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 ran-
dom 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 distribu-
tion of sums of independent (asymptotically) unitarily invariant large random
matrices: assume thatAN, BN are twoN×N selfadjoint random matrices, in-
dependent 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

the eigenvalues distribution of A_{N} (respectively B_{N}) converges to µ (respec-
tively 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 probabil-
ity 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 sup- port 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}+y^{2}dµ(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) = _{G}^{1}

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_{µ}is an analytic self-map of the (open) upper half-plane C^{+}. Moreover,

=F_{µ}(z)≥ =zfor allz∈C^{+}. If there is az_{0}∈C^{+}so that=F_{µ}(z_{0}) ==z_{0},

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 concen-
trated 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 con-
tinuous 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 ^{1}_{0} =∞ 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):

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 con- stant, 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 ω_{1}, ω_{2}:C^{+}→C^{+} with the property that:

(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 c_{1} < c_{2} 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) |z_{1}^{(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)})−c_{1}|>|f(zn^{(2)})−c_{2}|>|f(z_{n+1}^{(1)} )−c_{1}|, for alln∈N.

Dene γ: [0,1]→C^{+}∪ {x}so that γ(0) =i, γ(1) =x,γ 1−_{2n}^{1}

=zn^{(2)},
γ

1−_{2n+1}^{1}

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

1−_{k}^{1},1−_{k+1}^{1} i

(note that we can consider linear to mean hyperbolic-linear, in the sense of
γ([1− ^{1}_{k},1− _{k+1}^{1} ]) 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+|c_{1}|)^{−1},(1+|c_{2}|)^{−1},^{c}^{2}_{100}^{−c}^{1} .
SinceC^{+}∩C(f, x) =∅, there exists ann()∈Nso that

f γ 1−_{2n}^{1}

−c2

+

f

γ

1−_{2n+1}^{1}

−c1

< ^{}_{2} andf γ

1−_{2n}^{1} ,1

⊂C^{+}\Xfor alln≥n(),
where

X_{}=

z∈C:|<z| ≤ 1

, ≤ =z≤ 1

.

We claim that there exist 1−_{2n()+1}^{1} ≤t_{}< s_{} ≤1−_{2n()+2}^{1} so that:

(α) |f(γ(t_{}))−c_{1}| ≤,|f(γ(s_{}))−c_{2}| ≤, and

(β) f(γ([t_{}, s_{}]))∩(c_{1}+i[0, )) =∅, f(γ([t_{}, s_{}]))∩(c_{2}+i[0, )) =∅.
Indeed, this is a simple connectedness argument: we know from our con-
struction that

f

γ

1−_{2n()+1}^{1}

−c_{1}

< /2and f

γ

1−_{2n()+2}^{1}

−c_{2}
<

/2. By the continuity of f and γ,f γh

1−_{2n()+1}^{1} ,1−_{2n()+2}^{1} 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))−c_{1}|<

,

s_{}= inf

t∈

t_{},1− 1
2n() + 2

:|f(γ(t))−c_{2}|<

,

choices obviously allowed by the analyticity of the correspondence(1−_{2n()+1}^{1} ,1−

1

2n()+2)3t7→f(γ(t))∈C^{+}\X_{}.The set(c1+i[0, ])∪(c_{2}+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\[c_{1}, c_{2}],

• 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}_{}
.

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 ≤ t_{}_{1} < s_{}_{1} ≤ 1 − _{2n(}^{1}

1)+2 so that f(γ([t_{}_{1}, s_{}_{1}])) 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 = c_{1}, b = c_{2},
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 thanc_{2}.

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(ω_{1}, x)∩C^{+}6=∅ or C(ω_{2}, x)∩C^{+}6=∅, then cardC(ω_{1}, x) =
cardC(ω2, x) =cardC(Fµν, x) = 1;

(2) If cardC(ω_{1}, x) > 1, then both C(ω_{1}, x) and C(ω_{2}, 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, thenω_{1}, ω_{2} both extend
continuously toR;

(5) If eitherR\supp(µ)orR\supp(ν)is empty and eitherC(ω_{1}, x)orC(ω_{2}, 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

(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 ∈ (c_{1}, c_{2}), there exists a
z^{(c)}n ∈[z_{n}, w_{n}],n∈N, so thatω_{1}(zn^{(c)})∈c+i(0,+∞)andlimn→∞ω_{1}(z^{(c)}n ) =c.
In particular, {ω_{1}(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}(ω_{1}(z_{n}^{(c)})) +z_{n}^{(c)}

c∈[c_{1}, c_{2}], h_{µ}has real nontangential limit in co
is also dense in R∪ {∞}. But by Theorem 4 (b,c), ω2(z) = z+F_{µν}(z)−
ω_{1}(z) =z+F_{µ}(ω_{1}(z))−ω_{1}(z) = h_{µ}(ω_{1}(z)) +z. Thus, the sequence of curves
{ω_{2}([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])∩X^{1}

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,_{M}^{1}

6=∅for alln≥n(M). Thus, for any suchdone can
nd zn^{(d)} ∈ [zn, wn] so that <ω_{2}(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 ω_{1}(z_{n}^{(d)}) = lim

n→∞z^{(d)}_{n} +h_{ν}(ω_{2}(z^{(d)}_{n} ))> x+|a|+|b|+|x|+ 10≥ |a|+|b|+ 10,
an obvious contradiction with the fact that, by item (vi) of Lemma 5,

<ω_{1}([zn, wn])∈[a−1, b+ 1]forn∈Nlarge enough. We conclude thatC(ω1, x)
and C(ω_{2}, x) must indeed contain only one point. Since x ∈R was arbitrarily
chosen, the functions ω_{1}, ω_{2} 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}) =∞.

By dropping if necessary to a subsequence, we may assume without loss of gen-
erality 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 ^{=e}_{3} . If {ω_{1}(z^{(e)}n )}_{n} has an accumulation point in C^{+}+ ^{e}_{4}, 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}(ω_{1}(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 {ω_{1}(zn^{(e)}) : n ∈ N} or {ω_{2}(zn^{(e)}) : n ∈ N} is un-
bounded, 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(ω_{1},∞) contains an innite length interval itself. Pick a d ∈ C(ω_{1},∞)\
{∞} so that ^F_{µ}(d) < ∞ exists and ([4], Lemma 2.18) applies to ω_{1} and
d. Taking a sequence {z_{n}^{(d)}}_{n} tending to innity so that <ω_{1}(z_{n}^{(d)}) = d and

=ω_{1}(z_{n}^{(d)}) → 0, we conclude by Theorem 4 (c,b) that limn→∞ω_{2}(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 ω_{1}(c) exists and equals zero.

A similar conclusion holds for F_{ν} and ω_{2}. Julia-Caratheodory's Theorem 3

implies that

^ lim

v→ω_{1}(c)

F_{µ}(v)

v−ω1(c) = lim inf

v→ω_{1}(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({ω_{1}(c)}) −1 = ^ lim

v→ω_{1}(c)

F_{µ}(v)
v−ω1(c) −1

= lim inf

v→ω_{1}(c)

=F_{µ}(v)

=v −1

≤ lim inf

z→c

=F_{µ}(ω1(z))

=ω_{1}(z) + =z

=ω_{1}(z) −1

= lim inf

z→c

=ω_{2}(z)

=ω_{1}(z)

=

lim sup

z→c

=ω_{1}(z)

=ω_{2}(z)
−1

=

lim sup

z→c

=F_{ν}(ω2(z))

=ω_{2}(z) + =z

=ω_{2}(z) −1
−1

≤

lim sup

z→c

=F_{ν}(ω2(z))

=ω_{2}(z) −1
−1

≤

lim inf

v→ω_{2}(c)

=F_{ν}(v)

=v −1 −1

=

^ lim

v→ω_{2}(c)

F_{ν}(v)
v−ω_{2}(c) −1

−1

=

1

ν({ω_{2}(c)}) −1
−1

.

This will provide us with the desired result. If µ({ω_{1}(c)}) = 0, then
we obtain immediately that we must have ν({ω_{2}(c)}) = 1, i.e. ν = δ_{ω}_{2}_{(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({ω_{2}(c)}) −1 1

µ({ω_{1}(c)})−1

, so that

1≤ν({ω_{2}(c)}) +µ({ω_{1}(c)}).

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

It should be noted that compactly supported measures satisfy the hy- pothesis 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 hy-
pothesis 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 theo-
rem. The continuity statement follows quite easily now: assume towards con-
tradiction 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 in-
terval. Then ω_{1}(c) = ω_{2}(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 equa-
tion (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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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