1. Backgrounds
The Riesz potential
(
) operator is defined by
where
f is a measurable function. It has been widely developed in harmonic analysis including function spaces, mathematical physics and partial differential equations (see [
1,
2,
3,
4]).
For the endpoint case
, it is trivial to study the limitation
Instead, the convolution kernel is usually changed in such a derivative way
This logarithmic kernel produces a corresponding logarithmic potential operator, which represents a the better complement for the endpoint case of Riesz potential operator by virtue of effective properties and applications. For example,
is harmonic on
, while for teh lower dimension
,
is studied since it is harmonic on
(see [
5,
6]).
Recently, both Riesz potential and logarithmic potential have been studied in an anisotropic way, which is closely related with convex geometry analysis and mathematical physics (see [
7,
8,
9,
10,
11]). Here we first recall some basic concepts and results in convex geometry.
If the intersection of each line through the origin with a set
is a compact line segment,
K is called star-shaped with respect to the origin. Let
where
o is the origin, be the radial function of the star-shaped set
K.
K is called a star body with respect to the origin, if
is positive and continuous. We assume that
K is a star body with respect to the origin and
E is a bounded measurable set in this paper. Note that the radial function
is positively homogeneous with degree
, i.e.,
Let
and
denote, respectively, the
n-dimensional volume of
E and the complement of
E. We assume
in this paper, since when
, some trivial result follows directly. Let
denote the natural spherical measure on the boundary
of the unit ball
centered at the origin. Then
Let
denote by the Minkowski functional of
K:
where
Note that
and
, where
denotes the Euclidean norm. We refer to [
12,
13] for more information on convex geometry.
Let
,
and denote by
the
K-annulus body centered at
y with outer radius
a and inner radius
. Then, by the definition of the Minkowski functional, it follows that
Several anisotropic Riesz potentials are introduced and their optimal extreme values estimates are systematically studied in [
10]. We omit the details here for the brevity of this paper. Let
be the anisotropic
m-log-potential of measurable set
E at
with respect to
K, and
be the mixed volume of
K and
E. We refer to [
11] for these definitions and [
14,
15] for their relations with engineering and mathematical physics.
Note that
is obviously an extreme value of the anisotropic
m-log-potential. It is also closely related to convex geometry analysis. In [
11], when
m is an odd number, the optimal estimate for
is established as follows:
When
, the equality in (
2) holds if and only if
E is a
-ball introduced in [
11] up to the difference of a measure zero set.
For the application of the sharp estimate in (
2), the dual polynomial log-Minkowski inequality is established in [
11]:
where
m is an odd number,
K,
L are two star bodies and
is the normalized cone-volume measure
The equality in (
3) holds if and only if there exists
such that
.
Note that (
3) generalizes the dual log-Minkowski inequality for a mixed volume of two star bodies (see [
12,
16]) and produces the polynomial dual for the conjectured log-Minkowski inequality (see [
17]).
In this paper, we study the other extreme value of the anisotropic
m-log-potential:
Note that because may be negative, is defined for integer m.
In
Section 2, some fundamental properties of
are established. Then, in
Section 3, we are able to introduce the new annulus body to solve the problem of optimal estimate for
in a precise analytic way. For the application, a polynomial dual log-mixed volume difference law is induced from the optimal estimate.
2. Fundamental Properties
First we recall a metric property in [
11] for the Minkowski functional of a star body with respect to the origin.
Proposition 1. Let be the unit ball andThenand a quasi-triangle inequality holds for If
m is an even number, the supremum of the anisotropic
m-log-potential
(see [
11]). For the infimum of the anisotropic
m-log-potential
, it follows
Proposition 2. for m as an odd number.
Proof. Note that
K is a star body with respect to the origin and
E is a bounded measurable set. Then
. For all
, let
,
where
is defined in (
5). Hence, for all
,
Since
m is odd, it follows that
which implies
□
has the following metric properties for the nontrivial case (m is an even number).
Proposition 3. Let m be an even number.
- (i)
Monotonicity: let and are bounded measurable sets and . Then .
- (ii)
Translation-invariance: for all , let . Then .
- (iii)
Homogeneity: for all , .
Proof. (i) Since
, then for all
,
Hence,
(ii) For all
, by changing the variables
and
, it follows
(iii) For all
, by changing the variables
and
and the definition of Minkowski functional in (
1), it follows that
□
The continuity of the anisotropic
m-log-potential
has already been proven in [
11]. From this, it follows that
Lemma 1. Let m be an even number. The infimum inis achieved at some . Proof. We first conclude that
Actually, note that
E is a bounded measurable set, then
. For all
, let
where
is defined in (
5). It follows from
m being an even number and (
6) that
which implies that (
7) holds.
In the following, we will show that
. As a matter of fact, for
and
,
where
is in (
5). Let
. Because of (
7), there exists
such that for all
,
, which implies that
Since
is continuous and
D is compact, it can attain its minimum at a point
. Then
which implies
□
4. Conclusions
Theorem 1 and its Remark 1 complete the systematic study of the optimal upper and lower bounds of the extreme value of the anisotropic
m-log-potential on a bounded measurable set (for the part of its supremum, we refer to [
11]). Note that the anisotropic
m-log-potential extends the classical logarithmic potential two-fold in anisotropic and higher order of
m ways. By virtue of the wide development of Riesz potential with its better complement logarithmic potential for the end point case in harmonic analysis including function spaces, mathematical physics and partial differential equations (see [
1,
2,
3,
4,
5,
6]), these optimal estimates can be further applied to these related topics.
On the other hand, Brunn–Minkowski inequality and Minkowski inequality including their dual versions and generalizations are main topics in convex geometry analysis (see [
12,
13,
16,
17] and their references). The dual log-Minkowski inequality deals with the optimal estimate for mixed volume of two star bodies (see [
12,
16]), which exists as the dual version for the conjectured log-Minkowski inequality (see [
17]). The polynomial dual log-mixed volume difference law in Theorem 2 deduced from the optimal estimate in Theorem 1, deals with the optimal estimate for the difference of mixed volumes of two star bodies, which is totally new and contributes to these theories.