Sobolev interpolation inequalities with weights
Authors: Gutiérrez Cristian E. , Wheeden Richard L. 
Published: June 2012

We study weighted local Sobolev interpolation inequalities of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row StartFraction 1 Over w 2 left-parenthesis upper B right-parenthesis EndFraction integral Underscript upper B Endscripts StartAbsoluteValue u left-parenthesis x right-parenthesis EndAbsoluteValue Superscript p h Baseline w 2 left-parenthesis x right-parenthesis d x less-than-or-equal-to c left-parenthesis StartFraction 1 Over v left-parenthesis upper B right-parenthesis EndFraction integral Underscript upper B Endscripts StartAbsoluteValue u left-parenthesis x right-parenthesis EndAbsoluteValue Superscript p Baseline v left-parenthesis x right-parenthesis d x right-parenthesis Superscript h minus 1 Baseline 2nd Row times left-parenthesis StartFraction StartAbsoluteValue upper B EndAbsoluteValue Superscript p slash n Baseline Over w 1 left-parenthesis upper B right-parenthesis EndFraction integral Underscript upper B Endscripts StartAbsoluteValue nabla u left-parenthesis x right-parenthesis EndAbsoluteValue Superscript p Baseline w 1 left-parenthesis x right-parenthesis d x plus StartFraction 1 Over v left-parenthesis upper B right-parenthesis EndFraction integral Underscript upper B Endscripts StartAbsoluteValue u left-parenthesis x right-parenthesis EndAbsoluteValue Superscript p Baseline v left-parenthesis x right-parenthesis d x right-parenthesis comma EndLayout">

1

w
2

(
B
)

∫
B

|

u
(
x
)

|

p
h

w
2

(
x
)
d
x
≤
c

(

1

v
(
B
)

∫
B

|

u
(
x
)

|

p

v
(
x
)
d
x

)

h
−
1

×

(

|

B

|

p

/

n

w
1

(
B
)

∫
B

|

∇
u
(
x
)

|

p

w
1

(
x
)
d
x
+

1

v
(
B
)

∫
B

|

u
(
x
)

|

p

v
(
x
)
d
x

)

,

\begin {gathered} \frac {1} {{{w_2}(B)}}{\int \limits _B {|u(x){|^{ph}}{w_2}(x)dx \leq c\left ( {\frac {1} {{v(B)}}\int \limits _B {|u(x){|^p}v(x)dx} } \right )} ^{h - 1}} \hfill \\ \times \left ( {\frac {{|B{|^{p/n}}}} {{{w_1}(B)}}\int \limits _B {|\nabla u(x){|^p}{w_1}(x)dx + \frac {1} {{v(B)}}} \int \limits _B {|u(x){|^p}v(x)dx} } \right ), \hfill \\ \end {gathered}

\]
, where

1
>
p
>
∞
,
h
>
1
,
B

1 > p > \infty ,h > 1, B

is a ball in

R

n

{{\mathbf {R}}^n}

, and

v
v

,

w
1

{w_1}

, and

w
2

{w_2}

are weight functions. The case

p
=
2

p = 2

is of special importance in deriving regularity results for solutions of degenerate parabolic equations. We also study the analogous inequality without the second summand on the right in the case

u
u

has compact support in

B
B

, and we derive global Landau inequalities
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar nabla u double-vertical-bar Subscript upper L Sub Subscript w Sub Superscript q Subscript Baseline less-than-or-equal-to c double-vertical-bar nabla u double-vertical-bar Subscript upper L Sub Subscript v Sub Superscript p Subscript Superscript 1 minus a Baseline double-vertical-bar nabla squared u double-vertical-bar Subscript upper L Sub Subscript v Sub Superscript p Subscript Superscript a Baseline comma 0 greater-than a greater-than 1 comma 1 greater-than p less-than-or-equal-to q greater-than normal infinity">

‖

∇
u

‖

L
w
q

≤
c

‖

∇
u

‖

L
v
p

1
−
a

‖

∇
2

u

‖

L
v
p

a

,
0
>
a
>
1
,
1
>
p
≤
q
>
∞

{\left \| {\nabla u} \right \|_{L_w^q}} \leq c\left \| {\nabla u} \right \|_{L_v^p}^{1 - a}\left \| {{\nabla ^2}u} \right \|_{L_v^p}^a,0 > a > 1,1 > p \leq q > \infty

, when

u
u

has compact support.