A new approach to the Laplace transform and the Hardy operator

MASTER’S THESIS

2005:329 CIV

ERIC SETTERQVIST

Unitary Equivalence

A new approach to the Laplace transform and the Hardy operator

MASTER OF SCIENCE PROGRAMME

Department of Mathematics

Unitary Equivalence: A New Approach to the Laplace transform and the Hardy operator

Eric Setterqvist

Master’s Thesis Supervisor: Natan Kruglyak

Department of Mathematics

Lule˚ a University of Technology

Abstract

The thesis consists of three parts. In Part 1 we compose the Laplace transform L,

L(f (x))(s) = Z∞

0

f(x)e^−sxdx,

with the involution T f (x) = ¹_xf(_x¹) on L²(0, ∞) and show that the re- sulting operators T L and LT are unitary equivalent to multiplication by, respectively, the gamma functions Γ(¹₂+ iw) and Γ(¹₂ − iw) on L²(R).

Further, with the unitary equivalence between the composition T L and multiplication by Γ( ¹₂+ iw) we are able to derive the exact constant of the norm of the Laplace transform on L²(0, ∞). We end Part 1 with an estimate for the Laplace transform on L^p(0, ∞), 1 ≤ p ≤ 2.

In Part 2 it is shown that the Hardy operator, Hf (x) =¹_x Rx

0

f(s)ds, on L²(0, ∞) is unitary equivalent to multiplication by the function 1¹

2+iw on L²(R). We then consider the Hardy minus Identity operator,

(H −I)f (x) =¹_x Rx

0

f(s)ds−f (x), and with the unitary equivalence between H and multiplication by 1 ¹

2+iw we prove the equality kαI + (1 − α)(H − I)k_L2→L² = 1, 0 ≤ α ≤ 1.

Thus, the unit sphere of the space of all bounded linear operators on L²(0, ∞) contains an interval with the ends I and H − I.

In addition, we show that I − H = −(H − I) on L²(0, ∞) is unitary equivalent to I − AΩ, where AΩ is a convolution operator, on L²(R).

Moreover, I − AΩis a shift isometry in an orthonormal basis {kn},

kn(u) =

 ln(u)χ_(0,∞), n≥ 0, l−n+1(−u)χ_(−∞,0), n <0,

constituted of Laguerre functions lm. In this basis we have the simple shift

(I − AΩ)kn= kn+1, n∈ Z.

Finally, in Part 3 we derive a sharp estimate of H − I on the cone of decreasing functions in L^p(0, ∞).

Contents

1 Introduction 1

2 The Laplace transform and the Gamma function 2 2.1 The Laplace transform and related unitary equivalent op-

erators . . . 2 2.2 An estimate for the Laplace transform on L^p(0, ∞) . . . 6 3 The Hardy minus Identity operator onL²(0, ∞) 8 3.1 An unitary equivalent form of the Hardy operator . . . 8 3.2 Laguerre functions and the Identity minus Hardy operator . 10 4 Estimates of the Hardy minus Identity operator on the

cone of decreasing functions 13

4.1 Operator norms of H − I and (H − I)⁻¹ on cones in L^p(0, ∞) 13 4.2 Further estimates of H − I and (H − I)⁻¹ . . . 19

5 Future work 23

Appendix: The Laplace transform as a convolution on the multiplicative group and the Mellin transform 24

References 26

Acknowledgements

First of all I am grateful to my supervisor Professor Natan Kruglyak for the excellent guidance and the inspiration he has provided throughout the completion of this thesis.

Also, I express my appreciation to Professor Svante Janson and Pro- fessor Lars-Erik Persson who read preliminary versions of the thesis and suggested some valuable improvements of it. Finally, I would like to ac- knowledge Professor Sten Kaijser whose paper, see [6], influenced the di- rection of research.

1 Introduction

This thesis is devoted to the study of some aspects of and interactions between the Laplace transform, Hardy operators and Laguerre polynomials. We give in this section a short introduction to these subjects.

The Laplace transform, defined by

L(f (x))(s) = Z∞

0

f (x)e^−sxdx,

is a widely used tool in science and engineering. Its perhaps most significant gain in practice is the transform of linear dynamical systems, which arise in different areas like control theory, signal processing and mechanics, to algebraic equations. Therefore it is important not only for mathematicans to understand the Laplace transform and its properties beyond the elementary level.

The Hardy operator H, defined by

Hf (x) = 1 x

Zx

0

f (s)ds,

and generalizations of this operator have been the object of extensive studies in mathematical analysis during the last century. Closely related to the Hardy operator is the Hardy inequality which in its original form states that for non- negative functions in L^p(0, ∞) and p > 1 we have

Z∞

0



1 x

Zx

0

f (s)ds





p

dx ≤

 p

p − 1

p ∞Z

0

(f (x))^pdx.

In this thesis we consider in principal the Hardy minus Identity operator;

(H − I)f(x) = Hf(x) − f(x) = 1 x

Zx

0

f (s)ds − f(x).

This operator was shown in [7] to satisfy the relation k(H − I)fkL² = kfkL², ∀f ∈ L²(0, ∞).

Moreover, in [8] the result above is generalized to weighted L²-spaces.

The classical Laguerre polynomials and functions, whose origins go back to the nineteenth century, are used in different areas such as quantum physics and electrical engineering. One example is the quantum-mechanical description of the hydrogen atom. In this model the Laguerre polynomials are part of the eigenfunctions that corresponds to the bounded states of the atom.

2 The Laplace transform and the Gamma func- tion

Let L denote the Laplace transform defined by

L(f (x))(s) = Z∞

0

f (x)e^−sxdx.

In this section we show that the compositions of a specified involution T and the Laplace transform on L²(0, ∞) are unitary equivalent to multiplication by the gamma functions Γ(¹₂± iw) on L²(R). We then give a new proof of the known fact that the Laplace transform is a bounded transformation from L²(0, ∞) to L²(0, ∞) and, more specific, show that the operator norm of L is

kLkL²→L² = π^1/2.

In addition, we interpret the compositions of L and T as convolutions on R and derive, due to these interpretations, the following result:

kLfkL^p′ ≤ (p(2 − p))^(2−p)/2p(π(p − 1))^(p−1)/pkfkL^p, 1 ≤ p ≤ 2, 1 p+ 1

p^′ = 1.

2.1 The Laplace transform and related unitary equivalent operators

Let us start with some necessary definitions.

Definition 2.1 (Surjective isometry) Let X and Y be normed spaces. An operator A from X onto Y is said to be a surjective isometry if

kAxk^Y = kxkX, ∀x ∈ X.

Note that when A is a surjective isometry then the inverse A⁻¹ exists and A⁻¹ is also a surjective isometry.

Definition 2.2 (Unitary operator) A surjective isometry U : X → Y , where X and Y are Hilbert spaces, we will call an unitary operator.

From the definition one can derive that for an unitary operator U we have U^∗= U⁻¹ where U^∗ is the adjoint operator.

Definition 2.3 (Unitary equivalence) Let X and Y be Hilbert spaces and let A : X → X and B : Y → Y be bounded linear operators. We say that B is unitary equivalent to A if there exists an unitary operator U : X → Y such that B = U AU⁻¹.

Notation 2.1 (Unitary equivalence) B ∼uA.

It is clear from definition that if A ∼^uB and B ∼^uC then A ∼^uC.

Definition 2.4 Let the operator T be defined by T f (x) = ¹_xf (¹_x).

It is easy to show that T is an unitary operator from L²(0, ∞) onto L²(0, ∞) and T^∗= T⁻¹= T and hence that T , in addition, is an involution.

Definition 2.5 (Gamma function) The gamma function Γ(z), z ∈ C, Re z > 0, is defined by

Γ(z) = ^∞R

0

t^z−1e^−tdt.

For more information about the gamma function see e.g. [1].

Let MΓ(1/2+iw)and M_Γ(1/2−iw)be operators corresponding to multiplication by, respectively, Γ(¹₂+ iw) and Γ(¹₂− iw) on R. We are now ready to state our main theorem in this section:

Theorem 2.1 T L ∼^uMΓ(1/2+iw) and LT ∼^uM_Γ(1/2−iw). Proof. Let f ∈ L²(0, ∞). We have

T Lf (s) = 1 s

Z∞

0

f (x)e^−xsdx.

We introduce the operator V defined by V f (u) = e^u/2f (e^u). It is straight- forward to show that V is an unitary operator from L²(0, ∞) onto L²(R). Now, we have

V T Lf (u) = e^u/2 e^u

Z∞

0

f (x)e^−e−uxdx = [x = e^v] =

= Z∞

−∞

e^v/2f (e^v)e^{−e−(u−v)−(u−v)/2}dv =

= Z∞

−∞

V f (v)Ω(u − v)dv = (V f ∗ Ω) (u),

where Ω(u) = e^{−e−u−u/2}. Thus, we arrive at a convolution. Next, we introduce the Fourier transform F defined by

Ff (w) = 1

√2π Z∞

−∞

f (x)e^−iwxdx.

It is well-known, by the Plancherel theorem, that F is an unitary operator on L²(R). We now consider FV T Lf :

FV T Lf (w) = F (V f ∗ Ω) (w) =√

2πFV f (w)FΩ(w).

Let us calculate the Fourier transform of Ω:

√2πFΩ(w) = Z∞

−∞

e^{−e−x−x/2}e^−iwxdx =

y = e^−x

= Z∞

0

e^−yy^−1/2+iwdy ≡

≡ Γ(1 2 + iw).

Note that  Γ(1

2+ iw)  

2

= Γ(1

2 + iw)Γ(1

2 − iw) = π(cosh(wπ/2))⁻¹≤ π which gives

kFV T LfkL²(R)≤ π^1/2kFV fkL²(R)

or, because F and V are unitary operators;

kT LfkL²(0,∞)≤ π^1/2kfkL²(0,∞). This shows that T L is a bounded operator on L²(0, ∞).

The boundedness of T L combined with the fact that the composition FV : L²(0, ∞) → L²(R) is an unitary operator gives that T L on L²(0, ∞) is unitary equivalent to multiplication by Γ(¹₂+ iw) on L²(R). The result for LT can be proved in a similar manner.

One interesting remark:

Remark 2.1 If we consider the Carleman integral operator S on L²(0, ∞) de- fined by

Sf (x) = Z∞

0

f (y) x + ydy

it is not difficult (see e.g. [2]) to verify that the unitary operator U = FV , which we used in Theroem 2.1, transform the operator S to multiplication by the function π(cosh(wπ/2))⁻¹ on L²(R).

Then, from the equality Γ(1

2 + iw)Γ(1

2− iw) = π(cosh(wπ/2))⁻¹ and Theorem 2.1 it follows that

S = LL^∗= L².

Definition 2.6 (Operator norm) The norm of a bounded linear operator T : X → Y is defined by kT k_X→Y= sup

kxkX=1kT xkY.

The result of the corollary below seems to first appear in the papers [3] and [4] by G.H. Hardy. However, our proof is different and based on Theorem 2.1.

Corollary 2.1 The Laplace transform L is a bounded transformation from L²(0, ∞) to L²(0, ∞) and more precisely kLkL²→L²= π^1/2.

Proof. We first consider kT LkL²→L². From Theorem 2.1 we have the estimate

kT LfkL²(0,∞)≤ π^1/2kfkL²(0,∞), ∀f ∈ L²(0, ∞), which gives

kT LkL²→L² ≤ π^1/2. (2.1)

The unitary equivalence between T L on L²(0, ∞) and MΓ(1/2+iw) on L²(R) gives that

kT LkL²→L² =

M_Γ(1/2+iw)

L²→L². We can construct a function eg ∈ L²(R) given by

eg(w) = 1

p2δ(ε)χ(1/2−δ(ε),1/2+δ(ε)),

for every 0 < ε < π^1/2. χ_(a,b)is the characteristic function on (a, b) and δ(ε) denote the minimum distance between w = 1/2 and roots to the equation Γ(¹₂+ iw) = π^1/2− ε. Observe that kegkL²(R)= 1.

We then are able to derive, because

Γ(¹₂+ iw)

 is a continuous function, the following inequality:

MΓ(1/2+iw)

L²→L² = sup

kgkL2(R)=1

Γ(

1 2+ iw)g

_L2

(R)

≥ (π^1/2− ε) kegkL²(R)

= π^1/2− ε. (2.2)

Let ε → 0⁺ and we arrive, in view of (2.1) and (2.2), at the equality kT LkL²→L² = π^1/2.

Finally, the fact that T is an isometry from L²(0, ∞) onto L²(0, ∞) results in

kLkL²→L² = π^1/2.

2.2 An estimate for the Laplace transform on L

(0, ∞)

We recall the classic Young’s convolution inequality:

Let 1 ≤ p, q, r ≤ ∞, 1/p + 1/q − 1/r = 1, f ∈ L^pand g ∈ L^q. Then kf ∗ gkL^r ≤ kfkL^pkgkL^q.

This inequality is sharp only for the case p = 1 or q = 1. W. Beckner stated and proved in [5] the following sharp form of Young’s inequality:

Let 1 ≤ p, q, r ≤ ∞, 1/p + 1/q − 1/r = 1, f ∈ L^pand g ∈ L^q. Then kf ∗ gkL^r ≤ A^pAqAr^′kfkL^pkgkL^q

where Am=

m^1/m m^{′ 1/m′}

1/2

and _m¹ +_m^1′ = 1. The sharp Young’s inequality will be used in a generalization of Corollary 2.1:

Theorem 2.2 Let f ∈ L^p(0, ∞), 1 ≤ p ≤ 2 and ¹p+_p^1′ = 1.Then kLfkL^p′ ≤ (p(2 − p))^(2−p)/2p(π(p − 1))^(p−1)/pkfkL^p.

Proof. We introduce the operators Tr and Vr, 1 ≤ r < ∞, defined by Trf (x) = 1

x^2/rf (1 x), and

Vrf (u) = e^u/rf (e^u).

One can show that Tris an isometry from L^r(0, ∞) onto L^r(0, ∞) and that Vr

is an isometry from L^r(0, ∞) onto L^r(R). Now,

Tp^′Lf (s) = 1 s^2/p′

Z∞

0

f (x)e^−xsdx,

and

Vp^′Tp^′Lf (u) = e^u/p′ e^2u/p′

Z∞

0

f (x)e^−e−uxdx = [x = e^v] =

= Z∞

−∞

e^v/pf (e^v)e^{−e−(u−v)−(u−v)/p′}dv =

= Z∞

−∞

Vpf (v)Ωp^′(u − v)dv = (Vpf ∗ Ωp^′) (u). (2.3)

where Ωr(u) = e^{−e−u−u/r}.

We recall the sharp form of Young’s inequality and apply it to (2.3):

kV^pf ∗ Ω^qkL^p′ ≤ A^pAqA(p^′)^′kV^pf kL^p

Ωp^′

_Lq = A²pAqkV^pf kL^p

Ωp^′

_Lq. Now, kV^pf kL^p(R)= kfkL^p(0,∞)because Vp is an isometry and

Ωp^′

q L^q =

Z∞

−∞

 e^−e

−u

−u/p^′

q

du = Z∞

−∞

e^{−qe−u−qu/p′}du =

y = qe^−u

=

= q^−q/p′ Z∞

0

e^−yy^q/p′−1dy = q^−q/p′Γ(q/p^′).

Further, the constraints 1/p + 1/p^′ = 1 and 1/p + 1/q − 1/p^′ = 1 gives that q = ^p₂^′ and therefore

kΩ^p′kL^q =

2 p^′

1/p^′

(Γ(1/2))^2/p

′

=

2π p^′

1/p^′

=

2π(p − 1) p

(p−1)/p

. Also, some algebraic manipulations gives

A²_pAq = p^1/2(2 − p)^(2−p)/2p 2^(p−1)/p . With some further calculations we derive

A²_pAqkΩ^p′kL^q = (p(2 − p))^(2−p)/2p(π(p − 1))^(p−1)/p.

From the isometric properties of Vp^′ and Tp^′ it follows that kV^p′Tp^′Lf kL^p′(R)= kLfkL^p′(0,∞). The condition q ≥ 1 and q = ^p₂^′ implies that p^′≥ 2 which in turn gives 1 ≤ p ≤ 2. Altogether, we have the result

kLfkL^p′ ≤ (p(2 − p))^(2−p)/2p(π(p − 1))^(p−1)/pkfkL^p, 1 ≤ p ≤ 2.

Hardy proved in [4] the following, less sharp, result:

kLfkL^p′ ≤

2π p^′

1/p^′

kfkL^p, 1 ≤ p ≤ 2, 1 p+ 1

p^′ = 1.

The proof of Hardy is based on a generalized form of H¨older’s inequality. Note that if we apply the classic form of Young’s inequality in our proof of Theorem 2.2 we obtain the same constant as in Hardy’s result.

3 The Hardy minus Identity operator on L

(0, ∞)

In this section we show that the Hardy operator H on L²(0, ∞), defined by

Hf (x) = 1 x

Zx

0

f (s)ds,

is unitary equivalent to a multiplicative operator on L²(R). A paper by S. Kai- jser, see [6], where the Hardy operator is interpreted as a convolution initiated our research in this direction.

For the Hardy minus Identity operator H − I, where

(H − I)f(x) = 1 x

Zx

0

f (s)ds − f(x),

we are able, with an unitary equivalent mulitplicative operator, to give new and elementary proofs of the following properties:

i) kαI + (1 − α)(H − I)kL²→L²= 1, 0 ≤ α ≤ 1, ii) k(H − I)fkL² = kfkL², ∀f ∈ L²(0, ∞).

For result ii) see [7] for orignal statement and proof and [8] for the weighted case.

Finally, in Section 3.2 we show that an operator I − A^Ω, where I − A^Ω∼^u I − H = −(H − I), is a shift isometry in an orthonormal basis in L²(R).

Moreover, this basis is constituted by Laguerre functions.

3.1 An unitary equivalent form of the Hardy operator

Let f ∈ L²(0, ∞) and V denote the operator defined by V f(x) = e^x/2f (e^x).

From Section 2.1 we know that V is an unitary operator from L²(0, ∞) onto L²(R). Also, we introduce the operator M₁¹

2+iw which corresponds to mulipli- cation by 1 ¹

2+iw on L²(R).

Now to our main result:

Theorem 3.1 H ∼^u M₁¹

2+iw. Proof. Let V act on Hf :

V Hf (u) = e^u/2 1 e^u

e^u

Z

0

f (s)ds = [s = e^v] = Zu

−∞

f (e^v)e^v/2e^−(u−v)/2du.

Let Ω(u) = e^−u/2χ_(0,∞). We get

V Hf (u) = Z∞

−∞

V f (v)Ω(u − v)du = V f ∗ Ω(u).

Thus, we arrive at a convolution. Next, we Fourier transform the expression above:

FV Hf (w) = F(V f ∗ Ω)(w) =√

2πFV f (w)FΩ(w).

An easy calculation gives

√2πFΩ(w) = 1

1 2+ iw.

Since both F and V are unitary operators the theorem is proved.

We proceed with corollaries to Theorem 3.1:

Corollary 3.1 Let the operator H − I be defined by (H − I)f(x) = x¹

Rx

0 f (s)ds − f(x) and 0 ≤ α ≤ 1.

Then kαI + (1 − α)(H − I)kL²→L² = 1.

Proof. As for H we let V and F act on H − I :

FV (H − I)f(w) = FV Hf(w) − FV f(w).

From Theorem 3.1 we have

FV Hf (w) = 1

1

2+ iwFV f (w).

Thus,

FV (H − I)f(w) =

1 2− iw

1

2+ iwFV f (w), and we see that H − I ∼u

1 2−iw

1

2+iw. This implies that kαI + (1 − α)(H − I)kL²→L² = sup

kfkL2(0,∞)=1kαf + (1 − α)(H − I)fkL²(0,∞)

= sup

kgkL2(R)=1

αg + (1 − α)

1 2− iw

1 2+ iwg

L²(R)

= sup

kgk_L2(R)=1

1

2+ (2α − 1)iw

1 2+ iw

 g

L²(R)

≤

1

2+ (2α − 1)iw

1 2+ iw

L^∞(R)

sup

kgk_L2(R)=1kgkL²(R)

=

1

2+ (2α − 1)iw

1 2+ iw

L^∞(R)

= 1. (3.1)

We construct a function eg ∈ L²(R) given by eg(w) = 1

p2δ(ε)χ(−δ(ε),δ(ε))

for every 0 < ε < 1 where δ(ε) denote the minimum distance between w = 0 and roots to the equation

1

2+(2α−1)iw

1 2+iw

 = 1 − ε. Note that kegkL²(R)= 1.

We then are able to derive, because 

1

2+(2α−1)iw

1 2+iw

 is a continuous function, the following inequality:

kαI + (1 − α)(H − I)kL²→L² = sup

kgkL2 (R)=1

1

2+ (2α − 1)iw

1 2+ iw

 g

L²(R)

≥ (1 − ε) kegkL²(R)= 1 − ε. (3.2) Let ε → 0⁺ and we arrive, in view of (3.1) and (3.2), at the equality:

kαI + (1 − α)(H − I)kL²→L² = 1.

Hence, by Corollary 3.1, the unit sphere of the space of all bounded linear operators on L²(0, ∞) contains an interval with the ends I and H − I.

The special case α = 0 in Corollary 3.1 above could be developed to the following result:

Corollary 3.2 Let the operator H − I be defined by (H −I)f(x) = _x¹

Rx

0 f (s)ds−f(x). Then k(H − I)fk2= kfk2, ∀f ∈ L²(0, ∞).

Proof. H − I ∼u

1 2−iw

1

2+iw gives k(H − I)fkL²(0,∞) =

1 2 − iw

1

2 + iwFV f

L²(R)

= kFV fkL²(R)=

= kfkL²(0,∞).

3.2 Laguerre functions and the Identity minus Hardy op- erator

Let Ln denote the Laguerre polynomial of degree n defined by Ln(x) = e^x

n!

dⁿ

dxⁿ(xⁿe^−x)

and let ln denote the n:th Laguerre function defined by ln(x) = e^−x/2Ln(x).

It can be shown that {Lⁿ}^∞n=0 is an orthonormal basis in L²(0, ∞) with measure e^−xdx and {ln}^∞n=0 is an orthonormal basis in L²(0, ∞) with measure dx. For more information about Laguerre functions and polynomials see e.g.

[9].

In [10] the following sequences of functions are considered:

en(x) = −Lⁿ(ln x)

x χ_(1,∞), n ≥ 0, fn(x) = Ln(− ln x)χ(0,1), n ≥ 0.

It is shown in [10] that the union {eⁿ}∪{fⁿ} is an orthonormal basis in L²(0, ∞) with measure dx. Moreover, the authors derive that the Identity minus Hardy operator

(I − H)f(x) = −(H − I)f(x) = f(x) − 1 x

Zx

0

f (s)ds is a shift operator in this basis:

(I − H)en= en+1, n ≥ 0, (I − H)f0= e0,

(I − H)fn= f_n−1, n ≥ 1.

Now we construct an orthonormal basis in L²(R) based on the Laguerre function ln. Let gn and hn be defined by

gn(u) = ln(u)χ_(0,∞), n ≥ 0, hn(u) = ln(−u)χ_(−∞,0), n ≥ 0.

It follows from the properties of the Laguerre functions that {gⁿ} and {hⁿ} are orthonormal bases with measure du in L²(0, ∞) and L²(−∞, 0) respectively.

Since the sequences {gⁿ} and {hⁿ} have disjoint supports we conclude that the union {gⁿ} ∪ {hⁿ} is an orthonormal basis in L²(R).

Let U denote the shift operator defined by the formulas:

U gn = gn+1, n ≥ 0, (3.3)

U h0 = g0, (3.4)

U hn = h_n−1, n ≥ 1. (3.5)

Also, we introduce the operator

(I − AΩ)ϕ(u) = ϕ(u) − Ω ∗ ϕ(u) where Ω(u) = e^−u/2χ_(0,∞).

Now, our main result in this section reads

Theorem 3.2 The operator I − AΩ on L²(R), which is unitary equivalent to I − H on L²(0, ∞), is a shift isometry in the basis {gn} ∪ {hn} and the shifting property is given by (3.3)-(3.5) above.

Proof. To prove I − A^Ω∼^uI − H we recall the unitary operator V f(u) = e^u/2f (e^u) from L²(0, ∞) onto L²(R) and Theorem 3.1 where we showed that V Hf (u) = V f ∗ Ω(u). Now, let V act on (I − H)f :

V (I − H)f(u) = V f(u) − V f ∗ Ω(u) ≡ (I − A^Ω)V f (u). (3.6) The unitary equivalence follows from (3.6) and the fact that V is an unitary operator which also proves the isometric property of I − AΩ.

What remains to prove is that V en = gn and V fn = hn. The shifting property of I − AΩis then given by the unitary equivalence I − AΩ∼uI − H .

Let us consider V en :

V en(u) = e^−u/2Ln(u)χ_(0,∞) = ln(u)χ_(0,∞) ≡ gn(u).

Similary V fn:

V fn(u) = e^u/2Ln(−u)χ_(−∞,0) = ln(−u)χ_(−∞,0) ≡ hⁿ(u) which ends the proof.

Finally, we observe that we can state the result in Theorem 3.2 in a different and perhaps more convenient manner. Let kn, n ∈ Z, be defined by

kn(u) =

 gn(u), n ≥ 0, h_−n−1(u), n ≤ −1.

From the properties of {gn} ∪ {hn} it follows that the sequence {kn} is an orthonormal basis in L²(R). The properties (3.3) − (3.5) of the operator I − AΩ

translated to the basis {kn} gives the simple shift (I − AΩ)kn= kn+1, n ∈ Z.

4 Estimates of the Hardy minus Identity oper- ator on the cone of decreasing functions

As mentioned in Section 3 the identity k(H − I)fkL² = kfkL² is true for every f ∈ L²(0, ∞). In this section we are concerned with estimates for H − I and its inverse (H − I)⁻¹ on certain cones of functions in L^p(0, ∞), 1 < p < ∞.

4.1 Operator norms of H − I and (H − I)

on cones in L

(0, ∞)

We start with the definitions of the distribution function µ_f of a function f and its decreasing rearrangement f^∗.

Definition 4.1 Let f be a measurable function on a measure space (Ω, µ). The distribution function µ_f : [0, ∞) → [0, ∞] is defined by

µ_f(λ) = µ({x ∈ Ω : |f(x)| > λ}).

Now the definition of the decreasing rearrangement of f :

Definition 4.2 The decreasing rearrangement f^∗ : [0, ∞) → [0, ∞] is defined by f^∗(x) = inf{λ ≥ 0 : µf(λ) ≤ x}.

Let f^∗∗ be defined by

f^∗∗(x) = Hf^∗(x) = 1 x

Zx

0

f^∗(s)ds.

Observe that if f is a decreasing nonnegative function we have almost every- where; f^∗ = f and f^∗∗= Hf . Hence, f^∗∗− f^∗ = (H − I)f almost everywhere in this case.

The following equivalence in Lorentz L^p,q-spaces is proved in [11, Proposition 7.12] :

kfkLp,q :=



 Z∞

0

x^1/pf^∗(x)qdx x





1/q

≈



 Z∞

0

x^1/p(f^∗∗(x) − f^∗(x)q dx x





1/q

(4.1) where f ∈ L^p,q, f^∗∗(∞) = lim

x→∞f^∗∗(x) = 0, 1 < p < ∞ and 1 ≤ q ≤ ∞. Note that when q = p in (4.1) above we achieve the L^p-norm of f^∗and f^∗∗− f^∗and if f , in addition, is a decreasing nonnegative function we have the equivalence



 Z∞

0

(f (x))^pdx





1/p

≈



 Z∞

0

((H − I)f(x))^pdx





1/p

. (4.2)

We will next consider the case q = p and search for a sharp estimate in (4.2) above.

Now, f is a decreasing nonnegative function and we therefore restrict our investigation to the following subset of L^p(0, ∞):

Definition 4.3 C^p= {f ∈ L^p(0, ∞) : f is a decreasing nonnegative function}.

We also define the operator norm of H − I on C^p: Definition 4.4 kH − IkC^p= sup

f ∈C^p,kfkLp=1k(H − I)fkL^p.

Note that determine kH − IkC^p is equal to derive a sharp estimate in (4.2).

We will next give an expression for the operator norm kH − IkC^p,

p ∈ {2, 3, 4, ...}. To get a nice survey of the result we start with the case p = 3.

Theorem 4.1 kH − IkC³ = √3¹

2.

Proof. With the fact that f ∈ C³ we know that there exists a simple function g such that kf − gkL³ < ε, ∀ε > 0, because simple functions are dense in L³(0, ∞). Thus, it suffices to consider the case of simple functions which on C³ can be written as

g(x) = Xn i=1

ciχ_(0,ai)(x), ci≥ 0, i = 1, 2, ..., n, 0 < a¹< a2< ... < an. (4.3)

Straightforward calculations gives

(H − I)g(x) =















0 , 0 < x < a1 c1a1

x , a1< x < a2 c1a1+c2a2

x , a2< x < a3

... ...

c1a1+c2a2+...+cnan

x , an< x.

Let us calculate the norm of g :

kgk³L³ = Z∞

0

|g(x)|³dx =

= (c1+ c2+ ... + cn)³a1+ (c2+ c3+ ... + cn)³(a2− a¹) + ... + +(c_n−1+ cn)³(an− an−1) + c³_n(an− an−1). (4.4) Similary, the norm of √³

2(H − I)g :

√³

2(H − I)f

3 L³ =

Z∞

0

 √³

2(H − I)f(x)

3

dx =

= c³₁a³₁( 1 a²₁ − 1

a²₂) + (c1a1+ c2a2)³(1 a²₂ − 1

a²₃) + ... + +(c1a1+ c2a2+ ... + c_n−1a_n−1)³( 1

a²_n−1 − 1 a²_n) + +(c1a1+ c2a2+ ... + c_n−1a_n−1+ cnan)³ 1

a²_n (4.5) We interpret kgk³L³ and √³

2(H − I)g ³_L3 as polynomials in ci. To make a comparison of kgk³L³ and √³

2(H − I)g ³_L3 we then simply compare the coeffi- cients for different powers of ci:

kgk³L³

√³

2(H − I)g ³

L³

c³_i : ai ai

c²_icj, i < j : 3ai 3^a_a²ⁱ_j cic²_j, i < j : 3ai 3ai

cicjck, i < j < k : 6ai 6^a_a^ia_k^j

To obtain these coefficients we use the multinomial theorem and note that in the case of kgk³L³only the last term in (4.4) with the presence of cigives a contri- bution to the coefficient of the considered power. In the case of √³

2(H − I)g ³

L³

only the first term in (4.5) with the presence of cj or ck gives a contribution to the coefficient of the considered power.

We see that the coefficients for c²_icj and cicjck differ between kgk³L³ and √³

2(H − I)g ³_L3. For c²_icj we have a²_i aj

< ai

because 0 < ai< aj and for cicjck the inequality aiaj

ak < ai

is true because 0 < aj < ak. Hence, all coefficients for kgk³L³are bigger or equal than those for √³

2(H − I)g ³

L³. This gives the inequality k(H − I)gkL³ ≤ 1

√3

2kgkL³

where g is of the form (4.3). The fact that simple functions are dense in L³(0, ∞) gives that this result is true for a general function f ∈ C³. Further, an easy

calculation shows that the case of equality is obtained when g(x) = χ_(0,a). We therefore have

kH − IkC³ = 1

√3

2.

For a general integer p ≥ 2 the calculations become more involving but in principle the same method as in Theorem 4.1 could be applied to prove the following result:

Corollary 4.1 kH − IkC^p=_(p−1)¹1/p , p ∈ {2, 3, 4, ...}.

Proof. Simple functions are dense in L^p(0, ∞). In analogy with Theorem 4.1 it suffices to consider simple functions which in C^p can be written in the form:

g(x) = Xn i=1

ciχ_(0,ai)(x), ci≥ 0, i = 1, 2, ..., n, 0 < a1< a2< ... < an. (4.6) The calculations are in principle the same as in Theorem 4.1 and the ob- servations about the contributions to the coefficients are valid also in this case.

In short we obtain the following coefficients for a general term c^j_i¹₁c^j_i²₂...c^j_im^m in the polynomials kgk^pL^pand

(p − 1)^1/p(H − I)g ^p

L^p with use of the multinomial theorem

kgk^pL^p

(p − 1)^1/p(H − I)g

p

L^p

c^j_i1¹c^j_i2²...c_i^j_m^m: _j1!j2^p!_!...jm!ai1

p!

j1!j2!...jm!

a^j1_i1a^j2_i2...a^jm_im a^p−1_im

where i1< i2< ... < im, j1+ j2+ ... + jm= p, j1> 0 and j2, j3,..., jm≥ 0.

c^j_i¹₁c^j_i²₂...c^j_im^m accounts for all possible terms in the polynomials kgk^pL^p and

(p − 1)^1/p(H − I)g

p

L^p because i1, i2,..., im, j1, j2,..., jm and m are chosen arbitrary.

We see that

a^j_i1¹a^j_i2²...a^j_im^m a^p−1_im ≤ ai1

for all possible choices of i1, i2,..., im, j1, j2,..., jmand m.

This gives the inequality

k(H − I)gkL^p≤ 1

(p − 1)^1/pkgkL^p

where g is of the form (4.6). Simple functions are dense in L^p(0, ∞) and the case of equality is obtained when g(x) = χ_(0,a). Altogether this gives the result

kH − IkC^p= 1

(p − 1)^1/p, p ∈ {2, 3, 4, ...}.

With the result of Corollary 4.1 we have derived a sharp estimate in the equivalence (4.1) when f ∈ C^p and q = p ∈ {2, 3, 4, ...}.

The restriction to functions in C^p, or more precisely that the function is decreasing on (0, ∞), is essential for the estimate

k(H − I)fkL^p≤ 1

(p − 1)^1/pkfkL^p

to be true. We illustrate with a counterexample:

Example 4.1 Let f (x) = χ_(1,3/2). This function is not decreasing on (0, ∞).

Straightforward calculations gives:

1

√3

2kfkL³= √3¹

4 and k(H − I)fkL³ =q³

11 36

Hence, k(H − I)fkL³> √3¹

2kfkL³.

We proceed with an estimate for the inverse operator (H − I)⁻¹. This operator is, see [12] for derivation, the following:

(H − I)⁻¹f (x) = Z∞

x

f (s)

s ds − f(x).

Let us introduce the subset eC^p ⊂ L^p(0, ∞) defined by Definition 4.5

Ce^p= {f ∈ L^p(0, ∞) : xf(x) is an increasing nonnegative function}.

In analogy with C^p we define the operator norm of (H − I)⁻¹ on eC^p: Definition 4.6

(H − I)⁻¹ _Cep= sup

f ∈ eC^p,kfkLp=1

(H − I)⁻¹f _Lp.

From [8] we know that H − I is an unitary operator on L²(0, ∞). Hence, (H − I)^∗ = (H − I)⁻¹ where (H − I)^∗ is the adjoint operator.

Also, the theorem below is implicitly used in our proof of the main result for

(H − I)⁻¹ _e

C^p:

Theorem 4.2 Let f ∈ L^p(a, b), −∞ < a < b < ∞, q = _p−1^p , 1 < p < ∞.

Then there exists a function g ∈ L^q(a, b) such that

kfkL^p= sup

kgkLq=1

   Zb

a

f (x)g(x)dx   

The proof can be found in a textbook in functional analysis, e.g. in [13, pp.

144-145].

Now to our main result for

(H − I)⁻¹ _e

C^p: Theorem 4.3

(H − I)⁻¹ _Cep = (q−1)^q−1q , where q = _p−1^p and p ∈ {2, 3, 4, ...}.

Proof. First we construct a function g that fulfills the requirements in Theorem 4.2 and use the fact that (H − I)^∗ = (H − I)⁻¹.

Let us consider g(x) = λ

(H − I)⁻¹f (x)^q−1sgn((H − I)⁻¹f (x))

where λ is a real constant, sgn is the signum function and f ∈ eC^q. Some calculations gives

λ =



 Z∞

0

(H − I)⁻¹f (x)^qdx





−1/p

in order to fulfill kgkL^p= 1.

Next, we verify that  

∞R

0((H − I)⁻¹f (x))g(x)dx   =

(H − I)⁻¹f _Lq : 

 R∞

0((H − I)⁻¹f (x))g(x)dx   =

= λ^∞R

0((H − I)⁻¹f (x))

(H − I)⁻¹f (x))^q−1sgn((H − I)⁻¹f (x)) dx =

=

_∞R

0

(H − I)⁻¹f (x)^qdx

1/q

=

(H − I)⁻¹f

L^q. We now show that g ∈ C^p where p =_q−1^q .

1) g ∈ L^p(0, ∞) which follows immediately from the definition of g.

2)

(H − I)⁻¹f (x) = Z∞

x

f (s)

s ds − f(x) = Z∞

x

sf (s)

s² ds − f(x) ≥

≥ xf(x) Z∞

x

1

s²ds − f(x) = 0

where we use the fact that f ∈ eC^q. Moreover, (H − I)⁻¹f (x) ≥ 0 implies g(x) ≥ 0.

3) Let 0 < x1< x2.

(H − I)⁻¹f (x1) = Z∞

x1

f (s)

s ds − f(x1) =

x2

Z

x1

f (s) s ds +

Z∞

x2

f (s)

s ds − f(x1) =

=

x2

Z

x1

sf (s) s² ds +

Z∞

x2

f (s)

s ds − f(x¹) ≥

≥ x¹f (x1)

x2

Z

x1

1 s²ds +

Z∞

x2

f (s)

s ds − f(x¹) =

= −x¹ x2

f (x1) + Z∞

x2

f (s)

s ds ≥ −f(x2) + Z∞

x2

f (s) s ds =

= (H − I)⁻¹f (x2)

where we use the fact that f ∈ eC^q. Hence, (H − I)⁻¹f is a decreasing function on (0, ∞) which gives that g is a decreasing function on (0, ∞).

Altogether 1), 2) and 3) gives that g ∈ C^p.

Next, it follows by use of the definition of an adjoint operator, H¨older’s inequality and Corollary 4.1:

(H − I)⁻¹f

L^q = Z∞

0

((H − I)⁻¹f (x))g(x)dx = Z∞

0

f (x)((H − I)g(x))dx ≤

≤



 Z∞

0

((H − I)g(x))^pdx





1/p

 Z∞

0

(f (x))^qdx





1/q

=

= k(H − I)gkL^pkfkL^q ≤ 1

(p − 1)^1/pkgkL^pkfkL^q =

= 1

(p − 1)^1/p kfkL^q = (q − 1)^q−1q kfkL^q, ∀f ∈ eC^q. We note, by straightforward calculations, that we have equality above when

f (x) = 1

xχ_(a,∞) ∈ eC^q

and therefore

(H − I)⁻¹ _Ceq = (q − 1)^q−1q .

4.2 Further estimates of H − I and (H − I)

We proceed with a lemma:

Lemma 4.1 H − I is a bijective mapping between C^p and eC^p for p > 1.

Proof. We investigate the properties for (H −I)⁻¹f where f is an arbitrary function in eC^p. In Theorem 4.3 it was derived that (H − I)⁻¹f is a decreasing and nonnegative function on (0, ∞). Let us now show that

(H − I)⁻¹f ∈ L^p(0, ∞) which then implies (H − I)⁻¹f ∈ C^p. With the use of Minkowski’s inequality and the inequality;

Z∞

0

   Z∞

x

f (s) s ds

p

dx ≤ p^p Z∞

0

|f(x)|^pdx,

see [14, p. 244] for derivation, we get

(H − I)⁻¹f

L^p =



 Z∞

0

   Z∞

x

f (s)

s ds − f(x)   

p

dx





1/p

≤



 Z∞

0

   Z∞

x

f (s) s ds

p

dx





1/p

+



 Z∞

0

|f(x)|^pdx





1/p

≤ p



 Z∞

0

|f(x)|^pdx





1/p

+



 Z∞

0

|f(x)|^pdx





1/p

= (p + 1)



 Z∞

0

|f(x)|^pdx





1/p

= (p + 1) kfkL^p

Hence, (H − I)⁻¹f ∈ L^p(0, ∞). Together with the results in Theorem 4.3 this gives that (H − I)⁻¹f ∈ C^p. Let g(x) = (H − I)⁻¹f (x). This relation can be written as (H − I)g(x) = f(x). Now f was chosen arbitrary, thus H − I maps C^p onto eC^p. The injective property follows from the existence of the inverse operator and H − I is therefore a bijective mapping between C^p and eC^p.

Now to some consequences of Lemma 4.1.

Corollary 4.2 Let f ∈ eC^p and p ∈ {2, 3, 4, ...}.

Then

(H − I)⁻¹f

L^p≥ (p − 1)^1/pkfkL^p. Proof. We recall Corollary 4.1 which stated

k(H − I)kC^p= 1

(p − 1)^1/p, p ∈ {2, 3, 4, ...},

or

k(H − I)gkL^p≤ 1

(p − 1)^1/p kgkL^p, ∀g ∈ C^p, p ∈ {2, 3, 4, ...}. (4.7) From Lemma 4.1 we know that H − I is a bijection between C^p and eC^p. Let (H − I)g = f in (4.7) above. We arrive at the inequality

(p − 1)^1/pkfkL^p≤

(H − I)⁻¹f

L^p, ∀f ∈ eC^p, p ∈ {2, 3, 4, ...}.

and the corollary is proved.

Also, with Lemma 4.1 we can derive the following result:

Corollary 4.3 Let f ∈ C^q where q = _p−1^p and p ∈ {2, 3, 4, ...}.

Then kfkL^q ≤ (q − 1)^q−1q k(H − I)fkL^q. Proof. From Theorem 4.3 above we have

(H − I)⁻¹g

L^q ≤ (q − 1)^q−1q kgkL^q, ∀g ∈ eC^q. (4.8) Because H − I is a bijective mapping between C^q and eC^q by Lemma 4.1 we can write (4.8) in terms of:

kfkL^q ≤ (q − 1)^q−1q k(H − I)fkL^q, ∀f ∈ C^q . The proof is complete.

Corollary 4.1 is proved for integers bigger or equal than 2. The question is if this result is true for all real numbers p ≥ 2. This leads us to

Conjecture 4.1 kH − IkC^p= ¹

(p−1)^1/p, p ≥ 2.

The conjecture gives the following generalizations of the results above:

Remark 4.1 If Conjecture 4.1 is true, then we have the following generaliza- tion of Theorem 4.3:

(H − I)⁻¹ _Cep= (q − 1)^q−1q where 1 < q ≤ 2.

Remark 4.2 If Conjecture 4.1 is true, then we have the following generaliza- tion of Corollary 4.2:

Let f ∈ eC^p, p ≥ 2. Then

(H − I)⁻¹f

L^p≥ (p − 1)^1/pkfkL^p.

Remark 4.3 If Conjecture 4.1 is true, then we have the following generaliza- tion of Corollary 4.3:

Let f ∈ C^q where 1 < q ≤ 2. Then kfkL^q≤ (q − 1)^q−1q k(H − I)fkL^q.