Publicacions
Ma emá iques,
Vol
35
(1991,
97-117
.
AN
EXPLICIT
EXPRESSION
FOR
THE
IV,
FUNCTIONALS
OF
INTERPOLATION
BETWEEN
LP
SPACES
JESÚS
BASTERO
*
,
YVES
RAYNAUD
AND
M
.
LUISA
REZOLA
**
In oduc ion
and
No a ion
When
dealing
wi h
in e pola ion
spaces
by
eal
me hods
one
is
lead o
com-
pu e
(o a
leas
o
es ima e)
he
K- unc ional
associa ed
o
he couple
o
in e pola ion spaces
.
This
concep
was
i s
in oduced
by
J
.Pee e
(see
[8],
[9])
and some
e o s
ha e been done
o
ind
explici
exp essions
o
i
o
he
case
o
Lebesgue
spaces
.
I is
well
known
ha
o
he couple
consis ing
o
Ll
and
L°°
on
[0,
oo)
K
is
gi en
by K(
;
,
L
1
,
L°°)
=
0
*
whe e
*
deno es
he
non
inc easing
ea angemen
o
he
unc ion
.
In
[7]
Nilsson
and
Pee e
compu ed
he
K- unc ional
also
be ween
spaces
LP
and
L9
when
1
_<
p <
q
<
oo
.
Mo e
ecen ly
he
wo
i s
named
au ho s
ob ained an
explici
exp ession
o
a
sui able
modi ica ion
o
he
K- unc ional
o
he
case
(LP,
LM)
whe e
L
M
s ands
o
an
O licz
space
(see
[1]
)
.
The
aim
o
his
pape
is
o
answe
a
ques ion
aised
by
J
.
Pee e
o
he
au ho s
and
o
ex end
he
esul s
in
[1]
and
[7]
o
he
mo e
gene al case
o
he
K,
.- unc ionals
be ween
LP
spaces
.
The
no ion
o
K,
.- unc ional
was
in oduced
in
[4]
by
Holms ed
and
Pee e ob aining
also
some
es ima es
o
hose
unc ionals
be ween
gene al
compa ible
couples
o
in e pola ion
spaces
.
We
shall
w i e
LP
o he
Lebesgue
space LP([0,
oo»
.
Fo
E
LP
+
L9,1
<_
p
<
q
<
oc,
>
0
and
1
<
<
oo
we
de ine
he
K
- unc ional
by
K,(
;
)
=
in
(¡¡gil'
+
11
hil
)
l/
whe e
he
in imum
uns o e
all
possible
decomposi ions
=
g
+
h
wi h
g
E
LP
and
h
E
L9
.
(Ob iously
K,,,
will
mean
K,
> ,
>
(
;
)
=in max{ligJ
I
P , i jhJ
l
.})
No e
ha
=
1
co esponda
o he
classical
de ini ion
o
K-
unc ional
.
The
eade
is
e e ed
o
[2], [3]
and
[9]
o
backg ound
on
in e pola ion
spaces
.
*Resea ch
pa ially
suppo ed by
CAICYT
0804-84
and
DGICYT
PS87-0338
**Resea ch
pa ially
suppo ed by
CAICYT,
PB
85-0338
98

J
.
BASTERO,
Y
.
RAINAUD,
M
.L
.
REZOLA
Clea ly
he
K,
.- unc ional
de ines a
ea angemen
in a ian
no m
on
LP
-F
L9
equi alen
o
he
na u al
one
.
Hence
K,(
;
)
=
K,(
;
*)
and
his
allows
us
o
es ic
ou sel es
o
non
nega i e
and non
inc easing
unc ions
E
Lp
- -
L9
.
The
pape
is
di ided
in o
h ee
sec ions
and
one
appendix
.
Sec ion
Iis
de o ed
o
show
he
exis ence
o
ex emal
unc ions
which minimize
he'K
-
unc ionals
.
In
sec ion
II
we
gi e a
p ocedu e
o ge
such ex emal
unc ions
o
he
cases
1
<
<
oo
and
inally
he
case
=
oo
is
conside ed
in
sec ion
III
.
In
he
appendix
we
in oduce
and compu e
he
unc ionals
K,
,
,,
and
K
9
.
We
apply
hese
esul s
o
he
isome ic
p oblem
o
in e pola ion
be ween
he
couple
o
spaces
(L
P
,L
9)
.
Le
us
men ion
ha
he
me hod
we
use
in
sec ions
II
and
III
is
ac ually a
simpli ied
e sion
o
he
calculus
o
a ia ions,
bu
we
shall
include
some
p oo s
o
he
sake
o
comple eness
.
As
i
happens
in
[1]
and
[7]
he
solu ions
o
p
=
1
o
p
>
1
a e
essen ially
di e en
.
In
he
case
p
=
1
he
ex emal decomposi ion
o
is
achie ed
by
a
ho izon al
slicing
o he
unc ion
.
I
.
Exis ence
o
ex emal
solu ions
In
he
sequel 1
_<
p
<
q
<
oo
and
will
deno e
a
non
nega i e
and non
inc easing
ixed
unc ion
on
[0,
oo)
.
I
is
e y
easy
o see
ha
whe e
he
in imum
is
aken
o e
all
unc ions
g
E
LP
wi h
0
<
g
<
.
Le
g
be
a non
nega i e
measu able
unc ion
on
[0,
oo)
such
ha 0
<_
We
de ine
he
unc ional
<P(g
;
;
;
)
o simply
~¿(g)
by
max{11911P, il
-911
9
},
I is
clea
ha 0
<
(D(g)
<
oo
and
K
(
;
)
=
in
(ligli
+
il
-
9,,y)lI
K (
;
. )
=
in
{¿(9)
1/
;0
<
9
<
. ,9
E
L
P }
The
main
esul
o
his
sec ion
is
he
ollowing
heo em
¡ 1< <oo
;
i
=
oo
.
9
<
.
I
.1
Theo em
.
Le
1
_<
<_
oo
and
E
LP
+
L9
.

The e
exis s
a
non
inc easing
unc ion
g
E
LP,
0
<_
g
<_
,
such
ha
K
(
;
)
=
$(
g)
l
/
.

I
1
<
<
oo
,
his
unc ion
g
is
unique
and
-
g
is
also
non
inc easing
.
In
o de
o
.
p o e
his
heo em
we
need
se e al
lemmas
.
We
begin
by
es ab-
lishing
he
exis ence
o
ex emal
solu ions
.
THE
K
FUNCTIONALS
OF
INTERPOLATION

99
1
.2
Lemma
.

The e
exis s
a
unc ion
g
E
LP, 0
<
g
<
,
such
ha
K,(
;
)
_
q>(
g
)1/
.
P oo
. .
Le
a
be
de ined
by a
=
in {,¿(g) ;0
_<
g
<_ ,
g
E
LP}
.

We
can
choose
a
sequence
(gn)n
in
LP
such ha
0
<
9n
< and
a=
lim~-,,
4>(g
n
)
.
As
(
-
g),,
is
a
bounded
sequence
in
L9
we
may
suppose,
by
passing
o
subsequence
i
necessa y,
ha
(
-g
n
)
n
con e ges
weakly
o
a
unc ion
h
E L
q
.
Then
(g n
)
n
is
a
weakly
Cauchy
sequence
in
LP and
so
w-limg
n
=
-h
E
LP
.
Hence
i
g
=
-
h we
ha e
~¿(g)
<
limin
119n11p
+ limin
1I
-9n1Iq
=
li,~¿(gn)
=
a
.
The same
a gumen s
can
be
modi ied
o
=
oo
.
Rema k
.
Le
us
poin
ou
ha
he
same
ideas
used
in
p eceding
lemma
may
be
applied
o
K,- unc ionals
be ween
a la ge
class
o
in e pola ion spaces
.
Ac ually,
i
(A
0
,
A
l
) is
a
compa ible
couple
o
Banach
spaces
le
us
de ine
K
on
A
o
+A
1
by
K,(
;
)
=
in
(11911Á',
+
ilhIIA,)1/
whe e
he
in imum
uns
all
possible
decomposi ions
=
g
+
h
wi h
g
E
A
o
and
h
E
A
l
.
I
we
suppose
ha
A
o
is
weakly
sequen ially
comple e,
A
1
is
e lexi e
and Aá n
Ai
is
dense
in
A*,
hen
he e
exis
g
E
A
o
and
hE
A
1
such
ha
=9+h
and
K,(
; )
=
(11911Á
o
+ 1Ih1i ,)
1/
1
.3
De ini ion
.
A
non
inc easing
unc ion
g
in
LP
is
an
ex ema]
solu ion
o he
unc ional
K,(
;
)
i
0
<_
g
<
and
K
(
;
)
=
¿(g)1/
(K
(
;
)
=
(p(9)
i
=
oo)
.
Nex
we
will
s udy
he
uniqueness
o he
ex ema]
solu ions
.
1 .4
Lemma
.
Le¡
1
<
<
oo
.

I
K
(
;
)
=
¿(
91
)1/
=
<P(
92
)1/
,
hen
91
=
92-
P oo
. .
I is
e y easy
o
check
ha
he
unc ion
g
=
91
2
g2
also
e i ies
K
(
;
)
=
~¿(g)1/
o
all 1 <_
<
oo
.
Indeed,
by
using
Minkowski's
inequali y
we
ha e
,
¿(9)
1/
_<
1
[(119111,+119211,) +( ll -911Iq+ ll -92jjq)
[
1/
<
1
[x(91
_
)
1/
+4
>(92)
1/
]
=
K (
;
)
and
hen
1191
+
9211,
=
119111,
+
119211

11(
-
91)
+
(
.
-
92)1I9
=
II
-
9111
9
+
Il
-9211q-
I 1
<
<
oo
he
ec o s
in
R2
(1191
11,,
I1
-
91
I1
q
)
and
(11921I,,
11
-
9211
q
)
a e
also colinea
.
Since
Lq
is
s ic 1y
con ex
(q
>
1)
we
ob ain
ha ,
o ins ance,
100

J
.
BASTERO,
Y
.
RAINAUD,
M
.L
.
REZOLA
-
91
=
A(
-
92)
o
some A
>
0
.
Thus,
11911¡p
=
X119211,
and
oD(g1)
=
A (P(92)
which
implies
ha
A
=
1
and
consequen ly
g1
=
92-
Suppose
now
ha
=
oo
.

We
ealize
ha
K,,~(
;
)
=
4>(g)
implies
ha
J
l9IJp
=
J
~
-
9
JJq
.
Indeed,
i i
we e
JigJip
>
J
I
-9Jj9,
we
choose
some
posi i a
a
such ha
he
se
A
=
{x
E
[0,
oo)
;
g(x)
>
a}
has
m(A)
>
0
.
Take
0
<
S
small
enough
e i ying
il
-
9liq
<
il
-
9+áXAliy
<
119
-6
XA11p
<
liglip
Hence
$(9
-
6XA)
=
119
-
6
XA1
¡p
<
(P(g)
and
consequen ly
K,,~(i
;
)
<
~¿(g)
.
(An
analogous
a gumen
wo ks
i
we
suppose
liglip
<
il
-
glw)
.
Le
91,92
be
wo
unc ions
in
Lp
such
ha
K~(
;
)
=
<P(91)
=
<P(92)
.
We
may
epea he
a gumen a
used
o
1
<
<
oo
and
we
ob ain
(
91
2
92
)
<1
max{li91ilp
+
119211p, il
-
91lw
+ il
-
9211}
We
he e o e
ha e
li
-
91
+
-
9211,

=

l
i
-
91
i
l
e
+
l
i
-
9211,
and
hen
-
91
=
A(
-
92)
which
also
implies
g1
=
92
.
Rema k
.
In he
case
=
1di e en
solu ions
could
be
ob ained
.
We
ealiza
ha
i
g1 and
g
2
a e
wo
di e en
solu ions
any
o he
unc ion
g
in
he
segmen
de ined
by g1
and
92
is also
a solu ion
.
1
.5
Lemma
.
i)
I
0
<_
g
<
hen
0
<_
g*
<_
and
(P(g*)
<
<P(g)
(g*
is
he
non
inc easing
ea angemen
o
g)
.
ü)
I
he
unc ional
K,(
;
)
has
only
one
ex emal
solu ion g
hen
-
g
is
non
inc easing
.
P oo
.
i)
We
apply
he
p oposi ion
1
o
[6]
(which
is
also
aluable
in
he
in e al
(0,
oo)
)
and
hen
we
ha e
a
This
implies
ha
Ji
-
g*
li
q<_
l i
-
9I
l,
and
so
<D(g*)
<
~¿(g)
.
ii) I
K,(
;
)
=
~¿(g) hen
g
=
g*
.
The
unc ion
g1
de ined
by
g1
=
-(
-g)*
sa is ies
-P(g1)
_<
$(g
*
)
by
i)
.
Thus g
1
=
g*
and
so
=
g* '= (
-
g)*
which
implies
ha
he
unc ion
-
g
is
non
inc easing
.
Rema k
.
The
lemma
is
also
ue
i
we
conside
he
co esponding
K
unc-
ional
be ween
a
couple
o
ea angemen
in a ian
unc ion
spaces
.
P oo
o
he
heo em
L1
:
I is
ob ious
om
he
p eceding
lemmas
.
9
THE
K,
.
FUNCTIONALS
OF
INTERPOLATION

101
II
.
De e mina ion
o
he
ex ema¡
solu ions
when
1< <oo
The
main
ool
o
his
pa
is
he
ollowing
lemma
.
II
.1
Lemma
.
Le
g
be
an
ex ema¡
solu ion o
K,(
;
.
)
and
assume
supp
=
[0, a],
0
<
a
<
oo
.

The
ollowing
asse ions
a e
íe
:
i)
Ei he
g
<
a
.e
.
on
supp
,
o
=
1
and
g
=
,
a
.e
.
on
supp
.
ii)
I
p
>
1,
ei he
0
<
g(x)
and
g(x)p
-1
jjgjj
-
=
[ (x)
-
g(x)]q-1[l
-
g
lI -q
a
.
e
.

on
supp
,
o
=
1
and
g
=
O
.
iii)
I
p
=
1,
a
>
0
and
g
:~
hen
a
<
oo
and
(x)
-
g(x)
=
cons an
=
A
>
0,
a
.
e
.

on
[0, a],
whe e
a
and
A
e i
y
( a

-1


00
.
( I9)-1
-
a~l

ñ9_

-1
I
a~q
+
1

q]
Fu he mo e,
i
=
1,
hen
a
<
q'
(q'
is
¡he
conjuga e
exponen¡
4+Q,
=
1)
and
a
=
[ q'
-
a]-1IgIl X[a,oo)lIq
i )
I
g
=
0,
hen
=
1
.

Mo eo e ,
i
p
=
1
hen
ELq
1
L°°
and
q
'
~~gll ll
.
<-
Il llq
.
)
I
g
=
(
=
1)
and
p
=
1
hen
leng h
supp
=
b
<
q'
.
P oo
.
i)
Assume
ha
m{x
;
(x)
=
g(x)}
>
0
.

Then
he e
exis s
n
E
[®[
such
ha
B
n
=
{x
E
supp
;
l/n
<
(x)
=
g(x)}
and
m(B
n
)
>
0
.
Le
cp
be
he
unc ion
de ined
by
cp(S)
=
4>(gXB~
+
(
-
b)XBn)
o
0
<_
b
<
1/n
.
As
c,(0)
=
4>(g)
and
W'(0+)
<
0
we
necessa ily
ha e
ha
>
1
and
=
g
.
ii)
Le
A
n
=
{x
E
supp
;
g(x)
=
0,
(x)
>
1/n}
and
le
W(ó)
=
(P(9
+
5XA

)
o
0
<
b
<
l/n,
nE
N
.
I
>
1
o
7~
g
he
same
easons
as
be o e
imply
ha
m(An)
=
0,
and
so,
0
<
g
a
.e
.
on
supp
.
Conside
now
any
measu able
se
A
con ained
in
supp
,
wi h
0
<_
m(A)
<
oo
.
The
unc ion
cp(b)
=
<I>(9
+
áXA)
has
i s
minimum
in
S
=
0,
hence
yp'(O)
=
0
(
no e ha
p
>
1
)
.
Thus
A
l
l9I
I
P
-I
lgil
-1
-
l
l
-
9l
l
q
-'
I
-
91q
-1
=
0
and
hen
ii)
ollows
since
A
is
a bi a y
.
iii)
Le
A
be
any
compac
in e al
con ained
in
[0,
a)
.
Le
cp
be he
unc ion
de ined
by
W(ó)
=
4>(g
+ÓXA)
o
Ial
<
in { (x)
;
xE
A}
.
I is
clea
ha
W'(O)
does
exis s
and
ac ually
W'(0)
=
0
.
The e o e
we
ob ain
-
ll
-
9119
-
ql
-
9]
q-1
=
0
.
Since
his
exp ession
is
ue
o
any
compac
in e al
con ained
in
[0,
a)
we
deduce
ha
-
g
=
cons an
,
a
.e
.x
E
[0
;
a]
.

I
A
=
(x)
-
g(x),
hen

10
2

J
.
BASTERO,
Y
.
RAINAUD,
M
.L
.
REZOLA
g
=
(
-
A)X(o,a]
and
consequen ly
a
and
A
ha e
o
e i y
he
equa ion
(*)
.
No e
ha a
<
oo as
he
unc ion
-
g
E
Lq
.
I
=
1
we
easily
compu e
ha
1

ía

0o
(,)
,
j
(119)
-
1
- -
~
and
hen
( q'
-
a)
Aq
=
:
°
q
which
implies
iii)
.
i )
I
g
=
0
is
an ex emal
solu ion,
hen
E
Lq
.
We
epea he p eceding
a gumen s
by
conside ing
now
he
unc ion
cp(ó)
_
~(SX ~
~)
whe e
[c,
d]
C_
supp
and
0
<
S
<
(d)
.
Since
cp
has
he
minimum
in
b
=
0,
cp'(0+)
>-
0
and
hence
d
0
<

lim
á -1 (d
-
c
) 1P
-

,-1
l
l
i l
q
-q
6-
.0+
c
wha
implies
ha
=
1
.

I
p
=
1
we
he e o e
ob ain
1
>
i
1
l
.iq-q
(x)
a
.e
.
on
supp
.
)
I
g
=
(
=
1)
simila
a gumen s
o
hose
appea ing
in
i )
show
)
.
This
concludes he
p oo
o he
lemma
.
Nex we
will
s udy
he
case
p
>
1
.
II
.2 In
iew
he
p eceding
lemma
we
ha e
o
conside
he'class
o
unc ions
g e i ying
:
i)
g
E
LP,
-
g
E
Lq,
ii)O<g<
and
ll
-
gllq
-q
[ (
x
)
-
g(x)lq-1
=
llgli -Pg(x)P-1
a
.e
.
on
supp
.
Le
A
deno e
he
class
o
unc ions
sa is ying
i)
and
ii)
.
I is
e y
simple
o
check
ha
o
hem
q
>
(g)
=
ll
-
9li
q
-
q
1
(
-
g)
q-1
=
llgl1P
-P
1
g
P-1
.
q-1
Using
he
s ic 1y
dec easing
unc ion
h
de ined
by
h(y)
-
(
(x)
-
y
)
q-1
y
P-1
0
<
y
<
(x),
we
see
ha
he
ollowing
ac s a e ue
:
i) I
9
E
A,
g
is
non
inc easing
.
ii)
A
is
o ally
o de ed
.
Indeed,
le
91,
92
be
wo
elemen s
in
A
and
w i e
Mi
=
1lgi
l
l
-P
l
i
-
9i
l
l9- ,
i
=
1,
2
.
I is
clea
ha
M1=M2
(
espec i ely
M1<
M2)
implies
g1(x)
=
92(x)
a
.e
.x
Esupp
( espec i ely
g1(x)
>
92(x)
a
.e
.
x
E
supp
)
.
Fu he mo e,
o
q
>
,
he
inequali y
g
1
>
g2
yields
M
1
>
M2
.
Hence
he
se
A
has
only
one
elemen
which
is
necessa ily
he
unique
ex emal
solu ion
o
K
(
;
)
.
Le
now
assume
p
<
<
q
.

I
g
deno es
he
in
A
we
a e
going
o
p o e
ha
g=
min
A
and
ha
g
is
he
unique ex emal
solu ion
o
K,(
;
)
.
Indeed,
we
ealize
ha
o
any wo
elemen s
o
A,
g1
<
92
we
ha e
4)(g1)
<
<P(92)
.
THE
K
FUNCTIONALS
OF
INTERPOLATION

10
.3
Le
g
be
de ined
by g(x)
=
in
{g(x)
;
g
E
A}
E
LP

(LP
is
o de
con inuous,
see
[5})
.
De ine
a
=
i
IIgil,
and
le
(gn)n be
a
sequence
in
A
such
ha
g
EA
I
I91
I
Ip
<
.....
-<
I
I9n
I
Ip
<
.... .
->
a
.

The e o e
91
>_
92
>-
.
.
........
>
9n
>
.
. .
.
>
0
.
Since
ll
-
gnll'
<4
)(gn)
<
<D(gl)
we
ha e
ha
g
o
=
limg,
n
E
LP
and
-
g
o
E
Lq
.
By
passing
o
he
limi
in 11 .2
.
i
is
easily
checked
ha
g
o
E
A
,
oo
.
E en ually
we
conclude
ha
g=
g
o
E
A
.
We
summa ize
all
hese
ac s
in
he
ollowing
heo em
11
.3
Theo em
.
The
case
p>
1
.
i)
I
>_
q
hen
he
caass
A
has
only
one
elemen
which
is
he
unique
ex emal
solu ion o
K
(
;
)-
i)
I
p
<
<
q
hen
he
leas
elemen
o
A
is
he
unique
ex emal
solu ion
o
K,
(
;
)
.
iii)
I
1
<
<
p
we know
ha
he
unique
ex emal
solu ion o
K
(
;
)
is
an
elemen
o
A
.
i )
I
1
=
<
p
¡he
solu ions
e i y
¡he
equa ion
[ (x)
-
9(x)]
q-1
I
Ig1Ip
-1
=
g(X)
P-,
I I
-
9I
I9
-1
Now
g
could
be
equal
o
0
o
.
Fu hemo e
he
solu ion
is
unique
excep
i
=
AX[O
a]
.
In
his
case
=
al/p -1
/q
and
K,
(
;
)
=
Aal/P
.
Mo eo e
in
¡he
ou
cases,
i
g
is
an
ex emal
solu ion
we
ha e
1/

1/
I1
( i )
=
II9IIp-(pl )
U
gp-11

=
ll
-9IIq-(ql )
(1
(
-
9)
q-11
P oo
. .
We
only
ha e
o
p o e
he
las
pa
o
i )
.
I
91
7É
92
a e ex emal
solu ion
o
K
l
(
;
) hen
all
he
poin s
o
he
segmen
[91,921
a e
ex emal
solu ions
:
So
we
may
suppose
gl
:~
0
7
É
92
and
-
gl
7
É
0
:~
-
92
.
Since LP
and
Lq a e
s ic ly
con ex
spaces
he
same
easons
appea ing
in
he
lemma
I
.4
say
ha
g
l
=
ag
2
,
-
9
1
=
b(
-
92) o
posi i e
a,
b
.
The e o e
=
cg2,
o
some
c
>
1,
and
hus
g2(x)q-p
=
1192119-1II92IIp-1
a
.e
.
on
supp
.
Hence
=
AX[0
a],
>
0
and
=
a
l/p-1/q,
M
11
.4
Le us
now
conside
he
case
p
=
1
.
We
shall
deno e
by
b
=leng h
o
supp
<
oo
.
As
i
appea s
in
lemma
II
.1
.,
he ex emal
solu ions
o
K,(
;
),
g,
ha e
he ollowing
exp ession
g
=
(
-
A)X[o,a]
wi h 0
<
a
<
oo
.
I
a >
0
hen
A
e i ies
he
equa ion
IL1-iii)-(*)
.
We
de ine
he
unc ion
H(a,
A)
by
a

-1

00
1( lq)
-
1
H(a,
A)
_
~~

-
aA~

-
Aq
-1
[aAq
-F
J

qJ

(
>
1)
o

a
00
(1/q)-1
H
(a,
A)
=
1
-
Aq-1
I
aAq
+
~

qJ

(
=
1)
a
10
4

J
.
BASTERO,
Y
.
RAINAUD,
M
.L
.
REZOLA
o 0<a<_b,0<
.~<_
1
o
, >1and0<A<oc, =1
.
I aisa
ixed posi i e
numbe
he
unc ion
H(a,
A)
is
s ic ly
dec easing
in
he
a iable
a
.~
and
H(a,
0)
>
0
.

On
he
o he hand,
H(a,
1

)
<
0
in
he case
>
1,
alo
and
o
=
1
lima_,

H(a,A) <
0
i
and
only
i
a
<
9'
.
Hence,
he
equa ion
H(a,
a)
=
0
has
only
one
solu ion
A
=
a
o
any
0
<
a
<_
b (case
>
1)
and
o
any 0
<
a
<
min{
9
',
b}
(case
=
1)
.
When
a
=
band
=
1
he
co esponding
equa ion
H(a,
A)
=
0is
hen
1
=
b
l
l9
-1
and
so,
his
equa ion
has
solu ion
i
and
only
i
9
'
=
b
.
Fu he mo e,
i
g
is
an ex emal
solu ion o
he
unc ional
K
I
(
;
)
wi h
a
=
b,
hen
b
<
oo,
9
'
=
b,
Kl
(
;
)
=
1
l
11
1
(all
he
unc ions
-
a
a e
ex emal
solu ions
o he
unc ional
o 0
<_
A
<_
(b
-
)
)
and
H(b,
(b))
=
0
.
In
o de
o
de e mina e
he
possible
ex emal
solu ions g
we
only
ha e
o
check
he
alue
o a
equal
o
he
leng h
o
supp
g
.
The
main
ool
o
compu ing
his
alue
a
is
he
ollowing
Lmma
11
.5
Lemma
.
Leí
g
be
an
ex emal
solu ion o
K,(
;
)
and
leí
[0,
a]
_
supp
g,
a
>
0
.

Then
(a+)
<
Aa
<
(-_)
.
P oo
.
_Since
g
=
(
-
Aa)X[a,a~
>
0
we
ind
Aa
<
(a
-
)
.
I
Aa
<
(a+),
we
would
ha e
a
<
(x)
o
all
x
E
(a,
c)
.
By
conside ing
he
auxilia y
unc ion
M)
=
'¿(9
+
6X[a,c])
de ined
o
0
<
b
<
a,
we
easily
would
check
ha
cp'(0+)
=
l
i
-
gil'
-9

(
,
á
-1
-
.
9-1
)
.<
0
a
and,
consequen ly,
g
would
no
be ex emal
solu ion
o
he
unc ional
K,
.
(
;
)
which
con adic s
ou
assump ion
.
Hence
he
Lmma
ollows
.
11
.6
Rema k
.
As an
immedia e
consequence
o
Lemma
11
.5
.
we
ha e
o
s udy
he
se
I
=
{x
E
(0,
b]
;
H(x,
(x
-
)) _<
0
<_
H(x,
(x+))},
because
he
leng h
o he
suppo
o
he
possible
non
null
solu ions
belongs
o I
.
Then
we
de ine he
unc ion
F(x)
=
H(x,
(x))
de ined
o
0
<x
_< b
.
The
ollowing
p ope ies
will
allow
us o
compu e
easily
he
ex emal
solu ions o
K,(
;
)
.
II
.7
Lemma
.
The
ollowing
p ope ies
a e
ae
:
i)
F(
.)
is
non
dec easing
.
Fu he mo e
i
F(xl)
=
F(x2)
wi h
x
1
<x2
hen
is
cons an
in
he
closed
in e al
[x1,
x
2
] .
ii)
F(x
-
)
=
H(x,
(x
-
»
and
F(x
1
)
=
H(x,
(x+))
.
iii)
I xl
<
x2,
H(xl,
. (xl»
:5
H(x2,
. (x2))-
i )
I
is
ei he
emp y
o
an
in e al
and
he
unc ion
is
cons an
on
i s
in e io
.
Fu he mo e
I
=
{x
;
F(x
-
)
<
0
<
F(x+)}
.
P oo
.
i)
Le
xl
<
x2
be
wo
posi i e
numbe s
.
Since
is
non
dec easing
we
ha e
ha
z
x2 (x2)
T
XI (x1)
C
(
x
2)(
x
2
-
xl)
<

xl
and
he e o e
I
>
q
i
is
also
clea
ha
xl (xl)
q
+

q
>
x
l
(x1)
q
+(x2
-
xl) (x2)
q
+

q
>
x2 (x2)
q
+

q
x

x
2
x
2
Hence
(X1)q-1
IX, (X1)q
+
THE
K,
.
FUNCTIONALS
OF
INTERPOLATION

105
x

-1
xl (xl)
0
q
In
he
case 1
<
<
qwe
ealize
ha
( Iq)
-
1
(xl)
q
1
Ixl (xl)q
+

q1

_
L

x
1
00
xl) -1
xl
+
oo
(
L

~q~( Iq)-1
Jx
(xl)
(x2) -1
1x,
+J
00

( l))q]( /q)-1
x2
)
-1
~x2
+
lo


ql
( lq)
-
1
x2
(xl)

2)
-1 1X2
+
Jx2

( (
2))q]( lq)-1
and
hus
he
i s
pa o
i) is
p o ed
.
I
F(x
1
)
=
F(x2)
and
>
1
we
ha e,
in
pa icula ,
ha
x
2 (
x
2)
-
xl (x1)
=
o
any
x,
x
l
<
x
<
x
2
.
Then
(x)
=
(X1)
=
(x2)
and
is
cons an
in
[XI,
x2]
.
J
.`1,2
q
Suppos
eno
w ha
=1
, he
nw
e in
d
J
(
(x1)
)=
x2
-x
1
and
he e o e
we
ha e
ha
is also
cons an
in [XI,
x
2
] .
ii)
This
p ope y
is
easily
compu ed om
he
special
exp ession
de ining
he
unc ion
F(
.)
.
iii)
H(x
l ,
(xi
))
=

lim
F(z)
<

lim
F(z)
=
H(x2,
(x2)
.
<_
[
¡x2
-
x2 (x2)
0
~
~
(x2)q
-1
IX2 (X2)'
+

qJ
x2
z
?
(x-xl) (x)+(x2
-x) (x2)
112

J
.
BASTERO,
Y
.
RAINAUD,
M
.L
.
REZOLA
P oo
.
We
shall
p o e
he
heo em
in
h ee s eps
.
To
s a
wi h
we
shall
es ablish
he
heo em
o
simple
unc ions
.
n
Siep
1
.
Suppose
ha
=
i
:
aiXI
i
whe e
al
>
. . .
>
a
n
>
0,
Ii's
a e
pai wise
i=1
disjoin in e als
wi h
leng h(Ii)
=
mi
>
0
.
I
is
qui e clea ha
1C
,
9
(
;
)
_
n
min
cp(x),
being
x
=
(xl,
.
. .,
x
n
),
C
=
Jj[0,
mil and W
he
con inuous
unc ion
8=1
de ined
by
n

)
/P

(
n

9
/9
x

aPxi

+

ay
(Mi
-
xi
Le
x =
(11,
. .
.,
-
in) a
minimum
o
his
unc ion
cp
in
C
.
I
x
EC
hen
a

(x)
_
áx
i
0,
o
all
1
<
i
<_
n
.
Easy
compu a ions
would
imply
ha
a
l
=
.
. .
=
a
n
which
is
a
con adic ion
.
Thus 1 E
áC
and,
o
ins an e,
we
may
assume
ha
he e
exis s
j (1
<_ j
<
n)
such
ha
Yj
=
Qj
wi h
Qj
=
0
o
=
mj
.
Now
we
conside
he
unc ion
W(xl1
. .
.
.
xj_1, x7+1,
".
.,
x
n
)
=
W(xly
..
.,
xj_1,Qj5
xj+1)
..
.,
xn)
de ined
o 0
<
xi
<
mi,
i
:~
j
.
This
unc ion
0
a ains
i s
minimum
in
(xl,
.
.
.,
1
7
)
+
l,
. .
.,
7n)
.
Repea ing
he
a gumen
we would
ob ain
ha
he e
exis s
ano he
coo dina e
j'
such
ha
Tp
=
Qj,
(£j,
=
0
o
mj,) and
so
on
.
E en ually
we
ob ain
ha
he e
is
a
na u al
numbe
k,
1
<_
k
<
n,
such
ha
j
=
(Q1,
. .
.,Qk_1,7k,Qk+1,
.
.
., n)
whe e
each
2i
=
0
o
mi
and xk
is
equal
o 0
a
o
mk
o
a
solu ion
o
he
equa ion
in
x
:
OX
(~1,
--
.A-1,x)4+1)
. .
.,
£n)
=
0
.
In
o de
o p ecise
he
exac
o m
o
x
we
ealize
ha
i
x =
(0,
. .
.,
0)
o
=_
(ml,
. .
.,
m
n
)
he
p oo
o he
S ep
1
would
be
inished
.
So,
we
may
suppose
-

(0,
. .
.,
0)
and
Y
7É
(ml,
..
.,
n
n
)
.
Le
A
be
he
well
de ined
posi i e
numbe
A
=

4
[ i_1
ael
ii
/P-1
Le
A
be
he
se
A
=
{i
;
a
;
-
P >_
A}
.
The
ollowing
lemma
is
he
main
ool
in
he
p oo
o
his
s ep
.
A
.5
.
Lemma
.

The
se
A
:~
0
.
I
k
o
=max
A
hen
1
=
(ml,
".
.,
mko_1,
xko,
0,
".
.,
0)e

whe e
xko
=mk
o
i
aka
P
>
A
and

8-
(x)
=
0
o he wise
.
ax
ko
P oo
..
We
know
ha
x=
(~
1,
.
.
.,Bk_1,xk,Qk+1,
.
.
.,£n)
o
some
k,1
_<
k
_<
n
.
I
j
:~
k
le
h
be
he
eal
unc ion
de ined
by
As
Pj
=
0 ( espec i ely
2j
=
mj)
implies
ha
h'(0+)
>
0
( espec
.
h'(m
i
)_<
0)
i is
easy
o
check
ha Pi
=
0
( espec
.
=
mj)
implies
ha
al -
P
_<
A
( espec
.
al
-
P
>
A)
.
In
he
same
way
,
i
jk
=
0
hen
ak
-
P
_<
A,
i
!k
=mk
hen
aq
-
P
>
A
and
o he wise
ak
_P
=
A
.
As
x
:~
(0,
.
. .,
0)
he
se
A

~
.

Hence
i
j

<
k
0
( espec
.

j

>
k
0
)
he
co esponding
j- h
coo dina e
has
o
be
mj
( espec
.
0)
.
Finally
we
only
ha e
o
conside
he
di e en
alues
o
ak
o
wi h
espec o
A
and
his
concludes
he
p oo
o
he
lemma
.
o
By
applying
he
lemma
we
ha e
ha
1C,,9(
;
)
=
il
II
Q
o
K,,s(
;
)
=
li
ji
Hence
o
some
xE
supp
.
THE
K
FUNCTIONALS
OF
INTERPOLATION

113
_
h(y)
=
~
P(~1,
. .
., i)
y,
.
.
., xk,
. . .,
~n)
.
n
9 Iq
+
Cak
o
(mku
_
xko)+
5
a9m4/
ko+1
-
I
I X[0,x]
I
IP
+
i
l
X[x,-)
I
Iq
whe e
x
=
m
l
+
. . .
+
mko_1
+xk
o
.
In
he
he e
cases
he
S ep
1 is
p o ed
.
S ep
2
.
Suppose
now
ha
E
L°°
and
he leng h
o
he
suppo
o
is
ini e
.
We
can
app oxima e
by
a
non
dec easing
sequence
o
simple unc ions
( n)n
con e ging
o
in
he
L°°-no m
.
We
apply
he
S ep
1
o
he e
simple
unc ions
n
and we
ge
ha
K ,s(
;
n)
=
II X[O,x
]IIp
+ ij X[x
,oo)IIq
whe e
x
n
E
supp
n
.
Since
JC
,
s
( ,
n
)
<
K,,s(
;
)
by
passing
o
a
subsequence
i
necessa y he e
exis s
x
=
lim
x
n
and
hus
we
ha e
llm
n
)
=
Ii X[O,x]IIp
+ iI X[x,oo)II`

)
.
K,,s(
;
)
=
I
I
X[O,x]
I
IP
+
i l
X[z,oo)
I
I
q
S ep
3
.
I
is
a
gene al
non
inc easing
and non
nega i e unc ion
in
LP
+
Lq
we
app oxima e
by a
sequence
o
unca ions
o
and
we
apply
he
ideas
o
he
S ep
2
.
114

J
.
BASTERO,
Y
.
RAINAUD,
M
.L
.
REZOLA
In
o de
e
de e mina e
he
poin
x
E
supp
o
which
1C"-'(
;
)
=
11 X[O,
.J
1I
P
+
ll
X[x,oo)
11
q
we
conside
he
auxilia y
unc ion
cp
de ined
by
Ux
P)
/P
+
C
¡oo
q)
sI
q
o
0
<
x
<
b
=
leng h
o he
suppo
o
.
This
unc ion
cp
is
con inuous
and
has
de i a i e
cp'(x)
almos
e e ywhe e
.
Mo e
p ecisely,
he e
exis
cp'(x+)
and
cp'(x
-
)
and
hey
a e
equal
excep
a
mos
in
he
discon inui y poin s
o
.
I
is
easy
o
check
ha
0<x
<
b,
whe e
A
.6
.
P oposi ion
.
~
I
(x})
-
q
ll x[x,oo)11
q
-q
(x )Pl~¿(x)
- (x
q
P
9
I1 x[0,
.]H
P
-P
ps
11
X[x,°°)
119
-q
(cp'(x
:1
)
deno es
each
one
o he
wo
hand
side de i a i es o
cp,
in
he
same
way
(x
:
~)
ep esen s
he
wo
la e al
limi s
o )
.
Hence we
ha e
ha
i
x
0
is
a
poin
o
minimum
o
he
unc ion
W
necessa ily
x0
=
0,
o
=
b
o
(x~
)q
-
P
>_
4>(x0)
>
(xó
)q
-
P
.
In pa icula
we
ha e
he
ollowing
i)
I
ei he
>_
p
and
s
>
q
o
>pand
s
>_
q he e
exis s
only
one
poin
x
0
E
supp
o
which
¡he
minimum
o
cp
is
a ained
.
Fu he mo e
xO
=
sup{x
;
<D(x)
<
(x)q-P}
and
K ,s(
; )
=
ll x[xo,
.)11
q
-
q
L
p
g
ll x[O,x"]
llp
p
(x)+ll x[xo~
.)ll
gJ
ii)
I
=p
ands=q
00

¡
min x
x
dx
.
0
P oo
..
i)
Unde
hese
hypo heses
~¿
is
a
s ic ely
inc easing
unc ion
.
Le
I
=
{x
E
(0,
b)
;
<P(x)
<
(x)q
-
P}
.
I
=
01
means
ha
E
Lq
l
L°°,
=
pand
ll
llóo
P

<

E
ll llq-9
.

Then
'p'(x
:~)
>
0 o
all
x
and
hen
IC ,,(
;
)
=
11
11
q
.
and
THE
I1',
.
FUNCTIONALS
OF
INTERPOLATION

115
I
I
=
(0,
b)
hen
b
<
oo
and
~p'(x})
<
0 o
all
0
<
x
<
b
.
So
IC,,,,(
;
)
_
P
.
O he wise
cp'(x )
<
0
o
all
xE
I
and
~o'(xl)
>
0
i
x
11
.
Hence
i)
holds
.
ii)
Now
~¿(x)
_

and
his
pa
o
he
p oposi ion
is
easily
checked
.
Rema k
.
We
no e
ha
ou
1C
p
,
Q
unc ional
is
exac ly
he
co esponding
)C
P
,
q
unc ional
used
by
Spa
in
[11,
de ini ion
(3
.1)]
La e
on
we
shall
p o e
ha
o
, s
>_
1 all
he
K,,,-mono onici ies a e
equi alen
in
any
in e media e
space
be ween
LP
and
L9
and
ha
a
weake
esul
is
ue
o
he co espondig
IC ,,,-mono onici ies
.
A
.7
.
P oposi ion
.
Le
X
be
an
in e media e
space
be ween
LP
and
L9
and
,
s
>_
1
.

The
ollowing
asse ions
a e
ae
:
i)
X
is
(1,K
,
.,)-mono one
q
X
is
(1,
K)
-mono one
.

ü)
I
>_
p
and
s
>_
q,
X
is
(1,
K,
,
,)-mono one
~-¿
X
is
(1,
/C,,,)-mono one
.
P oo
.
.
i)
Se
E
he
unc ional de ined
o
E
LP+L
9
and
X
>
0
by
E(,
;
)
_
in

11
-
g11
a
.
I is
clea
ha
I
Ig11
P
<_a
K ,s(
;
)
=in
[A
+
E(i`l/ j)9J

(A
.7
.1)
K,,
(
;
)
-
A

1/s
(see
[11],
lemma
3
.3)
.
Then
K
,,(
;
)
>
K ,,(
;9)
o
all
>
0
i
and
only
i
E(X
;
)
>
E(X
;
g)
o
all
A
>
0
and
hence
i
and
only
i
Kj,l(
;
)
>_
Kj,l(
;
g)
o all >0
.
ii)
The
p oo
o
his
pa
is
simila o
he
p e ious
one
by
using
a
sui able
modi ica ion
o
he
unc ional
E,
namely
£,
de ined
by
£(X
;
)
=

in

1I X[=,
.)1w
I1No, 311
P
<_a
Now
he
co esponding
simila
exp essions
(A
.7
.1)
and (A
.7
.2)
o
he
unc ion-
als
K ,,
and £
occu
.
Ac ually,
he
only
hing
we
ha e
o
compu e
is
ha
gi en
A
>
0
wi h
£(A
;
)
>
0
he e
exis s
>
0
such
ha
K,,,(
;
)
=
A
+
£(,X
;
)s
.
I is
clea
ha
he e
exis s
only
one
poin
y,
0
<
y
<
leng h
supp
,
such
ha
li
X[o,yj
I
P
=
A
and
li
X[y,,, .)
lI
q
=
£(
A
;
)
.
Since
>p
and
s
>
q
we
ake
=

qA
'
-P

(y)
P-9
and we
apply
P oposi ion
A
.6
.
ps£(A
;
)
8-
q
Rema k
.

Fo
,
s >_ 1 i
is
clea
ha
i
X
is
(1,
1C
,
,)-mono one
=>
X
is
(1,
K, )-mono one
.
Now
we
ecall
ha
a
la ice
homomo phism
is
a
linea
bounded
ope a o
be-
ween
Banach
la ices
which
maps
disjoin ly
suppo ed
unc ions
in o
disjoin ly
suppo ed
unc ions
.
11
6

J
.
BASTERO,
Y
.
RAINAUD,
M
.L
.
REZOLA
A
.8
.
Theo em
.
Le
X
be
an
in e media e
space
be ween
LPand
Lq
.
X
is
(1,Kp,q)-mono one
i
and
only
i
he
ollowing
in e pola ion
esul
is
ue
:
I
T
E
C(LP)
n
G(Lq)
is
a
la ice
homomo phism
hen
T
E
.C(X)
and
IITIIx-x
_<
max{IITIILy-L",
IITIILI--
.La}
.
P oo
.
Fi s
o
all
we
ema k
ha
bo h
asse ions
imply
ha
X
is
a
ea -
angemen
in a ian
unc ion
space
.
Suppose
ha
X
is
(1,
IC
p
q
)-mono one
and
ha
T
is
a
la ice
homomo phism
such
ha
max{IITIILD-DI,
IITIIL9-L9}
_< 1
.
I
EX,
since
T
is
a
la ice
homomo phism
we
ha e
Kp,q(
;T )

a
nT,
IIgIIP+ IlhII9
<
a
+
h
,
IIgIII+ ilhII9
qAh=0

q^h=0
=
Kp,q(
;
)
o
all i
>
0
.
Thus
T
E
X
and
IIT
IIx
<
Il lix
.
On
he
con e se
hand,
le
E
X
and
g
E
LP
+
Lq
such
ha
o
all
>
0
K
p
q (
;
)
>_
Kp,q(
;
g)
.
Applying
he
lemma
4
.2
o
[11]
we
ob ain
ha
o
each
e
>
0
he e
is
a
la ice
homomo phism
T
E
E.C(LP)
n
C(Lq)
which
e i ies
TE( )
=
g
.
Then
by
using
he
hypo heses
we
ha e
g
E
X
and
IIgllx
:5
IITIIx-xll
IIx
<
(1
+
E)II
IIx
Since
his
is
ue
o
all e
>
0
hus
he
p oo
o
he
heo em
is
comple e
.
As a
consequence
o
his
esul
we
can
es ablish
he
ollowing cha ac e iza ion
o
in e pola ion
spaces
wi h
espec
o
he couple
(LP,
Lq)
;
his
co olla y
is
in
essence
an
in e pola ion
esul
A
.9
.
Co olla y
.
Le
X
be
an
in e media e
space
be ween
LP
and
Lq
.
The
ollowing
sia emenis
a e
equi alen
:
i)
X
is
an
in e pola ion
space,
ii)
X
is
an
in e pola ion
space
o
la ice
homomo phisms
.
P oo
.
We
only
ha e
o
show
ii)
=>
i)
.
We
de ine
a
new
equi alen
no m
II
.
II
on
X
in
he
ollowing
way
M 1II
=supJIT 1I
whe e
he
in imum
uns
o e
all
possible
la ice
homomo phisms
T
E
£(LP)
n
,c(Lq)
such
ha
max{IITIILD-LI,
IITIIL9-L9}
_< 1
.
I is
clea
ha
(X,111
.111)
is
an
exac
in e pola ion
space
o
la ice
homomo phisms
.
Thus
(X,111
.111)
ha e
o be
(1,
IC
p
,
q
)-mono one
.
Hence
he
esul
ollows
by
applying
he
heo em
5
.2
o
[11]
.
Acknowledgemen s
.
The
au ho s
a e
indeb ed
o
P o esso s
J
.
Be gh,
J
.
L8 5m
and J
.
Pee e
o
he
help ul
co espondence
which
mo i a ed
his
wo k
and
o
P o esso
N
.
Kal on
o
some
in e es ing
ema ks
on
he
inal
pa
o
his
pape
.
THE
K,
.
FUNCTIONALS
OF
INTERPOLATION

117
Re e ences
1
.

J
.
BASTERO
AND
Y
.
RAYNAUD,
On
he
K- unc ional
o
in e pola ion
be ween
LP and
O licz spaces,
Jou nal
o
App oxima ion
Theo y
60,
1
(1990),11-23
.
2
.

C
.
BENNETAND
R
.
SHARPLEY,
"In e pola ion
o
Ope a o s,"
Academic
P ess,
1988
.
3
.

J
.
BERGH
AND
J
.
LOFTROM,
.
"In e pola ion spaces
.
An
in oduc ion,"
Sp inge
Ve lag,
1976
.
4
.

J
.
HOLMSTEDTAND
J
.
PEETRE,
On
ce ain
unc ionals
a ising in
he
heo y
o
in e pola ion
spaces,
Jou nal
o
Func
.
Analysis
4
(1969),
88-94
.
5
.

J
.
LINDENSTRAUSS
AND
L
.
TZAFRIRI,
"Classical
Banach
Spaces
II,"
Sp in-
ge
Ve lag,
1979
.
6
.

G
.
LORENTZ
AND
T
.
SHIMOGAKI,
In e pola ion
heo ems
o
Ope a o s
in
Func ion
Spaces,
Jou nal
o
Func
.
Analysis
2
(1968),
31-51
.
7
.

P
.
NILSSON
AND
J
.
PEETRE,
On
he
K- unc ional
be ween
L
l
and
LZ
and
some
o he
K- unc ionals,
Jou nal
o
App oxima ion
Theo y
48
(1986),
322-327
.
8
.

J
.
PEETRE,
Nou elles
p op ié és
d'espaces
d'in e pola ion,
C
.R .A .S
.
Pa is
256
(1963),
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Jesús
Bas e o
:
Depa amen o
de Ma emá icas
Facul ad
de
Ciencias
Uni e sidad
de
Za agoza
50009
Za agoza
SPAIN
M
.
Luisa
Rezola
:
Depa amen o
de
Ma emá icas
Facul ad
de
Ciencias
Uni e sidad
de Za agoza
50009
Za agoza
SPAIN
Y es
Rainaud
:
Equipe
d'Analyse
Tou
46,
4éme
e age
Uni e si é
Pa is
VI
75252
Pa is
Cedex
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