optmal value of K. Ths we can do under the condtons of Corollary 2.3. However, as the value of K(m; a,..., a m for Lemma 4. case ( seems to be unknown when m 3 and s >, the value of K gven by Proposton 2.2 wll not n general be optmal.

12 5 Acknowledgement The author s most grateful to the referee for a number of helpful suggestons. References [] Yu-Mng Chu and We-Feng Xa, Necessary and suffcent condtons for the Schur harmonc convexty of the generalzed Murhead mean, Proc. A. Razmadze Math. Inst. 52 ( [2] K.J. Falconer, Generalzed dmensons of measures on almost self-affne sets, Nonlnearty 23 ( [3] K.J. Falconer and Ymn Xao, Generalzed dmensons of mages of measures under Gaussan processes, Adv. Math. 252 ( [4] D.J.H. Garlng, Inequaltes: a Journey nto Lnear Analyss, Cambrdge Unversty Press (2007. [5] G.H. Hardy, J.E. Lttlewood and G. Pólya, Inequaltes, Cambrdge Unversty Press, Pbk. Ed. (988. [6] Mao-Kun Wang, Yu-Mng Chu and Ye-Fang Qu, Some comparson nequaltes for generalzed Murhead and dentrc means, J. Inequal. Appl. 200 (200 Art. ID