Preliminaries and Examples
Main result
Isometries on L2 (X) and monotone functions
Santiago Boza
[email protected]. UPC
joint work with
Javier Soria
University of Barcelona
Positivity VII, Leiden. July 2013
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Índex
1
Preliminaries and Examples
2
Main result
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In [B-Soria,(2011)] optimal estimates concerning the norm of the
Hardy averaging operator
Z
1 x
Sf (x) =
f (t) dt,
x 0
minus the identity I, have been established on Lp (R+ ) on the cone
of decreasing functions, for the inequality:
kSf − f kLp (R+ ) ≤ Ckf kLp (R+ ) .
However, one of the most remarkable results is that S − I is, in fact,
an isometry on L2 (R+ )
k(S − I)f kL2 (R+ ) = kf kL2 (R+ ) ,
(1)
and the same holds for the adjoint operator S ∗ , defined by
Z ∞
f (t)
∗
dt,
S f (x) =
t
x
Isometries on L2 (X) and monotone functions
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Preliminaries and Examples
Main result
Preliminaries and Examples
In [B-Soria,(2011)] optimal estimates concerning the norm of the
Hardy averaging operator
Z
1 x
Sf (x) =
f (t) dt,
x 0
minus the identity I, have been established on Lp (R+ ) on the cone
of decreasing functions, for the inequality:
kSf − f kLp (R+ ) ≤ Ckf kLp (R+ ) .
However, one of the most remarkable results is that S − I is, in fact,
an isometry on L2 (R+ )
k(S − I)f kL2 (R+ ) = kf kL2 (R+ ) ,
(1)
and the same holds for the adjoint operator S ∗ , defined by
Z ∞
f (t)
∗
dt,
S f (x) =
t
x
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Preliminaries and Examples
The corresponding property fails for the discrete counterpart of the
Hardy operator
m
S d ({an }n∈N )(m) =
1 X
aj .
m j=1
In [Brown, Halmos and Shields, (1965)], [Kaiblinger, Maligranda,
Persson, (2000)] the isometry (1) was extended to the generalized
Hardy operator acting on L2 (w)
Z x
1
f (t)w(t) dt,
Sw f (x) =
W (x) 0
Rx
where w is a weight on R+ and W (x) = 0 w(t) dt.
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Preliminaries and Examples
The corresponding property fails for the discrete counterpart of the
Hardy operator
m
S d ({an }n∈N )(m) =
1 X
aj .
m j=1
In [Brown, Halmos and Shields, (1965)], [Kaiblinger, Maligranda,
Persson, (2000)] the isometry (1) was extended to the generalized
Hardy operator acting on L2 (w)
Z x
1
f (t)w(t) dt,
Sw f (x) =
W (x) 0
Rx
where w is a weight on R+ and W (x) = 0 w(t) dt.
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What about this question in a more general setting (the discrete
case, other kind of operators, etc.)?:
The isometric property on L2 can be obtained just by looking at the
action of the operator on certain classes of functions (e.g., in R+ , to the
cone of positive and decreasing functions).
Kalton and Randrianantoanina (1994) proved that L2 is, essentially, the
only r.i. space for which there exist non-trivial isometries.
Isometries on L2 (X) and monotone functions
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Preliminaries and Examples
Main result
Preliminaries and Examples
What about this question in a more general setting (the discrete
case, other kind of operators, etc.)?:
The isometric property on L2 can be obtained just by looking at the
action of the operator on certain classes of functions (e.g., in R+ , to the
cone of positive and decreasing functions).
Kalton and Randrianantoanina (1994) proved that L2 is, essentially, the
only r.i. space for which there exist non-trivial isometries.
Isometries on L2 (X) and monotone functions
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Preliminaries and Examples
Main result
Preliminaries and Examples
What about this question in a more general setting (the discrete
case, other kind of operators, etc.)?:
The isometric property on L2 can be obtained just by looking at the
action of the operator on certain classes of functions (e.g., in R+ , to the
cone of positive and decreasing functions).
Kalton and Randrianantoanina (1994) proved that L2 is, essentially, the
only r.i. space for which there exist non-trivial isometries.
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Preliminaries and Examples
For this purpose, we will work on a measure space (X, dµ) and consider
integral operators TK bounded on L2 (X) given by a kernel K defined on
X × X; that is,
Z
TK f (x) =
K(x, y)f (y) dµ(y), f ∈ L2 (X),
X
under the assumption that its absolute value |K(x, y)| (for simplicity we
will take K to be real) defines also a bounded operator T|K| on L2 (X).
Our aim is to study conditions on K so that
kTK f − f kL2 (X) = kf kL2 (X) .
We will study also how the isometric property behaves when the operator
TK is projected to the subspace L2 (Y ), where Y ⊂ X. Brown et all
considered both the Hardy operator and its adjoint, and the cases
X = R+ and Y = (0, 1).
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Preliminaries and Examples
For this purpose, we will work on a measure space (X, dµ) and consider
integral operators TK bounded on L2 (X) given by a kernel K defined on
X × X; that is,
Z
TK f (x) =
K(x, y)f (y) dµ(y), f ∈ L2 (X),
X
under the assumption that its absolute value |K(x, y)| (for simplicity we
will take K to be real) defines also a bounded operator T|K| on L2 (X).
Our aim is to study conditions on K so that
kTK f − f kL2 (X) = kf kL2 (X) .
We will study also how the isometric property behaves when the operator
TK is projected to the subspace L2 (Y ), where Y ⊂ X. Brown et all
considered both the Hardy operator and its adjoint, and the cases
X = R+ and Y = (0, 1).
Isometries on L2 (X) and monotone functions
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Preliminaries and Examples
Main result
Preliminaries and Examples
For this purpose, we will work on a measure space (X, dµ) and consider
integral operators TK bounded on L2 (X) given by a kernel K defined on
X × X; that is,
Z
TK f (x) =
K(x, y)f (y) dµ(y), f ∈ L2 (X),
X
under the assumption that its absolute value |K(x, y)| (for simplicity we
will take K to be real) defines also a bounded operator T|K| on L2 (X).
Our aim is to study conditions on K so that
kTK f − f kL2 (X) = kf kL2 (X) .
We will study also how the isometric property behaves when the operator
TK is projected to the subspace L2 (Y ), where Y ⊂ X. Brown et all
considered both the Hardy operator and its adjoint, and the cases
X = R+ and Y = (0, 1).
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One classical result:
Lemma Let H be a real Hilbert space with the norm kxk =< x, x >1/2 .
Let T : H −→ H be a bounded operator and T ∗ its adjoint. Then,
(i) the operator T is an isometry; i.e., kT xk = kxk for every x ∈ H if
and only if T ∗ ◦ T = I;
(ii) the operator T is a surjective isometry if and only if
T ∗ ◦ T = T ◦ T ∗ = I.
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As a consequence, we get
Theorem Let TK be an integral operator as before. Then, the following
facts are equivalent:
(i) TK − I is an isometry on L2 (X);
(ii) for almost all (x, y) ∈ X × X,
Z
K(z, x)K(z, y) dµ(z) = K(x, y) + K(y, x).
X
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Again, we can formulate an analogue to the previous theorem which
establishes necessary and sufficient conditions on K to obtain a surjective
isometry:
Theorem The following facts are equivalent:
(i) TK − I is a surjective isometry on L2 (X);
∗
(ii) TK
− I is a surjective isometry on L2 (X);
(iii) for almost all (x, y) ∈ X × X,
Z
Z
K(z, x)K(z, y) dµ(z) =
K(x, z)K(y, z) dµ(z)
X
X
= K(x, y) + K(y, x).
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As a corollary, if the previous theorem is applied to the kernel
K(x, y) =
1
1
χ(0,x) (y)
,
x
w(y)
with respect to the measure dµ(y) = w(y) dy on R+ , we obtain the
following.
Corollary Let S be the Hardy operator and w be a weight on R+ . Then,
S − I is an isometry on L2 (w) if and only w is constant almost
everywhere.
The same result holds for the adjoint operator S ∗ − I.
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The previous corollary shows taking w = χ(0,1) that, for f ∈ L2 (R+ ), the
restriction of either Sf − f , or S ∗ f − f , to the interval (0, 1) is not an
isometry on L2 (0, 1), namely, there exists f ∈ L2 (R+ ) such that
2
Z 1Z ∞
Z 1
ds
f (s) − f (t) dt 6=
f 2 (t) dt,
s
0
t
0
(and similarly for S − I). However, it was proved by Brown et all that
the restriction of the adjoint operator S ∗ , minus the identity, to functions
f ∈ L2 (0, 1) is indeed an isometry:
2
Z 1Z 1
Z 1
ds
f (s) − f (t) dt =
f 2 (t) dt.
s
0
t
0
Again, this result is false for S − I: there exists f ∈ L2 (0, 1) such that
2
Z 1 Z t
Z 1
1
f (s) ds − f (t) dt 6=
f 2 (t) dt,
t 0
0
0
it suffices to take constant functions to get a counterexample.
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The previous corollary shows taking w = χ(0,1) that, for f ∈ L2 (R+ ), the
restriction of either Sf − f , or S ∗ f − f , to the interval (0, 1) is not an
isometry on L2 (0, 1), namely, there exists f ∈ L2 (R+ ) such that
2
Z 1Z ∞
Z 1
ds
f (s) − f (t) dt 6=
f 2 (t) dt,
s
0
t
0
(and similarly for S − I). However, it was proved by Brown et all that
the restriction of the adjoint operator S ∗ , minus the identity, to functions
f ∈ L2 (0, 1) is indeed an isometry:
2
Z 1Z 1
Z 1
ds
f (s) − f (t) dt =
f 2 (t) dt.
s
0
t
0
Again, this result is false for S − I: there exists f ∈ L2 (0, 1) such that
2
Z 1 Z t
Z 1
1
f (s) ds − f (t) dt 6=
f 2 (t) dt,
t 0
0
0
it suffices to take constant functions to get a counterexample.
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Preliminaries and Examples
The previous corollary shows taking w = χ(0,1) that, for f ∈ L2 (R+ ), the
restriction of either Sf − f , or S ∗ f − f , to the interval (0, 1) is not an
isometry on L2 (0, 1), namely, there exists f ∈ L2 (R+ ) such that
2
Z 1Z ∞
Z 1
ds
f (s) − f (t) dt 6=
f 2 (t) dt,
s
0
t
0
(and similarly for S − I). However, it was proved by Brown et all that
the restriction of the adjoint operator S ∗ , minus the identity, to functions
f ∈ L2 (0, 1) is indeed an isometry:
2
Z 1Z 1
Z 1
ds
f (s) − f (t) dt =
f 2 (t) dt.
s
0
t
0
Again, this result is false for S − I: there exists f ∈ L2 (0, 1) such that
2
Z 1 Z t
Z 1
1
f (s) ds − f (t) dt 6=
f 2 (t) dt,
t 0
0
0
it suffices to take constant functions to get a counterexample.
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Assume TK : L2 (X) −→ L2 (X) is bounded, and Y ⊂ X is a given
measurable subset. Using the orthogonal decomposition
L2 (X) = L2 (Y ) ⊕ L2 (Y c ), we define the operator
Z
TK,Y f (x) =
K(x, y)f (y) dµ(y),
x ∈ Y.
Y
Theorem. Let TK be an operator such that TK − I is an isometry in
L2 (X). Then, the following facts are equivalent:
(i) For every f ∈ L2 (Y ),
kTK,Y f − f kL2 (Y ) = kf kL2 (Y ) ;
(ii) Ker(PY c ◦ TK ◦ iY ) = L2 (Y ).
(iii) K(x, y) = 0, a.e. (x, y) ∈ Y c × Y .
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Assume TK : L2 (X) −→ L2 (X) is bounded, and Y ⊂ X is a given
measurable subset. Using the orthogonal decomposition
L2 (X) = L2 (Y ) ⊕ L2 (Y c ), we define the operator
Z
TK,Y f (x) =
K(x, y)f (y) dµ(y),
x ∈ Y.
Y
Theorem. Let TK be an operator such that TK − I is an isometry in
L2 (X). Then, the following facts are equivalent:
(i) For every f ∈ L2 (Y ),
kTK,Y f − f kL2 (Y ) = kf kL2 (Y ) ;
(ii) Ker(PY c ◦ TK ◦ iY ) = L2 (Y ).
(iii) K(x, y) = 0, a.e. (x, y) ∈ Y c × Y .
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Some other examples besides the Hardy operator and its adjoint:
Convolution operators: TK f (x) = (K ∗ f )(x), with K ∈ L1 (R). We
denote by m = K̂, TK − I is an isometry in L2 (X) if and only if
k(TK − I)f kL2 (R) = k(m − 1)fˆkL2 (R) = kfˆkL2 (R) , f ∈ L2 (R),
and, hence
|m(ξ) − 1| = 1,
for every ξ ∈ R.
which is equivalent to
m(ξ)m̄(ξ) = m(ξ) + m̄(ξ),
for every ξ ∈ R,
or to
Z
K(z − x)K(z − y) dz = K(x − y) + K(y − x)
R
An example of kernel K ∈ L1 (R) verifying this condition is
K(x) =
sin(2π 2 x)
.
πx(2πx + 1)
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Some other examples besides the Hardy operator and its adjoint:
Convolution operators: TK f (x) = (K ∗ f )(x), with K ∈ L1 (R). We
denote by m = K̂, TK − I is an isometry in L2 (X) if and only if
k(TK − I)f kL2 (R) = k(m − 1)fˆkL2 (R) = kfˆkL2 (R) , f ∈ L2 (R),
and, hence
|m(ξ) − 1| = 1,
for every ξ ∈ R.
which is equivalent to
m(ξ)m̄(ξ) = m(ξ) + m̄(ξ),
for every ξ ∈ R,
or to
Z
K(z − x)K(z − y) dz = K(x − y) + K(y − x)
R
An example of kernel K ∈ L1 (R) verifying this condition is
K(x) =
sin(2π 2 x)
.
πx(2πx + 1)
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Preliminaries and Examples
Some other examples besides the Hardy operator and its adjoint:
Convolution operators: TK f (x) = (K ∗ f )(x), with K ∈ L1 (R). We
denote by m = K̂, TK − I is an isometry in L2 (X) if and only if
k(TK − I)f kL2 (R) = k(m − 1)fˆkL2 (R) = kfˆkL2 (R) , f ∈ L2 (R),
and, hence
|m(ξ) − 1| = 1,
for every ξ ∈ R.
which is equivalent to
m(ξ)m̄(ξ) = m(ξ) + m̄(ξ),
for every ξ ∈ R,
or to
Z
K(z − x)K(z − y) dz = K(x − y) + K(y − x)
R
An example of kernel K ∈ L1 (R) verifying this condition is
K(x) =
sin(2π 2 x)
.
πx(2πx + 1)
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Preliminaries and Examples
Some other examples besides the Hardy operator and its adjoint:
Convolution operators: TK f (x) = (K ∗ f )(x), with K ∈ L1 (R). We
denote by m = K̂, TK − I is an isometry in L2 (X) if and only if
k(TK − I)f kL2 (R) = k(m − 1)fˆkL2 (R) = kfˆkL2 (R) , f ∈ L2 (R),
and, hence
|m(ξ) − 1| = 1,
for every ξ ∈ R.
which is equivalent to
m(ξ)m̄(ξ) = m(ξ) + m̄(ξ),
for every ξ ∈ R,
or to
Z
K(z − x)K(z − y) dz = K(x − y) + K(y − x)
R
An example of kernel K ∈ L1 (R) verifying this condition is
K(x) =
sin(2π 2 x)
.
πx(2πx + 1)
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Separate variables kernels For integral operators whose kernel is a
tensor product on each variable K(x, y) = a(x)b(y),
(x, y) ∈ R+ × R+ , TK − I is an isometry implies that
b(y) =
2a(y)
.
kak2L2 (R+ )
In this case, the same characterization is obtained if we replace the
condition
Z
K(z, x)K(z, y) dµ(z) = K(x, y) + K(y, x).
X
on the kernel by the weaker diagonal condition:
Z
K 2 (z, x) dµ(z) = 2K(x, x), a.e. x ∈ X.
X
However, we will see that, in general, these two conditions are not
equivalent.
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Separate variables kernels For integral operators whose kernel is a
tensor product on each variable K(x, y) = a(x)b(y),
(x, y) ∈ R+ × R+ , TK − I is an isometry implies that
b(y) =
2a(y)
.
kak2L2 (R+ )
In this case, the same characterization is obtained if we replace the
condition
Z
K(z, x)K(z, y) dµ(z) = K(x, y) + K(y, x).
X
on the kernel by the weaker diagonal condition:
Z
K 2 (z, x) dµ(z) = 2K(x, x), a.e. x ∈ X.
X
However, we will see that, in general, these two conditions are not
equivalent.
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K(x, y) = χE (x, y)
Let us consider K(x, y) = χE (x, y), for some measurable set
E ⊂ R+ × R+ , TK − I is a surjective isometry if and only if
|E x ∩E y | = |Ex ∩Ey | = χE (x, y)+χE (y, x),
a.e. (x, y) ∈ R+ ×R+ ,
where Ex0 and E y0 denote the sections on each variable; that is,
Ex0 = {y ∈ R+ : (x0 , y) ∈ E}
and
E y0 = {x ∈ R+ : (x, y0 ) ∈ E}.
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K(x, y) = χE (x, y)
Let us consider K(x, y) = χE (x, y), for some measurable set
E ⊂ R+ × R+ , TK − I is a surjective isometry if and only if
|E x ∩E y | = |Ex ∩Ey | = χE (x, y)+χE (y, x),
a.e. (x, y) ∈ R+ ×R+ ,
where Ex0 and E y0 denote the sections on each variable; that is,
Ex0 = {y ∈ R+ : (x0 , y) ∈ E}
and
E y0 = {x ∈ R+ : (x, y0 ) ∈ E}.
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..
.
3
..
.
3
2
2
1
1
E3,1
1
2
3 ···
E3,2
1
2
3 ···
1
Sets generated by a 3 × 3 period corresponding to a surjective isometry.
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..
.
..
.
..
.
4
4
4
2
2
E4,1
2
4
2
E4,2
2
···
4





..
.
..
.
..
.
4
4
2
E4,4
4
2
E4,5
2
···
4





..
.
..
.
..
.
4
4
E4,7
···
···

4
4
4


2
E4,6
2
···

2
···

4
2
4



2
E4,3
2
···
2
E4,8
2
4
···
2
E4,9
2
4
···
Sets generated by a 4 × 4 period corresponding to a surjective isometry.
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n-dimensional Hardy operator The Hardy operator has an
n-dimensional extension Sn .
For simplicity, let us assume n = 2, and for s, t > 0 define,
Z Z
1 s t
S2 f (s, t) =
f (x, y) dy dx.
st 0 0
Contrary to the 1-dimensional case, it is easy to see that S2 − I is
not an isometry on L2 (R2+ ). In fact, for 0 < s1 < s2 and
0 < t2 < t1 , our condition reads,
Z
1
1
1
χ(0,x)×(0,y) (s1 , t1 ) χ(0,x)×(0,y) (s2 , t2 ) dy dx =
,
2
xy
s 2 t1
R+ xy
but
1
1
χ(0,s1 )×(0,t1 ) (s2 , t2 ) +
χ(0,s2 )×(0,t2 ) (s1 , t1 ) = 0.
s 1 t1
s 2 t2
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n-dimensional Hardy operator The Hardy operator has an
n-dimensional extension Sn .
For simplicity, let us assume n = 2, and for s, t > 0 define,
Z Z
1 s t
S2 f (s, t) =
f (x, y) dy dx.
st 0 0
Contrary to the 1-dimensional case, it is easy to see that S2 − I is
not an isometry on L2 (R2+ ). In fact, for 0 < s1 < s2 and
0 < t2 < t1 , our condition reads,
Z
1
1
1
χ(0,x)×(0,y) (s1 , t1 ) χ(0,x)×(0,y) (s2 , t2 ) dy dx =
,
2
xy
s 2 t1
R+ xy
but
1
1
χ(0,s1 )×(0,t1 ) (s2 , t2 ) +
χ(0,s2 )×(0,t2 ) (s1 , t1 ) = 0.
s 1 t1
s 2 t2
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Definition Let (X, Σ, µ) be a measure space. We say that E ⊂ Σ is
isoadmissible if:
(i) µ(E) < ∞, for every E ∈ E;
(ii) ∅ ∈ E;
(iii) E is stable under finite unions and intersections;
(iv) UE = {χE : E ∈ E} is a total set on L2 (X, dµ); that is,
Z
f (x) dµ(x) = 0, for all E ∈ E ⇒ f (x) = 0, a.e. x ∈ X.
E
Isometries on L2 (X) and monotone functions
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Preliminaries and Examples
Main result
Main result
For E ⊂ Σ isoadmissible, we denote by DE = {χE1 + χE2 : E1 , E2 ∈ E}
and
n
o
FE = f ∈ L2 (X) : x ∈ X : |f (x)| > t ∈ E, for every t > 0 .
THEOREM. Let E be an isoadmissible set on (X, Σ, µ) and let TK be an
integral operator. Then, the following facts are equivalent:
(i) TK − I is an isometry on L2 (X);
(ii) TK − I is an isometry restricted to FE ;
(iii) TK − I is an isometry restricted to DE .
Isometries on L2 (X) and monotone functions
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Preliminaries and Examples
Main result
Main result
For E ⊂ Σ isoadmissible, we denote by DE = {χE1 + χE2 : E1 , E2 ∈ E}
and
n
o
FE = f ∈ L2 (X) : x ∈ X : |f (x)| > t ∈ E, for every t > 0 .
THEOREM. Let E be an isoadmissible set on (X, Σ, µ) and let TK be an
integral operator. Then, the following facts are equivalent:
(i) TK − I is an isometry on L2 (X);
(ii) TK − I is an isometry restricted to FE ;
(iii) TK − I is an isometry restricted to DE .
Isometries on L2 (X) and monotone functions
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Preliminaries and Examples
Main result
Main result
Remark. It is relevant to observe that, in general, the diagonal condition
Z
K 2 (z, x) dµ(z) = 2K(x, x), a.e. x ∈ X,
X
is not enough to get an isometry for TK − I.
Moreover, we need to evaluate TK − I on, at least, the sum of two
characteristic functions of sets in E and this condition cannot be replaced
by testing TK − I on just the characteristic functions of one set in E.
Counterexample To see this, let us consider X = {1, 2} endowed with the
2
2
counting
measure on
Σ = P(X), so that L (X) = R . Choose
E = ∅, {1}, {1, 2} , which is clearly an isoadmissible set.
Isometries on L2 (X) and monotone functions
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Preliminaries and Examples
Main result
Main result
Remark. It is relevant to observe that, in general, the diagonal condition
Z
K 2 (z, x) dµ(z) = 2K(x, x), a.e. x ∈ X,
X
is not enough to get an isometry for TK − I.
Moreover, we need to evaluate TK − I on, at least, the sum of two
characteristic functions of sets in E and this condition cannot be replaced
by testing TK − I on just the characteristic functions of one set in E.
Counterexample To see this, let us consider X = {1, 2} endowed with the
2
2
counting
measure on
Σ = P(X), so that L (X) = R . Choose
E = ∅, {1}, {1, 2} , which is clearly an isoadmissible set.
Isometries on L2 (X) and monotone functions
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Preliminaries and Examples
Main result
Main result
Example. X = R+ and E = {(0, r) : r > 0}. It isRclear that to show that
r
E is isoadmissible because {χE } is a total set: if 0 f (x) dx = 0, for
+
every r > 0, then f ≡ 0, a.e. in R . In this case FE is the cone of
positive and decreasing functions in R+ .
THEOREM. Let TK be an operator given by a regulated function K.
The following facts are equivalent:
(i) TK − I is an isometry on L2 (R+ );
(ii) TK − I is an isometry on the cone of decreasing functions L2dec (R+ );
(iii) for almost all (r, s) ∈ R+ × R+ ,
Z ∞
K(x, r)K(x, s) dx = K(r, s) + K(s, r).
0
Isometries on L2 (X) and monotone functions
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Preliminaries and Examples
Main result
Main result
Example. X = R+ and E = {(0, r) : r > 0}. It isRclear that to show that
r
E is isoadmissible because {χE } is a total set: if 0 f (x) dx = 0, for
+
every r > 0, then f ≡ 0, a.e. in R . In this case FE is the cone of
positive and decreasing functions in R+ .
THEOREM. Let TK be an operator given by a regulated function K.
The following facts are equivalent:
(i) TK − I is an isometry on L2 (R+ );
(ii) TK − I is an isometry on the cone of decreasing functions L2dec (R+ );
(iii) for almost all (r, s) ∈ R+ × R+ ,
Z ∞
K(x, r)K(x, s) dx = K(r, s) + K(s, r).
0
Isometries on L2 (X) and monotone functions
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Preliminaries and Examples
Main result
Main result
Remark
Note that, in general, the equivalence between (i) and (ii) in the
previous theorem is not a direct consequence of the invariance by
rearrangements property of the L2 (R+ )-norm since, for an arbitrary
integral operator TK , bounded on L2 (R+ ), the equality
kTK f − f kL2 (R+ ) = kTK f ∗ − f ∗ kL2 (R+ )
fails in general for an f ∈ L2 (R+ ).
Isometries on L2 (X) and monotone functions
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Preliminaries and Examples
Main result
Main result
Some other examples
X = Rn+ and
n
E = D ⊂ Rn+ : if (x1 , . . . , xn ) ∈ D, then (y1 , . . . , yn ) ∈ D,
o
whenever 0 < yj ≤ xj , j = 1, . . . , n .
Now, it is enough to observe that the condition of being a total set holds
just considering sets that are intervals of the form:
D = (0, r1 ) × · · · × (0, rn ),
r1 , . . . , rn > 0.
It is easy to see that FE is the cone of positive functions in Rn+ ,
decreasing on each variable.
We observe that if we restrict to the family of cubes
Ecubes = {(0, r)n : r > 0}, then UEcubes is not total in Rn+ .
This is an example that shows how all this theory can be extended to a
more general context where an ordered structure can be defined: discrete
setting like N or ordered trees ([B.,Soria](2008))
Isometries on L2 (X) and monotone functions
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Preliminaries and Examples
Main result
Main result
Some other examples
X = Rn+ and
n
E = D ⊂ Rn+ : if (x1 , . . . , xn ) ∈ D, then (y1 , . . . , yn ) ∈ D,
o
whenever 0 < yj ≤ xj , j = 1, . . . , n .
Now, it is enough to observe that the condition of being a total set holds
just considering sets that are intervals of the form:
D = (0, r1 ) × · · · × (0, rn ),
r1 , . . . , rn > 0.
It is easy to see that FE is the cone of positive functions in Rn+ ,
decreasing on each variable.
We observe that if we restrict to the family of cubes
Ecubes = {(0, r)n : r > 0}, then UEcubes is not total in Rn+ .
This is an example that shows how all this theory can be extended to a
more general context where an ordered structure can be defined: discrete
setting like N or ordered trees ([B.,Soria](2008))
Isometries on L2 (X) and monotone functions
24/25
Preliminaries and Examples
Main result
Main result
Some other examples
X = Rn+ and
n
E = D ⊂ Rn+ : if (x1 , . . . , xn ) ∈ D, then (y1 , . . . , yn ) ∈ D,
o
whenever 0 < yj ≤ xj , j = 1, . . . , n .
Now, it is enough to observe that the condition of being a total set holds
just considering sets that are intervals of the form:
D = (0, r1 ) × · · · × (0, rn ),
r1 , . . . , rn > 0.
It is easy to see that FE is the cone of positive functions in Rn+ ,
decreasing on each variable.
We observe that if we restrict to the family of cubes
Ecubes = {(0, r)n : r > 0}, then UEcubes is not total in Rn+ .
This is an example that shows how all this theory can be extended to a
more general context where an ordered structure can be defined: discrete
setting like N or ordered trees ([B.,Soria](2008))
Isometries on L2 (X) and monotone functions
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Preliminaries and Examples
Main result
Some references
S. Boza and J. Soria, Rearrangament transformations on general
measure spaces, Indag. Math. 19 (2008), 33–51.
S. Boza and J. Soria, Isometries on L2 (X) and monotone functions,
Accepted for publication in Math. Nach.
A. Brown, P. R. Halmos, and A. L. Shields, Cesàro operators, Acta
Sci. Math.(Szeged) 26 (1965), 125–137.
N. Kaiblinger, L. Maligranda, and L. E. Persson, Norms in weighted
L2 -spaces and Hardy operators, in Function Spaces, The Fifth
Conference (Poznań, 1998), Lecture Notes in Pure and Appl. Math.,
vol. 213, Dekker, New York, 2000, 205–216.
N.J. Kalton and B. Randrianantoanina, Surjective isometries on
rearrangement-invariant spaces, Quart. J. Math. Oxford Ser.(2) 45
(1994), no. 179, 301–327.
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