This paper is dedicated to the memory of Donna L. Wright.

Transcription

1 Matheatical Reseach Lettes 8, (2 DISTRIBUTIONAL AND L NORM INEQUALITIES FOR POLYNOMIALS OVER CONVEX BODIES IN R n Anthony Cabey and Jaes Wight This pape is dedicated to the eoy of Donna L Wight Intoduction Let P d,n be the vecto space of all polynoials of degee at ost d in R n Let be a convex body of volue in R n and let Since P d,n is finite ( diensional, the nos p ae all euivalent to each othe Recently thee has been consideable inteest in the behaviou of the constants in these euivalences as vaies when we conside abitay unit-volue convex bodies See fo exaple the wok of Budnyi and Ganzbug [BG], Goov and Milan [GM], Bougain [Bou], Bobkov [Bobk] and Nazaov, Sodin and Volbeg [NSV] In this pape, we wish to coplete the analysis of the constants in these euivalences as well as to extend these esults to the vecto-valued setting Fo a (eal o coplex Banach space X with no and a polynoial p : R n X of degee at ost d, we define the functional p # (x = p(x d Fo a convex body in R n of volue, we conside the usual L nos of p # ove ; that is, p # = ( p# (x dx = ( p(x d dx When =, we set p # = exp log p# (x dx and p # is the usual L no of p # Let Hölde s ineuality gives a tivial ineuality fo the L nos with (best possible constant and fo the evese ineuality we have: Theoe Let p : R n X be a polynoial of degee at ost d, let be a convex body in R n of volue and let Then thee exists an absolute constant C independent of p, d,, n,, and X such that p # C [nb(n, + ] [nb(n, + ] whee B denotes the classical Beta function p # Received Novebe 7, 2 Suppoted by a Levehule Fellowship Suppoted in pat by an ARC gant 233

2 234 ANTHONY CARBERY AND JAMES WRIGHT Recall that nb(n, + = u d( u n ; in the liiting cases = and =, the uantity [nb(n, + ] is to be undestood as /n and espectively In paticula we note that the estiate in Theoe is independent of the no fo X By standad estiates fo the Beta function we obtain: Coollay Let p : R n X be a polynoial of degee at ost d, let be a convex body in R n of volue and let Then thee exists an absolute constant C independent of p, d,, n,, and X such that (a if n then p # C p #, (b if n then p # n C ax(, p#, (c if n then p # C ax(, ax(, p# Up to the nueical constant C, the constant on the ight hand side of Theoe is optial if one seeks an ineuality valid fo abitay convex bodies One siply takes p(x =x d and = {(x,x R n : x, x x } The scala-valued case =, (in which case the constant on the ight hand side is essentially n is due to Budnyi and Ganzbug [BG] Fo diensionless bounds, the scala-valued cases =, and = d, 2d ae due to Bobkov [Bobk] (in these cases the diensionless bound on the ight hand side is essentially One can then extapolate these bounds to get shap diension fee hinchine-ahane type ineualities in the exponential class This efined ealie wok of Bougain [Bou] which in tun extended a esult of Goov and Milan [GM] to the geneal degee d case fo the linea case d = Nazaov, Sodin and Volbeg [NSV] have also obtained Bobkov s diensionless bound in the case = and (by diffeent ethods, as well as othe inteesting esults Ou Theoe ay be viewed as a copletion of all these esults, giving the pecise behaviou in all the paaetes, d, n, and The case and geneal has a stonge foulation in tes of distibutional ineualities fo vecto-valued polynoials ove convex bodies in R n (which ay be of independent inteest fo cetain pobles in eal and haonic analysis In fact, we have: Theoe 2 Let p : R n X be a polynoial of degee at ost d, and let be a convex body in R n of volue Let Then thee exists an absolute constant C independent of p, d,, n, and X so that fo any α>, p # α {x : p # (x α} Cn(nB(n, +

3 INEQUALITIES FOR POLYNOMIALS OVER CONVEX BODIES IN R n 235 In paticula, we have: Coollay Let p : R n X be a polynoial of degee at ost d, let be a convex body in R n of volue and let Then thee exists an absolute constant C independent of p, d,, n, and X so that fo any α>, (a if n then (b if n then p # α {x : p # (x α} Cn; p # α {x : p # (x α} C ax(, As befoe, up to the constant C, the ineualities ae shap (to see this we use the sae exaple as fo Theoe The scala-valued case = is due to Budnyi and Ganzbug [BG] Nazaov, Sodin and Volbeg [NSV] have obtained Theoe 2 independently by soewhat diffeent ethods In 6, Reak 2 below, we shall show how one can obtain the case and geneal in Theoe fo Theoe 2 In coon with Bobkov s wok [Bobk] (and that of Nazaov, Sodin and Volbeg [NSV] the ain tool in this cuent wok is the utilisation of a cetain poweful exteal esult of annan, Lovász and Sionovits which we now state Fo a, b R n and λ define the easues µ a,b,λ by φ, µ a,b,λ = φ(a( t+bt(λ t n dt Theoe ([LS] Suppose f,f 2,f 3,f 4 ae continuous nonnegative integable functions on R n and α, β > Suppose that fo evey a, b R n and λ, ( f dµ a,b,λ α ( f 2 dµ a,b,λ β ( f 3 dµ a,b,λ α ( f 4 dµ a,b,λ β Then fo evey convex open set in R n ( α ( β ( f f 2 f 3 α ( f 4 β (Note that the evese iplication is staightfowad Finally, C will denote a geneic absolute constant whose pecise value ay change fo line to line 2 Reduction to weighted ineualities in diension We shall fist pove the esults in the scala-valued setting and then show in 5 how one can extend the aguents to the vecto-valued setting In the scala-valued setting, by the annan, Lovász and Sionovits theoe of the intoduction, Theoes and 2 ae euivalent (afte a liiting aguent because χ {x : p(x α} is not a continuous function to Theoes 3 and 4 espectively:

4 236 ANTHONY CARBERY AND JAMES WRIGHT Theoe 3 Let p : R C be a polynoial of degee at ost d, n N, λ and Then thee exists an absolute constant C independent of the above paaetes such that ( p(t d (λ t n dt (λ t n dt ( C [nb(n, + ] [nb(n, + ] p(t d (λ t n dt (λ t n dt Theoe 4 Let Let p : R C be a polynoial of degee at ost d, n N, λ and Then thee exists an absolute constant C independent of the above paaetes so that fo any α>, ( p(t d (λ t n dt (λ t n dt α d χ { p(t α} (λ t n dt (λ t n dt Cn(nB(n, + Although the fos of the ineualities in Theoes 3 and 4 ake sense only fo < <, it is clea how to extend the when =, = and/o =, = Fo instance, when =, the conclusion of Theoe 4 takes the fo ( exp [log p(t ](λ t n dt α d χ { p(t α} (λ t n dt C d (λ t n dt (λ t n dt To pove Theoes 3 and 4, we ay of couse assue that > and < and then pass to the liit 3 Poof of Theoe 3 We begin with the poof of Theoe 3 We fist need soe peliinay leas The fist is a well-known eleentay Reez type ineuality It is also a siple conseuence of the case n = of Theoe o Theoe 3 and as such is aleady contained in [BG], fo instance We include a siple poof fo the convenience of the eade Lea Thee is an absolute constant C so that if p : R C is a polynoial of degee at ost d, if, and if t u, then ( t t p d C u( t u u p d (We have the usual intepetation in the liiting cases, =,

5 INEQUALITIES FOR POLYNOMIALS OVER CONVEX BODIES IN R n 237 Poof of Lea to show We ay assue that =, = and u = So we want p d L [,t] Ctexp d log p(s ds fo t Clealy we ay also assue that p(z = (z ζ j is onic Now ax p(s d = ax s ζj d = ax st ζj d t ax s ζj /t d s t s t s s Moeove t and ζ j 2 iplies s ζ j /t 2 ζ j 4 s ζ j fo s ; so that we ae left with poving ax s ζ j 2 { s ζ j /t d C exp d ζ j 2 } log s ζ j ds The te on the left of this ineuality is bounded by 3, while the te on the ight is bounded below by exp γ whee γ = inf log s ζ ds The lea is established with C = 2e γ ζ 2 Lea 2 Thee is an absolute constant C so that if < 2, ( 2 ( t/ t + dt C ( ( t/ t + dt We eak that the te on the ight hand side of Lea 2 is itself bounded below by [( +B( +,+ ] Poof of Lea 2 2 ( t/ t + dt 4 ( t/ t + dt 4 e 2t t + dt ( +2e But ( t/ t + dt ( + Taking th oots establishes the lea ( t/ t dt ( + e t t dt =( +! Lea 3 Thee is an absolute constant C so that if p : R C is a polynoial of degee at ost d, if < < and if 2 t x, then [ t ] p C t + x d ( u/ ( u p [( +B( +,+ ] d du

6 238 ANTHONY CARBERY AND JAMES WRIGHT Poof By Lea, we have fo t u p d C t + u +( t u p d Multiplying this ineuality by ( u/ and integating with espect to u fo to t yields [ t u + ( u/ du ]( t p t d C t + ( u/ ( u p d du Lea 2 and the eak following its stateent now iply that ( t p d C t + t ( u/ ( u p d du [( +B( +,+ ] Lea 3 now follows upon taking th oots Poof of Theoe 3 We ay assue that < < Fo ease of notation we wite fo n, and denote [( +B( +,+ ] by A (fo fixed We assue 2 (othewise the poof siplifies, and changing vaiables we see that we have to show, fo each λ and all polynoials p of degee at ost d λ λ ( p(t d ( t/ dt ( C A p(t d ( t/ dt λ A λ ( t/ dt ( t/ dt Case : λ Notice that if t, e 2 ( t/ fo 2 Moeove, ( λ λ p(t d dt C ( λ λ p(t d dt fo <,< by Lea Finally, since A is an inceasing function of this case is coplete Case 2: >λ Let x = λ ; then x as λ Fo x, x ( t/ dt is bounded above and below by absolute constants So we wish to see that fo x and <, ( x ( p d ( t/ dt C A ( x p d ( t/ dt A Now x x p d ( t/ dt = ( t/ ( t x p d dt + ( x/ p d We shall concentate on the fist te, the aguents fo the second being siila but easie We distinguish two subcases of (:

7 INEQUALITIES FOR POLYNOMIALS OVER CONVEX BODIES IN R n 239 Subcase (i: /2 x In this subcase, 2 and x ( t/ ( t p d dt = x 2 ( t/ ( t p d dt + 2 ( t/ ( t p d dt The estiate fo the second te hee is a special case (x = /2 of subcase (ii below, so it suffices to deal with the fist te Using Lea 3 we have x 2 ( t t ( p d dt C [ x A ( t t + dt ][ x ( u u ( 2 C ( + + A A ( + ( x Taking th oots establishes subcase (i Subcase (ii: x /2 p d ( t dt ] p d du (2 x ( t/ ( t p d dt = ( t/ ( t x p d dt + ( t/ ( t p d dt The fist te is easy to deal with since by Lea t ( t/ p d dt p d C ( p d C ( Fo the second te, Lea iplies that fo t u p u d C t + p d u +( t p d ( t/ dt Multiplying this ineuality by ( u/ and integating with espect to u fo to t yields [ t u + ( u/ du ][ t ] p t d C t + ( u p d ( u/ du C t + t p d ( u/ du C t + x p d ( u/ du

8 24 ANTHONY CARBERY AND JAMES WRIGHT povided t x But t Thus fo t x 2, t p d u + ( u/ du e t t 2 C e t t 2 [ x p d ( u/ du ] u + du = e t t ( +2 C [ x p d ( u/ du ] Now ultiplying both sides of this ineuality by ( t/ and integating with espect to t fo to x gives x ( t/ ( t p d dt C [ x ( t/ dt ][ x C [ x p d ( u/ du ] p d ( u/ du ] (as ( t/ and x 2 Taking th oots finishes subcase (ii of (2, and hence (, poving Theoe 3 4 Poof of Theoe 4 The fist step in poving Theoe 4 is the special case n =, = : Lea 4 Thee is an absolute constant C so that fo all polynoials p : R C of degee at ost d and all intevals I, p d L (I α d {x I : p(x α} C I This lea is an old esult and in fact the best constant C is known to be 4 This is due to Dudley and Randol, [DR] Howeve this esult fo soe absolute constant C is an easy conseuence of a classical ineuality of H Catan [C] which we now state: Catan s lea Let w,w 2,,w d be d points in the coplex plane C and let h> Then the set of points z C such that the ineuality d z w j h d j= holds can be coveed by at ost d cicles, the su of whose adii is 2eh Note, in paticula, Catan s lea iplies the coesponding stateent of Lea 4 fo onic (as opposed to L -noalised polynoials We povide a poof of Lea 4 fo copleteness Poof of Lea 4 We ay assue that I =[, ] by tanslating and dilating the polynoial p Obseve that the stateent of the lea is invaiant unde ultiplication of p by any nonzeo constant, and (up to changing the value of C unde ultiplication of p by a function, whose d th oot is bounded above and

9 INEQUALITIES FOR POLYNOMIALS OVER CONVEX BODIES IN R n 24 below by absolute constants So if p(z =A (z ζ j, we ay ultiply p by ζ j (z ζ j and then by (A ζ j without changing attes Thus ζ j 2 ζ j 2 we ay assue that p(z = (z ζ j This odified p(z is now onic, ζ j 2 has degee k d say, and when esticted to the unit inteval [, ] satisfies p d 3 We ay theefoe assue α and Catan s ineuality tells us that {x [, ] : p(x α} Cα k Cα d, copleting the poof of the lea Note the case = of Theoe 4 and thus Theoe 2 is now an iediate conseuence: we eely have to obseve that fo t and λ, we have (λ t n n (λ sn ds Poof of Theoe 4 Again we ay assue that << Fo ease of notation we again wite fo n and assue 2 (the cases = and = ae easie Let I = p(t d (λ t dt (λ t dt and II = α d χ { p α} (λ t dt (λ t dt We wish to show that I II C(+[(+B(+,+] We iediately ake the change of vaiables t λ t in all integals, so that I = λ p(t d ( t dt λ ( t dt and II = α d λ χ { p α} ( t dt λ ( t dt (fo a possibly diffeent polynoial p Note that if D := λ ( t dt, then fo λ, we have D λ ( / dt λ 2e while fo λ, D ( t/ dt ( / dt 2e Case : λ In this case we have I 2eλ λ p(t d ( t/ dt 2eλ λ p(t d dt,

10 242 ANTHONY CARBERY AND JAMES WRIGHT while II 2eλ α d so that ( I II (2e + λ λ λ χ { p α} ( t/ dt 2eλ α d p(t d dt α d λ χ { p α} dt λ λ χ { p α} dt C(2e + by Lea 4 Thus I II is bounded above by an absolute constant in this case Case 2: >λ In this case, since D is unifoly bounded below and the nueatos of I and II ae deceasing with λ, we ay take λ = and educe attes to showing that Ĩ ĨI C( + [( +B( +,+ ] whee Ĩ = p(t d ( t/ dt and ĨI = α d χ { p α} ( t/ dt Now Ĩ = ( t/ d { t p(s d ds }dt = ( t/ { t p(s d ds }dt dt which in tun is less than Hdt + C H ( t/ t + dt whee H = p(s d ds, by Lea Hence Ĩ H [ +C ] ( t/ t + dt H = H [+C +2 [ ] +C + ( +B( +,+ ] ( s s + ds Theefoe Ĩ CH ( + [( +B( +,+ ] On the othe hand, t ĨI = α d χ { p α} ( t/ dt = α d ( t/ χ { p(s α} ds dt α d [ χ { p(s α} dsdt + t ] ( t/ χ { p(s α} dsdt C + C ( t/ tdt C e t/2 tdt C by Lea 4, whee = p d L [,] Thus [ ] p Ĩ ĨI C d L [,] ( + [( +B( +,+ ] p d L [,]

11 INEQUALITIES FOR POLYNOMIALS OVER CONVEX BODIES IN R n 243 which in tun is less than C( + [( +B( +, + ] as euied, copleting the poof of Theoe 4 (Note that we have used in passing that n[nb(n, + ] is bounded below unifoly in n and 5 The vecto-valued case To extend Theoes and 2 to the vecto-valued setting, we fist obseve that ou aguents extend to a wide class of functions than polynoials of degee at ost d Following a peliinay vesion of [NSV], we say that a function u : R n R is of class L if it is the estiction to R n of a pluisubhaonic function ũ(z log z ũ : C n R such that li sup When n =,u(x = d log p(x is z of class L if p : R C is a polynoial of degee d We can wite such a p as p(x =A d j= (x ζ j, so that d log p(x = d log A + d d j= log x ζ j, and the distinguishing featue of a function of class L (when n = is that it can be witten as u(x = constant + log x ζ dµ(ζ whee µ is a positive easue of ass at ost one in the plane This is the well-known Riesz epesentation fo subhaonic functions, see fo exaple Hayan s book [H] In paticula, it is not difficult to see that the key leas, Lea and Lea 4, eain valid if one eplaces p(x d with exp u(x, whee u is a geneal function of class L in one diension With these eaks in ind the eade will have no touble extending Theoes and 2 to functions of class L to obtain the following: Theoe 5 Let u : R n R be a function of class L, and be a convex body in R n of volue Then thee exists an absolute constant C independent of,,, n and u so that e u L ( C [nb(n, + ] e u [nb(n, + ] L ( Theoe 6 Let u : R n R be a function of class L, and be a convex body in R n of volue Then thee exists an absolute constant C independent of,, n and u so that e u L ( e u L, ( Cn[nB(n, + ] To obtain the vecto-valued extension of Theoes and 2, we siply obseve that wheneve p : R n X is a polynoial of degee at ost d with values in a Banach space X, u(x = d log p(x is a function of class L Indeed, ũ(z the estiate li sup log z is staightfowad, and using the fact w = z sup l X, l l(w fo any w X, one easily sees that ũ(z is pluisubhaonic

12 244 ANTHONY CARBERY AND JAMES WRIGHT 6 Futhe eaks If we let in ineuality (, we have, since ( t/ e t fo > and <t, ( x p(t d e t dt ax(, ( x C p(t d e t dt ax(, by the doinated convegence theoe, whee C is absolute: cobining this with Leas and 4 yields the following esults Poposition Thee exists an absolute constant C such that if p is a polynoial of degee at ost d, N>, and < <, ( N p(t d e t dt ( N p(t d e t dt ax(, C N ax(, N e t dt e t dt Poposition 2 Thee exists an absolute constant C such that if p is a polynoial of degee at ost d, N>, and <<, ( N N p(t d e t dt α d χ { p(t α} e t dt Cax(, N N e t dt e t dt Popositions and 2 ae also tue if one eplaces p(t d with exp u, whee u is any function of class L Using anothe theoe of annan, Lovász and Sionovits [LS] (which is siila to thei theoe stated in the intoduction except that the easues µ ae eplaced by easues with exponential densities we then obtain Theoe 7 Let X be a Banach space and let p : R n X be a polynoial of degee at ost d Suppose < < and µ is a log-concave pobability easue on R n Then thee is an absolute constant C such that ( p(x ax(, ( d dµ(x C p(x d dµ(x, ax(, and fo the sublevel set estiate: Theoe 8 Thee exists an absolute constant C such that if p : R n X is a polynoial of degee at ost d, <<, and µ is a log-concave pobability easue on R n, then ( p(x d dµ(x α d µ{x R n : p(x α} C

13 INEQUALITIES FOR POLYNOMIALS OVER CONVEX BODIES IN R n 245 A easue is said to be log-concave if it is suppoted by an affine subspace L of R n, and with espect to Lebesgue easue on L has a density of the fo e g(x whee the set = {x : g(x < } and g ae convex In addition to chaacteistic functions of convex bodies, these easues include gaussians e x 2 dx Of couse we can let o in Theoes 7 and 8 to obtain estiates in the exp log class L 2 To see why Theoe 2 iplies the case and geneal of Theoe, we fist obseve that Theoe 2 has a tivial evese ineuality Consideing the sublevel set fo p(x with α d =2 p d, we have /4 p d L ( sup α d {x : p(x α} unifoly fo > Hence, by Theoe 2 α> /4 p d L ( sup α d {x : p(x α} Cn(nB(n, + α> In paticula, using these ineualities with, we see that the nos p d L ( fo and [sup α> α d {x : p(x α} ] = p /d L, ( ae unifoly euivalent Theefoe fo any, we have ( p(x d dx Cn(nB(n, + ( p(x d dx 3 It is easy to see that the conclusion of Theoe 2 has the following euivalent foulation fo geneal finite-volue convex bodies : ( p d Cn(nB(n, + E p d L (E unifoly ove all closed subsets E of (with the sae constant C Soewhat supisingly, one can eplace the L no on the ight side with the salle L no, ( E E p d, incuing only an exta facto of 2 in the estiate This was obseved in [BG] fo the case = and follows by consideing the non-deceasing eaangeent of p ove E, p (τ (ie, p is the invese of the easue of the sublevel sets of p esticted to E The estiate in Theoe 2 iplies a lowe bound fo p, naely ( p τ d Cn(nB(n, + [p (τ] d fo τ E Raising this to the th powe, integating in τ and then taking the th oot gives the desied bound 4 The convexity of the set is cucial in obtaining the fo of the constant in Theoe If instead one asks fo the fo of the constant B in the ineuality ( p ( d B p d F F

14 246 ANTHONY CARBERY AND JAMES WRIGHT whee F is now an abitay (unit-volue copact set in R n and, one ay see that not only ust B contain a facto of cvxf (whee cvxf denotes the convex hull of F but also a facto n To see this, conside the exaple p(x =x d as befoe and F = {(x,x R n : x (, /n ( ɛ,, x x } fo suitable ɛ uch salle than /n The poof of the esulting ineuality ( F p d Cn cvxf ( F p d is due to Budnyi and Ganzbug [BG] (at least in the case = To see this, we fist obseve p(x d p(x d p d L (F p(x d p d L (cvxf F F F p(x d { p F (nb(n, + d L (cvxf } whee the last ineuality follows fo the case = and geneal of Theoe Next, using the following euivalent foulation of Theoe 2 (which we deived in Reak 3 above ( (3 p d Cn(nB(n, + inf E ( E E E p d when =, E = F and = cvxf, we obtain the esult Inteestingly, (3 can be thought of as a way to foulate the analogue of Lea in the highediensional context, and it is natual to enuie as to whethe the constant n(nb(n, + can be ipoved upon if we estict E to ange ove convex subsets of This howeve is not the case To see this, take = {(x,x R n :<x <n, x < n x }, E = {(x,x R n :<x <, x < n x }, and p(x =x d (Of couse, X = R hee 5 If p : R n C is a polynoial of degee at ost d, it is well known that ω = p is an A weight when >d+ with A bounds independent of the coefficients of p ; see [RS] Theoe 2, when = d, can be viewed as a shap endpoint esult of this natue Recall that a weight ω is in A if B B [ ω(xdx B B ω(x / dx ] / A< fo all balls B in R n The sallest constant A fo which the above holds is called the A bound, A (ω, fo ω Using Theoe 2 with = d, we see that thee is an absolute constant C such that if p : R n X is a polynoial of degee at ost d with values in a Banach space X and >d+, [ A ( p (Cd d ] (d +

15 INEQUALITIES FOR POLYNOMIALS OVER CONVEX BODIES IN R n 247 We eak that this estiate eains valid when we allow the A bound to also vay ove all convex bodies in R n, not just Euclidean balls B See also [NSV] 6 The theoe of annan, Lovász and Sionovits which we used elies heavily on the non-negativity of the functions involved Howeve thee ae phenoena, closely elated to sublevel set pobles fo polynoials, which ae highly oscillatoy in natue; ost notably estiates fo oscillatoy integals Fo exaple, it follows fo Theoe 72 of [CCW] that if Q =[, ] n, p : Q R is a polynoial of degee at ost d so that Q p = and p L (Q =, then fo λ lage and eal, (4 e iλp(x dx C d,n Q λ d Can we expect ipoveent to this along the lines enjoyed by sublevel sets? In paticula if Q p = and p L (Q =, can we take C d,n in (4 to be C in(d, n? On aveage the answe is yes, because a diect conseuence Theoe 2 is that fo p L ( =, a convex body of volue, φ S(R with ˆφ χ [,] µ { } e iλp(x dx φ(λ/µdλ C in(d, n µ d with C absolute To see this, note that the left side is eual to ˆφ(µp(xdx which is in tun euivalent to {x : p(x µ } (This well-known aguent also deonstates the fact that oscillatoy integal estiates iply sublevel set estiates 7 Acknowledgeents We would like to thank A Giannopoulos and A Volbeg fo binging these pobles to ou attention In fact, the pesent authos inteest in these pobles pincipally aose upon eceiving in July 2 a pepint Diensionless A p bounds and the distibution of polynoials of any eal vaiables by M Sodin and A Volbeg, which late evolved into [NSV] In this pepint the poble of tansfeence of A p estiates though diensions was consideed, and a vesion of ou cuent Theoe 2 in the case =, but with exta logaithic tes in α was given We ecod ou debt of thanks to the authos of [NSV] fo geneously shaing thei wok with us and also fo dawing to ou attention seveal efeences Finally we thank the efeee fo vaious helpful coents Refeences [Bobk] S G Bobkov, Reaks on the gowth of L p - nos of polynoials, Lectue Notes in Math 745 (2, Spinge, [Bou] J Bougain, On the distibution of polynoials on high diensional convex sets, Lectue Notes in Math 469 (99, Spinge, 27 37

16 248 ANTHONY CARBERY AND JAMES WRIGHT [BG] Yu A Budnyi and M I Ganzbug, A cetain exteal poble fo polynoials in n vaiables, (Russian Izv Akad Nauk SSSR Se Mat 37 (973, , English tanslation: Math USSR-Izv 7 (973, [CCW] A Cabey, M Chist and J Wight, Multidiensional van de Coput and sublevel estiates, J Ae Math Soc 2 (999, 98 5 [C] H Catan, Su les systèes de fonctions holoophes à vaiétés linéaies lacunaies et leus applications, Ann Sci École No Sup 45 (928, [DR] R M Dudley and B Randol, Iplications of pointwise bounds on polynoials, Duke Math J 29 (962, [GM] M Goov and V D Milan, Bunn theoe and a concentation of volue fo convex bodies, Isael seina on geoetical aspects of functional analysis (983/84, Tel Aviv Univesity, Tel Aviv, 984 [H] W H Hayan, Subhaonic Functions, vol 2, London Matheatical Society Monogaphs 2, Acadeic Pess, London, 989 [LS] R annan, L Lovász and M Sionovits, Isopeietic pobles fo convex bodies and a localization lea, Discete Coput Geo 3 (995, [NSV] F Nazaov, M Sodin, and A Volbeg, The geoetic LS lea, diension-fee estiates fo the distibution of values of polynoials and distibution of zeoes of ando analytic functions, pepint [RS] F Ricci and E M Stein, Haonic analysis on nilpotent goups and singula integals I: Oscillatoy integals, J Funct Anal 73 (987, Depatent of Matheatics and Statistics, Univesity of Edinbugh, JCMB, ing s Buildings, Mayfield Road, Edinbugh EH9 3JZ, Scotland E-ail addess: cabey@athsedacuk E-ail addess: wight@athsedacuk