The proof turns out to be a fairly straightforward extension of the proof of the main result in [4]

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L^∞-BOUNDEDNESS OF DENSITY FOR FREE ADDITIVE CONVOLUTIONS

SERBAN TEODOR BELINSCHI

We show that the density of the free additive convolution of two Borel probability measures µ, ν on Rwhose Cauchy transforms have reasonably good behaviour at innity is bounded whenever µ({a}) +ν({b})<1for alla, b∈R.

AMS 2010 Subject Classication: 46L54, 30E20.

Key words: free convolution, analytic transforms, regularity.

In this note, we prove that the density of the free additive convolution of two Borel probability measures supported on R whose Cauchy transforms behave well at innity can be unbounded only if there exist points u, v∈R so thatuis an atom of the rst measure,vof the second, and the weight of the two atoms adds up to at least one. In particular, ifµ, νare two compactly supported Borel probability measures onRandµ({u}) +ν({v})<1for all pairsu, v∈R, thenµν is absolutely continuous with respect to the Lebesgue measure and its densityt7→ ^d^(µν)(t)_d_t ∈C_c(R), the space of compactly supported continuous functions on R. Our result is in fact slightly stronger: we refer the reader to Corollary 8 for the complete statement. The proof turns out to be a fairly straightforward extension of the proof of the main result in [4].

Free probability, introduced by Voiculescu in the '80s for the purpose of studying free group factors, turned out to have important applications in random matrix theory and related elds. This line of research was pioneered by the groundbreaking work [10] (for an introduction and more recent develop- ments, see [2] and references therein). In particular, free additive convolution allows one to investigate the asymptotic behaviour of the eigenvalue distribution of sums of independent (asymptotically) unitarily invariant large random matrices: assume thatAN, BN are twoN×N selfadjoint random matrices, independent from each other, and so that the distribution ofA_N does not change under conjugation with unitary matrices: AN

=D V^∗ANV for all V ∈ U_N. If

REV. ROUMAINE MATH. PURES APPL. 59 (2014), 2, 173184

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the eigenvalues distribution of A_N (respectively B_N) converges to µ (respectively ν), then the eigenvalue distribution of A_N +B_N converges toµν, the free additive convolution of µand ν [10]. This is one of the main motivations behind the study of properties of free additive convolutions of Borel probability measures on the real line. (Remarkably, the above-mentioned property can be taken also as a denition of the operation ; for an introduction to free probability, we refer to [12] and references therein.) In fact, our investigation was motivated by a question of Alice Guionnet, originating in problems from random matrix theory.

The rest of the paper is organized as follows: in Section 1 we gather a few auxiliary results and show that Biane's subordination functions extend continuously toR(an extension of ([4], Theorem 3.3) and in Section 2 we prove our main result and indicate some applications.

1. PRELIMINARY RESULTS: CONTINUITY OF THE SUBORDINATION FUNCTIONS

1.1. Analytic transforms

Consider two arbitrary Borel probability measures µ, ν supported on R, neither of them a point mass. We shall denote by supp(µ)the topological support ofµ. As in [4], our main tools in the study of the free additive convolution µν will be analytic transforms related to the Cauchy (or Cauchy-Stieltjes) transform: the Cauchy transform of µis dened by

Gµ(z) = Z

R

dµ(t)

z−t, z∈C\supp(µ).

The importance of Gµ stems from, among others, its imaginary part:

=G_µ(x+iy) =R

R

−y

(x−t)²+y²dµ(t). It is a well-known result in classical analysis that one can recoverµas the limit in the weak topology of−π⁻¹=G_µ(x+iy)dx asy↓0 :

(1) Z

R

f(x)dµ(x) =−1 πlim

y↓0

Z

R

f(x)=G_µ(x+iy)dx, for allf ∈C_c(R), where, as in the introduction,C_c(R)denotes the space of compactly supported continuous functions on the real line. We shall dene as well the reciprocal of the Cauchy transformFµ(z) = _G¹

µ(z) and the h-transform ofµ,hµ(z) =Fµ(z)−z. Some properties of these transforms that will be relevant for our purposes are listed below. For further details and proofs we refer to ([1], Chapter III).

(1) F_µis an analytic self-map of the (open) upper half-plane C⁺. Moreover,

=F_µ(z)≥ =zfor allz∈C⁺. If there is az₀∈C⁺so that=F_µ(z₀) ==z₀,

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then there is a constant a∈ Rso that F_µ(z) =z−a, z ∈C, and thus, µ=δ_a, the Dirac distribution concentrated ina.

(2) In particular, h_µ(z) = F_µ(z)−z:C⁺ → C⁺ whenever µ is not concentrated in one point.

(3) limy→+∞F_µ(iy)/iy= 1 and limz→∞,=z>cF_µ(z) =∞ for any xed c >0. (4) Conversely, a map F satisfying (1)(3) above is of the form F = F_µ;

moreover, equation (1) allows us to explicitly recover µfromF.

(5) Let µ= µ^ac+µ^s be the Lebesgue decomposition of µ in the absolutely continuous, and respectively singular, components with respect to the Lebesgue measure. If we denote by g the density of the absolutely continuous part ofµ, dµ^ac(x) =g(x)dx, then

• g(x) = limy↓0−π⁻¹=G_µ(x+iy),for Lebesgue-almost allx∈R;

• limy↓0−π⁻¹=G_µ(x+iy) = +∞, forµ^s-almost allx∈R;

• Finally, with the conventions ¹₀ =∞ and _∞¹ = 0, 1

µ({x}) = lim

y↓0

Fµ(x+iy) iy = lim

y↓0

1 iyG_µ(x+iy).

(6) For any function F as in (4), there exist a ∈ R and ρ a nite Borel measure onRso that

(2) F(z) =a+z+

Z

R

1 +tz

t−z dρ(t), z∈C⁺.

Next to last item above makes clear the importance of the boundary behaviour of analytic maps for our study. As in [4], it turns out that the most appropriate language for this study is that of cluster sets [7]. If a functionf is dened on a domainD⊆C,x∈D andA⊆D∪ {x}, then

C_A(f, x) ={w | ∃{z_n}_n∈_N⊂A\ {x} such that lim

n→∞zn=x, lim

n→∞f(zn) =w}

is the cluster set of f at xalong A. IfA=D, we shall writeC(f, x)instead of C_D(f, x).Note thatC_A(f, x)is empty wheneverxis not a point of accumulation of A. It follows quite easily (see [7], Theorem 1.1) that if f:C⁺ → C and x∈R∪{∞}, thenC(f, x)is either one point or a continuum. If for any angleΓ with vertex atx, bisectorx+i[0,+∞)and measure strictly less thanπ,C_Γ(f, x) contains exactly one point and the point is independent of the measure ofΓ in [0, π), we say that f has nontangential limit at x. We dene the nontangential limit of f at x to be the one point inC_Γ(f, x) and we denote it by ^f(x). (If x=∞,the denition is the same, with the bisector being i(0,+∞).) We shall write ^limz→xf(z) to indicate that we take the nontangential limit off at x. The following theorem is one of the fundamental results in complex analysis ([7], Theorems 2.2 and 2.3):

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Theorem 1. Assume that f:C⁺ → C⁺ is analytic. Then for Lebesgue- almost all x ∈ R∪ {∞}, ^f(x) exists and is nite. Moreover, if there exists a path γ included in C⁺∪ {x} terminating at x so that Cγ(f, x) contains only the pointa (i.e. the limit off atx alongγ equals a), then f has nontangential limit at x and ^f(x) =a.

In particular, all limits in the list (1)(5) of properties of F_µ and G_µ can be taken nontangentially.

It is known that an analytic function on a given domain inCis determined by its values on any set with accumulation points inside its domain. The following theorem of Privalov ([7], Theorem 8.1) gives a boundary version of this fact:

Theorem 2. Letf:C⁺−→Cbe an analytic function. Assume that there exists a set E of nonzero linear measure inRsuch that the nontangential limit of f exists at each point ofE, and equals zero. Then f(z) = 0 for all z∈C⁺.

Finally, as the last relation in item (5) above suggests, it will be useful to have a version of the derivative of an analytic function at a point in its boundary. This is provided by the Julia-Caratheodory Theorem [8], which we shall reproduce here in its upper half-plane version, as given in [4]:

Theorem 3. Let f:C⁺ → C⁺ be analytic, and let a ∈R. Assume that

^f(a) =c∈R.Then

^lim

z→a

f(z)−c

z−a = lim inf

z→a

=f(z)

=z , where the equality is considered in C∪ {∞}. Conversely, if

lim inf

z→a

=f(z)

=z <∞,

then ^limz→af(z) exists and belongs to R∪ {∞}. Moreover, if f is not constant, then we have lim infz→a=f(z)/=z >0.

1.2. The subordination phenomenon

The importance of the Cauchy transform in free probability became ev- ident in the pioneering work [9] of Voiculescu, in which he found an analytic transform, dened in terms of Gµ, which linearizes free convolution, namely the R-transform. However, in this paper we shall only make use of another important analytic tool, Biane's subordination functions [6, 11]:

Theorem 4. Given µ, ν two Borel probability measures on R, there exist unique analytic functions ω₁, ω₂:C⁺→C⁺ with the property that:

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(a) ^limz→∞ω_j(z)/z= 1, j∈ {1,2};

(b) Fµ(ω1(z)) =Fν(ω2(z)) =F_µν(z), z∈C⁺; (c) ω1(z) +ω2(z) =F_µν(z) +z, z∈C⁺.

The property in item (c) has been proved in [5].

In ([4], Theorem 3.3 (3)) it is shown that, under certain mild restrictions on the supports of µ and ν, the subordination functions extend continuously to R. Below we shall remove these restrictions. In order to do that, we need a strenghtened version of ([4], Lemma 2.18):

Lemma 5. Let f:C⁺ →C⁺ be a nonconstant analytic function. Assume that x∈R∪ {∞} is so that C(f, x) =R∪ {∞}. Then there exists a sequence of mutually disjoint segments {[z_n, wn]}_n∈_N⊂C⁺ so that:

(i) limn→∞zn= limn→∞wn=x;

(ii) limn→∞f(z_n) =a < b= limn→∞f(w_n);

(iii) The setsf([z_n, w_n]), n∈N are mutually disjoint in C⁺; (iv) For anyc∈(a, b), there is an n_c∈N so that

f([zn, wn])∩(c+i(0,+∞))6=∅ for all n≥nc; (v) For any[c, d]⊂(a, b),

n→∞lim supn

max{=v|v∈f([z_n, w_n])∩(t+i(0,+∞))}

t∈[c, d]o

= 0;

(vi) For anyc∈R\[a, b], there is an nc∈Nso that:

f([z_n, w_n])∩(c+i(0,+∞)) =∅ for all n≥n_c.

Before starting the proof of the lemma, we should rst note that, while this make no dierence when x is nite, it is most appropriate to consider hyperbolic, rather than Euclidean, segments [zn, wn]; then, there is no loss of generality in assuming that x is nite. Second, that (as it will be even clearer from the proof) it is only necessary to assume thatC(f, x)⊆R∪{∞}. However, only the case of equality will be relevant for us.

Proof. The proof follows by and large the proof of ([4], Lemma 2.18).

Pick c₁ < c₂ in the interior of C(f, x) (relative to the topology of R). By the denition of C(f, x), there exist sequences {z⁽¹⁾_n }_n∈_N,{z_n⁽²⁾}_n∈_N⊂C⁺ so that:

(a) |z₁⁽²⁾−x|<1;

(b) |z_n⁽²⁾−x|>2|z_n⁽¹⁾−x|>4|z⁽²⁾_n+1−x|, for all n∈N,n >0; (c) |f(z_n^(j))−c_j|< ¹_n, for all n∈N, n >0, j ∈ {1,2};

(d) |f(z_n⁽¹⁾)−c₁|>|f(zn⁽²⁾)−c₂|>|f(z_n+1⁽¹⁾ )−c₁|, for alln∈N.

Dene γ: [0,1]→C⁺∪ {x}so that γ(0) =i, γ(1) =x,γ 1−_2n¹

=zn⁽²⁾, γ

1−_2n+1¹

= zn⁽¹⁾ and γ is linear on each of the intervals h

1−_k¹,1−_k+1¹ i

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(note that we can consider linear to mean hyperbolic-linear, in the sense of γ([1− ¹_k,1− _k+1¹ ]) following the hyperbolic geodesic between endpoints, but, as mentioned before, this eectively makes no dierence for nite x). Condi- tion (b) above guarantees thatγ is a nite-length piecewise linear simple path in C⁺∪{x}andlimt→1γ(t) =x. Fix now∈ 0,min

(1+|c₁|)⁻¹,(1+|c₂|)⁻¹,^c²₁₀₀^−c¹ . SinceC⁺∩C(f, x) =∅, there exists ann()∈Nso that

f γ 1−_2n¹

−c2

+

f

γ

1−_2n+1¹

−c1

< ₂ andf γ

1−_2n¹ ,1

⊂C⁺\Xfor alln≥n(), where

X=

z∈C:|<z| ≤ 1

, ≤ =z≤ 1

.

We claim that there exist 1−_2n()+1¹ ≤t< s ≤1−_2n()+2¹ so that:

(α) |f(γ(t))−c₁| ≤,|f(γ(s))−c₂| ≤, and

(β) f(γ([t, s]))∩(c₁+i[0, )) =∅, f(γ([t, s]))∩(c₂+i[0, )) =∅. Indeed, this is a simple connectedness argument: we know from our con- struction that

f

γ

1−_2n()+1¹

−c₁

< /2and f

γ

1−_2n()+2¹

−c₂ <

/2. By the continuity of f and γ,f γh

1−_2n()+1¹ ,1−_2n()+2¹ i

is a con- nected set included inC⁺\X. We pick

t= sup

t∈

1− 1

2n() + 1,1− 1 2n() + 2

:|f(γ(t))−c₁|<

,

s= inf

t∈

t,1− 1 2n() + 2

:|f(γ(t))−c₂|<

,

choices obviously allowed by the analyticity of the correspondence(1−_2n()+1¹ ,1−

1

2n()+2)3t7→f(γ(t))∈C⁺\X.The set(c1+i[0, ])∪(c₂+i[0, ])disconnects C⁺\X so a connectedness argument shows thatf(γ([t, s]))is a (usually not simple) curve included in only one of the following two sets:

z∈C⁺\X:=z < ,<z∈(c1, c2) , C⁺\X

\

z∈C⁺\X:=z < ,<z∈(c1, c2) .

Trivially now, exactly one of the following two alternatives must occur:

(A) • f(γ([t, s]))∩(c+i(0, ))6=∅for allc∈[c1+, c2−],

• f(γ([t, s]))∩(c+i(0,+∞)) =∅for all c∈R\[c₁, c₂],

• f(γ([t, s]))∩ c+i[¹,+∞)

=∅for all c∈R, or

(B) • f(γ([t, s]))∩(c+i[0, ]) =∅for all c∈(c1, c2),

• f(γ([t, s]))∩(c+i(0, ))6=∅for allc∈

−¹, c1−

∪

c2+,¹ ,

• f(γ([t, s]))∩ c+i[¹,+∞)

6=∅for all c∈

−¹,¹ .

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By continuity, f(γ([t, s])) is a compact set in C⁺, hence, at strictly positive distance, say ₁, from R. By repeating the previous argument with replaced by the (necessarily smaller) 1 > 0, we nd n(1) > n() and s <

1 − _2n(¹

1)+1 ≤ t₁ < s₁ ≤ 1 − _2n(¹

1)+2 so that f(γ([t₁, s₁])) falls under either(A1)or(B1). Iterating this procedure, we construct a sequence{_n}_n∈_N decreasing to zero and disjoint compact intervals [tn, sn]⊂(sn−1,1)⊂(0,1) so that{f(γ([t_n, s_n]))}_n is a sequence of disjoint paths satsfying either (A_n) or (Bn). Of course, innitely many paths will fall in at least one of the two categories. The proof of our lemma is now almost completed: if for innitely many n the path f(γ([t_n, s_n])) falls under (A_n), then pick a = c₁, b = c₂, zn = γ(tn), wn = γ(sn) (along a subsequence, if necessary). If only nitely many paths satisfy(An), then picka=c2andbto be any real number strictly larger thanc₂.

The reader will notice that the statement of the lemma can be somehow improved, at the cost of little eort, but much space. Using this lemma, we can prove the statement of ([4], Theorem 3.3 (3)) in full generality:

Theorem 6. Let µ, ν be two Borel probability measures on R, neither concentrated in a single point. If ω1, ω2 are the two subordination functions provided by Theorem 4, then ω1, ω2 extend continuously toC⁺∪Ras functions with values in C∪ {∞}.

Proof. The vast majority of the proof is contained in the proof of ([4], Theorem 3.3 (3)). We shall outline its steps below, and prove in detail only the case not covered by ([4], Theorem 3.3 (3)). It is shown in the proof of ([4], Theorem 3.3 (3)) that:

(1) If either C(ω₁, x)∩C⁺6=∅ or C(ω₂, x)∩C⁺6=∅, then cardC(ω₁, x) = cardC(ω2, x) =cardC(Fµν, x) = 1;

(2) If cardC(ω₁, x) > 1, then both C(ω₁, x) and C(ω₂, x) are innite, i.e.

either closed intervals in R with nonempty interiors, or complements in R∪ {∞}of open (possibly empty) intervals;

(3) IntC(ω1, x)∩supp(µ^ac) =IntC(ω2, x)∩supp(ν^ac) =∅,where Int denotes the interior inR∪ {∞};

(4) IfR\supp(µ)andR\supp(ν)are both nonempty, thenω₁, ω₂ both extend continuously toR;

(5) If eitherR\supp(µ)orR\supp(ν)is empty and eitherC(ω₁, x)orC(ω₂, x) contains more than one point, thenC(ω1, x) =C(ω2, x) =R∪ {∞}. Assume towards contradiction that C(ω1, x), C(ω2, x) contain more than one point. By item (4) above, it follows that we only need to consider the case when at least one of R\supp(µ), R\supp(ν) is empty, and by items (5) and

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(3) it follows that µ = µ^s, ν = ν^s. Without loss of generality, assume that supp(µ) =supp(µ^s) =R. Pick two points a < bin Rfor which the conclusion of Lemma 5 applies to ω1 and x. Sinceµ=µ^s and [a, b]⊂supp(µ), the set

N = n

^lim

z→chµ(z) :c∈(c1, c2) so thathµ has real nontangential limit in c o

is dense inR∪ {∞}for anyc1 < c2 ∈[a, b]([3], Proposition 1.9). By Lemma 5 (and with the notations from Lemma 5), for any c ∈ (c₁, c₂), there exists a z^(c)n ∈[z_n, w_n],n∈N, so thatω₁(zn^(c))∈c+i(0,+∞)andlimn→∞ω₁(z^(c)n ) =c. In particular, {ω₁(z^(c)_n )}_n tends to c nontangentially. Since by Theorem 1,h_µ has nite nontangential limit for almost all pointsc∈[a, b], we obtain that the set

O =n

n→∞lim h_µ(ω₁(z_n^(c))) +z_n^(c)

c∈[c₁, c₂], h_µhas real nontangential limit in co is also dense in R∪ {∞}. But by Theorem 4 (b,c), ω2(z) = z+F_µν(z)− ω₁(z) =z+F_µ(ω₁(z))−ω₁(z) = h_µ(ω₁(z)) +z. Thus, the sequence of curves {ω₂([zn, wn])}_n∈_N has as limit all of R∪ {∞}, in the sense that for anyM >1 there exists n(M) ∈ N so that (i) ω2([zn, wn])∩X¹

M = ∅ for all n ≥ n(M), and (ii) for anyd∈[−M, M]with the possible exception of at most one point, ω2([zn, wn])∩ d+i

0,_M¹

6=∅for alln≥n(M). Thus, for any suchdone can nd zn^(d) ∈ [zn, wn] so that <ω₂(zn^(d)) =d and ω2(z^(d)n ) → d as n→ ∞. Since ν is purely singular as well, by [3, Proposition 1.9], there is an M > 0 large enough so that one can ndd∈[−M, M]so that^h_ν(d)>|a|+|b|+|x|+ 10.

This implies that

n→∞lim ω₁(z_n^(d)) = lim

n→∞z^(d)_n +h_ν(ω₂(z^(d)_n ))> x+|a|+|b|+|x|+ 10≥ |a|+|b|+ 10, an obvious contradiction with the fact that, by item (vi) of Lemma 5,

<ω₁([zn, wn])∈[a−1, b+ 1]forn∈Nlarge enough. We conclude thatC(ω1, x) and C(ω₂, x) must indeed contain only one point. Since x ∈R was arbitrarily chosen, the functions ω₁, ω₂ extend continuously to R.

2. BOUNDEDNESS OF THE CAUCHY TRANSFORM OF µν

We are now ready to prove the main result of this paper.

Theorem 7. Let µ, ν be two Borel probability measures on R with the property that Fµ, Fν are continuous at innity. Assume that the setG_µν(C⁺) is unbounded. Then there exists u, v∈R so that µ({u}) +ν({v})≥1.

Proof. We mostly use the results and methods from [4, 5]. Let us start by assuming that there exists a sequence{z_n}n∈N∈C⁺, so thatlimn→∞G_µν(z_n) =∞.

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By dropping if necessary to a subsequence, we may assume without loss of generality that limn→∞z_n = c ∈ supp(µν)∪ {∞}. Of course, then we have limn→∞F_µν(zn) = 0. It will be necessary rst to show that c6=∞. Assume towards contradiction that c = ∞. Then it follows by item (3) of Subsec- tion 1.1 that 0,∞ ∈ C(F_µν,∞). It is clear that C(F_µν,∞)∩C⁺ = ∅ : if not, let e ∈ C(F_µν,∞)∩C⁺ and nd {z_n^(e)}_n ⊂ C⁺ converging to ∞ so that F_µν(z^(e)n ) → e. By relation (c) in Theorem 4 there is a subsequence of {z_n^(e)}_n on which at least one of ω1, ω2, say ω1, has imaginary part strictly greater than ^=e₃ . If {ω₁(z^(e)n )}_n has an accumulation point in C⁺+ ^e₄, then F_µ extends analytically around it, providing an obvious contradiction. If not, by item (3) of Subsection 1.1, we obtain C⁺ 3 e = limn→∞F_µν(z_n^(e)) = limn→∞F_µ(ω₁(zn^(e))) = limz→∞,=z>=e/4F_µ(z) = ∞, a contradiction. Thus, C(F_µν,∞) ⊆ R∪ {∞}. Since 0,∞ ∈ C(F_µν,∞), it follows that at least one of [−∞,0],[0,+∞] is included in C(F_µν,∞). Consider {z_n^(e)}_n tending to innity so that =F_µν(z^(e)n ) → 0 and <F_µν(z^(e)n ) = e 6= ∞ (it follows from ([4], Lemma 2.18) that this is possible for all but nitely many points e ∈ C(F_µν,∞)). If either {ω₁(zn^(e)) : n ∈ N} or {ω₂(zn^(e)) : n ∈ N} is unbounded, then, by the hypothesis of continuity at innity of F_µ, F_ν, one of limn→∞Fµ(ω1(zn^(e))),limn→∞Fν(ω2(zn^(e)))cannot be nite, a contradiction. If ω1, ω2 are bounded on {z^(e)_n }_n, then, as, by Theorem 4, (a), ∞ ∈ C(ω1,∞), C(ω₁,∞) contains an innite length interval itself. Pick a d ∈ C(ω₁,∞)\ {∞} so that ^F_µ(d) < ∞ exists and ([4], Lemma 2.18) applies to ω₁ and d. Taking a sequence {z_n^(d)}_n tending to innity so that <ω₁(z_n^(d)) = d and

=ω₁(z_n^(d)) → 0, we conclude by Theorem 4 (c,b) that limn→∞ω₂(z_n^(d)) = limn→∞Fµ(ω1(zn^(d)))− ω1(zn^(d)) +z^(d)n = ^Fµ(d) − d+∞ = ∞, so, since

^Fµ(d) = limn→∞Fµ(ω1(zn^(d))) = limn→∞Fν(ω2(zn^(d))), by item (3) of Sub- section 1.1,Fν is not continuous at innity, which contradicts our hypothesis.

So c∈R. Using Theorem 6 above, we conclude that

z→climωj(z) = lim

n→∞ωj(zn) =ωj(c)

exists for j = 1,2. Next, observe that ω_j(c)6=∞. Indeed, otherwise we would have

∞= lim

v→∞Fµ(v) = lim

z→cFµ(ω1(z)) = lim

n→∞Fµ(ω1(zn)) = lim

n→∞F_µν(zn) = 0, an obvious contradiction. A similar reasoning for Fν andω2 proves our claim.

By Theorem 1, the nontangential limit of F_µ at ω₁(c) exists and equals zero.

A similar conclusion holds for F_ν and ω₂. Julia-Caratheodory's Theorem 3

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implies that

^ lim

v→ω₁(c)

F_µ(v)

v−ω1(c) = lim inf

v→ω₁(c)

=F_µ(v) exists in[1,+∞)∪ {∞}. Same holds for ν. =v

Now, since it was covered in [5], we can refrain from considering the case when(µν)({c})>0. So assume without loss of generality that(µν)({c}) = 0. As we have seen in item (5) of Subsection 1.1, this implies that the Julia- Caratheodory derivative ofF_µν at cis innite.

Let us now put together the previous formulas as in [4].

1

µ({ω₁(c)}) −1 = ^ lim

v→ω₁(c)

F_µ(v) v−ω1(c) −1

= lim inf

v→ω₁(c)

=F_µ(v)

=v −1

≤ lim inf

z→c

=F_µ(ω1(z))

=ω₁(z) + =z

=ω₁(z) −1

= lim inf

z→c

=ω₂(z)

=ω₁(z)

=

lim sup

z→c

=ω₁(z)

=ω₂(z) −1

=

lim sup

z→c

=F_ν(ω2(z))

=ω₂(z) + =z

=ω₂(z) −1 −1

≤

lim sup

z→c

=F_ν(ω2(z))

=ω₂(z) −1 −1

≤

lim inf

v→ω₂(c)

=F_ν(v)

=v −1 −1

=

^ lim

v→ω₂(c)

F_ν(v) v−ω₂(c) −1

−1

=

1

ν({ω₂(c)}) −1 −1

.

This will provide us with the desired result. If µ({ω₁(c)}) = 0, then we obtain immediately that we must have ν({ω₂(c)}) = 1, i.e. ν = δ_ω₂_(c). Similarly, if we interchange µ with ν. So both measures must in fact have nontrivial atoms at the corresponding ω_j(c). Multiplication will give us

1≥

1

ν({ω₂(c)}) −1 1

µ({ω₁(c)})−1

, so that

1≤ν({ω₂(c)}) +µ({ω₁(c)}).

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This, together with Bercovici and Voiculescu's result [5] on atoms, provides us with the claimed conclusion.

It should be noted that compactly supported measures satisfy the hypothesis of the above theorem, but they are by no means the only ones; the Gaussian and the Cauchy distributions are two other obvious examples of such measures.

Corollary 8. For any two probability measures µ, ν satisfying the hypothesis of the previous theorem, if we haveµ({a}) +ν({b})<1for alla, b∈R, then µν is absolutely continuous with respect to the Lebesgue measure and the density t7→ ^d(µν)(t)_dt is uniformly bounded and continuous.

Proof. The boundedness is a trivial consequence of the above theorem. The continuity statement follows quite easily now: assume towards contradiction that C(F_µν, c) is nontrivial for some c ∈ R. As seen before, this necessarily implies that C(F_µν, c) ⊆ R∪ {∞} contains a nontrivial interval. Then ω₁(c) = ω₂(c) = ∞ by Theorems 4 and 6. However, since Fµ(∞) = Fν(∞) = ∞, this provides readily a contradiction. Thus, F_µν extends continuously to R. This implies that G_µν also extends continuously, and thus, G_µν(C⁺ ∪R) being bounded, the density is continuous by equation (1).

Acknowledgments. The author is grateful to Alice Guionnet for having asked him the question that motivated this research. Most of the tools and methods used in this paper originate in the author's PhD work under the supervision of Hari Bercovici.

The author is deeply grateful for his support and teaching, during the PhD and after.

Supported in part by a Discovery Grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada.

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[8] J.H. Shapiro, Composition operators and classical function theory. Springer, New York, 1993.

[9] D. Voiculescu, Addition of certain noncommuting random variables . J. Funct. Anal. 66 (1986), 323346.

[10] Dan Voiculescu, Limit laws for random matrices and free products. Invent. Math. 104 (1991), 201220.

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Received 1 January 2014 CNRS, Institut de Mathematiques de Toulouse, Equipe de Statistique et Probabilites,

F-31062 Toulouse Cedex 09, 118 Route de Narbonne, France

and

Queen's University, Jerey Hall, Department of Mathematics and Statistics,

Kingston, ON K7L 3N6, Canada and

Simion Stoilow Institute of Mathematics, of the Romanian Academy serban.belinschi@math.univ-toulouse.fr,

sbelinsch@mast.queensu.ca