The best constant for 𝐿^{∞}-type Gagliardo-Nirenberg inequalities
Authors: Liu Jian-Guo , Wang Jinhuan 
Published: March 2023

In this paper we derive the best constant for the following

L

∞

L^{\infty }

-type Gagliardo-Nirenberg interpolation inequality
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar u double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript Baseline less-than-or-equal-to upper C Subscript q comma normal infinity comma p Baseline double-vertical-bar u double-vertical-bar Subscript upper L Sub Superscript q plus 1 Subscript Superscript 1 minus theta Baseline double-vertical-bar nabla u double-vertical-bar Subscript upper L Sub Superscript p Subscript Superscript theta Baseline comma theta equals StartFraction p d Over d p plus left-parenthesis p minus d right-parenthesis left-parenthesis q plus 1 right-parenthesis EndFraction comma">

‖
u

‖

L

∞

≤

C

q
,
∞
,
p

‖
u

‖

L

q
+
1

1
−
θ

‖
∇
u

‖

L
p

θ

,

θ
=

p
d

d
p
+
(
p
−
d
)
(
q
+
1
)

,

\begin{equation*} \|u\|_{L^{\infty }}\leq C_{q,\infty ,p} \|u\|^{1-\theta }_{L^{q+1}}\|\nabla u\|^{\theta }_{L^p},\quad \theta =\frac {pd}{dp+(p-d)(q+1)}, \end{equation*}

where parameters

q
q

and

p
p

satisfy the conditions

p
>
d
≥
1

p>d\geq 1

,

q
≥
0

q\geq 0

. The best constant

C

q
,
∞
,
p

C_{q,\infty ,p}

is given by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript q comma normal infinity comma p Baseline equals theta Superscript minus StartFraction theta Over p EndFraction Baseline left-parenthesis 1 minus theta right-parenthesis Superscript StartFraction theta Over p EndFraction Baseline upper M Subscript c Superscript minus StartFraction theta Over d EndFraction Baseline comma upper M Subscript c Baseline colon-equal integral Underscript double-struck upper R Superscript d Baseline Endscripts u Subscript c comma normal infinity Superscript q plus 1 Baseline d x comma">

C

q
,
∞
,
p

=

θ

−

θ
p

(
1
−
θ

)

θ
p

M
c

−

θ
d

,

M
c

≔

∫

R

d

u

c
,
∞

q
+
1

d
x
,

\begin{equation*} C_{q,\infty ,p}=\theta ^{-\frac {\theta }{p}}(1-\theta )^{\frac {\theta }{p}}M_c^{-\frac {\theta }{d}},\quad M_c≔\int _{\mathbb {R}^d}u_{c,\infty }^{q+1} dx, \end{equation*}

where

u

c
,
∞

u_{c,\infty }

is the unique radial non-increasing solution to a generalized Lane-Emden equation. The case of equality holds when

u
=
A

u

c
,
∞

(
λ
(
x
−

x
0

)
)

u=Au_{c,\infty }(\lambda (x-x_0))

for any real numbers

A
A

,

λ
>
0

\lambda >0

and

x

0

∈

R

d

x_{0}\in \mathbb {R}^d

. In fact, the generalized Lane-Emden equation in

R

d

\mathbb {R}^d

contains a delta function as a source and it is a Thomas-Fermi type equation. For

q
=
0

q=0

or

d
=
1

d=1

,

u

c
,
∞

u_{c,\infty }

have closed form solutions expressed in terms of the incomplete Beta functions. Moreover, we show that

u

c
,
m

→

u

c
,
∞

u_{c,m}\to u_{c,\infty }

and

C

q
,
m
,
p

→

C

q
,
∞
,
p

C_{q,m,p}\to C_{q,\infty ,p}

as

m
→
+
∞

m\to +\infty

for

d
=
1

d=1

, where

u

c
,
m

u_{c,m}

and

C

q
,
m
,
p

C_{q,m,p}

are the function achieving equality and the best constant of

L
m

L^m

-type Gagliardo-Nirenberg interpolation inequality, respectively.