AN ISOPERIMETRIC INEQUALITY FOR RIESZ
CAPACITIES
PEDRO J. MENDEZ-HERN
ANDEZ
Let A be a compact set of Rn , and A be the ball centered at the origin with the same measure as A. We prove that, if
C is the -Riesz capacity with 0 < < 2, then C (A) C (A ).
We also prove an isoperimetric inequality for the expected measure
of the stable sausage generated by A. Our results also yield isoperimetric inequalities for the relativistic -stable processes, and other
Levy processes.
Abstract.
1. Introduction
It is well known that, among all compact sets of equal measure, the
ball has the smallest Newtonian Capacity. This is one of the classical
generalized isoperimetric inequalities of G. Polya and G. Szego [11].
In [10], J. M. Luttinger provided a new method, based on multiple
integrals inequalities, to prove this and many other isoperimetric inequalities. In this paper we adapt the method of Luttinger [10] to
obtain isoperimetric inequalities for Riesz capacities. The M. Riesz
kernel is
(n2)
1
k (x y ) = n=2 1
(2) 2
jx yjn ;
where n 2 and 0 < < n. Let A be a compact set in Rn, the -Riesz
capacity of A is dened by
h
C (A) = inf
ZZ
i 1
k (x y ) d(x) d(y )
;
where the inmum is taken over all probability Borel measures supported in A. If = 2 and n 3, then this is the classic Newtonian
capacity. Let jAj be the Lebesgue measure of A, and let A be the ball
in Rn centered at the origin such that jA j = jAj. Then Polya and
Szego's classical result is:
(1)
C2 (A) C2 (A ):
1991 Mathematics Subject Classication. 30C45.
Key words and phrases. Symmetric stable processes, isoperimetric inequalities,
Riesz capacities.
1
PEDRO J. MENDEZ-HERN
ANDEZ
2
Naturally one might ask if (1) holds for all Riesz capacities. This
problem was stated by P. Mattila in [8], where the author nds lower
bounds for the Hausdor measure of the projection of A on an mdimensional subspace in terms of Cm (A) . In this paper we present a
very short proof of the following result:
Let
jAj > 0. Then,
Theorem 1.
(2)
where
2 (0; 2), and A be a compact set of Rn such that
C (A) C (A );
A is the ball, centered at the origin, such that jAj = jA j.
Theorem 1 was previously proved by D. Betsakos. His proof relies
on some polarization inequalities for transition densities of killed symmetric stable processes and a well-known relationship between Green's
functions and Riesz capacities.
Luttinger obtained (1) from a probabilistic representation of the
Newtonian capacity, due to F. Spitzer [14], and the following rearrangement inequality, proved by R. Friedberg and J. M. Luttinger [7].
Let F0 ; : : : ; Fm : Rn ! [0; 1], and let H0 ; : : : ; Hm be
nonnegative nonincreasing radially symmetric functions in Rn. Then,
Theorem 2.
Z
Z
:::
Z h
:::
1
Z h
1
m
Y
j =0
(1 Fj (zj ) )
m
Y
m
i Y
(1 Fj (zj ) )
j =0
Hi (zj
m
i Y
Hi (zj
zj +1 ) dz0 dzm zj +1 ) dz0 dzm ;
j =0
j =0
where Fi is the symmetric decreasing rearrangement of Fi , and zm+1 =
z0 .
Actually Friedberg and Luttinger proved Theorem 2 for the Steiner
symmetrization of functions in R. However, it was proved in [4] that
the symmetric decreasing rearrangement of a function in Rn can be
obtained as the limit of a sequence of Steiner symmetrizations with
respect to dierent planes. It is known that such rearrangement inequalities, combined with a probabilistic representation of the heat
kernel, imply the classical Raleigh{Faber{Krahn inequality and many
other generalized isoperimetric inequalities for heat kernels and Green's
functions of the Laplacian and fractional Laplacian; see [1], [9], [10].
The other key result, in Luttinger's proof of (1), is Spitzer's study
of the expected volume of the Brownian sausage generated by A. The
AN ISOPERIMETRIC INEQUALITY FOR RIESZ CAPACITIES
3
Markov processes associated to Riesz kernels are the symmetric stable processes. Let Xt be an n-dimensional symmetric -stable process of order 2 (0; 2), and let TA = inf ft > 0 : Xt 2 Ag be the rst
time Xt hits A. Dene
EA (t)
=
Z
P x(TA t)dx;
this quantity can be interpreted as the expected Lebesgue measure of
[st[Xs + A].
In the case that = 2,
EA (t)
jAj =
Z
Ac
P x(TA t)dx
can be interpreted as the total heat ow, up to time t, from A to the
surroundings. This was the original motivation of Spitzer to study the
behavior of EA (t). R. K. Getoor [6] proved that
EA (t)
= C (A);
(3)
lim
t!1 t
extending Spitzer's result to all symmetric stable processes.
Not only do Theorem 2 implies isoperimetric inequalities for the
expected area of the stable sausage, but this method applies, without
change, to any Levy processes whose transition probability densities are
radially symmetric and nonincreasing. This class of processes includes
the relativistic -stable processes and any processes of the form BAt ,
where Bt is a Brownian motion and At is a subordinator. We will prove
Theorem 1 in x2, and we will discuss extensions of Theorem 1 to other
processes in x3.
2. Proof of Theorem 1
Recall that the process Xt has right continuous sample paths and
stationary independent increments. Its innitesimal generator is
=2 ;
where 0 < 2 and is the Laplacian in Rn. When = 2 the
process Xt is just an n-dimensional Brownian motion Bt running at
twice the speed. If 0 < < 2, then Xt = B2t , where t is a stable
subordinator of index =2 that is independent of Bt ; see [2]. Thus,
Z 1
h jx y j2 i
1
p (t; x; y ) =
exp
g=2 (t; u)du;
4u
0 (4u)n=2
where g=2 (t; u) is the transition density of t . Hence, for every positive
t, p (t; x; y ) = ft (jx y j) and the function ft (r) is decreasing.
PEDRO J. MENDEZ-HERN
ANDEZ
4
Let Ak be a decreasing sequence of compact sets such that the interior
of Ak contains A for all k, and \1
k=1 Ak = A. By the right-continuity
of the sample paths and the Markov property of stable processes, we
have
Z
P z0
f
TA
t gdz0 =
Z h
lim lim
lim lim
Z
k!1 m!1
lim lim
Z
k!1 m!1
:::
:::
P z0
1
1
Z h
1
1
f
TA
i
> t g dz0 =
i
Z h
Z h
P z0
P z0 fXs 2 Ac ; 0 s tg dz0 =
1
k!1 m!1
Z h
m
Y
j =1
f X mjt 2
m
iY
IAck (zj )
i
= 1; : : : ; m g dz0 =
Ack ; j
j =1
p (t=m; zj
m
Y
m
i Y
j =1
j =1
[ 1 IAk (zj ) ]
zj 1 ) dz0 dzm =
p (t=m; zj zj 1 ) dz0 dzm ;
where IAk is the indicator function of Ak . Since ft (x) is nonincreasing
and radially symmetric, we can take Hm = 1 and F0 = 0 in Theorem
2 to obtain
lim lim
Z
k!1 m!1
lim lim
k!1 m!1
Z
:::
:::
Hence,
EA (t) =
Z
Z h
1
Z h
1
Pz0 f
m
Y
m
i Y
j =1
j =1
[ 1 IAk (zj ) ]
m
Y
m
i Y
j =1
j =1
TA
[ 1 IAk (zj ) ]
p (t=m; zj zj 1 ) dz0 dzm p (t=m; zj zj 1 ) dz0 dzm :
Z
t gdz0 Pz f TA t gdz0 = EA (t):
0
Finally we conclude from (3) that
C (A) = tlim
!1
EA (t)
t
EA (t)
tlim
= C (A ):
!1 t
AN ISOPERIMETRIC INEQUALITY FOR RIESZ CAPACITIES
5
3. Isoperimetric inequalities for radial Levy processes
Let Xt be a Levy process whose transition density is radially symmetric and decreasing. This class includes any process of the form BAt ,
where Bt is a Brownian motion and At is a subordinator. An important example of such processes is the relativistic -stable processes.
The innitesimal generator of the relativistic -stable process is
[m ]=2 m;
where m > 0 is the mass of a relativistic particle. These processes
arises in the study of relativistic Hamiltonian systems in physics; see
[2], [5], [13] and the references there.
Let TA = inf ft > 0 : Xt 2 Ag be the rst time Xt hits A, and
consider
Z
EA (t) = P x(TA t)dx:
This quantity can be interpreted as the expected Lebesgue measure
of [st [Xt + A], the Xt -sausage generated by A. The following result
generalizes the results of the previous section:
Theorem 3. Let Xt be a transient L
evy processes whose transition
density is radially symmetric and decreasing. Let A be a compact set
of Rn with positive measure. If A is the ball centered at the origin
with jAj = jA j, then for all t 0
EA (t) EA (t);
(4)
and
C (A) C (A );
where C (A) is the capacity associated to Xt .
An examination of the proof of Theorem 1 shows that (4) follows
from Theorem 2 and the fact that the transition densities of Xt are
radially symmetric and decreasing. On the other hand S. C. Port and
C. J. Stone [12] proved that
EA (t)
(6)
lim
= C (A);
t!1 t
thus (5) follows from (4). We include a proof of (6) for the convenience
of the reader. The capacity, associated to Xt , of a compact set A is
dened as
Z
Z
(5)
C (A) = lim !0
E x[ e
TA ]dx
= lim !0
1
0
e
t E
A (dt):
PEDRO J. MENDEZ-HERN
ANDEZ
6
On the other hand the Markov property and the symmetry of Xt imply
EA (t) EA (t h) =
=
Z Z
lim EA (t)
t!1
P x (t h < TA t)dx
P x(TA > t h; Xt
=
Then
Z
Z
h
2 dy)P y (TA h)dydx
P y (TA > t h)P y (TA h)dy:
EA (t h) =
Z
P y (TA = 1)P y (TA h)dy:
This is an additive function of h which is bounded and measurable.
Hence it is a linear function of h and there exists a constant (A) such
that
lim EA (t) EA (t h) = h (A):
t!1
It follows that
1
EA (t) = (A);
lim
t!1 t
and integration by parts implies that
C (A) = lim !0
which proves (6).
Z 1
0
e
t E
A (dt)
= (A);
References
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AN ISOPERIMETRIC INEQUALITY FOR RIESZ CAPACITIES
7
[9] P. J. Mendez-Hernandez, Brascamp-Lieb-Luttinger Inequalities for Convex Domains of Finite Inradius, Duke Math. J. 113 (2002), 93-131.
[10] J. M. Luttinger, Generalized isoperimetric inequalities. II , J. Math. Phys. 14
(1973), 586{593.
[11] G. Polya and G. Szego, Isoperimetric Inequalities in Mathematical Physiscs,
Princeton University Press, Princeton, (1951)
[12] S. C. Port, C.J. Stone, Innite Divisible Processes and their Potential Theory
I, Ann. Inst. Fourier (Grenoble) 21 (1971), 157-275.
[13] M. Ryznar, Estimates of Green Function for Relativistic -Stable Processes,
Potential Analisis 17 (2002), 1-23.
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Mathematics Department, University of Utah, Salt Lake City, UT
84112
E-mail address : mendez@math.utah.edu