441
Internat. J. Math. & Math. Sci.
Vol. 8 No. 3 (1985) 441-448
NORM-PRESERVING L-L INTEGRAL TRANSFORMATIONS
YU CHUEN WEI
Department of Mathematics
University of Wisconsin-Oshkosh
Oshkosh, Wisconsin 54901 U.S.A.
22,
(Received March
In this paper we consider an L-L integral transformation
ABSTRACT.
fG(x,y)f(y)dy,
F(x)
f(y)
1984)
is defined on
G
transformation
G(x,y)
where
[0,).
is only a necessary condition, where
f01G,(x,t) Idx
G,(x,t)
fIG,(x,t) Idx
=limh+0inf
O}
O, y
for almost all
i
i
of the form
and
For an L-L integral
The following results are proved:
to be norm-preserving,
G’s.
{(x,y): x
D
is defined on
G
[t+h
t
G(x,y)dy
t
t
for each
I for almost all
t
0
is a necessary
and sufficient condition for preserving the norm of certain
f
L.
In this paper
0.
x
For certain
-
G
the analogous result for sum-preserving L-L integral transformation
method. L-L integral transformation.
Fubini-Tonelli Theorem.
KEY WORDS AND PHRASES.
of integrals.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE.
I.
into
2.
Absolutely continuity
-
The well-known summability method defined by a
matrix A
g into E, is sum-preserving if and only if for each k,
G
present study we also discuss conditions under which
L
L,
is proved.
44A02, 44A06, 42A76.
INTRODUCTION.
from
0
(ank),
mapping
i.
In>lank
G(x,y),
defined by
In our
mappings
is norm-preserving or sum-preserving.
NOTATION.
The notation and terms used are:
The statement that
U
> 0,
limit as
f
if
u
f
is Lebesgue integrable on
[0,u]
is Lebesgue integrable on
.
means that for every
ff(x)dx
L
the space of functions that are Lebesgue integrable on
D
the first quadrant of the plane, i.e., D
all
G
GL
L
x > 0,
where
f
is defined on
C
the collection of all
the subcollection o[
G
[0,)
of the form
such th,t
tends to a finite
[0,)
{(x,y): x
0, y
of the form (*) F(x)
F,
G- an integral transformation, G: f
[or
[0,)
and that
F
and
G(x,y)
with norm
0}.
fG(x,y)f(y)dy,
defined on
D.
(*).
L
whenever
f
L.
and essentially bounded on
the space of functions which ar measurable
ess
fi
[0,) with norm
s,,Px>Olf(x)
442
3.
Y.C. WEI
MAIN THEOREM
THEOREM i.
If
G c GL
f c L
and for every
fIF(x) ]dx fIf(y)Idy
then for almost all
y > 0,
fIG,(x,y) lax
{y+h
lira inf
G(x,t)dt, for each x > O.
-Y
h-O
O} satisfying
{y
Suppose that there is a set A
G,(x,y)
where
PROOF.
that either
=
fIG,(x,y) Idx
L,
> i
for each
x
for every measurable set
A
Since
G c
1
y c A
for all
O,
it follows from Theorem
generality, we can assume that
A
Case i).
y c A,
Suppose that for all
such
for all
see
[i],
y c A.
that
without loss of
is a bounded measurable set.
fo[G,(x,y) Idx
f0]G,(x,y)]dx
c
,
i
< i.
Without loss of generality
y c A
we assume that for each
where
(T.S.T.),
fAG(X,y)dy
of finite measure,
0 < mA
<fIG,(x,y)
Idx
or
is a small positive number.
I
<
Let
c,
A(y)
f(y)
then
f G(x,y) A(Y)dY
F(x)
[A G(x,y)dy
and
I111- fIf o G(x,y)A(Y)dyldx
<-- f0 fA IG(x’y) Idydx
Since for each
x
>_ O, G,(x,y)
G(x,y)
for almost all
y > O,
see [2,
Theorem 5. P. 255], so it follows from the Fubini-Tonelli Theorem that
F[[ <_.
f .{A[G(x,y)[dydx
A f]G,(x,Y) ldxdy
fA(l
Hence, for case i),
G
c)dy
is not norm-preserving.
> i.
Suppose that for all y
generality, we assume that for each y e A
Case ii).
fo[G(x,y) [dx
where
is a small positive number.
e
for all
x
[0,).
If
F(x)
0
F
and
+
f(y)
for almost all
XA(y);
x
F(x)
then
[0,),
AG(X,y)dy
then
0 < mA
we’ re done.
Suppose that
G c
Let
> i
Without loss of
GL,
F(x)
0
for all
x
in some set with positive measure.
so it follows from Theorem (T. S. T) by author, see [I],
G,(x,y)
Since
is measur-
443
NORM-PRESERVING L-L INTEGRAL TRANSFORMATIONS
D
able on
foIG,(x,y) Idx
and
< M
y > O,
for almost all
M
where
is a constant.
Thus
f0"f [G*(x’y) ldydx
A
A
0
[G*(x’y)]dxdy
>
O,
there is an
i > /2 >
Given
Idydx
XO AlG,(x,y)
=.
<
X
n
<
mA/2
O’
for all
A
y
A0,
y
..fO
Jolc.,(x,y) Idx
and from
y
+
o
[C,(x,y) [dx
for each
< e
mA/4
having positive measure and for
< /8
A
y
that
]G,(x,y)Idx i / 3/4
{(x,y)
[0,X0]
0}
AO: IG,(x,y) < /24X
E}.
Then 0 <mEy < X
[0,X0]: (Xl,Y)
0
Let E
{x
0.
E
let
i
X
such that
A c A,
0
It follows that there is at least a subset
A
y
satisfying
all
> 0
0
1
and for any
y c
for all
A0"
S ince
JA SO ’G,(x,y)Idxdy SAf<=O IG,(x,y> ,dxdy
<_
0
so it follows from the absolute continuity of the integral that there is a
x A
satisfying mH < 6, and
that for every measurable set H
0
6 > 0
such
_= [O,xo]
ffHiG,(x,y) [dydx <
JA
If
G(x,y)dy
0
0
x > 0,
for almost all
FII
mAo/4
G(x,y) X
then
A0(Y)dY
J’IIAoO-IIAOII
G(x,y)dyldx
-0 <A
and we’re done.
By the
__= [O,Xo] AO
set
F
and
G,(x,y)
our on
A
0
x E
[0,X0]
Y
of
H
such that if
F.
is continuous over
uiN__
F
[0,X0]
1
the value of
(x,y")
’)
Generalizati0n0
0 for some set of x >_ 0 with
G(x,y)dy
of Luzin’s Theorem we can choose a closed
[O,Xo] A0 F then mH < 6 and
It is clear that G,(x,y)
N of subsets A
Thus we can have a finite number
such that
(x,y’);
IG,(x,
E
F.
A
So we suppose that
positive measure.
[O,Xo]
-G,(x,y")
x e [O,X
0],
<
x
A
i
G,(x,y)
X2xio
i
and within each strip
[0,X0]
,
such that
G,(x,y)
>
O,
< O,
F 0
(A Ai
if
(x,y)
if
(x,y) c F 0 (A
Ai)
i
and
,..,[G,(x,y[
i
i
E,
(x,y")
A
each
there are three sets
Then for
i
and
/
G,(x,y)
mA
x A
124X0
if
(x,y) e E
y
> 0
of set
for each
More precisely, for
are close to one another.
(x,y’),
is uniformly continu-
of
A
i
A+i,
A
and
Y.E. WEI
444
(x,y) e F N
Hence, if
[0,X0]
then
Ai,
IIA.
01 0 G(x,y) A.(y)dyldx
G(x,y)dyldx
y
JA+. f
A
i
G, (x
y) dydx +
i
f
f
(A
A
i
G,(x,y)dy)dx
i
Y
=JAi fA+
+IAiA(-G,(x,y))dydx
G.(x,y)dxdy
1
y
f + UA
A
.
i
r
y
l
13.(x y)l
dydx
i
+rE IfAi 13.
(x y) dy dx
y
i
Xo
fA ]G(x,y) ldydx_ JE SA ]G(x,y) idydx fE IS G, dyi
S S’< *<x’y’ I<’>’<’x <S, S,< I,<x.y, <,>, S,< AI,<x.>,,<,yl <,x
I. fOiG,(x,Y) idxdy- SE /A.iG,<x,y) idydx
I0
i
+
Y
i
y
>_.
y
i
dx
i
2
y
1
(I + 3e/4)mA
i
If
(x ,y)
i
m{H N [0,X0]
0
AiJ
F
II--
(slnce
mAi/8
e
for some
A
e
i
N
{Ai} I,
Y
< X
0)
then for such an
A i,
SiS G(x,y) XA i(y)dyldx
fol.f. G(x’y)dyldx
I,s,,,. o,<x,>,><,>,.<,x
>
mE
flS.
+
G.(x,y)dyl=i
Slio .S, ,ISA
G,(x y).yldx
(x,y) e H
(x,y) e F
0[Ai G,(x,y)dyldx
> (i
>
If
m{H
such that
[0 X 0]
Ai}
+ 3gl4)mA i
mAi
# 0
XA
II,
for all
IJ
emA./8z
and we’re done.
A
H0[0,X0]xA i
i
e
N
{Ai} I
IG, (x’y)
then there is at least an A.
dydx < (
mA0/4)
mAi/mA0
NORM-PRESERVING L-L INTEGRAL TRANSFORMATION
and for such an
445
A i,
[,F.,
=0 [y:
A.(y)dyIdx
G(x,y)
I s=0 G,(x,y) A (y)dyldx + s;
s:
1
<o IJAi
(x,y)
f0 I
i
G, (x,Y) dY
iSAi
dx
0
G,(x,y)
XA i (y) dy
dx
G,(,y)dyl d
(x,y) : H
F
G,(x y)dYldx- n.
mAi/4
(x,y) e F
>_
+ 3e/4)mA i
(i
mAi/8
2 e
(I + el2)mA.
> mA
i
:11A
G
Hence, case (ii) we have proved that
norm-preserving and so the proof is
is not
complete.
Theorem I shows us that if
G e
GL,
for almost all
y
>_
0,
f0iF(x) ldx J0[f(y)lay
y > O, J[G,(x,y) [dx
S:iF(x)idx flf(Y)Idy for every f e L.
is a necessary condition for
whenever
example will tell us that for almost all
cient condition for
JoIG,(x,y) Idx
f e
I
e.
1
The next
is not a suffi-
fiG,(x,y)
But the following theorem will show that for certain G’s,
[dx =i is
a necessary and sufficient condition for preserving the norms of certain f e L.
Example.
Define
-i/4x I/2
if
x
if
x e [i, ),
(0,i),
G(x,y)
for all
i/2x
2
and
-2/(y + i) 2
[0 I)
if
y
f
y e [i =,)
f (y)
10/(y
Then
if (y)[dy
1)
2
1
2/(y +
0
=-2(y +
l)2dy
+ I 101(y +
l)2dy
1)-llo
=-1+2+5=6
and
f0[G,(x,y) ldx f
i
0
I/4xll2dx +
1
i/2x
2
dx
2xi/2/4i0
1_ l/2x i<,>
i
112 + 112
But
F(x)
0
G(x,y) f (y)dy
f (-I /4xi/2)
0
0
1
f (y) dy,
(i/2x2) f (Y) dY’
if
x e (0,i)
if
x
[i, (R))
y > 0;
Y.C. WEI
446
where
(i/4xi12
0
f(y)dy
I
=-i/4xl/2[f-2/(y + l)2dy + lO/(y + l)2dy]
/4xl/2 -2 (-l) (y + i)-i[0i + lO(-l(y + i)-iii
_i
=-i/4xi/2[i
_l/x
2 + 5]
1/2
g
x
if
(0,i)
and
(l12x 2)(I
2/x 2
+ 5)
2
x e [i )
if
Therefore
-i
+ 2(-l)x
2xi/2! 0i
=2+2
THEOREM 2.
G e
GL;
Jolf(y) ly
6
4
i)
F
f
whenever
f e L
and
ii)
IIF
f
whenever
f
e
and
iii)
F
f
whenever
iv)
f
G,(x,y)dx
PROOF
it is clear that
iv).
i)
f(y) > 0
f(y) < 0
D
and
]0
G(x,y) f(y)dy
and
JoIF(x)Idx
is equivalent to
i).
G(x,y) > 0
D
on
O.
y
F(x)
[0,);
[O,m);
on
on
0 G(x,y)If(y)
F(x)
L, if
f
If(y) Idy,
Assuming that
f(y)G(x,y) > 0
have
is a nonnegative function on
for almost all
i,
f
Since
g
IIF II=
to
G(x,y)
Suppose that
then the folowing are equivalent;
on
We now prove that
and
f(y) > 0
i)
for all
is equivalent
y e [0,),
Hence
D.
F(x)
G(x,y)f(y)dy _0
0
so
IF (x)
_
F (x)
Therefore
and
IIF I1: S0IF(x)ldx
S
F
G(x,y)f(y)dydx
0
By the Fubini-Tonelli Theorem and for each
all
y
O,
IIF
0
F(x)dx
f(Y)
0
x
>_ O,
C,(x,y)
G,(x,y)dxdy
Hence
F
)0 G,(x,y)dx
f
i
if and only if
for almost all
y
_>_
0
C(x,y)
for almost
we
NORM-PRESERVING L-L INTEGRAL TRANSFORMATIONS
Next we prove th
447
Let
iii) is equivalent to iv).
f(y),
if
f(y) > 0
0
if
f(y) < 0
-f(y), if
f(y) < 0
f+
f-0
Since
G(x,y)
0
F+
D,
on
0
F
f (y) > 0
f e L
so whenever
G(x’Y)f+(y)dy
if
0
for all
x
C(x,y)f (y)dy >_ 0
for all
x > 0
0
Z
and
f(y)
/ G(x,y) If(y)Idy,
F(x)
if
f+
f-
F+
+ F-
then
IF(x)
It follows from i) that
SO 1F (x)
F(x)l
flf(Y)IdY
if and only if
0
G,(x,y)dx
y > 0
for almost all
i
We are also interested in the analogous sum-preserving question for L-L integral
f(y)dy whenever f e L?
F(x)dx
transformations, viz., when is
0
Next we give the definition of sum-preserving for L-L integral transformations
s
and aresult concerning it.
DEFINITION.
GL
G
The integral transformation
is said to be sum-preserving
if and only if
0
for all
f(y) e L,
COROLLARY.
then
f
G
where
0
f(y)dy
fO G(x,y)f(y)dy.
F(x)
Suppose that
F(x)dx
G(x,y)
is a nonnegative function on
is a sum-preserving transformation whenever
G,(x,y)dx
i
for almost all
PROOF.
Since
L,
f
y
>_
f+
f e L
where
(y),
if
f(y)
0
if
f (y) < 0
0
f+
-f(y),
if
f(y) < 0
if
f(y)
f
0
and
IO
f(y)dy
f] f+dy- f] f-dy
and
if and only if
0.
f-,
D
>_
0
G e
GL;
Y.C. WEI
448
Then
I
I G(x’Y)[f+I G(x’Y)f+(y)dy- I
F(x)
G(x,y)f(y)dy
f-]dy
and
01 6(x’Y)f+(y)dy-
F(x)dx
6(x,y)f-(y)dy]dx
0
G(x’Y)f+(y)dydx-
J0
G(x,y)f-(y)dy
0
G(x,y)f (y)dydx
By the Fubini-Tonelli Theorem
0
G(x’Y)f+(y)dydx
(Y)
0
and
s f;
0
O(x,y)f (y)dydx
f (y)
0
0
s=
0
G,(x,y)dxdy
G,(x,y)dxdy
Thus
O(x,y)
f+(y)dydx
f+(y)dy
0
and
J0
O(x,y) f (y)dydx
f (y)dy
0
if and only if
0
G,(x,y)dx
i
for almost all
y
0
Therefore
if and only if
f7
f
0
if and only if
I
(::,,:)d
0
I
O,(x,y)dx
F(x)dx
G,(x,y)dx
0
0
f dy
for almost all
y >_0
f(y)dy
1
for almost all
y >_0
The proof is completed.
REFERENCES
[.
WEI, I.C.
2.
NATONSON, I.P. Theory of functions of a real variable.
Unger Publishing1961.
3.
SUNOUCHI, G.[. and TSUCHIKURA, T. Absolute regularity for convergent integrals,
Tohoku Math. J. (2) 4 (1952), 153-156.
4.
TATCHELL, J.B. On some integral transformation, Proc. London Math. Soc. (3)
(1953), 257-266.
5.
HARDY, G.H.
6.
FRIDY, J.A. Absolute summability matrices that are stronger than the identity
mapping, Proc. Amer. Math. Soc. 47 (1975), I|2-118.
A property of L-L integral transformation, (will be published).
Vol. I, Frederick
Divergent series, Clarendon Press, Oxford, 1949.