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On the other hand, it is well known that in higher complex dimension n 2, plurisubharmonic functions are rather connected to the complex Monge-Amp`ere≥ operator which is a fully non linear second order differential operator. Therefore Pluripotential theory cannot be completely described by logarithmic potentials as shown by Magnus Carlehed in [Carlehed]. How- ever the class of logarithmic potentials provides a natural class of plurisub- harmonic functions which turns out to be included in the local domain of definition of the complex Monge-Amp`ere operator. This study was started by Carlehed ([Carlehed99, Carlehed] in the case when the measue is com- pactly supported on Cn with a locally bounded potential. Our main goal is to extend this study to the case of arbitrary probability measures on the complex projective space Pn. The motivation for this study comes from the fact that the complex Monge-Amp`ere operator plays an important role in K¨ahler geometry when dealing with the Calabi conjecture and the problem of the existence of a K¨ahler-Einstein metric (see [GZ17]). Indeed in geometric applications one needs to consider degenenerate complex Monge-Amp`ere equations. To this end a large class of singular potentials on which the complex Monge-Amp`ere operator is well defined was introduced (see [GZ07], [BEGZ10]). This leads naturally to a general definition of the complex Monge-Amp`ere operator (see [CGZ08]). However the global domain of definition of the complex Monge-Amp`ere operator on compact K¨ahler manifolds is not yet well understood. Besides this, thanks to the works of Cegrell and Blocki (see [Ce04], [Bl04], [Bl06]), the local domain of definition is characterized in terms of some local integrability conditions on approximating sequences. Our goal here is then to study a natural class of projective logarithmic potentials and show that it is contained in the (local) domain of definition of the complex Monge-Amp`ere operator on the complex projective space Pn, giving an interesting subclass of the domain of definition of the complex Monge-Amp`ere operator on the complex projective space and exhibiting many different interesting behaviours. More precisely, let µ be a probability measure on the complex projective space Pn. Then its projective logarithmic potential is defined on Pn as follows:
ζ η n Gµ(ζ) := log | ∧ | dµ(η), ζ P . Pn ζ η ∈ Z | || | Our first result is a regularizing property of the operator G acting on the convex compact set of probability measures on Pn.
Theorem A. Let µ be a probability measure on Pn (n 2). Then ≥ 1) G DMA (Pn,ω) and for any 0 c n 3) the Monge-Amp`ere measure (ω +dd Gµ) is absolutely continuous with respect to the Lebesgue measure on (Pn,ω) if the measure µ has no atom in Pn; conversely if the measure µ has an atom at some point a Pn, the c n ∈ complex Monge-Amp`ere measure (ω +dd Gµ) also has an atom at the same point a. n Here ω := ωFS is the Fubini-Study metric on the projective space P n and DMAloc(P ,ω) denotes the local domain of definition of the complex Monge-Amp`ere operator on the K¨ahler manifold (Pn,ω) (see Definition 2.2 below). The fact that the projective logarithmic potential of any measure belongs to the domain of definition of the complex Monge-Amp`ere operator as well as the third property was proved earlier by the second named author ([As17]). These results generalize and improve previous results by M. Carlehed who considered the local setting (see [Carlehed, Carlehed99]). When the measure has a ”positive dimension”, we obtain a strong reg- ularity result in terms of this dimension as defined by the formula (3.2) in section 3.2. Theorem B : Let µ be a probability measure on Pn (n 2) with positive dimension γ(µ) > 0. Then the following holds : ≥ 1) G W 1,p(Pn) for any p such that 1 p< (2n γ(µ))/(1 γ(µ)) ; µ ∈ ≤ − − + in particular Gµ is H¨older continuous of any exponent α such that (1 γ(µ)) 0 <α< 1 2n − + ; − 2n γ(µ) − 2) G W 2,p(Pn) for any µ ∈ 2n γ(µ) 1 Recall that for a real number s R, we set s+ := max s, 0 . Observe that in the last two statements, the cirtical∈ exponent is + when{ } 2 γ(µ) 2n. ∞ ≤ ≤ 2. Preliminaries 2.1. Lagrange identities. Let us fix some notations. For a = (a0, , an) Cn+1 and b = (b , , b ) Cn+1 we set · · · ∈ 0 · · · n ∈ n n a ¯b := a ¯b , a 2 := a 2. · j j | | | j| Xj=0 Xj=0 4 S. ASSERDA, FATIMA Z. ASSILA AND A. ZERIAHI We can define the wedge product a b in the vector space 2 Cn+1. The hermitian scalar product on Cn+1 induces∧ a natural hermitian scalar product 2 n+1 V on C . If(ej )0 j n is the associated canonical orthonormal basis of n+1 ≤ ≤ C , then the sequence (ei ej)0 i 2.2. The complex projective space. Let Pn = Cn+1 0 /C be the complex projective space of dimension n 1 and \ { } ∗ ≥ π : Cn+1 0 Pn, \ { } −→ the canonical projection. which sends a point ζ = (ζ , ,ζ ) Cn+1 0 0 · · · n ∈ \ { } to the complex line C ζ which we denote by [ζ] = [ζ0, ,ζn]. ∗ · · · · By abuse of notation we will denote by ζ = [ζ0, ,ζn] and call the ζj’s the homogenuous coordinates of the (complex line)·ζ · ·. As a complex manifold Pn can be coved by a finite number of charts given by := ζ Pn; ζ = 0 , 0 k n. Uk { ∈ k 6 } ≤ ≤ For a fixed k = 0, ,n, the corresponding coordinate chart is defined on by the formula · · · Uk k k k k z (ζ)= z := (zj )0 j n,j=k, where zj := ζj/ζk for j = k. ≤ ≤ 6 6 LOGARITHMICPOTENTIALS 5 n The map γk : k C is an homomeorphism and for k = ℓ the transition functions (changeU ≃ of coordinates) 6 1 zk (zℓ)− : zℓ( ) zk( ) ◦ Uℓ ∩Uk −→ Uℓ ∩Uk is given by 1 w = zℓ (zk)− (z , , z ), for (z , , z ) Cn, z = 0, ◦ 1 · · · n 1 · · · n ∈ ℓ 6 where wi = zi/zℓ for i / k,ℓ , wk = 1/zℓ and wℓ = zk/zℓ. The (1, 1)-form ddc log∈ {ζ is} smooth, d-closed on Cn+1 0 and invariant under the C -action. Therefore| | it descends to Pn as a smooth\ { } closed (1, 1)- ∗ n form ωFS on P so that c n+1 dd log ζ = π∗(ω ), in C 0 . | | FS \ { } In the local chart ( , zk) we have Uk 1 ω = ddc log(1 + zk 2) |Uk 2 | | Observe that the Fubini-Study form ω is a K¨ahler form on Pn and the n correponding Fubini-Study volume form dVFS = ω /n! is given in the chart ( , zk) (Cn, z) by the formula Uk ≃ dV (z) dV Cn = c 2n , FS | n (1 + z 2)n+1 | | n n where dV2n(z) = βn = β /n! is the euclidean volume form on C , β := ddc z 2 being the standard K¨ahler metric on Cn. | | 2.3. The complex Monge-Amp`ere operator. Here we recall some defi- nitions and give a useful characterization of the local domain of definition of the complex Monge-Amp`ere operator given by Z. Blocki (see [Bl04], [Bl06]). Definition 2.2. Let X be a complex maniflod of dimension n an η a smooth closed (semi)-positive (1, 1)-form on X. 1) We say that a function ϕ : X R is η-plurisubharmonic in X if it is locally the sum of a plurisubharmonic−→ ∪ {−∞} function and a smooth function and η + ddcϕ is a positive current on X. We denote by PSH(X, η) the convex set of all of η-plurisubharmonic functions in X. 2) By definition, the set DMA (X, η) is the set of functions ϕ PSH(X, η) Loc ∈ for which there exists a positive Borel measure σ = σϕ on X such that for all open U Ω and (ϕj) PSH(U, η) C∞(U) ϕ in U, the sequence ⊂⊂ ∀ ∈ c∩ n ց of local Monge-Amp`ere measures (η + dd ϕj) converges weakly to σ on U. c n In this case, we set (η + dd ϕ) = σϕ and call it the complex Monge-Amp`ere measure of the function ϕ in the manifold (X, η). When η = 0, we write DMAloc(X)= DMAloc(X, 0). In the case when X Cn is an open subset and η = 0, Bedford and Taylor [BT76, BT82] extended⊂ the complex Monge-Amp`ere operator (ddcu)n to plurisubharmonic functions which are locally bounded in X. Moreover, 6 S. ASSERDA, FATIMA Z. ASSILA AND A. ZERIAHI they showed that this operator is continuous under decreasing sequences in PSH(X) LLoc∞ (X). In [De87],∩ Demailly extended the complex Monge-Amp`ere operator (ddc)n to plurisubharmonic functions locally bounded near the boundary ∂X i.e. in the complement of a compact subset of X. n When X = P and η = ωFS, it was proved in [CGZ08] that if ϕ n n ∈ PSH(P ,ωFS) is bounded in a neighbourhood of a divisor in P , then ϕ n ∈ DMAloc(P ,ωFS). n When Ω ⋐ C is a bounded hyperconvex domain, the set DMAloc(Ω) coincides with the Cegrell class (Ω) (see [Ce04], [Bl06]). Moreover for E any u1, , un DMAloc(Ω), it is possible to define the intersection current ddcu · · · ddcu as a positive Borel measure on Ω ([Ce04]). 1 ∧···∧ n Therefore for any ϕ1, , ϕn DMAloc(X, η), it is possible to define the intersection current · · · ∈ (ϕ , , ϕ ) := (η + ddcϕ ) (η + ddcϕ ) M 1 · · · n 1 ∧···∧ n as a positive Borel measure on X. In particular if X is compact and η is a smooth (1, 1) form on X which is closed semi-positive and big i.e. n − X η > 0, then R (η + ddcϕ ) (η + ddcϕ )= ηn. 1 ∧···∧ n ZX ZX Moreover the operator is continuous for monotone convergence of se- M quences in DMAloc(X, η). In dimension two, a simple characterization of the local domain of defini- tion of (ddc)2 was given by [Bl04]. Theorem 2.3. [Bl04] Suppose Ω is an open subset of C2, then DMA (Ω) = PSH(Ω) W 1,2(Ω), Loc ∩ loc 1,2 where Wloc (Ω) is the usual Sobolev space. In higher dimension, the characterization of the set DMALoc(Ω) is more complicated (see [Bl06]). Theorem 2.4. Let u be a negative plurisubharmonic function on Ω Cn, n 3. The following are equivalent: ⊂ ≥1- u DMA (Ω), ∈ Loc 2- z Ω, Uz Ω an open neighborhood of z such that for any sequence u PSH∀ ∈ C∃ (U⊂ ) u in U , the sequences j ∈ ∩ ∞ z ց z n p 2 c c p n p 1 u − − du d u (dd u ) ω − − , p = 0, 1, ..., n 2 | j| j ∧ j ∧ j ∧ − are locally weakly bounded in Uz. It is clear that these results extend to the case of DMAloc(X, η). Remark 2.5. We can define the global domain of defintion DMA(X, η) by taking only bounded global approximants. Therefore DMA (X, η) loc ⊂ LOGARITHMICPOTENTIALS 7 DMA(X, η) but the inclusion is strict (see [GZ07]) at least in the case when (X, η) is a compact K¨ahler manifold. Very little is known about the space DMA(X, η) (see [CGZ08]). 2.4. The Lelong Class. Recall that the Lelong class (Cn) is defined as the convex set of plurisubharmonic functions in Cn satisfyingL the following growth condition at infinity : 1 (2.4) u(z) C + log(1 + z 2), z Cn, ≤ u 2 | | ∀ ∈ where Cu is a constant depending on u (see [Le68]). There are two important subclasses of (Cn). Set L 1 (Cn) := u (Cn); u(z)= log(1 + z 2)+ O(1), as z + . L+ { ∈L 2 | | | |→ ∞} Associated to a function u (Cn) we define its Robin function of u ([BT88]) ∈ L 1 2 ρu(ξ) := lim sup(u(λξ) log λξ ) = limsup(u(λξ) log(1 + λξ ). C λ − | | C λ − 2 | | ∋ →∞ ∋ →∞ Cn When ρu∗ , then it is a homogenous function of order 0 on such 6≡ −∞ Cn that ρu∗ (z) + log z is plurisubharmonic in . This implies that ρu∗ is a well | | n 1 defined function on the projective space P − which is actually an ωFS-psh n 1 n 1 on P − , where ωFS is the Fubini-Study metric on P − . Following ([BT88]), we can define the subclass of logarithmic Lelong po- tentials as follows : (Cn) := u (Cn); ρ L⋆ { ∈L u 6≡ −∞}· n n n 1 n 1 Observe that +(C ) ⋆(C ) and u ⋆(C ) iff ρu L (P − ). We refer to [Ze07] for moreL properties⊂L of this class.∈L ∈ Then there is a 1-1 correspondence between the Lelong class (Cn) and the set PSH(Pn,ω) of ω-plurisubharmonic functions on Pn. IndeedL we will n write P = 0 ˙ H , where U ∪ ∞ n n 1 H := ζ P ; ζ0 = 0 P − , ∞ { ∈ }≃ n is the hyperplane at infinity and observe that 0 = P H . Given u (Cn), we associate the functionUϕ defined\ on∞ by ∈L U0 1 φ (ζ) := u(z , , z ) log(1 + z 2), with z := z0 = (ζ /ζ , ,ζ /ζ ). u 1 · · · n − 2 | | 1 0 · · · n 0 n By definition of the class (C ), the function φu is locally upper bounded L n 1 in 0 near H . Since H = 0 P − is a proper analytic subset (hence U ∞ ∞n { }× a pluripolar subset) of P , the function φu can be extended into an ω- plurisubharmonic function on Pn by setting n 1 φu(0,ζ′)= limsup ϕ(ζ), ζ′ P − . 0 ζ (0,ζ ) ∈ U ∋ → ′ 8 S. ASSERDA, FATIMA Z. ASSILA AND A. ZERIAHI Then we have a well defined ”homogenization” map (Cn) φ PSH(Pn,ω ). L ∋7−→ u ∈ FS This is a bijective map and its inverse map is defined as follows : given φ PSH(Pn,ω ), we can define ∈ FS u (z) := φ([1, z]) + (1/2) log(1 + z 2). φ | | n n 1 It is clear that uφ (C ) and H(uφ)= φ. Observe that for ζ′ P − , we have ∈L ∈ ρu(ζ′)= φu(0,ζ′). Example 2.6. Let P Cd[z1, , zn] be a polynomial of degree d 1. Then u := (1/d) log P ∈ (Cn).· · It · is easy to see that its homogeneization≥ | |∈L φ = φu is given by φ(ζ) := (1/d) log Q(ζ) log ζ , ζ Pn, | |− | | ∈ where Q is the unique homogenuous polynomial on Cn+1 of degree d such that by Q(1, z)= P (z) for z Cn. ∈ n Proposition 2.7. Let u1, , un +(C ) and for each i = 1, ,n set · · · ∈ L Pn · · · φi := φui the homogeneization of ui on . Then c c c c (2.5) dd u1 dd un = (ω + dd φ1) (ω + dd φn) = 1. Cn ∧···∧ Pn ∧···∧ Z Z n n Proof. Observe that if u +(C ), its homogeneization φ = φu PSH(P ,ω) is a bounded ω-psh function∈L in a neighbourhood of the hyperplane∈ at in- n n finity H . Hence by [CGZ08], +(C ) DMAloc(P ,ω) and its complex Monge-Amp`ere∞ measure is well definedL and⊂ puts no mass on H . Hence ∞ 1= (ω + ddcφ)n = (ω + ddcφ)n. Pn Cn Z Z Since ω + ddcφ = ddcu in the weak sense on Cn, the formula (2.5) follows. 3. Projective logarithmic Potentials on Cn 3.1. The euclidean logarithmic potential in Cn. Our main motivation is to study projective logarithmic potentials on Pn. It turns out that when localizing these potentials in affine coordinates, we end up with a kind of projective logarithmic potential on Cn which we shall study here. We first introduce the normalized logarithmic kernel on Cn Cn defined by × 1 z w 2 K(z, w) := log | − | , (z, w) Cn Cn. 2 1+ w 2 ∈ × | | Let µ be a probability measure on Cn. We define its logarithmic potential as follows, for z Cn ∈ 1 z w 2 (3.1) Uµ(z)= K(z, w)dµ(w)= log | − |2 dµ(w) Cn 2 Cn 1+ w Z Z | | LOGARITHMICPOTENTIALS 9 n It is well known that for any w C , the function Kw := K( , w) is plurisub- n n∈ · harmonic in C , Kw +(C ) and satisfies the complex Monge-Amp`ere equation ∈ L (ddcK( , w))n = δ , · w n in the sense of currents on C , where δw is the unit Dirac mass at w. Theorem 3.1. Let µ be a probability measure on Cn. Then for any z Cn, ∈ 1 U (z) log(1 + z 2). µ ≤ 2 | | 2n 1 and if σ2n 1 is the normalized Lebesgue measure on the unit sphere S − , − Uµ(z)dσ2n 1(z) log √5. z =1 − ≥− Z{| | } n In particular Uµ (C ). If moreover µ ∈Lsafisfies the following logarithmic moment condition at in- finity i.e. log(1 + w 2)dµ(w) < + , Cn | | ∞ Z then U (Cn) and its Robin function ρ := ρ satisfies the lower bound µ ∈L⋆ µ Uµ 2 ρµ(ζ) ωn 1 (1/2) log(1 + w )dµ(w), Pn 1 − ≥− Cn | | Z − Z n 1 where ωn 1 is the Fubini-Study volume form on P − . − Proof. By (2.3), for any z, w Cn, ∈ z w 2 1+ z w¯ 2 | − | 1 | · | . (1 + z 2)(1+ w 2) ≤ − (1 + z 2)(1+ w 2) | | | | | | | | Then for z, w Cn, ∈ 1 z w 2 1 K(z, w) := log | − | log(1 + z 2). 2 1+ w 2 ≤ 2 | | | | For any fixed w Cn, the function z K(z, w) is plurisubharmonic in n ∈ n −→ C . It follows that Uµ (C ) provided that we prove that Uµ . To prove the last statement,∈L write for z Cn 6≡ −∞ ∈ 2 Uµ(z)= I1(z)+ I2(z), where z w 2 I1(z) := log | − |2 dµ(w), w 2 1+ w Z| |≤ | | and z w 2 I2(z) := log | − |2 dµ(w). w >2 1+ w Z| | | | 10 S. ASSERDA, FATIMA Z. ASSILA AND A. ZERIAHI Let us prove that I1 and I2 . For the first integral observe that for z 3, we have6≡ −∞ 6≡ −∞ | |≥ I (z) µ(B(0, 2)) log 5 > . 1 ≥− −∞ ( w 1)2 1 For the second integral, observe that for z 1, | |− for w 2. | | ≤ 1+ w 2 ≥ 5 | | ≥ Hence for z < 1, | | | | ( w 1)2 I2(z) log | |− 2 dµ(w) log 5 > . ≥ w >2 1+ w ≥− −∞ Z| | | | Cn 1 Cn Cn Therefore I1 and I2 belongs to PSH( ) Lloc( ) and then Uµ PSH( ). n ⊂ ∈ This proves that Uµ (C ). Then for z, w Cn, ∈L ∈ z w 2 z w 2 log | − | log | | | − | | | . 1+ w 2 ≥ 1+ w 2 | | | | Fix 0 <δ< 1 and z Cn such that z = 1. Then splitting the integral ∈ | | defining Uµ in two parts, we get : 2 2 (1 δ) B z w 2Uµ(z) log − 2 µ( (0, δ)) + log | − | 2 dµ(w) ≥ 1+ δ w δ 1+ w Z| |≥ | | Hence by submean value inequality, we obtain 2 2 (1 δ) B w 2 Uµ(ξ)dσ2n 1(ξ) log − 2 µ( (0, δ)) + log | | 2 dµ(w) z =1 − ≥ 1+ δ w δ 1+ w Z| | Z| |≥ | | (1 δ)2 δ2 log − µ(B(0, δ)) + µ( w δ ) log ≥ 1+ δ2 {| |≥ } 1+ δ2 Then taking δ = 1/2 we obtain Uµ(z)dσ2n 1(z) log √5. z =1 − ≥− Z| | This also implies that Uµ . To prove the last statement6≡ −∞observe that ρ 0 on Cn and µ ≤ lim (log λz w 2 log λz 2) = 0. λ + | − | − | | | |→ ∞ Then apply Fatou’s lemma to obtain the conclusion. 3.2. Riesz potentials. Here we prove a technical lemma which is certainly well known but since we cannot find the right reference for it, we will give all the details needed in the sequel. Here we work in the euclidean space RN with its usual scalar product and its associated euclidean norm . Let µ be a probability measure with compact support on RN . Definek · k its Riesz potentials by dµ(w) RN Jµ,α(x) := α = µ ⋆ Jα(x), x , Cn x y ∈ Z | − | LOGARITHMIC POTENTIALS 11 where 1 J (x) := , x RN . α x α ∈ | | Observe that J Lp (RN ) if and only if 0 < p < N/α. α ∈ loc Given a probability measure µ on RN , there are many different notions of dimension for the measure µ. Here we use the following one (see [LMW02]). We define the L´evy concentration functions of µ as follows: µ(x, r) := µ(B(x, r)), Q (r) := sup µ(x, r); x Suppµ , µ { ∈ } where B(x, r) is the euclidean (open) ball of center x and radius r> 0. The lower concentration dimension of µ is given by the following formula: log Q (r) (3.2) γ(µ)= γ (µ) := lim inf µ − r 0+ log r · → We will call it for convenience the dimension of the measure µ. The following property is well known. Lemma 3.2. Let µ be a probability measure µ with compact support on Cn. Then its dimension γ(µ) is the supremum of all the exponents γ 0 for which the following estimates are staisfied: there exists C > 0 such≥ that x RN , r ]0, 1[, ∀ ∈ ∀ ∈ µ(B(x, r)) Crγ. ≤ Moreover we have 0 γ(µ) N. ≤ ≤ Proof. The first statement is obvious and we have γ(µ) 0 since µ has a finite mass. To prove the upper bound, observe that for any≥ fixed r> 0, the function x µ(x, r) is a non negative Borel function on Rn. By Fubini’s Theorem, we7−→ have for any r> 0, µ(x, r)dx = dµ(y) dx RN RN y B(x,r) Z Z Z ∈ ! = dx dµ(y). RN Z ZB(y,r) ! N If we denote by τN the volume of the euclidean unit ball in R , we obtain N (3.3) µ(x, r)dx = τN r . RN Z Therefore for any r> 0, we have τ rN Q (r) µ(RN )= Q (r), N ≤ µ µ which implies immediately that γ(µ) N. ≤ Example 3.3. 1. Then the measure µ has no atom in RN if and only if γ(µ) > 0. 2. Let 0 < k N be any real number and let A RN be any Borel subset such that its k-dimensional≤ Hausdorff measure satisfies⊂ 0 < λ (A) < + . k ∞ 12 S. ASSERDA, FATIMA Z. ASSILA AND A. ZERIAHI Then the restricted measure λk,A := 1Aλk is a Borel measure of dimension k. In particular the N-dimensional Lebesgue measure λk,A has dimension N. Lemma 3.4. Let µ be a probability measure with compact support on RN and 0 < α < N and γ := γ(µ) its dimension. Then N γ J Lp (RN ) if 1 p p µ( , r) p = µ(x, r) dx k · k RN Z p 1 min Qµ(r), 1 − µ(x, r)dx ≤ { } RN p Z1 2n C min Q (r), 1 − r . ≤ { µ } Therefore for any fixed κ> 0, we have κ (p 1)/p 2n/p dr µ ⋆ J Cα Q (r) − r k αkp ≤ µ rα+1 Z0 dr + ∞ r2n/p . rα+1 Zκ Using the estimate on Qµ in terms of the dimension and minimizing in κ> 0, we get the required result. LOGARITHMIC POTENTIALS 13 3.3. The projective logarithmic kernel. Our main motivation is to study projective logarithmic potentials on Pn. It turns out that when localizing these potentials in affine coordinates, we end up with a kind of projective logarithmic potential on Cn which we shall study first. Recall the definition of the logarithmic kernel from the previous section, 1 z w 2 K(z, w) := log | − | , 2 1+ w 2 | | The projective logarithmic kernel on Cn Cn is defined by the following formula: × 1 z w 2 + z w 2 N(z, w) := log | − | | ∧ | , (z, w) Cn Cn. 2 1+ w 2 ∈ × | | Lemma 3.5. 1. The kernel N is upper semi-continuous in Cn Cn and smooth off the diagonal of Cn Cn. × n × 2. For any fixed w C , the function Nw := N( , w) : z N(z, w) is plurisubharmonic in C∈n and satisfies the following inequality· 7−→ K(z, w) N(z, w) (1/2) log(1 + z 2), (z, w) Cn Cn ≤ ≤ | | ∀ ∈ × n n hence for any w C , N( , w) +(C ). 3. The kernel ∈N has a logarithmic· ∈L singularity along the diagonal i.e. for any (z, w) Cn Cn, ∈ × 1 0 N(z, w) K(z, w) log(1 + min z 2, w 2 ). ≤ − ≤ 2 {| | | | } 4. For any w Cn, ∈ c n (dd Nw) = δw, n where δw is the unit Dirac measure on C at the point w. Proof. The first statement is obvious. Let us prove the second one. By the formula 2.3 we have for any z, w Cn, ∈ z w 2 + z w 2 1+ z w¯ 2 | − | | ∧ | = 1 | · | . (1 + z 2)(1+ w 2) − (1 + z 2)(1+ w 2) | | | | | | | | Thus for z, w Cn, ∈ 1 z w 2 + z w 2 1 N(z, w)= log | − | | ∧ | log(1 + z 2). 2 1+ w 2 ≤ 2 | | | | This yields z w 2 N(z, w) K(z, w) = (1/2) log 1+ | ∧ | . − z w 2 | − | Now observe that by Lemma 2.3, we have z w 2 = (z w) w 2 z w 2 w 2. | ∧ | | − ∧ | ≤ | − | | | By symmetry we obtain the required inequality. 14 S. ASSERDA, FATIMA Z. ASSILA AND A. ZERIAHI To prove the third property, observe that for a fixed w Cn, the two functions ∈ u(z) := N(z, w), v(z) := K(z, w), n belongs to +(C ). Hence by (2.5) they have the same total Monge-Amp`ere mass in CnLi.e. (ddcu)n = 1. Cn Z On the other we know that c n (dd v) = δw. u(z) Since limz w = 1, by the comparison theorem of Demailly [De93], they → v(z) have the same residual Monge-Amp`ere mass at the point w. Therefore the total Monge-Amp`ere mass of the measure (ddcu)n is concentrated at the point w, which proves our statement. 3.4. The projective logarithmic potential. Let µ be a probability mea- sure with compact support on Cn. We define the projective logarithmic potential of µ as follows : 1 z w 2 + z w 2 (3.4) Vµ(z)= log | − | | 2∧ | dµ(w), 2 Cn 1+ w Z | | It follows that V (Cn) provided that we prove that V . µ ∈L µ 6≡ −∞ Theorem 3.6. Let µ be a probability measure with compact support on Cn. Then for any z Pn, ∈ 1 U (z) V (z) log(1 + z 2). µ ≤ µ ≤ 2 | | and Vµ(z)dσ2n 1(z) log √5. z =1 − ≥− Z{| | } Cn Moreover Vµ ⋆( ) and its Robin function ρµ := ρVµ = ρUµ satisfies the lower bound ∈L 2 ρµ(ξ) ωn 1 (1/2) log(1 + w )dµ(w), Pn 1 − ≥− Cn | | Z − Z n 1 where ωn 1 is the Fubini-Study volume form on P − . − Proof. The right hand side estimate in the first inequality follows from lemma 3.5, while the other statements follow from Theorem 3.1 since Vµ n ≥ Uµ in C . LOGARITHMIC POTENTIALS 15 4. The projective logarithmic potential on Pn 4.1. The projective logarithmic kernel. The projective logarithmic ker- nel on Pn Pn is defined by the following formula : × ζ η 2 G(ζ, η) := (1/2) log | ∧ | , (ζ, η) Pn Pn. ζ 2 η 2 ∈ × | | | | Lemma 4.1. 1. The kernel G is a non positive upper semi-continuous function in Pn Pn and smooth off the diagonal of Pn Pn. 2. For any× fixed η Pn, the function G( , η) : ζ × G(ζ, η) is a non positive ω-plurisubharmonic∈ in Pn which is smooth· in P7−→n η i.e. G( , η) PSH(Pn,ω). \ { } · ∈ 3. The kernel G has a logarithmic singularity along the diagonal. More precisely for a fixed η , in the coordinate chart ( , zk) we have ∈Uk Uk zk(ζ) zk(η) 2 0 G(ζ, η) (1/2) log | − | log(1+min zk(ζ) 2, zk(η) 2 ). ≤ − (1 + zk(ζ) 2)(1 + zk(η)) 2 ≤ {| | | | } | | | | 4. For any η Pn, G( , η) DMA (Pn,ω) and ∈ · ∈ loc (ω + ddcG( , η))n = δ , · η n where δη is the unit Dirac measure on P at the point η. Proof. This lemma follows from Lemma 3.5 by observing that in each open chart ( , zk) we have for (ζ, η) , Uk ∈Uk ×Uk 1 G(ζ, η)= N(z, w) log(1 + z 2), − 2 | | where z := zk(ζ) and w := zk(η). 4.2. The projective logarithmic potential. Let Prob(Pn) be the convex compact set of probability measures on Pn. Given µ Prob(Pn) we define its (projective) logarithmic potential as follows ∈ Gµ(ζ): = G(ζ, η)dµ(η), Pn Z ζ η = log | ∧ | dµ(η), Pn ζ η Z | || | As observed in ([As17]) the projective kernel G can be expressed in terms of n the geodesic distance d on the K¨ahler maniflod (P ,ωFS). Namely we have d(ξ, η) Gµ(ζ)= log sin dµ(η). Pn √2 Z Thanks to this formula , we see that if f is a radial function on Pn i.e. f(ζ) := g(d(ζ, a)) for a fixed point a Pn. Then choosing polar coordinates around a we have ∈ π/√2 (4.1) f(ζ)dV (ζ)) = g(r)A(r)dr, Pn Z Z0 16 S. ASSERDA, FATIMA Z. ASSILA AND A. ZERIAHI where A(r) is the ”area” of the sphere of center a and radius r. The expres- sion of A(r) is given by the formula 2n 2 A(r)= cn sin − (r/√2) sin(√2r), 2n where cn a constant depending on the volume of the unit ball in R . For more details, see [Rag71, Page 168],[AB77, Section 3] or [Hel65, Lemma 5.6]. This allows to give a simple example. Proposition 4.2. 1. Let σ be the Lebesgue measure asociated to the Fubini- Study volume form dV . Then for any ζ Pn, ∈ G (ζ)= α , σ − n where αn > 0 is a numerical constant given by the formula (4.3) below. n 2. For any µ Prob(P ), Gµ is a negative ω-plurisubharmonic function in Pn such that ∈ (4.2) Gµ(ζ)dV (ζ)= αn. Pn − Z Proof. We use the same computations based on the formula (4.1) as in ([As17]). 1. By (4.1) and the co-area formula d(ζ, η) Gσ(ζ) = log sin dV (η) Pn √2 Z π √2 r = log sin A(r)dr √2 Z0 π √2 r 2n 2 r = cn log sin sin − sin(√2r)dr √2 √2 Z0 π 2 2n 1 = 2√2cn (log sin(t)) sin − (t) cos(t)dt 0 Z1 2n 1 = 2√2 u − log udu. Z0 Therefore the statement follows if we set 1 2n 1 (4.3) αn := √2 u − log udu Z0 2. To prove the second statement it is enough to observe the following symetry n Gµ(ζ)dσ(ζ)= Gσ(η)dµ(η)= αnµ(P )= αn, Pn Pn − − Z Z thanks to the formula (4.2). LOGARITHMIC POTENTIALS 17 5. The Monge-Ampere` measure of the potentials n 5.1. The Monge-Amp`ere measure of Vµ in C . We begin this section by showing that Vµ belongs to the domain of definition of the complex Monge-Amp`ere operator for (n 3). ≥ Theorem 5.1. Let µ be a probability measure on Cn (n 2) with compact support. Then ≥ n 1) Vµ DMAloc(C ). ∈ 2,p Cn 2) Vµ Wloc ( ) for any 0 Proof. The first part of the theorem was proved in [As17] but for convenience we reproduce the proof here since we will use the same compuations to prove the other statements. 1. We will use the characterization due to B locki ([Bl06]). For ε> 0, set 2 2 2 ε 1 z w + z w + ε Vµ (z) := log | − | | ∧ 2 | dµ(w). 2 Cn 1+ w Z | | ε Cn Cn It is clear that Vµ ( ) C∞( ) and decreases towards Vµ as ε decreases ∈L ∩ n to 0. Since Vµ in plurisubharmonic in C , we have ε n p 2 r1 n V − − L (C ) for any r > 0. | µ | ∈ Loc 1 On the other hand, recall that z w 2 = z w z w 2, | ∧ | | i k − k i| 1 i ∂ ( z w 2+ z w 2)= z w + w (z w z w ) w (z w z w ). ∂z | − | | ∧ | m − m j m j − j m − i i m − m i m m ∂ ε 2 Vµ (z) ∂zm zm wm + m ε 1 z w + w z w Vµ (z) | − | 2 | || ∧ 2|dµ(w) |∇ | ≤ 2 Cn z w + z w Z | − | | ∧ | √2 (1 + w ) z w 2 + z w 2 | | | 2− | | 2 ∧ | dµ(w) ≤ 2 Cn z w + z w Z | −p | | ∧ | √2 (1 + w ) | | dµ(w) ≤ 2 Cn z w Z | − | √2 √2 dµ(w) + (1 + z ) . ≤ 2 2 | | Cn z w Z | − | In conclusion we have for z Cn, ∈ √2 √2 (5.1) V ε(z) + (1 + z )J (z). k∇ µ k≤ 2 2 | | µ,1 By Lemma 3.4 we conclude that V ε Lq (Cn), q < 2n, |∇ µ |∈ Loc ∀ and so V ε(z) 2 Lr2 (Cn) for r ε n p 2 ε c ε c ε p n p 1 V − − dV d V (dd V ) ω − − | µ | µ ∧ µ ∧ µ ∧ ZK ε n p 2 ε 2 c ε c ε n p 1 (Vµ ) − − Vµ dd Vµ ... dd Vµ ω − − ≤ K | ||∇ | ∧ ∧ ∧ ∧ Z p times − n 2 2 ε n p 2 | ε 2 ∂{z ε } ∂ ε n C (Vµ ) − − Vµ Vµ (z) ... Vµ (z) ω ≤ K | ||∇ | ∂zk1 ∂zl1 ∂zkp ∂zlp l,kX=1 Z n 2 2 ε n p 2 ε 2 ∂ ε ∂ ε C (Vµ ) − − r1 Vµ r2 Vµ (z) ... Vµ (z) , ≤ k k k∇ k ∂zk ∂zl s1 ∂zkp ∂zlp sp l,k=1 1 1 X where denotes the norm in Lt( K). k · kt LOGARITHMIC POTENTIALS 19 Since 0 p n 2, in this estimates we have a product of p + 2 n ≤ ≤ − k ≤ terms such that p + 1 n 1 terms are in Lloc with k Indeed since p + 1 We note for later use that the kernel 1 z w 2 + z w 2 N(z, w) := log | − | | ∧ | , (z, w) Cn Cn. 2 1+ w 2 ∈ × | | can be approximated by the smooth kernels 1 z w 2 + z w 2 + ǫ2 N (z, w) := log | − | | ∧ | , (z, w) Cn Cn. ǫ 2 1+ w 2 ∈ × | | n Since each function Nε( , wj) +(C ), we know by Lemma 3.4, that the measures · ∈L ddc N (z, w ) ... ddc N (z, w ) z ε 1 ∧ ∧ z ε n n are well defined probability measures on C . Hence we have for all w1, ..., wn Cn ∈ c c ddzNǫ(z, w1) ... ddzNǫ(z, wn) = 1. Cn ∧ ∧ Z Proposition 5.2. If µ be a probability measure on Cn then the complex Hessian currents associated to Vµ are given by the following formula: for any 1 k n, ≤ ≤ c k c c (dd Vµ) = dd N(., w1) ... dd N(., wk)dµ(w1)...dµ(wk), Cn k ∧ ∧ Z( ) in the sense of (k, k)-currents on Cn. n Proof. Since Vµ belongs to the domain of definition DMAloc(C ), the com- c k plex Monge-Amp`ere current (dd Vµ) is well defined. 20 S. ASSERDA, FATIMA Z. ASSILA AND A. ZERIAHI We assume that 2 k n. Set ≤ ≤ 1 1 z w 2 + z w 2 + ǫ2 Vµ,ǫ(z) := log | − | | ∧2 | dµ(w) 2 Cn 2 1+ w Z | | = Nǫ(z, w)dµ(w) Cn Z Let χ 0 be a positive smooth (n k,n k)-form with compact support n ≥ − (j)− j in C and denote for j = 2, ,n, by µ = µ⊗ the product measure on (Cn)j. Then applying Fubini’s· · · theorem and integration by parts formula, we obtain c c (k) Aε := χ(z) ∧ dd Nǫ(z,w1) ∧ ... ∧ dd Nǫ(z,wk)dµ (w) Zz Cn Z(Cn)k ∈ c c c (k) = Nǫ(z,w1)dd χ(z) ∧ dd Nǫ(z,w2) ∧ ... ∧ dd Nǫ(z,wk) dµ (w), Z(Cn)k Zz Cn ∈ n k where w := (w1, , wk) (C ) . Integrating by· parts · · and∈ applying again Fubini’s theorem we obtain Aε c c (k 1) = Nǫ(z,w1)dµ(w1) ddzχ(z) ∧···∧ ddzNǫ(z,wk)dµ − (w′) Zz Cn ZCn Z(Cn)k−1 ∈ c c c (n 1) = Vǫ(z) dd χ(z) ∧ dd Nǫ(z,w2 ∧···∧ dd Nǫ(z,wn)dµ − (w′). Zz Cn Z(Cn)k−1 ∈ k 1 where w′ := (w2, . . . , wk) C − ). Using Fubini’s theorem∈ and integrating parts once again, we obtain when k 2 ≥ Aε c c c c (k 2) = Nǫ(z,w2)dd χ ∧ dd Vǫ ∧ dd Nǫ(z,w3 ∧···∧ ddzNǫ(z,wk) dµ − . Z(Cn)k−2 Zz Cn ∈ Repeating this process k times we get the final equation c c (k) (5.5) χ ∧ dd Nǫ(·,w1) ∧ ... ∧ dd Nǫ(·,wk)dµ (w) ZCn Z(Cn)k c ǫ k = χ ∧ (dd Vµ ) . ZCn Now we want to pass to the limit as ε 0. The first term can be written as follows: ց c c (k) χ dd Nǫ( , w1) ... dd Nǫ( , wk)dµ (w1, , wk) Cn ∧ Cn k · ∧ ∧ · · · · Z Z( ) (k) = Iε(w1, , wk)dµ (w1, , wk), Cn k · · · · · · Z( ) where c c Iε(w1, , wk) := χ dd Nε( , w1) dd Nε( , wk). · · · Cn ∧ · ∧···∧ · Z LOGARITHMIC POTENTIALS 21 Observe that for any fixed (w , , w ) (Cn)k 1 · · · k ∈ ddcN ( , w ) ... ddcN ( , w ) ddcN( , w ) ... ddcN( , w ) ǫ · 1 ∧ ∧ ǫ · k → · 1 ∧ ∧ · k weakly in the sense of currents on Cn as ε 0. Hence the family of functions ց n n Iε(w1, , wk) are uniformly bounded on (C ) and by Fubinis’theorem, it converges· · · as ε 0 pointwise to the function → c c I(w1, , wk) := χ dd N(., w1) dd N(., wk). · · · Cn ∧ ∧···∧ Z Therefore by Lebesgue convergence theorem we conclude that (k) (k) lim Iε(w1, , wk)dµ = I(w1, , wk)dµ ε 0 Cn k · · · Cn k · · · → Z( ) Z( ) c c (k) = χ dd N( , w1) ... dd N( , wk)dµ (w1, , wk). Cn ∧ Cn k · ∧ ∧ · · · · Z Z( ) For the second term in (5.5), observe that since Vµ,ε Vµ as ε 0, it follows by the convergence theorem that the second termց converges alsoց and c k c k χ (dd Vµ,ǫ) χ (dd Vµ) Cn ∧ → Cn ∧ Z Z as ε 0. Nowց passing to the limit in (5.5), we obtain the required statement. We will need a more general result. Let φ be a plurisubharmonic function n n in C such that φ DMAloc(C ). We define the twisted potential associated to a Probability measure∈ µ by setting φ Vµ := Vµ + φ. Then we have the following representation formula: φ φ Vµ (z)= N (z, w)dµ(w), Cn Z where N φ(z, w) := N(z, w)+ φ(z), z Cn. ∈ We can prove a similar representation formula for the Monge-Amp`ere measure of the twisted potential. n n Proposition 5.3. If µ be a probability measure on C and φ DMAloc(C ) φ ∈ then the complex Monge-Amp`ere currents associated to Vµ are given by the following formula: for any 1 k n, ≤ ≤ c φ k c φ c φ (dd Vµ ) = dd N (., w1) dd N (., wk)dµ(w1) dµ(wk), Cn k ∧···∧ · · · Z( ) in the sense of (k, k)-currents on Cn. The proof is the same as above. 22 S. ASSERDA, FATIMA Z. ASSILA AND A. ZERIAHI 5.2. The regularizing property. We prove a regularizing property of the operator V which generalizes and imporves a result of Carlehed [Carlehed99]. Recall that a positive measure µ on Cn is said to have an atom at some point a Cn if µ( a ) > 0. ∈ { } n n Theorem 5.4. Let µ be a probability measure on C and φ DMAloc(C ) which is smooth in some domain B Cn. Then the Monge-Amp`ere∈ current c φ n ⊂ (dd Vµ ) are absolutely continuous with respect to the Lebesgue measure on B if and only if µ has no atoms on B. The proof of the ”if part” of the theorem is based on the following lemma. Lemma 5.5. Assume that φ is smooth in some domain B Cn and let w , w Cn be such that w = w . Then the Borel measure⊂ 1 · · · n ∈ 1 6 2 ddcN φ( , w ) ddcN φ( , w ), · 1 ∧···∧ · n is absolutely continuous with respect to the Lebesgue measure on B. Proof. The proof is based on an idea of Carlehed [Carlehed99]. We are n reduced to the proof of the following fact: let a, b1, . . . , bm C with 1 m n k such that b = a, for 1 j m, then the following∈ current ≤ ≤ − j 6 ≤ ≤ k µ := ddcN φ( , a)) ddcN φ( , b )). m,k · · j 1 j m ≤^≤ is absolutely continuous with respect to the Lebesgue measure on B. Indeed, since the current µ is smooth in B a, b , . . . , b , it is enough k \ { 1 m} to show that µm,k puts no mass at the points a, b1, . . . , bm that belong to B. Assume for simplicity that a B and define the function ∈ m u(z) = log( z a 2 + z a 2)+ φ(z)+ N φ(z, b ). | − | | ∧ | j Xj=1 n n Observe that u is a plurisubharmonic function in C such that u DMAloc(C ). Since u(z) log( z a 2 + z a 2) as z a, it follows from∈ Demailly’s comparison≃ theorem| − ([De93])| | that∧ | → n (ddcu)n = ddc log( a 2 + a 2) = 1. a a |·− | |·∧ | Z{ } Z{ } On the other hand, performing the exterior product, we obtain n (ddcu)n ddc log( a 2 + a 2) + µ δ + µ , ≥ |·− | |·∧ | k ≥ a m,k n in the weak sense on C . Therefore µm,k( a ) = 0, which proves the required statement and the Lemma follows. { } We now prove the theorem. LOGARITHMIC POTENTIALS 23 Proof. Assume first that the measure µ has an atom at some point w Cn ∈ then µ cδw, where c := µ( w ) > 0. Therefore from the definition we see that z≥ Cn { } ∀ ∈ V φ(z) cN(z, w)+ cφ(z). µ ≤ φ This implies that Vµ has a positive Lelong number at w at least equal to c φ n n c and then the Monge-Amp`ere mass satisfies (dd Vµ ) ( w ) c > 0 (see [Ce04]). { } ≥ c φ n Assume now that µ has no atoms in B. We want to show that (dd Vµ ) is absolutely continuous with respect to Lebesgue measure on B. Indeed, let K B be a compact set such that λ2n(K) = 0 and let us prove that c φ⊂n (dd Vµ ) (K) = 0. Set ∆ := (w, ..., w) : w B Cn ... Cn. B { ∈ }⊂ × × By Fubini theorem, denoting by w = (w , , w ) (Cn)n, we get 1 · · · n ∈ c φ n (dd Vµ ) ZK c φ c φ (n) = dd N (·,w1) ∧···∧ dd N (·,wn)dµ (w) ZK Z(Cn)n c φ c φ (n) = dd N (·,w1) ∧···∧ dd N (·,wn)dµ (w) n n ZK Z(C ) ∆B \ c φ c φ (n) + dd N (·,w1) ∧ ... ∧ dd N (·,wn)dµ (w). ZK Z∆B Since µ puts no mass at any point in B, it follows from Fubini’s theorem that (n) ∆B has a zero measure with respect to the product measure µ = µ ... µ on (Cn)n. Hence we have ⊗ ⊗ c φ n (dd Vµ ) ZK c φ c φ (n) = dd N (·,w1) ∧ ... ∧ dd N (·,wn)dµ (w1, · · · ,wn) n n ZK Z(C ) ∆B \ c φ c φ (n) = dd N (·,w1) ∧ ... ∧ dd N (·,wn)dµ (w1, · · · ,wn). n n Z(C ) ∆B ZK \ We set f(w , ..., w )= ddcN φ( , w ) ... ddcN φ( , w ). 1 n · 1 ∧ ∧ · n ZK Then using the previous lemma,we see that if (w1, , wn) / ∆B, the mea- sure · · · ∈ ddcN φ( , w ) ... ddcN φ( , w ), · 1 ∧ ∧ · n is absolutely continuous with respect to the Lebesgue measure on B. Hence f(w , . . . , w ) = 0 if (w , , w ) / ∆ and then 1 n 1 · · · n ∈ B c φ n (n) (dd Vµ ) = f(w1, . . . , wn)dµ (w1, , wn) = 0 n n K (C ) ∆B · · · Z Z \ and the theorem is proved. 24 S. ASSERDA, FATIMA Z. ASSILA AND A. ZERIAHI 6. Proofs of Theorems 6.1. Localization of the potential. To study the projective logarithmic potential we will localize it in the affine charts and use the previous re- sults. Let (χj)0 j n a fixed partition of unity subordinated to the covering ≤ ≤ ( j)0 j n. We define mj := χjdµ and I = Iµ := j 0, ,n ; mj = 0 . U ≤ ≤ { ∈ { · · · } 6 } Then I = and for j I, the measure µj := (1/mj )χjµ is a probability 6 ∅ n ∈ R measure on P supported in the chart j and we have a the following convex decomposition of µ U µ = mjµj. j I X∈ Therefore the potential Gµ(ζ) can be written as G (ζ)= m G (ζ), ζ Pn, µ j µj ∈ j I X∈ so that we are reduced to the case of a compact measure supported in an affine chart. Without loss of generality we may always assume the µ is compactly supported in . Then the potential G can be written as follows U0 µ 2 G ζ η µ(ζ) := G(ζ, η)dµ(η) = (1/2) log | ∧2 |2 dµ(η). 0 0 ζ η ZU ZU | | | | Since we, integrate on 0 we have η0 = 0 and we can use the affine coordi- nates U 6 w := (η /η , , η /η ). 1 0 · · · n 0 Therefore ζ η 2 = ζ w ζ 2 + ζ w ζ w 2. | ∧ | | 0 j − j| | i j − j i| 1 j n 1 i z w 2 + z w 2 (6.1) G(ζ, η) = (1/2) log | − | | ∧ | (1 + z 2)(1 + w 2) | | | | z w 2 + z w 2 (6.2) = (1/2) log | − | | ∧ | (1/2) log(1 + z 2) 1+ w 2 − | | | | = N(z, w) (1/2) log(1 + z 2), − | | LOGARITHMIC POTENTIALS 25 where z w 2 + z w 2 N(z, w) := (1/2) log | − | | ∧ | 1+ w 2 | | is the projective logarithmic kernel on Cn which was studied in the previous sections. 6.2. Proof of Theorem A. 1. From the previous localization, it follows that for each j I, we have for ζ ∈ ∈Uj G (ζ)= V (z) (1/2) log(1 + z 2), µj µj − | | j where z := z (ζ) are the affine coordinates of ζ in j. By Theorem 2.10 U G and Theorem 5.2, it follows that for each j I, the function µj is ωFS- p Pn ∈ plurisubharmonic and Gµj L ( ) for any 0 Aknowlegements : This paper is dedicated to the memory of Professor Ahmed Intissar who passed away in July 2017. He was a talented mathemati- cian who had a big scientific influence among the mathematical community in Morocco. We are grateful to him for his generosity, mathematical rigor and all what he taught us. We would like to thank Professor Vincent Guedj for many interesting discussions and useful suggestions on the subject of this paper. We also thank the referee for his careful reading. By pointing out several typos and making useful suggestions, he helped to improve the presentation of the final version of this article. References [AB77] R.Askey, H.Binghani, Gaussian processes on compact symetric spaces. Z.Wahrscheinchkeitstheorie und Verw.Gebiete 37 (1976-1977) no 2, 127-145. [As17] F. Z. Assila, Logarithmic potentials on Pn. 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ZERIAHI Universite´ Ibn Tofail, Kenitra´ Maroc, E-mail address: [email protected] Universite´ Ibn Tofail, Kenitra´ Maroc, E-mail address: [email protected] IMT; UMR 5219, Universite´ de Toulouse; CNRS, Paul Sabatier, F-31400 Toulouse, France E-mail address: [email protected]