The <span class="mathjax-formula">$p$</span>-adic measure on the orbit of an element of <span class="mathjax-formula">${\mathbb {C}}_p$</span>

The p-adic Measure on the Orbit of an Element of C

.

V. ALEXANDRU(*) - N. POPESCU(**) - M. VAÃJAÃITU(**) - A. ZAHARESCU(***)

ABSTRACT- Given a prime numberpand the Galois orbitO(x) of an elementxofCp, the topological completion of the algebraic closure of the field ofp-adic numbers, we study functionals on the algebraC(O(x);Cp) with values in a subfield ofCp.

Introduction.

Let pbe a prime number,Q_pthe field of p-adic numbers,Q_p a fixed algebraic closure ofQp, andCpthe completion ofQpwith respect to thep- adic valuation. LetO(x) denote the orbit of an elementx2Cp, with respect to the Galois groupGˆGalcont(Cp=Qp). We are interested in the behavior of rigid analytic functions defined on E(x)ˆ(C_p[ f1g)nO(x), the com- plement ofO(x). In the present paper we provide several results concerned with functionals defined on the algebraC(O(x);C_p) with values in a suitable subfield ofCp. This investigation is needed in the more general attack on the problem of explicit description of rigid analytic functions onE(x), since, as we shall see below, there is a close relationship between these func- tionals and certain classes of rigid analytic functions. The paper consists of seven sections. The first one contains notations and some basic results. The

(*) Indirizzo dell'A.: Department of Mathematics, University of Bucharest, Romania.

E-mail: vralex@k.ro

Partially supported by Grant no. 27.694/2005, cod CNCSIS 849.

(**) Indirizzo degli A.: Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-014700, Bucharest, Romania.

E-mail: Nicolae.Popescu@imar.ro Marian.Vajaitu@imar.ro

Partially supported by CEEX Program of the Romanian Ministry of Education and Research, Contract No. CEx 05-D11-11/2005.

(***) Indirizzo dell'A.: Department of Mathematics, University of Illinois at Urbana-Champaign, Altgeld Hall, 1409 W. Green Street, Urbana, IL, 61801, USA.

E-mail: zaharesc@math.uiuc.edu

2000 Mathematics Subject Classification: 11S99.

second section is concerned with linear functions and functionals on var- ious fields. Section 4 studies p-adic measures and p-adic equivariant measures on O(x). Theorem 1 shows that the study of all functionals on C(O(x);C_p) can be reduced to the study of all functionals onC_G(O(x);C_p).

Then, using results from the previous sections, the functionals on C(O(x);Cp) are closely related to the trace. Theorems 2 and 3 are useful complements to Theorem 1. In order to further clarify the relation between functionals and rigid analytic functions, in Section 5 we investigate the Cauchy transform of a function with respect to a measure. In the last section we present an analogue at an important theorem of Barsky [B], which relates the measures onO(x) with a suitable class of rigid analytic functions on the complement ofO(x). We remark that some of the results of this paper can be extended to a wider class of compact subsets ofCp.

1. Notations and basic results.

1. Letpbe a prime number,Qpthe field ofp-adic numbers,Qpa fixed algebraic closure of Q_p, andC_p the completion of Q_p (see [Ar], [APZ1], [APZ2]). Denote by G the Galois group Gal(Q_p=Q_p) endowed with the Krull topology. One knows that G is canonically isomorphic to GˆGalcont(Cp=Qp), the group of all continuous automorphisms ofCp. We shall identify these two groups.

For any closed subgroupHofGdenoteFix(H)ˆ fx2C_p:s(x)ˆxfor all s2Hg. Then Fix(H) is a closed subfield of C_p. If x2C_p, denote H(x)ˆ fs2G:s(x)ˆxg. ThenH(x) is a subgroup ofG, andFix(H(x))ˆ

ˆQgp[x] is the closure of the polynomial ringQp[x] inCp. We say thatxis a topological generic elementofQgp[x]. Moreover, by [APZ1] one knows that any closed subfieldK ofCp has a topological generic element, i.e. there existsx2Ksuch thatK ˆQg_p[x].

2. Let x2C_p. Denote O(x)ˆ fs(x):s2Gg the orbit of x. The map s?s(x) from Gto O(x) is continuous, and it defines a homeomorphism fromG=H(x) (endowed with the quotient topology) toO(x) (endowed with the induced topology fromCp) (see [APZ1]). In such a way,O(x) is a closed compact and totally disconnected subspace of C_p, and the group G acts continuously onO(x): ifs2G,t(x)2O(x) thens?t(x)ˆ(st)(x). One has the following result:

PROPOSITION1. 1)The subfieldQgp[x]is canonically isomorphic to the set of all equivariant continuous functionsf :O(x)!C_p, i.e. the contin-

uous functions which verify the condition:f(s? y)ˆs(f(y))for alls2G andy2O(x).

2)There exists a familyfMn(x)g_n0of polynomials inQp[x]such that i) degMn(x)ˆn for all n0,

ii) 1

p<jMn(x)j 1,

iii) Any element f 2Qgp[x] can be written uniquely in the form:

f ˆ P

n0anMn(x)where fang_n is a sequence of elements in Qp such that limn a_nˆ0. Moreover one has:jfj ˆsup

n0ja_nM_n(x)j.

3)If KxˆQgp[x]\Qp, thenKfxˆQgp[x]and Gal(Qp=Kx)is canonically isomorphic to H(x).

2. Linear functions and functionals.

1. Let K=k be a (not necessarily finite) algebraic extension, and fK_ng_n0,K₀ˆk, K_nK_n‡1 for anyn, a family of subfields of K, finite overksuch that[_nK_nˆK. Denote byL(K=k) the set of allk-linear maps ofKintok. ThenL(K=k) is in a canonical way aK-vector space. Namely if W2 L(K=k) andx2KthenxWis the linear map defined by (xW)(a)ˆW(xa) for alla2K.

Now assume thatkis of zero characteristic and for alla2Kdenote by Tr_K=k(a) or simply by Tr(a) the element of k defined as follows: if kK⁰K is a finite intermediate extension and a2K⁰, then Tr(a)ˆ

ˆ 1

[K⁰:k]tr_K⁰_=k(a), wheretr_K⁰_=k(a) denotes the usual relative trace ofaover k. It is clear thatTr(a) is independent of the choice ofK⁰.

If K=k is a finite extension, then for any W2 L(K=k) there exists a uniquea2Ksuch thatWˆaTr. Now assume [K:k]ˆ 1and letW_nbe the restriction of W to Kn. Then W_n2 L(Kn=k) and so there exists a unique elementan 2Knsuch that for alla2Kn one has:

W_n(a)ˆTr(a_na):

…1†

Thus for any a2Kn, one has: W_n‡1(a)ˆW_n(a), and so by (1) one has:

Tr(an‡1a)ˆTr(ana). This means that 1

[K_n‡1 :K_n]tr_Kn‡1=Kn(an‡1)ˆan; n0:

…2†

Conversely, letAˆ fang_nbe a sequence of elements ofKsuch thatan 2Kn

for alln0, and that condition (2) is accomplished. Then for any n0 denote byW^A_n :Kn!kthe mapW^A_n(a)ˆTr(ana),a2Kn. By (2) there re- sults that one can define a linear mapW^A:K!ksuch that the restriction of W^AtoKnis justW^A_n. DenotePˆlim

Kn; 1

[Kn‡1:Kn]trKn‡1=Kn

n0. Then by the above considerations there results that the map A?W^Adefines a k- isomorphism between thek-vector spacePandL(K=k).

2. Now assume that K is endowed with an ultrametric absolute value j j:K!R. We consider the functionals W:K!k, i.e. allk-linear maps which are continuous.

A linear mapW:K !kis continuous if and only if there exists a positive real numberMsuch thatjW(a)j Mjaj, for anya2K. SinceWis defined uniquely by a sequence Aˆ fa_ng_n, a_n2K for all n0, which verify condition (2), thenWis continuous if and only if there exists a positive real numberMsuch that

jTr(a_ny)j Mjyj;

for alln, and ally2Kn. Now one can writeyˆ x

a_n, wherex2Kn, and so the above condition can be written as:

jTr(x)j janj Mjxj;

for alln0 and anyx2K_n.

LetLˆK_n, be fixed. We want to relate sup

x2Kn

tr(x)

[L:k]jxjand D_L

[L:k], where D_Lis the different ofLwith respect tok(see [Ar]).

REMARK 1. Any x2K_n can be represented as xˆp^lxa, where 1 jaj>1

pandlxis integer. Then one has: max

x2Kn

jtr(x)j

jxj ˆ max

x2Kn;_p¹<jxj1

jtr(x)j jxj , and the quotient between this number and MKn ˆmaxn

jtr(x)j:x2 2K_n;1

p<jxj 1o

is a real number whose module belongs to the real intervalh

1;1 p

.

If a2K_n then jaj jD_Knj ¹ if and only if jtr(aO_Kn)j 1. (Here jtr(aO_Kn)j ˆmaxfjtr(ax)j:x2O_Kn; the ring of integers of K_ng.) Let us denote by mKn the integer part of log_1=pjDKnj. If aˆp ^mKn, then jtr(p ^mKnOKn)j 1, or equivalently jtr(OKn)j (1=p)^mKn. This means that M_Kn(1=p)^mKn.

Letaˆp ^(mKn‡1). Then there existsy2OKn such thatjtrp ^(mKn‡1)yj>1 or equivalently log_1=pjtryj<mKn‡1. Therefore one obtains

log_1=p(MKn)<mKn‡1 or1 p

_mKn‡1

<MKn:

Thus we have m_Knlog_1=p(M_Kn)<m_Kn‡1. In conclusion, one has the following result:

PROPOSITION2. The linear mappingW:K!k,WˆW^A, is continuous if and only if there exists a positive real numberM such that:

jDKnjjanj

[K_n:k] M; for all n1:

…3†

REMARK 2. Letx2Cp such thatTr(x) is defined (see [APZ2]). One has the following result. LetKˆQg_p[x]\Q_p, andKˆ [_nK_nbe the union of a filtered family of finite extensions ofQ_p. The following assertions are equivalent:

1) The sequencefjTr(Mn(x))jg_nis bounded (see Proposition 1).

2) The sequencen jD_Kn=Qpj j[K_n :Q_p]j

o

n is bounded.

3) The linear mapTr:K!Qpis continuous.

These results are related to some results from [APP].

3. Denote byAthe set of all elementsA2Pwhich verify condition (3) for a suitableM>0. It is easy to see thatAis ak-vector subspace ofP.

Denote by K⁰ thek-vector space of all k-functionals onK. By the above considerations one has:

PROPOSITION3. The mapping A?W^A defines an isomorphism ofk- vector spaces betweenAandK⁰.

Now letAˆ fang_n be an element ofAsuch thatan6ˆ0 for alln0.

Letc2K⁰,cˆW^B,Bˆ fb_ng_n2 A. Denoteu_nˆb_n

a_n,n0. We assert that the sequencefu_ng_nis bounded, i.e. there exists a real numberM>0 such thatjunj Mfor alln0. Indeed, assume that there exists a subsequence fuqngsuch thatjuqnj ! 1. Thenj1=uqnj !0 and so lim

n c(1=uqn)ˆ0. But c(1=u_qn)ˆTr(a_qn), and thusW^A(1)ˆ0, a contradiction.

If for anyn0, one denotesW_nˆunW^A, then one has:c(x)ˆlim

n W_n(x) for any x2K. Indeed, for n large enough one has: W_n(x)ˆ(u_nW^A)(x)ˆ

ˆW^A(u_nx)ˆTr(a_nu_nx)ˆTr(b_nx)ˆc(x). This shows that the sequence fW_ng_n converges pointwise to c in K⁰ and so by the Banach-Steinhaus Theorem (see [R]) one hascˆlim

n W_n ˆlim

n (unW^A).

PROPOSITION4. Notations and hypotheses are as above. LetWˆW^A. Denote byS_Athe family of all sequencesuˆ fu_ng_nsuch that:

1)un2Kn for all n1.

2)The sequence of real numbersju_njis bounded.

3)One has: 1

[K_n‡1:K_n]tr_Kn‡1=Kn(un‡1)ˆun, n0.

For any uˆ fu_ng_n2S_A, denote by u_W the mapping defined by:

(u_W)(x)ˆlim

n W(u_nx). Then u_W2K⁰and ifW6ˆ0, then for anyc2K⁰there exists a unique u2S_Asuch that u_Wˆc.

We leave the details to the reader.

3. Integration onC(X;K).

1. Let K be a field which is complete with respect to an ultrametric absolute valuej j, and letX be a compact ultrametric space. Denote by C(X;K) theK-algebra of all continuous functions fromXtoK(see [Sch]).

Also, denote byV(X) the collection of all the open compact subspaces ofX.

By aK-valued measureonXwe mean a functionm:V(X)!Ksuch that:

(i) If U;V 2V(X); U\V ˆ ;, then m(U[V)ˆm(U)‡m(V) (ad- ditivity).

(ii) kmk ˆsupfjm(U)j:U2V(X)g<1(boundedness).

The K-valued measures on K form a normed vector space M(X;K) under the obvious operations and with the normk kdefined by (ii).

The following statement (see [Sch]) can be viewed as the ultrametric analog of Riesz representation theorem:

PROPOSITION5. For eachW2C(X; K)⁰(the space of all functionals on C(X; K)), the mappingU?W(j_U)ˆm_W(U),U 2V(X), is a measurem_Won X(herej_U denotes the characteristic function ofU). The mappingW?m_W is aK-linear isometry ofC(X; K)⁰ontoM(X; K).

4. Equivariant measures onO(x).

1. Let x2Cp. The subset of C(O(x);Cp) consisting of all equivariant elements (see Proposition 1) is denoted by C_G(O(x);C_p). The mapping f?f(x) defines an isomorphism betweenC_G(O(x);C_p) andQg_p[x]. We shall identify theseQp-algebras via this isomorphism.

Letf 2C(O(x);Cp) ands2G. Denote bys? fthe function defined by:

(s?f)(y)ˆf(s ¹(y)) for all y2O(x). In this way the group G acts con- tinuously on the algebraC(O(x);C_p).

2. LetKbe a closed and normal subfield ofCp(i.e. for anys2G, one hass(K)ˆK). By anequivariant K-functionalonC(O(x);C_p) we mean a linear and continuous map W:C(O(x);C_p)!K such that W(s?f)ˆ

ˆs(W(f)) for alls2Gand allf 2C(O(x);Cp). It is easy to see that by the correspondence stated in Proposition 5, the equivariant K-functionals on C(O(x);Cp) are in one to one correspondence with the so calledequivariant measures onO(x), i.e. the elementsm2M(O(x);C_p), such that m(sU)ˆ

ˆs(m(U)) for allU2V(O(x)) ands2G. In this paper we consider mainly the caseKˆQ_pand shall investigate the above correspondence between functionals and measures.

THEOREM 1. Any Qp-functional W:C_G(O(x);Cp)!Qp can be un- iquely extended to a C_p-functional eW:C(O(x);C_p)!C_p and this func- tional provides us with a measure m_W on O(x) with values in Q_p. The measurem_Wis equivariant.

Next, we recall an important criterion for the existence of a measure with given properties.

PROPOSITION6 (The abstract Kummer congruences, [Ka]). LetXbe a compact ultrametric space, letOpbe the ring of integers ofCpand letffig be a system of continuous functions fromC(X; Op). If theCp-linear span of ff_igis dense inC(X;C_p)andfl_igis an arbitrary system of elements ofO_p, then the following assertions are equivalent:

a) There ism2 M(X;O_p)with the propertyR

X f_idmˆl_i.

b) For an arbitrary choice of elements g_i2Cp almost all of which vanish,

X

i

g_if_i(x)2pⁿO_p for all x2 X implies X

i

g_il_i2pⁿO_p:

Now, let fang_n0 be a bounded sequence of Qp. Let us define W:C_G(O(x);Cp)!Qp such thatW(Mn(x))ˆan. As we know from Propo- sition 1 any f 2C_G(O(x);Cp) can be written as f ˆ P

n0anMn(x), with an!0. We define W(f):ˆ P

n0anan. Using Theorem 1 we have a similar result of abstract Kummer congruences. More precisely we have:

PROPOSITION7. Letfa_ng_n0be a bounded sequence ofQ_p. There exists a unique functional W:C(O(x);C_p)!C_p such that W(M_n(x))ˆa_n, for anyn0.

THEOREM2. Ifmis an equivariant measure on O(x)with values inQ_p then the mapping f? R

O(x)f(t)dm(t)is a functional on C_G(O(x);C_p) with values inQp.

PROOF OFTHEOREM1. We use the notations from Proposition 1. For any s0 let us denote A_sˆW(M_s(x)). Then for anyuˆP

s a_sM_s(x), one has W(u)ˆP

s asAs. Since the equality is true for any u2Qgp[x], there results that the sequencefA_sg_s ofp-adic numbers is bounded and so the set of all Qp-functionals on Qgp[x]ˆC_G(O(x);Cp) is in one to one corre- spondence with the set of bounded sequencesfA_sg_sofp-adic numbers.

Now let U be an open ball on O(x), x2U, and denote H(U)ˆ fs2G:sUˆUg. ThenH(U) is a subgroup ofG, and denote by K_U the subfield of Q_p fixed by H(U). Since H(x)H(U), then K_U K_xˆQg_p[x]\Q_p. It is clear that [G:H(U)]<1 and let

S_nˆ fs₁ˆe;s₂;. . .;s_ngbe a system of representatives for the right co-

sets ofGwith respect toH(U). Also choosea2Qpsuch thatKuˆQp(a). It is clear that the elements fs1(a);. . .;sn(a)g are distinct, and the balls fs_i(U)g_1in are pair-wise disjoint and coverO(x). Let us put

f₀^(U) ˆ P

si2Sn

s_i(1)j_si(U)ˆj_O(T) f₁^(U) ˆ P

si2Sn

s_i(a)j_si(U) . . . . f_n^(U)₁ˆ P

si2Sn

si(a^{n 1})j_si(U): 8>

>>

>>

>>

>>

<

>>

>>

>>

>>

>: …4†

We assert that the function f_i^(U) is equivariant for all i, 0i<n, i.e.

f_i^(U)(s(y))ˆs(f_i^(U)(y)) for all y2O(x) and any s2G. For that it is enough to assume iˆ1. Firstly, we remark that one has: f_i^(U)(y)ˆ

ˆs_i(y) fory2s_i(U); 1in:Now we can write:f_s;1^(U)ˆsf₁^(U)(s ¹(y))ˆ

ˆss_i(a) ifs ¹(y)2s_iU(or equivalentlyx2ss_i(U)).

It is clear that the following permutations

s₁U s₂U . . . s_nU

ss₁U ss₂U . . . ss_nU

; s₁(a) s₂(a) . . . s_n(a)

ss₁(a) ss₂(a) . . . ss_n(a)

…5†

coincide, so the applications

s₁U s₂U . . . s_nU

s₁(a) s₂(a) . . . s_n(a)

; ss₁U ss₂U . . . ss_nU

ss₁(a) ss₂(a) . . . ss_n(a)

also coincide. This shows thatf₁^(U)ˆf_s;1^(U), i.e.f₁is equivariant.

Furthermore (4) is a Cramer system whose determinant is different from zero. It follows that

X^{n 1}

iˆ0

Qp(a)f_i^(U)ˆXⁿ

iˆ1

Qp(a)j_siU; …6†

and this sum is direct. This shows that the functionalW can be extended uniquely to a functional

e

W:C_G(O(x);C_p)_QpQ_p!Q_p:

According to ([Sch], page 273), the C_p-algebra C(O(x);C_p) has an orthonormal basis consisting of characteristic functions of balls. Then the Qp-subalgebraCG(O(T);Cp)QpQpis dense in theCp-algebraC(O(T);Cp).

But theneWcan be extended uniquely to aCp-functional:

e

W:C(O(T);C_p)!C_p:

Then by Riesz's Theorem (Proposition 5) there exists a unique measurem_Won O(x) associated toeW. It is clear that for any ballUinO(x),m_W(U)ˆeW(j_U), can be obtained from (4) by applying the functionaleW. We must show that the measurem_Wis equivariant. For that let as aboveU be an open ball ofO(x) which contains x. Denote Aˆ(a_ij) the nn matrix, where a_ijˆs_i(a^j), 1in; 0j<n 1. Then by (4) there results:

j_s1(U) j_sn..._(U) 0 B@

1 CAˆA ¹

f₀^(U) ...

f_n^(U)₁ 0 BB

@ 1 CC A

and so by applyingeWone has:

m_W(s1(U)) m_W(sn...(U)) 0

B@

1 CAˆA ¹

W(f₀^(U)) ...

W(f_n^(U)₁) 0 BB

@

1 CC A:

Now fors2Gdenote byA_sthe matrix obtained fromAby applyingsto all the entries ofA. Then (A_s) ¹ˆA_s¹.

Sincef_i^(U)are equivariant functions, one has:f_i^(s(U))ˆf_i^(U)for alls2G.

Then one obtainsm_W(sU)ˆsm_W(U), for alls2G, as claimed. p PROOF OFTHEOREM2. Letmbe an equivariant measure onO(x) with values inQ_p. Denote byW:C(O(x);C_p)!C_pthe functional associated tom.

We must show that for anyf 2 C_G(O(x);C_p) one hasW(f)2Q_p. Since any elementf 2 C_G(O(x);Cp)ˆQgp[x] is a limit of a sequencefang_nof elements ofKx(see [APZ1]) it is enough to show that fora2Kx, one hasW(a)2Qp. But this follows by the proof of Proposition 5 (see [Sch]) sincemis equivariant.

Some details are left to the reader (see also the proof of Theorem 3). p 3. By the above considerations there results that ifW:C_G(O(x);Cp)!Qp

is a functional then the associated measurem_W(Proposition 5) takes values in K_xˆQg_p[x]\Q_p. At this point we describe the equivariant measures onO(x) with values inQ_p.

THEOREM 3. For the element x2Cp the following assertions are equivalent:

1) There exists a functional W:C_G(O(x);Cp)!Qp such that for any open ball U of O(x)one has:m_W(U)2Q_p.

2)The function Tr:K_x!Q_pdefined by Tr(a)ˆ 1

degatr_Qp(a)=Qp(a)is continuous and one has:W(a)ˆTr(a)for alla2K_x.

PROOF. Leta2Kx, andaˆs1(a); ;sn(a) be all the conjugates ofa over Q_p, nˆdeg(a). Since a2Qg_p[x], denote a:O(x)!C_p the local constant function defined bya(s(x))ˆs(a). One has:W(a)ˆW(a). The ele- mentssi(x), 1inare all distinct and lete>0 be a real number such that all the balls B(s_i(x);e) are pairwise disjoint. Denote H(x;e)ˆ

ˆ fs2G:js(x) xj<eg. ThenH(x;e) is a subgroup ofGof finite index. If Nˆ[G:H(x;e)] andft_ig_1iNis a set of right representatives for the co- sets ofGwith respect toH(x;e), then the ballsfB(ti(x);e)g_1iNcoverO(x)

and are any two disjoint. One has B(ti(x);e)ˆti(B(x;e)), and so by hy- pothesis there results thatm_W(B(ti(x);e))ˆm_W(B(x;e)). SinceW(1)ˆ1, one has:

m_W(s(x))ˆ1ˆX

i

m_W(B(t_i(x);e))ˆNm_W(B(x;e)):

According to our choice ofe, there results that on any ballB(t_i(x);e) the functionais constant andNˆkdeg (a), wherekis an integer. Then

W(a)ˆX

i

a(ti(x))m_W(B(ti(x);e)):

Since for exactlykballsB(ti(x);e) the functionatakes the same valuessj(a), by the above considerations one further obtains

W(a)ˆW(a)ˆXⁿ

jˆ1

ks_j(a)mW(B(x;e))ˆnkm_W(B(x;e)1 n

Xⁿ

jˆ1

s_j(a)

!

ˆTr(a):

The implication from 2) to 1) is left to the reader. p REMARK 3. Let W be as in Theorem 3. By the above considerations there results that for any e>0, one has: m_W(B(x;e))ˆ 1

N, where Nˆ[G:H(x;e)]. This shows that thep-adic measurem_Wcoincides with the p-adic Haar measurep_xdefined in [APZ2].

Under the hypothesis of Theorem 3 there results that if QpK1 Kn Kxis a tower of finite extensions ofQp such that [nKn ˆKx, then by considerations from Section 3, one hasWˆW^A, whenAˆn 1

[Kn :Qp] o

n1.

5. Cauchy Transforms onO(x).

1. Let x2C_p. For any real number e>0 denote B(x;e)ˆ

ˆ fy2C_p:jx yj<eg and B[x;e]ˆ fy2C_p:jx yj eg. Also denote E(x;e)ˆ fy2Cp:jy tj e; for allt2O(x)g. The complement ofE(x;e) in Cp[ f1g is denoted byV(x;e). Both setsE(x;e) andV(x;e) are open and closed, and one has: \_eV(x;e)ˆO(x). Denote E(x)ˆ [_eE(x;e)ˆ

ˆC_p[ f1g nO(x).

For anyx2C_pande>0 denoteH(x;e)ˆ fs2G:js(x) xj<egand H[x;e]ˆ fs2G:js(x) xj eg. Let Se (respectively Se) be a complete

system of representatives for the right cosets ofGwith respect toH(x;e) (respectivelyH[x;e]). ThenE(x;e)ˆ [s2SeB(s(x);e).

2. LetW:C(O(x);Cp)!Kbe a functional, whereKis a suitable closed subfield ofCp. Also denote bymthe measure associated toWaccording to Proposition 5. Then for anyf 2C(O(x);Cp) one has:

W(f)ˆ Z

O(x)

f(t)dm(t):

For anyz2E(x) and anyf 2C(O(x);C_p), the functionT(f;z):O(x)!C_p defined by

T(f;z)(t)ˆ f(t)

z t; t2O(x)

belongs toC(O(x);Cp). Hence for anyz2E(x) one can define the element F_K(m;f;z)ˆ

Z

O(x)

f(t) z tdm(t);

called theCauchy transform of f with respect tom(orW).

Now letz₀2E(x). Then for anyz2E(x) andt2O(x) one can write:

f(t)

z₀ tˆ f(t)

z t‡f(t)(z z0)

(z t)² ‡ ‡f(t)(z z0)ⁿ (z t)^n‡1 ‡ : …7†

Generally the series (7) does not converge, but it converges for a suitable choice of z. Indeed, let jz0 tj e for all t2O(x), and let a be a real number such that 0<e

a<1. Then for any z2B(z₀;e²=a), the series (7) converges since one has:z z0

z t ⁿe a

_n 1

e<1, for anyt2O(x). Hence for anyz2B(z0;e²=a) one can write:

Z

O(x)

f(t)

z0 tdm(t)ˆ Z

O(x)

f(t)

z tdm(t)‡ ‡ Z

O(x)

f(t)(z z₀)ⁿ

(z t)^n‡1 dm(t)‡ …8†

and this series is also convergent (since W is continuous) for all z2B(z0;e²=a).

3. Letfeng_nbe a strictly decreasing sequence of positive real numbers with limit zero. One can assume that SnSn‡1 for all n0, where S_nˆS_en

For anyn0, choose an elementan2Kxsuch thatjx anj<en. Then by ([Sch], page 276) it follows that

FK(m;f;z)ˆlim

n

X

s2Sn

m(B(s(x);en)) f(s(x)) z s(x): By this equality it also follows that

F_K(m;f;z)ˆlim

n

X

s2Sn

m(B(s(x);e_n)) f(s(an)) z s(a_n): …9†

By the above considerations one obtains the following result.

THEOREM4. Let x2C_p, and letW:C(O(x);C_p)!K be a functional, where K is a closed subfield ofCp. Then for any f 2C(O(x);Cp)the function FK(m;f;z):E(x)!Cpdefined by(8)is a rigid analytic function on E(x), and lim

z!1F_K(m;f;z)ˆ0.

4. If the field K and the functional Ware fixed we shall write simply F(f;z) instead ofFK(m;f;z).

6. Equivariant Cauchy Transforms.

1. Forx2C_pconsider an equivariant functionalW:C_G(O(x);C_p)!Q_p. By Theorem 1 the associate measuremˆm_Wis equivariant. On has the fol- lowing result:

THEOREM5. For any nonzero element f 2C_G(O(x);C_p), the function F(f;z), the Cauchy transform of f with respect toW, is an equivariant rigid analytic function defined on E(x). Any element of O(x)is a singular point for F(f;z), if F(f;z)6ˆ0.

PROOF. We observe that for any z2E(x) and any s2G, one has s(z)2E(x). Recall that F(f;z) is equivariant if for any s2G one has:

F(f;s(z))ˆs(F(f;z)). Now this equality is true sinceWis equivariant (see (8)). Furthermore ifF(f;z)6ˆ0, it has a singular point (a pole) which must belong to O(x). Then by equivariance all elements ofO(x) are poles for

F(f;z). p

COROLLARY 1. For an element x2Cp the following assertions are equivalent:

a) O(x)is a finite set.

b) x2Qp.

c) There exists an element f 2C_G(O(x);Cp), f 6ˆ0 such that F(f;z)2Qp(z).

2. Forf 2C_G(O(x);Cp) the elementf(x) belongs toQgp[x], and so with notations as in Proposition 1 one has: f(x)ˆP

n a_nM_n(x). Then for any t2O(x) it followsf(t)ˆP

n a_nM_n(t). Then by (8) one can write:

F(f;z)ˆX

n0

Z

O(x)

f(t)(z z0)ⁿ

(z t)^n‡1 dm(t)ˆX

n0

X

m0

Z

O(x)

amMm(t)(z z0)ⁿ (z t)^n‡1 dm(t):

7. Cauchy Transforms of measures.

1. In what follows we shall prove that Barsky's Theorem (see [B]) is valid forE(x),x2C_p. Namely, we shall prove that there exists a bijective mapping between the measures on O(x) and rigid analytic functions on E(x) with residue zero at infinity and which verify some boundary condi- tions.

LetF:E(x)!Cpbe a rigid analytic function such thatF(1)ˆ0 and that the set of real numbers fekFk_E(x;e)g_e>0 is bounded. (HerekFk_E(x;e)ˆ

ˆ sup

z2E(x;e)jF(z)j.)

One knows that E(x;e)ˆC_p[ f1g nV(x;e), and V(x;e)ˆ

ˆ [_s2SeB(s(x);e). By the Mittag-Leffler Theorem (see [FV]) one can write:

F(z)ˆX

s2Se

X

n1

a^(e)_n;s

(z s(x))ⁿ; z2E(x;e); a^(e)_n;s2C_p: One also has

F(z)ˆX

s2Se

F^(e)_s (z);

where F_s^(e)(z)ˆ P

n1

a^(e)_n;s

(z s(x))ⁿ,ja^(e)_n;sj

eⁿ !0. Then by Cauchy's inequalities, one obtains

ja^(e)_n;sj eⁿkFk_E(x;e); n1:

…10†

By the unicity of Mittag-Leffler's conditions for any 0<e⁰<eone has F^(e)_s (z)ˆX

t2Se0

^ tˆ^s

F^(e_t⁰⁾(z); z2E(x;e)

(here^tˆ^smeanst2sH(x;e)) and so X

n1

a^(e)_n;s

(z s(x))ⁿˆX

t2Se0

^ tˆ^s

X

m1

a^(e_m;t⁰⁾ (z t(x))^m: …11†

Since jz s(x)j e, and js(x) t(x)j<e, it follows that jz t(x)j ˆ

ˆ jz s(x)jand so:

a^(e_m;t⁰⁾

(z t(x))^mˆ a^(e_m;s⁰⁾ (z s(x))^m

1 ^t(x)_{z s(x)}^s(x)_m

ˆ a^(e_m;t⁰⁾ (z s(x))^m

X

k0

m‡k 1 k

ht(x) s(x) z s(x)

i_k : …12†

If we denotem‡kˆn, then by identifying the coefficients of the terms of degreenin (11) and (12) one obtains:

a^(e)_n;sˆX

t2Se0

^ tˆ^s

X^{n 1}

kˆ0

n 1 k

a^(e_{n k;t}⁰⁾ (t(x) s(x))^k; …13†

wheren1.

Now for anyn1 one defines a sequencefm_n;eg_n;eof measures onO(x) by the equality

m_n;eˆX

s2Se

a^(e)_n;sds(x); …14†

whered_ydenotes the Dirac measure concentrated aty2C_p. By (13) one obtains, fornˆ1:

a^(e)_1;sˆX

t2Se0

^ tˆ^s

a^(e_1;t⁰⁾; 0<e⁰e:

This equality further implies that for any ballBof radiusd,ed, one has m_1;e(B)ˆm_1;e0(B) whereas e⁰e. Then by (10) and the Banach-Steinhaus Theorem (see [R]) there results that the mapping

B?m₁(B)ˆlim

e m_1;e(B)

(whereBruns over all the open balls ofO(x)) defines ap-adic measure on O(x). One says thatmˆm₁is themeasure associatedto the rigid analytic functionF.

Furthermore, fornˆ2, by (13) one obtains a^(e)_2;sˆX

t2Se0

^ tˆ^s

[a^(e_2;t⁰⁾‡a^(e_1;t⁰⁾(t(x) s(x))]:

IfBis an open ball ofO(x) of radiusd, by the previous equality and (14) there results that

m_2;e m_2;e0 ˆX

s(x)2Bs2Se

X

t2Se0

^ tˆ^s

a^(e_1;t⁰⁾(t(x) s(x))

and so by (10) one has:

jm_2;e m_2;e0j ee⁰j jj jF _E(x;e0)eM whereMˆsup

e>0 ej jj jF _E(x;e)<1, by hypothesis. Then by (10) and the Banach- Steinhaus Theorem, there exists a measurem₂onO(x) defined by

m₂(B)ˆlim

e m_2;e(B);

for all the open ballsBofO(x). In the same manner for alln3 one can define a measurem_nonO(x) by:

m_n(B)ˆlim

e m_n;e(B):

Next, by an easy computation it follows that for all n2, one has m_n;e

Me^{n 1}, and som_n ˆ0 for alln2. In what follows we shall prove that

F(z)ˆ Z

O(x)

1

z tdm(t); z2E(x):

…15†

Indeed, with the above notations and using (10) one has:

F(z)

Z

O(x)

1

z tdm_1;e(t) ˆ

F(z) X

s2Se

a^(e)_1;s z s(x)

e²j jj jF _E(x;e)Me;

forjzjsufficiently large. It is clear, by the definition ofmˆm₁, that (see (14))

one has: Z

O(x)

1

z tdm(t)ˆlim

e!0

Z

O(x)

1

z tdm_1;e(t)

and so by the last inequality one obtains (15) forjzjsufficiently large. Be- causeE(x) is infra-connected, by analytic continuation one obtains (15) for allz2E(x). Finally one has the following result:

THEOREM6. For any x2Cp, there exists a bijective mapping between the p-adic measures on O(x)and the functions F:E(x)!C_pwhich verify the following conditions:

i) F is rigid analytic on E(x), and F(1)ˆ0.

ii) The set of real numbersfej jj jF _E(x;e)g_e>0 is bounded.

Moreover, by this bijective mapping the rigid analytic and equivar- iant functions are in one to one correspondence with equivariant mea- sures on O(x).

8. Cauchy Transforms of Lipschitzian Distributions.

LetXbe an arbitrary compact subset ofC_pwithout isolated points. A mapf :X!Cpis said to bel-Lipschitzian provided there exists a positive real number lsuch thatjf(x) f(y)j ljx yj, for any x;y2X. A map f :X!Cpis said to be Lipschitzian provided there exists a real numberl for which f isl-Lipschitzian. Let us remark that Lip(X;C_p), the set of Lipschitzian functions, is aC_p-vector space. Moreover, it is a Banach space with the norm

f j j j j ˆsup

x6ˆy

jf(x) f(y)j jx yj ‡sup

x2Xjf(x)j:

Let us denote byV(X) the set of open compact subsets ofX. It is clear that any element ofV(X) is a disjoint finite union of open balls ofX.

A distributionmonXwith values inCpis a mapm:V(X)!Cpwhich is finitely additive, thus if Dˆ [ⁿ

iˆ1D_i with D_i2V(X) for 1in and D_i\D_j ˆ ;for 1i6ˆjn, thenm(D)ˆPⁿ

iˆ1m(D_i).

We call a distributionm:V(X)!CpLipschitzian provided that for any e>0 there is a de>0 such that for any 0<dde and any x2X, djm(B(x;d))j e.

One knows (see [VZ, Theorem 1]) that any Lipschitzian function f :X!Cp is integrable with respect to any Lipschitzian distribution m:V(X)!Cp. Moreover, by the proof of this result, for anye>0 there existsd_e>0 such that if

S(m;f;B₁;. . .;Bn;x₁;. . .;xn)ˆXⁿ

iˆ1

f(x_i)m(B_i)

is an arbitrary Riemann sum withx_i2B_iandB_iopen ball of radiusd_ide

for 1in, the following inequality holds:

S(m;f;B₁;. . .;B_n;x₁;. . .;x_n) Z

X

f(t)dm(t)emaxf1;2lg;

…16†

wherelis the Lipschitzianity constant with respect tof.

Let us consider now an element z2P¹(Cp)nX, and f 2Lip(X;Cp).

DefineT(f;z):X!Cpby

T(f;z)(t)ˆ f(t)

z t; t2X:

…17†

It is easy to see that T(f;z) is well defined and Lipschitzian. In fact, if dˆd(z;X) is the distance fromztoX, andt₁;t₂2Xwe have

jT(f;z)(t₁) T(f;z)(t₂)j ˆ

f(t₁) z t₁

f(t₂) z t₂

Ajt₁ t₂j;

…18†

whereAˆA(z;f;X)ˆ 1

d²maxfljzj; lsup

t2X jtj; sup

t2X jf(t)jg.

We can integrate this function with respect to any Lipschitzian dis- tribution, so the map

F_X(m;f;z)ˆ Z

X

T(f;z)(t)dm(t)

is well defined. It is the Cauchy transform of the Lipschitzian function f with respect to the Lipschitzian distributionm. Using a similar argument as in the proof of Theorem 6.1 from [APZ2] one obtains the following result.

THEOREM7. Let X be a compact subset ofCpwithout isolated points,m a Lipschitzian distribution on X, and f 2Lip(X;Cp). Then

F_X(m;f;z):ˆ

Z

X

f(t) z tdm(t) …19†

is well defined, rigid analytic on P¹(Cp)nX, and vanishes at infinity.

Moreover, any element of X is a singular point of FX(m;f;z).

Next, let f 2Lip(X;C_p) and z2P¹(C_p)nX. From (16), with T(f;z) instead off andSfor the corresponding Riemann sum, one has

jS FX(m;f;z)j emaxf1;2Ag;

…20†

whereAis defined above. If (B_i)_1inis a covering ofXwith disjoint open balls of radiusd_e, we have from the definition ofmthat

jSj eM dde; …21†

whereMˆsup

x2Xjf(x)j. From (20) and (21) one has jFX(m;f;z)j emax

1;2A;M dde

; …22†

and so

d²jF_X(m;f;z)j emax

d²;2d²A;Md d_e

: …23†

Denoting byEd(X) the complement inP¹(Cp) of ad-neighborhood ofX, and using the definition ofA, from (23) we obtain

limd!0d²jjFX(m;f;z)jj_Ed(X)Be;

…24†

whereBis an absolute constant that does not depend ond. Lettinge!0, we obtain the following result.

THEOREM8. Let X be a compact subset ofC_pwithout isolated points.

Let f 2Lip(X;C_p)and letmbe a Lipschitzian distribution on X. To any pair(m;f)as above, we can associate a rigid analytic function F_X(m;f;z)in such a way that it vanishes at infinity, and satisfies the boundary condi- tion:

d!0limd²jjF_X(m;f;z)jj_Ed(X)ˆ0:

…25†

If X,mand f are equivariant then FX(m;f;z)is also equivariant.

A natural question that arises is to provide a converse to the statement of this theorem.

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Manoscritto pervenuto in redazione il 12 settembre 2006.