MARIAN V ˆAJ ˆAITU and ALEXANDRU ZAHARESCU
We investigate integration of Lipschitzian functions on a compact ultrametric spaceX with values in a Banach space over a local field, with respect to a Lips- chitzian distribution onX.
AMS 2000 Subject Classification: 11S99.
Key words: Lipschitzian function, unbounded distribution.
1. INTRODUCTION
The notion of distribution, as defined in [3], plays an important role in problems concerned with integration on ultrametric spaces, with special em- phasis on spaces derived in one way or another from the theory of local fields.
In the case of bounded distributions, one can integrate any continuous func- tion. However, there exist continuous functions that are not integrable with respect to a general, unbounded distribution. Manin (see [4] and [2]) con- sidered a notion of Lipschitzian distribution in a p-adic context which allows integration of Lipschitzian functions with respect to such distributions. These notions also play an important role in the process of constructing a trace for Lipschitzian elements over a p-adic field (see [1]). Further properties of Lips- chitzian elements are established in [5]. In the present paper, we study such notions in a more general setting. In Section 2 we consider Lipschitzian distri- butions on a general compact ultrametric spaceX, and prove the integrability with respect to a Lipschitzian distribution on X of Lipschitzian functions on X with values in a Banach space over a local field. Further properties of such integrals and a transformation formula are then established in Section 3.
2. LIPSCHITZIAN FUNCTIONS AND LIPSCHITZIAN DISTRIBUTIONS
We will work in the following context. Let (X, d) be a ultrametric space, that is, a metric space for which d(x, z) ≤ max{d(x, y), d(y, z)}, for any
REV. ROUMAINE MATH. PURES APPL.,53(2008),1, 79–88
x, y, z ∈ X. In what follows we will assume that X is compact. Let K be a field, complete with respect to a nonarchimedean absolute value | · |, and let A be a K-vector space, complete with respect to a nonarchimedean norm that we will denote by k · k. Thus, for any α, β ∈ K and any u, v ∈ A we have |α| ≥ 0, and |α| = 0 if and only if α = 0; |α +β| ≤ max{|α|,|β|};
|αβ|=|α| · |β|; kuk ≥ 0, and kuk= 0 if and only if u = 0; kαuk =|α| · kuk;
ku+vk ≤ max{kuk,kvk}. In particular, the considerations below apply to the case when A is a Banach algebra overK. In that case, besides the above properties we also have kuvk ≤ kuk · kvk for any u, v∈A.
Given a mapf :X → A and a positive numberλ, the mapf is said to beλ-Lipschitzian provided thatkf(x)−f(y)k ≤λd(x, y),for anyx, y∈X. A mapf :X→Ais called Lipschitzian provided there exists a positive numberλ for whichf isλ-Lipschitzian. Let us remark that iff :X →A, andg:X→A are Lipschitzian and α ∈K, then αf and f +g are Lipschitzian. Also,f g is Lipschitzian if A is a K-algebra. Thus, the Lipschitzian maps from X to A form a K-vector space, and in caseA is aK-algebra they form a K-algebra.
Given an elementx∈X and a numberδ >0, we denote by B(x, δ) and B[x, δ], respectively, the open and the closed ball inX of radiusδ centered at x. Thus, B(x, δ) = {y ∈ X : d(x, y) < δ}, B[x, δ] = {y ∈ X :d(x, y) ≤ δ}.
The ball B(x, δ) is an open subset of X and, sinceX is a ultrametric space, B[x, δ] is also an open subset of X. Moreover, the complementary of each of these two balls inXis an open subset ofX, asX\B(x, δ) = S
y∈X\B(x,δ)
B(y, δ), and X\B[x, δ] = S
y∈X\B[x,δ]
B(y, δ). So, both B(x, δ) and B[x, δ] are closed subsets of X, hence compact, X being compact. Let us denote by Ω(X) the set of subsets of X which are open and compact. Thus, X ∈ Ω(X) and any open or closed ball in X belongs to Ω(X). Any finite union D=
n
S
i=1
Di with Di ∈ Ω(X) for each i ∈ {1,2, . . . , n}, also belongs to Ω(X). Note that if in particular, one takes, eachDito be an open or closed ball inX, then, removing if necessary any ball Di which is contained in another ball Dj, one can write D as a finite union of pairwise disjoint balls. This follows from the fact that in a ultrametric space any two balls are either disjoint or one is contained in the other. Let us also remark that any D ∈Ω(X) can be written as a finite union of pairwise disjoint balls. Indeed,Dbeing open, for eachx∈Dthere is δx >0 for whichB(x, δx) ⊆D. The ballsB(x, δx), x ∈D, cover D, which is compact, so there is a finite covering D=B(x1, δx1)∪ · · · ∪B(xm, δxm), say.
Removing some of these balls if necessary, one finally obtains a partition of D with finitely many pairwise disjoint open balls. Note that if in the above argument D∈Ω(X) and δ > 0 is such that B(x, δ)⊆D for anyx∈D, then
D can be partitioned into finitely many open balls of radius δ. In particular, any open or closed ball of radius δ0 > δ can be written as a finite union of pairwise disjoint open balls of radius δ.
In what follows, by a distribution µ on X with values in K we mean a finitely additive map µ: Ω(X)→K. Thus, ifD=
n
S
i=1
Di withDi∈Ω(X) for 1≤i≤nandDi∩Dj =∅for 1≤i6=j≤n, thenµ(D) =
n
P
i=1
µ(Di). We note in passing that an alternative definition of a distribution onX with values in K is as follows. Let (εn)n∈N be a strictly decreasing sequence of positive real numbers with lim
n→∞εn = 0. For each n, write X as a finite union of pairwise disjoint open balls of radiusεn, and denote byXnthe set of these balls. Each element of Xn+1 is an open ball of radius εn+1, and this ball is contained in a unique open ball of radius εn, which is an element of Xn. In this way one obtains a canonical map ϕn+1 :Xn+1 → Xn. A distribution on X, which is identified with the projective limit of the Xn,n≥1, with values inK, say, is then defined by a sequence of mapsµn:Xn→K, such that for anynand any element z ∈ Xn one has µn(z) = P
u∈ϕ−1n+1(z)
µn+1(u). Returning to the above definition of a distribution onXwith values inK, let us fix such a distribution µ : Ω(X) → K. Given a map f :X → A, for any partition of X with open balls, X =
n
S
i=1
Bi, Bi ∩Bj = ∅ for i 6= j, and any choice of points xi ∈ Bi, 1 ≤ i ≤ n, consider the Riemann sum S(µ, f, B1, . . . , Bn, x1, . . . , xn) :=
n
P
i=1
µ(Bi)f(xi)∈A.We say thatf is integrable with respect to the distribution µ provided there exists Λ∈ A with the property that for any ε > 0 there is δ > 0 such that for any partition X =
n
S
i=1
Bi, Bi∩Bj =∅ for i 6= j, where each Bi is an open ball in X of radius less than δ, and any choice of points xi ∈Bi, 1≤i≤n, we havekS(µ, f, B1, . . . , Bn, x1, . . . , xn)−Λk< ε. In that case, we denote, as usual, Λ =R
Xfdµand call it the integral of f on X with respect to the distribution µ.
Note that any locally constant function f : X → A is integrable with respect to any distribution µ: Ω(X)→K.
A distributionµ: Ω(X)→ K is said to be a measure if and only if it is bounded.
We call Lipschitzian a distributionµ: Ω(X)→K if for anyε >0 there is δε > 0 such that for any 0 < δ ≤ δε and any x ∈ X, δ|µ(B(x, δ))| ≤ ε,
where | · | denotes the absolute value on K. Let us show that any Lips- chitzian function f : X → A is integrable with respect to any Lipschitzian distribution µ : Ω(X) → K. Fix such a µ and a f. We first construct a sequence (Sr)r∈N of Riemann sums as follows. For each positive integer r, write X as a finite union of pairwise disjoint open balls of radius 21r, and denote them by Br,1, Br,2, . . . , Br,Nr. Next, choose points xr,j ∈ Br,j, for 1 ≤ j ≤ Nr, and denote the corresponding Riemann sums, by Sr = S(µ, f, Br,1, . . . , Br,Nr, xr,1, . . . , xr,Nr).We claim that (Sr)r∈N is a Cauchy se- quence in A. In order to prove this, let us fix r and compare Sr with Sr+1. Each open ball of radius 21r in X can be written as a finite union of pairwise disjoint open balls of radius 2r+11 . Therefore, there are disjoint nonempty sets J1, . . . , JNr, with J1∪ · · · ∪JNr ={1,2, . . . , Nr+1}, such that Br,i=S
j∈JiBr+1,j for any i∈ {1,2, . . . , Nr}. We now writeSr−Sr+1 in the form
Sr−Sr+1 =
Nr
X
i=1
µ(Br,i)f(xr,i)−
Nr+1
X
i=1
µ(Br+1,i)f(xr+1,i)
=
Nr
X
i=1
µ(Br,i)f(xr,i)−X
j∈Ji
µ(Br+1,j)f(xr+1,j) . Here, we may rewrite µ(Br,i) as P
j∈Ji
µ(Br+1,j) by the additivity ofµ. Hence
Sr−Sr+1=
Nr
X
i=1
X
j∈Ji
µ(Br+1,j) f(xr,i)−f(xr+1,j) . It follows that
kSr−Sr+1k ≤ max
1≤i≤Nr
maxj∈Ji
|µ(Br+1,j)| · kf(xr,i)−f(xr+1,j)k.
Since f is λ-Lipschitzian for some λ >0, we get kSr−Sr+1k ≤λ max
1≤i≤Nrmax
j∈Ji |µ(Br+1,j)| ·d(xr,i, xr+1,j).
Here both xr,i andxr+1,j belong to the open ballBr,i of radius 21r. So, kSr−Sr+1k ≤ λ
2r max
1≤i≤Nr
maxj∈Ji |µ(Br+1,j)|= 2λ max
1≤i≤Nr
maxj∈Ji
|µ(Br+1,j)|
2r+1 . Here, on the far right side Br+1,j is an open ball of radius 2r+11 , therefore the ratio |µ(B2r+1r+1,j)| converges to 0 as r → ∞, uniformly with respect to j ∈ Ji, 1≤i≤Nr. Precisely, for anyε >0 there isδε>0 such that for any 0< δ≤δε and any open ball B of radius δ one has δ|µ(B)| ≤ ε. Then, for any r ≥
[log2(1/δε)], we have |µ(B2r+1r+1,j)| ≤εfor anyj, and sokSr−Sr+1k ≤2λε. Since the norm k · k is nonarchimedean, we havekSn−Smk ≤2λε for any n, m≥ [log2(1/δε)],which proves our claim that (Sr)r∈N is a Cauchy sequence. Since A is complete, the sequence (Sr)r∈N converges to an element Λ ∈ A. Fix now a ε > 0 and let δε > 0 be defined as above. As a consequence of the above relations, we have kSr−Λk ≤ 2λε for any r ≥ h
log2 1
δε
i
.Let now S(µ, f, B1, . . . , Bn, x1, . . . , xn) be an arbitrary Riemann sum with xi ∈ Bi
and Bi open ball of radius δi < δε for 1 ≤ i ≤ n. Choose r such that
1
2r < min{δε, δi,1 ≤ i ≤ n}. Since kSr −Λk < ε, in order to show that S(µ, f, B1, . . . , Bn, x1, . . . , xn) is close to Λ it is enough to show that it is close to Sr. Due to the possible huge size of some of the numbers 2rδi we cannot obtain directly a good enough bound forkS(µ, f, B1, . . . , Bn, x1, . . . , xn)−Srk.
What we do is to construct a sequence Σ0,Σ1, . . . of Riemann sums in the following way. First, put Σ0 = S(µ, f, B1, . . . , Bn, x1, . . . , xn). Assume now that Σ0,Σ1, . . . ,Σj are already constructed, and construct Σj+1 as follows.
Suppose Σj =S(µ, f, D1, . . . , Dm, y1, . . . , ym), where each Di is an open ball of radius δi ≥ 21r, andyi ∈Di for 1≤i≤m. Consider each i∈ {1,2, . . . , m}
and distinguish three cases. If δi = 21r keep the ball Di and the point yi, and use them in the partition and respectively the choice of points for Σj+1. If 2r−11 ≥ δi > 21r break the ball Di as a finite disjoint union of open balls of radius 21r, Di =
Li
S
l=1
Dil, say, and choose points yil ∈ Dil, for 1 ≤ l ≤ Li. Here, each Dil is an open ball in X of radius 21r, so it coincides with one of the balls Br,1, . . . , Br,Nr which appear in the partition from the definition of Sr. If t ∈ {1,2, . . . , Nr} is such that Dil = Br,t, choose yil to coincide with the point xr,t from the definition of Sr. We then use the balls Dil and the points yil, 1 ≤ l ≤ Li, in the partition and respectively the choice of points for Σj+1. Last, if δi > 2r−11 write Di as a finite disjoint union of open balls of radius δ2i, Di =
Li
S
l=1
Dil say, and choose arbitrary points yil ∈ Dil. Then use the balls Dil and the points yil, 1 ≤ l ≤ Li, for Σj+1. This completes the construction of Σj+1. In this manner, we obtain a sequence Σ0,Σ1, . . . of Riemann sums. Clearly, by its construction, this sequence stabilizes to Sr, in the sense that Σj =Srfor alljlarge enough. Note also that the radius of any open ball used in the partition of X from the definition of any Σj, belongs to the interval 1
2r, δε
. Moreover, we have the following boundedness condition.
For any j, any open ball Di from the partition associated with Σj, and any ball Dil from the partition associated with Σj+1 which is contained in Σj, the radius of Di is not larger than two times the radius of Dil. We now proceed
to bound kΣj−Σj+1k for all j. Fixj and denote as above Σj =S(µ, f, D1, . . . , Dm, y1, . . . , ym) =
m
X
i=1
µ(Di)f(yi).
By the construction above of Σj+1, if a ball Di has radius 21r then Di and yi
appear in both Σj and Σj+1. So, the termµ(Di)f(yi) cancels in the difference Σj −Σj+1. For the other values of i, we group together the termµ(Di)f(yi) from Σj with the partial sum P
1≤l≤Li
µ(Dil)f(yil) from Σj+1, and use the fact that µis additive andk · k is nonarchimedean, to get
kΣj−Σj+1k=
X
i
µ(Di)f(yi)− X
1≤l≤Li
µ(Dil)f(yil)
=
X
i
X
1≤l≤Li
µ(Dil)(f(yi)−f(yil))
≤max
i max
1≤l≤Li
|µ(Dil)| · kf(yi)−f(yil)k.
Next, using the Lipschitzianity offand the Lipschitzianity ofµwe deduce that kΣj −Σj+1k ≤max
i max
1≤l≤Li
ε
δil ·λd(yi, yil),
where δil denotes the radius of the open ball Dil. Last, we combine the fact that for any i and l both points yi and yil lie inside the ball Di, with the boundedness condition above which in particular states that the radius δi of the ball Di is at most 2δil. Consequently, d(yδi,yil)
il ≤2 for any iand l, and we conclude that kΣj −Σj+1k ≤ 2ελ for all j ≥0. Therefore, choosing j0 large enough such that Σj0 =Sr, we find that
kS(µ, f, B1, . . . , Bn, x1, . . . , xn)−Srk=kΣ0−Σj0k ≤ max
0≤j≤j0−1kΣj−Σj+1k ≤2ελ, and combining this with the inequality kSr−Λk ≤εwe finally obtain
kS(µ, f, B1, . . . , Bn, x1, . . . , xn)−Λk ≤εmax{1,2λ}.
This completes the proof that f is integrable with respect to µ. We state below this result.
Theorem 1. Let X be a compact ultrametric space,(K,| · |) a complete nonarchimedean valued field and(A,k·k)a complete nonarchimedeanK-vector space. Then any Lipschitzian function f :X →A is integrable with respect to any Lipschitzian distribution µ: Ω(X)→K.
3. SOME PROPERTIES OF INTEGRALS
WITH RESPECT TO LIPSCHITZIAN DISTRIBUTIONS
The integrability of Lipschitzian functions with respect to Lipschitzian distributions imply some properties of the corresponding integrals by selecting convenient Riemann sums. For instance, the map (µ, f)7→R
Xfdµis bilinear.
Theorem 2. Let X, K andA be as in the statement of Theorem 1. For any Lipschitzian distributions µ1, µ2 : Ω(X)→ K, any Lipschitzian functions f1, f2 :X→A, and any α1, α2, β1, β2∈K, we have
Z
X
(α1f1+α2f2)d(β1µ1+β2µ2) =
=α1β1
Z
X
f1dµ1+α1β2
Z
X
f1dµ2+α2β1
Z
X
f2dµ1+α2β2
Z
X
f2dµ2. Proof. Note first that the distributionµ:=β1µ1+β2µ2 and the function f := α1f1+α2f2 are Lipschitzian. So, f is integrable with respect to µ by Theorem 1. Next, take a smallδ >0, writeX as a finite disjoint union of open balls of radius δ, X =
n
S
i=1
Bi say, and choose points xi ∈ Bi, for 1 ≤i ≤ n.
Then
S(µ, f, B1, . . . , Bn, x1, . . . , xn) =
n
X
i=1
µ(Bi)f(xi) =
=
n
X
i=1
(β1µ1(Bi) +β2µ2(Bi))(α1f1(xi) +α2f2(xi)) =
=β1α1S(µ1, f1, B1, . . . , Bn, x1, . . . , xn)+β1α2S(µ1, f2, B1, . . . , Bn, x1, . . . , xn)+
+β2α1S(µ2, f1, B1, . . . , Bn, x1, . . . , xn) +β2α2S(µ2, f2, B1, . . . , Bn, x1, . . . , xn).
Asδ→0, each Riemann sum converges to the corresponding integral and this proves the theorem.
Suppose now that (X, dX) and (Y, dY) are compact ultrametric spaces, and a Lipschitzian map Ψ : X → Y is given. Then for each distribution µ : Ω(X) → K one has a distribution ν : Ω(Y) → K, defined by ν(D) = µ(Ψ−1(D)), for allD∈Ω(Y). Here the continuity of Ψ assures that Ψ−1(D) is an open compact subset of X for any open compact subset D of Y, so ν is well defined. Also, the additivity of ν follows from that of µ. We claim that if the distribution µ is Lipschitzian then, ν is Lipschitzian. In order to prove the claim, choose first λ > 0 such that Ψ is λ-Lipschitzian. Next, fix ε > 0 and apply the Lipschitzianity assumption on µ with ε replaced by λε.
Thus, there is δε/λ > 0 such that for any 0 < δ ≤ δε/λ and any open ball B in X of radius δ one has δ|µ(B)| ≤ λε. Let now D be an open ball in Y of radius η, with 0 < η ≤ λδε/λ. We proceed to find an upper bound for η|ν(D)|=η|µ(Ψ−1(D))|. We know that Ψ−1(D), as an open compact subset of X, can be written as a finite union of open balls inX. Let us remark that for any point x ∈Ψ−1(D) the whole open ball in X of radius λη centered at x is contained in Ψ−1(D). Indeed, if z ∈ X is such that dX(x, z) < λη, then dY(Ψ(x),Ψ(z))≤λdX(x, z) < η and, since Ψ(x)∈D, we have Ψ(z) ∈D, so z∈Ψ−1(D). Therefore, all the open balls of radius λη centered at points from Ψ−1(D) are contained in Ψ−1(D). We deduce that Ψ−1(D) can be written as a finite disjoint union of open balls of radius ηλ, Ψ−1(D) = ∪ni=1Bi, say.
Then η|ν(D)| = η
n
P
i=1
µ(Bi)
≤ η max
1≤i≤n|µ(Bi)|. For each i, Bi is an open ball of radius ηλ ≤ δε/λ. It follows that ηλ
µ(Bi)
≤ λε, which in turn gives η|ν(D)| ≤ε. This proves the claim thatν is a Lipschitzian distribution.
One may now integrate any Lipschitzian function f : Y → A with re- spect to ν. Note that for such an f the composition f ◦Ψ : X → A is also Lipschitzian, so it is integrable with respect to µ, and a natural question that arises is whether there is any relation between the integrals R
Xf ◦Ψdµ and R
Y fdν. To answer this question, choose a small η > 0, write Y as a finite disjoint union of open balls of radiusη,Y =
m
P
j=1
Dj, say, choose pointsyj ∈Dj
for 1≤j≤m, and consider the Riemann sum
S(ν, f, D1, . . . , Dm, y1, . . . , ym) =
m
X
j=1
ν(Dj)f(yj) =
m
X
j=1
µ(Ψ−1(Dj))f(yj).
Reasoning as above, one finds that for each j all the open balls of radius ηλ centered at points from Ψ−1(Dj) are contained in Ψ−1(Dj). Thus, for each j, Ψ−1(Dj) can be written as a finite disjoint union of open balls of radius
η
λ, Ψ−1(Dj) =
Hj
S
h=1
Bj,h, say. Choose points xjh ∈Bjh, 1≤h ≤Hj, 1≤j ≤ m, and form the Riemann sum S(µ, f ◦Ψ, B11, . . . , BmHm, x11, . . . , xmHm) =
m
P
j=1 Hj
P
h=1
µ(Bjh)f(Ψ(xjh)).We know that forηsmall enough the Riemann sums S(ν, f, D1, . . . , Dm, y1, . . . , ym) andS(µ, f◦Ψ, B11, . . . , BmHm, x11, . . . , xmHm) are close to the integrals R
Y fdν and R
Xf ◦Ψdµ, respectively. Therefore, if we show that the two Riemann sums are close for small η, it will follow that
R
Y fdν=R
Xf ◦Ψdµ. We have
kS(ν, f, D1, . . . , Dm, y1, . . . , ym)−S(µ, f ◦Ψ, B11, . . . , BmHm, x11, . . . , xmHm)k=
=
m
X
j=1
µ(Ψ−1(Dj))f(yj)−
Hj
X
h=1
µ(Bjh)f(Ψ(xjh))
=
=
m
X
j=1 Hj
X
h=1
µ(Bjh)
f(yj)−f(Ψ(xjh)) ≤
≤ max
1≤j≤m max
1≤h≤Hj
|µ(Bjh)| · kf(yj)−f(Ψ(xjh))k.
If we fix ε >0, chooseδε>0 such thatδ|µ(B)| ≤εfor any open ball B inX of radiusδ ≤δε, and assume thatηis small enough such that λη ≤δε, then we have λη|µ(Bjh)| ≤ε, for allj andh. On the other hand, for alljandhone has
kf(yj)−f(Ψ(xjh))k ≤λfkyj−Ψ(xjh)k ≤λfη,
where the first inequality follows from the λf-Lipschitzianity of f for some positive number λf depending on f while the second one is implied by the fact that both yj and Ψ(xjh) belong to Dj, which is a ball in Y of radius η.
Combining the above relations we find thatkS(ν, f, D1, . . . , Dm, y1, . . . , ym)− S(µ, f◦Ψ, B11, . . . , BmHm, x11, . . . , xmHm)k ≤λλfε,providedηis small enough in terms of ε. Since ε was arbitrary, we conclude that the integrals R
Y fdν and R
Xf◦Ψdµare equal. We state below this result theorem.
Theorem 3. Let X, Y be compact ultrametric spaces,Ψ :X→Y a Lip- schitzian map, (K,| · |) a complete nonarchimedean valued field, and(A,k · k) a complete nonarchimedeanK-vector space. Then for any Lipschitzian distri- bution µ : Ω(X) → K, the map ν : Ω(Y) → K given by ν(D) = µ(Ψ−1(D)) for D∈Ω(Y) is a Lipschitzian distribution, and for any Lipschitzian function f :Y →A one has
Z
Y
fdν = Z
X
f ◦Ψdµ.
Acknowledgement. This work was partially supported by Contract 2-CEx06-11- 20/2006 with the Romanian Ministry of Education and Research.
REFERENCES
[1] V. Alexandru, N. Popescu and A. Zaharescu,Trace onCp. J. Number Theory88(2001), 1, 13–48.
[2] N. Koblitz,p-adic Numbers,p-adic Analysis and Zeta Functions. Springer-Verlag, 1977.
[3] B. Mazur and P. Swinnerton-Dyer,Arithmetic of Weil curves. Invent. Math.25(1974), 1–61.
[4] Ju. I. Manin,Periods of cusp forms, andp-adic Hecke series. Mat. Sb. (N.S.)92(134) (1973), 378–401, 503. (Russian)
[5] A. Zaharescu, Lipschitzian elements over p-adic fields. Glasg. Math. J. 47 (2005), 2, 363–372.
Received 2 March 2006 Romanian Academy
“Simion Stoilow” Institute of Mathematics P.O.Box 1-764
RO-70700 Bucharest, Romania Marian.Vajaitu@imar.ro
and
University of Illinois at Urbana-Champaign Department of Mathematics Altgeld Hall, 1409 W. Green Street
Urbana, IL, 61801, USA zaharesc@math.uiuc.edu
