A duality approach for optimal decomposition in real interpolation

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Department of Mathematics

A duality approach for optimal decomposition

in real interpolationer

Japhet Niyobuhungiro

LiTH-MAT-R–2013/08–SE

Department of Mathematics

Linkping University

S-581 83 Linkping, Sweden.

A duality approach for optimal

decomposition in real interpolation

Japhet Niyobuhungiro

Abstract

We use our previous results on subdifferentiability and dual character-ization of optimal decomposition for an infimal convolution to establish mathematical properties of exact minimizers (optimal decomposition) for the K–,L–, and E– functionals of the theory of real interpolation. We char-acterize the geometry of optimal decomposition for the couple(`p_{, X)on}

Rn_{and provide an extension of a result that we have establshed recently}

for the couple(`2_{, X)onR}n_{. We will also apply the Attouch–Brezis}

theo-rem to show the existence of optimal decomposition for these functionals for the conjugate couple.

1

Introduction

In this paper,(X0, X1)will be a regular Banach couple, i.e., X0and X1are both

Banach spaces which are linearly and continuously embedded in the same Hausdorff topological vector space and moreover the intersection X0∩X1 is

dense in both X0and X1. Several functionals such as L–, K– and E– functionals

are very important in the theory of real interpolation. A more or less detailed theory on these functionals can be found for example in the books Bergh and Löfström (1976); Brudnyi and Krugljak (1991). Another good reference is the book Bennett and Sharpley (1988). Given a couple of Banach spaces(X0, X1),

an element x ∈ X0+X1 and a positive parameter t, the K– functional is

defined by the formula K(t, x; X0, X1) = inf w∈X1  kx−wk_X 0+tkwkX1  .

The K– functional is at the center of the so–called K– method of real interpo-lation that is basically concerned with the construction of suitable families of

∗_{Department of Mathematics, Linköping University, SE–581 83 Linköping, Sweden. E-mail:}

japhet.niyobuhungiro@liu.se

†_{Department of Applied Mathematics, National University of Rwanda, P.O. Box: 56 Butare}

real interpolation spaces between X0and X1. The K– functional is a particular

case of the more general L– functional which is defined by Lp0,p1(t, x; X0, X1) =_w∈Xinf 1  kx−wkp0 X0+tkwk p1 X1  , (1)

for 1≤p0, p1<∞. The E– functional is basically seen as a distance functional

and is defined by the expression E(t, x; X0, X1) = inf

kwk_X1≤tkx−wkX0.

Definition 1.1 (Exact and near minimizers). We say that the element (which depends on x and t) wt∈X1is a near minimizer for the functional (1) if there exists

C>0 independent of x and t such that

kx−wtkpX00+tkwtk

p1

X₁ ≤CLp0,p1(t, x; X0, X1).

If C =1, then wt is called exact minimizer. If wt ∈ X1 is an exact minimizer, then

we will call

x=wt+ (x−wt), (2)

optimal decomposition for (1) corresponding to x.

Remark 1. It is important to note that the optimal decomposition does not always

exist.

Given an element x in X0+X1 and some parameter t > 0, we consider the

following L– functional Lp0,p1(t, x; X0, X1) =_x=xinf 0+x1  1 p0 kx0kp_X0₀+ t p1 kx1k p₁ X1  , (3) for 1≤ p0, p1<∞.

Problem. Suppose that the optimal decomposition x = x0,opt+x1,opt for K–

func-tional (respectively for L– and E– funcfunc-tionals) corresponding to the element x exists. Give a characterization of this decomposition. In other words, what are the mathemat-ical properties of x0,optand x1,opt.

For example, in the case of the L– functional (3), we want the mathematical properties of the decomposition x=x0,opt+x1,optsuch that

Lp0,p1(t, x; X0, X1) = 1 p0 x_0,opt p0 X0+ t p1 x_1,opt p1 X1.

In Section 2, we will provide some results which will be used in the subse-quent sections. In Section 3, we state and prove theorems on dual characteriza-tion of optimal decomposicharacteriza-tion for the E–, L– and K– funccharacteriza-tionals. In Seccharacteriza-tion 4, we characterize the geometry of optimal decomposition for the couple(`p, X)

on Rn. The result here will be an extension of a theorem that we have es-tablshed in (Niyobuhungiro, 2013a) for the couple(`2, X) onRn. In Section 5 we will apply the Attouch–Brezis theorem to show the existence of optimal decomposition for the L–, K– and E– functionals for the conjugate couple, and finally the last section outlines some final remarks and discussions.

2

Preliminaries

Let(X0, X1)be a regular Banach couple and let E be a Banach space. We will

consider two functions ϕ0 : X0 −→ R∪ {+∞}and ϕ1 : X1 −→ R∪ {+∞}

both assumed to be convex, lower semicontinuous and proper. Furthermore we consider their respective extensions ϕ0 : X0+X1 −→ R∪ {+∞} and

ϕ1: X0+X1−→R∪ {+∞}defined on the whole sum X0+X1as follows

ϕ0(u) =  ϕ0(u) if u∈X0; +∞ if u∈ (X0+X1) \X0. (4) and ϕ1(u) =  ϕ1(u) if u∈X1; +∞ if u∈ (X0+X1) \X1. (5)

For a function F defined on E with values in R∪ {+∞}, the set ∂F(x) will denote the subdifferential of F at x. i.e., the set defined by

∂F(x) ={y∈E∗: F(z) ≥F(x) +hy, z−xi, ∀z∈ E},

where E∗ is the space of all bounded linear functionals y : E−→ R. If ∂F(x)

is not empty, then F is said to be subdifferentiable at x.

The effective domain or simply domain of F will be denoted dom F and defined by

dom F={x∈E : F(x) < +∞}. We will call F proper if dom F6=∅.

Proposition 2.1. Any points y ∈ E∗and x ∈ E satisfy the inequality, well–known as Young–Fenchel inequality

F(x) +F∗(y) ≥hy, xi, (6)

where F∗is the conjugate of F. i.e., the function defined from E∗intoR∪ {+∞}by F∗(y) = sup

x∈dom E

{hy, xi −F(x)}, for y∈E∗. Equality in (6) holds if and only if y∈∂F(x):

y∈∂F(x) ⇐⇒F(x) +F∗(y) =hy, xi.

The following theorem establishes sufficient conditions for the infimal convo-lution(ϕ0⊕ϕ1)to be subdifferentiable.

Theorem 2.1(Subdifferentiability of infimal convolution). Let there be given x ∈ X0+X1 such that (ϕ0⊕ϕ1) (x) < +∞ and a0 ∈ X0, a1 ∈ X1 such that

a0+a1=x. Let the functions S and R be defined by

S(y) =ϕ0(a0−y) and R(y) =ϕ1(a1+y), y∈ (X0∩X1), (7)

be convex, lower semicontinuous and proper. Let ϕ∗₀and ϕ∗₁ be the respective conju-gate functions of ϕ0and ϕ1. Suppose that

(1) the sets dom S and dom R satisfy

[

λ≥0

λ(dom S−dom R) =X0∩X1 (8)

(2) The conjugate function S∗of S is given by

S∗(z) =



ϕ∗₀(−z) +hz, a0i if z∈X0∗;

+∞ if z∈ X₀∗+X₁∗

\X∗₀. (9) (3) The conjugate function R∗of R is given by

R∗(z) =



ϕ∗₁(z) +h−z, a1i if z∈X1∗;

+∞ if z∈ X₀∗+X₁∗

\X∗₁. (10) Then the function(ϕ0⊕ϕ1)is subdifferentiable on its domain in X0+X1.

In (Niyobuhungiro, 2013b), we have used the (Attouch and Brezis, 1986) theo-rem to prove Theotheo-rem 2.1. The theotheo-rem by H. Attouch and H. Brezis provides a sufficient condition for the conjugate of the sum of two convex lower semi-continuous and proper functions to be equal to the exact infimal convolution of their conjugates.

Theorem 2.2 (Attouch-Brezis). Let ϕ, ψ : E −→ R∪ {+∞}be convex, lower semicontinuous, and proper functions such that

[

λ≥0

λ(dom ϕ−dom ψ) is a closed vector space. (11)

Then

(ϕ+ψ)∗=ϕ∗⊕ψ∗on E∗, (12)

and, moreover, the infimal convolution is exact.

The following lemma, proof of which can be found in (Niyobuhungiro, 2013b), establishes a useful characterization of optimal decomposition for infimal con-volution.

Lemma 1 (Key lemma). Let F0 and F1 be convex and proper functions from a

Banach space E with values in R∪ {+∞}such that their infimal convolution F= (F0⊕F1)defined by

F(x) = (F0⊕F1) (x) =_x=xinf

0+x1

{F0(x0) +F1(x1)} (13)

is subdifferentiable at the point x ∈ dom(F0⊕F1). Then the decomposition x =

x0,opt+x1,optis optimal for (13) if and only if there exists y∗∈E∗that is dual to both

x0,optand x1,optwith respect to F0and F1respectively. i.e.,



F0 x0,opt
= hy∗, x0,opti −F₀∗(y∗);

F1 x1,opt
= hy∗, x1,opti −F1∗(y∗).

Lemma 2. Let 1 ≤ p0 < +∞ and let a0 ∈ X0 be given. Then the function S :

X0∩X1−→R∪ {+∞}defined by

S(y) = 1

p0

ka0−ykpX00,

is convex, proper and lower semicontinuous.

Proof. It is clear that the function S is convex and proper. Let us show that

it is lower semicontinuous. Suppose that uj
+∞_j=1 ∈ X0∩X1converges to y in

the norm of X0∩X1: lim j→+∞ uj−y _X 0∩X1 =0. (15)

From definition of the function S, we can write S(y) = 1 p0 ka0−ykpX00 = 1 p0 a₀−y+u_j−u_j p0 X0.

It follows from triangle inequality that S(y) ≤ 1 p0  a₀−u_j X0+ u_j−y X0 p₀ . (16) Since by definition u_j−y X0∩X1 =max  u_j−y X0, u_j−y X1  , it follows that u_j−y X0 ≤ u_j−y X0∩X1.

Putting this into (16) we have S(y) ≤ 1 p0  a₀−u_j X0+ u_j−y X0∩X1 p₀ . This is equivalent to S(y) ≤ 1 p0   p0S(uj)1/p0+ u_j−y X0∩X1 p₀ . It follows that [p0S(y)]1/p0 ≤ p0S(uj)1/p0+ u_j−y X0∩X1.

Taking limit, it follows from (15), that S(y) ≤lim inf

j→+∞ S uj
,

In a similar way, one can prove that

Lemma 3. Let 1 ≤ p1 < +∞, and let t > 0 and a1 ∈ X1 be given. Then the

function R : X0∩X1−→R∪ {+∞}defined by

R(y) = t

p1

ka1+ykp_X1₁, (17)

is convex, proper and lower semicontinuous.

Lemma 4. LetBX1(a1; t)denote the ball of X1centered at a1 ∈ X1and of radius t

and let R : X0∩X1−→R∪ {+∞}be the function defined by

R(u) =



0 if u∈ BX1(a1; t) ∩ (X0∩X1);

+∞ otherwise.

Then its conjugate function R∗is equal to

R∗(z) =

 _tkzk

X∗₁ +hz, a1i if z∈X1∗;

+∞ if z∈ X∗₀+X∗₁

\X₁∗. For the proof, we will need two claims.

Claim 2.1. Let us assume that z∈X₁∗. Then sup y∈X0∩X1 ka1−ykX₁ ≤t hz, yi = sup y∈X1 ka1−ykX₁ ≤t hz, yi

Proof of Claim 2.1. It is clear that

sup y∈X0∩X1 ka1−ykX1 ≤t hz, yi ≤ sup y∈X1 ka1−ykX1 ≤t hz, yi.

We only need to show that sup y∈X0∩X1 ka1−ykX1 ≤t hz, yi ≥ sup y∈X1 ka1−ykX1 ≤t hz, yi.

In this end, it is enough to prove that for y∈X1such thatka1−ykX1 ≤t, we

have that sup y∈X0∩X1

ka1−ykX1 ≤t

hz, yi ≥ hz, yi.

Put a1−y=u and take the an element

 1− 1 n  u in X1. Then  1− 1 n  u _X 1 =  1− 1 n  kuk_X 1 ≤  1−1 n  t.

By denseness of X0∩X1in X1, take un ∈X0∩X1and a1,n∈ X0∩X1such that un−  1− 1 n  u _X 1 ≤ t 2n and ka1−a1,nkX1 ≤ t 2n. Let us pick yn =a1,n−un. We must check that

lim n→+∞ky−ynkX1 =0 and ka1−ynkX1 ≤t. Indeed, ky−ynkX1 =ka1−u− (a1,n−un)kX1 ≤ ka1−a1,nkX1+kun−ukX1 =ka1−a1,nk_X1+ un −  1−1 n  u+  1−1 n  u−u _X 1 ≤ ka1−a1,nk_X1+ un −  1−1 n  u _X 1 +  1−1 n  u−u _X 1 ≤ t 2n+ t 2n+ t n = 2t n. Therefore lim n→+∞ky−ynkX1 =0.

We also have that

ka1−ynkX1 =ka1− (a1,n−un)kX1 ≤ ka1−a1,nkX1+ un −  1− 1 n  u+  1−1 n  u _X 1 ≤ ka1−a1,nk_X1+ un −  1− 1 n  u _X 1 +  1−1 n  u _X 1 ≤ t 2n+ t 2n+  1− 1 n  t=t. Therefore ka1−ynkX1 ≤t.

Then we conclude that sup y∈X0∩X1

ka1−ykX1 ≤t

hz, yi ≥ lim

Claim 2.2. Suppose that z∈ X∗₀+X∗₁
\X₁∗. Then sup y∈X0∩X1 ka1−ykX1 ≤t hz, yi = +∞

Proof of Claim 2.2. The fact that z ∈ X₀∗+X₁∗

\X∗₁ means z is defined on X0∩X1and is unbounded on the setBX1(0; 1) ∩ (X0∩X1). In fact, if it were

bounded onBX1(0; 1) ∩ (X0∩X1), then it would be possible to extend it as a

bounded linear functional on X1. Put the setΩ=

n

y∈ X0∩X1:ka1−ykX1 ≤t

o . We need to prove that

sup

y∈Ωhz, yi = +∞.

Notice that hz, yiis well defined because z ∈ (X0∩X1)∗ = X0∗+X1∗
.

As-sume by contradiction that sup

y∈Ωhz, yi =C< +∞.

Choose a1,ε∈X0∩X1such thatka1−a1,εkX1 <

t

2 and put the set

Ωt 2 =  u∈X0∩X1:ku−a1,εkX1 ≤ t 2  . We have in that case, that

ka1−ukX1 =ka1−a1,ε+a1,ε−ukX1 ≤ ka1−a1,εkX1+ka1,ε−ukX1 ≤ t 2+ t 2 =t, and thatΩt

2 ⊂Ω. Condider the setBX1(0;

t

4) ∩ (X0∩X1)and take an element

v∈ BX1(0; t 4) ∩ (X0∩X1). Note that u= (2v+a1,ε) ∈Ωt 2. In fact k2v+a1,ε−a1,εk_X1 =2kvkX1 ≤2 t 4 = t 2. For such an element v we have that

hz, vi =Dz,−a1,ε 2 E +1 2hz, 2v+a1,εi ≤C0+ 1 2C=C1< +∞, where C0and C1are some constants. We conclude that

sup

v∈B_X1(0;1)∩(X0∩X1)

hz, vi ≤4C1

t < +∞.

This is a contradiction of the fact that z is unbounded onBX1(0; 1) ∩ (X0∩X1).

Therefore sup

Proof of Lemma 4. Let z∈X₁∗. By defintion of the function R, we have that R∗(z) = sup

y∈X0∩X1

ka1−ykX1 ≤t

hz, yi.

From Claim 1, we have that R∗(z) = sup y∈X1 ka1−ykX1 ≤t hz, yi = sup y∈X1 ka1−ykX1 ≤t hz, y−a1i + hz, a1i. Therefore R∗(z) =tkzk_X∗ 1 +hz, a1i. If z∈ X₀∗+X₁∗

\X∗₁, then it follows from Claim 2 that sup

y∈X0∩X1

ka1−ykX1 ≤t

hz, yi = +∞.

This completes the proof of Lemma 4.

Lemma 5. Let a0∈ X0and a1∈ X1be given. Let functions S and R be defined on

X0∩X1by S(y) = 1 p0 ka0−ykp_X0₀ and R(y) = t p1 ky+a1k p₁ X1, (18)

and let p0_i, i=0, 1 be such that _p1_i + _p10

i =1. Then

(1) In the case 1<p0, p1< +∞, we have that

S∗(z) = ( 1 p0₀ kzk p0₀ X∗ 0 +hz, a0i if z∈X ∗ 0; +∞ if z∈ X₀∗+X₁∗
\X₀∗. (19) and R∗(z) = ( t p0 1 z_t p0₁ X∗ 1 +h−z, a1i if z∈X ∗ 1; +∞ if z∈ X₀∗+X₁∗
\X∗₁. (20) (2) In the case p0= p1=1, we have that

S∗(z) = ( hz, a0i if kzkX₀∗ ≤1, z∈ BX∗₀(0; 1); +∞ if z∈ X₀∗+X₁∗
\BX∗₀(0; 1). (21) and R∗(z) = ( h−z, a1i if kzkX₁∗ ≤t, z∈ BX∗₁(0; t); +∞ if z∈ X∗₀+X∗₁
\BX∗₁(0; t). (22)

Proof. We will prove (19) and (21) only. The case of (20) and (22) is similar.

(1) Case 1< p0< +∞.

Assume that z ∈ X∗₀. Let us define the function ϕ : R −→ R+ in the following way

t7−→ϕ(t) = 1

p0

|t|p0_.

It is clear that this function is convex, lower semicontinuous, proper and even. The dual ϕ∗of ϕ is by definition

ϕ∗(t∗) =sup t∈R  t∗· t− 1 p0 |t|p0  .

The supremum is attained at t∈R satisfying t∗=|t|p0−1_{sgn(t)and we}

get

ϕ∗(t∗) = 1

p0₀|t

∗_|p0₀

. (23)

We define another function T : X0−→R+ defined by

T(y) = 1

p0

kykp0

X0.

Let us calculate its conjugate. T∗(z) =sup y∈X0 n hz, yi −ϕ(kyk_X0) o . This can be rewritten as follows

T∗(z) =sup t≥0 sup kyk_X0=t {hz, yi −ϕ(t)} =sup t≥0    t sup kyk_X0=1 hz, yi −ϕ(t)    =sup t≥0 n tkzk_X∗ 0 −ϕ(t) o .

Since ϕ is an even function then T∗(z) =sup t∈R n tkzk_X∗ 0 −ϕ(t) o = ϕ∗(kzk_X∗ 0). Hence T∗(z) =ϕ∗(kzk_X∗ 0).

From the definition of conjugate of ϕ (see (23)), we get T∗(z) = 1

p0₀kzk

p0₀ X∗₀.

Let us now calculate S∗. We have that S∗(z) = sup y∈X0∩X1 {hz, yi −S(y)} = sup y∈X0∩X1  hz, yi − 1 p0 ky−a0kp_X0₀  . From definition of the function T, this can be written as

S∗(z) = sup

y∈X0∩X1

{hz, yi −T(y−a0)}.

Since our couple(X0, X1)is regular, i.e., X0∩X1is dense in both X0and

X1then as T is a continuous function with respect to the norm of X0, we

have that S∗(z) = sup y∈X0 {hz, y−a0i −T(y−a0)} + hz, a0i. This is equal to S∗(z) =T∗(z) +hz, a0i. (24) Whence S∗(z) = 1 p0₀kzk p0₀ X₀∗+hz, a0i.

Now let us assume that z∈ X∗₀+X∗₁

\X₀∗.

In this case z is unbounded onBX0(0; 1) ∩ (X0∩X1). Therefore we must

have that sup_y∈X0{hz, yi} = +∞ (see Claim 2.2). It follows that

S∗(z) = +∞,

(2) Case p0=1.

In this case the function T becomes T(y) =kyk_X

0.

Its conjugate function is by definition T∗(z) =sup y∈X0 n hz, yi − kyk_X 0 o .

By the same reasoning as above, we have that

T∗(z) =sup t≥0 sup kyk_X0=t {hz, yi −t} =sup t≥0    t sup kyk_X0=1 hz, yi −t    Equivalently T∗(z) =sup t≥0 tnkzk_X∗ 0 −1 o

It is clear that T∗(z) = ( 0 if kzk_X∗ 0 ≤1, z∈ BX ∗ 0(0; 1); +∞ if z∈ X₀∗+X₁∗
\B_X∗ 0(0; 1).

It follows from (24), that S∗(z) = ( hz, a0i if kzkX₀∗≤1, z∈ BX∗ 0(0; 1); +∞ if z∈ X₀∗+X₁∗
\B_X∗ 0(0; 1).

3

Dual characterization of optimal decomposition

for general Banach couples

In this section we state and prove theorems on dual characterization of optimal decomposition for the E–, L– and K– functionals. The idea of the proofs can be summarized in three steps: First we consider the concerned functional as an infimal convolution and reformulate the infimal convolution at hand as a minimization of a sum of two specific functions S and R defined on the intersection X0∩X1 of the couple. Next we prove that it is subdifferentiable

by verifying requirements in Theorem 2.1. Finally use the characterizaton of its optimal decomposition by applying Key Lemma for specific functions.

3.1

Optimal decomposition for the E– functional

Given x∈ X0+X1, 1≤ p0< +∞ and a parameter t>0, the E– functional is

defined by Ep₀(t, x; X0, X1) = inf kx1k_X1≤t 1 p0 kx−x1kpX00. (25) 3.1.1 Case: p0>1

For p0>1, we can express the E– functional (25) as the infimal convolution

Ep0(t, x; X0, X1) = (ϕ0⊕ϕ1) (x),

where ϕ0and ϕ1are functions defined on the sum X0+X1by

ϕ0(u) = ( 1 p0 kuk p0 X0 if u∈X0; +∞ if u∈ (X0+X1) \X0. (26)

and ϕ1is defined by ϕ1(u) =  0 if u∈ BX1(0; t); +∞ if u∈ (X0+X1)\BX1(0; t). (27) whereBX₁(0; t)is the ball of X1of radius t and centered at 0:

B_X1(0; t) =nx∈X0+X1: kxkX₁ ≤t

o .

In this case the functions ϕ0 : X0−→ R∪ {+∞}and ϕ1 : X1 −→R∪ {+∞}

are defined by ϕ0(u) = 1 p0 kukp0 X0 and ϕ1(u) =  0 if u∈ BX1(0; t); +∞ if u∈X1\BX1(0; t).

Let us fix a0∈ X0and a1∈X1such that x=a0+a1. Then we can rewrite the

E– functional (25) as follows Ep0(t, x; X0, X1) = inf ka1−yk_X1≤t 1 p0 ka0+ykXp00, y∈X0∩X1. Equivalently, Ep0(t, x; X0, X1) =_y∈Xinf 0∩X1 {S(y) +R(y)},

where S, R : X0∩X1−→R∪ {+∞}are functions defined by

S(y) = 1 p0 ka0+ykpX00 and R(y) =  0 if y∈ B_X1(a1; t) ∩ (X0∩X1); +∞ otherwise. (28) The following theorem gives the dual characterization of optimal decomposi-tion for the Ep0– functional. i.e., the decomposition x =x0,opt+x1,opt, where

x0,opt ∈X0and x1,opt∈ BX1(0, t)such that

Ep0(t, x; X0, X1) = 1 p0 x−x1,opt p0

X0, x1,opt∈ BX1(0, t), x0,opt=x−x1,opt.

Theorem 3.1. Let1 < p0<∞. Then the decomposition x= x0,opt+x1,opt, where

x1,opt ∈ BX₁(0, t), is optimal for the Ep0– functional if and only if there exists y∗ ∈

X₀∗∩X∗₁such that:    1 p0 x_0,opt p0 X0 = hy∗, x0,opti − 1 p₀0 ky∗k p0₀ X∗₀; hy∗, x1,opti =tky∗k_X∗ 1.

Proof. We have shown that for fixed a0 ∈ X0 and a1 ∈ X1 such that x =

a0+a1, the Ep0– functional can be defined by

Ep0(t, x; X0, X1) = inf

y∈X0∩X1

where functions S and R are defined by (28). Moreover Ep0(t, x; X0, X1) = (ϕ0⊕ϕ1) (x),

where functions ϕ0 and ϕ1 are defined by (26) and (27) respectively. It

fol-lows from Lemma 2, that the function S is convex, lower semicontinuous and proper. The function R is convex and lower semicontinuous as an indicator function of a convex and closed setBX₁(a1; t) ∩ (X0∩X1). It is proper because

BX1(a1; t) ∩ (X0∩X1) 6= ∅ because the space X0∩X1is dense in X1.

More-over, since dom S=X0∩X1and dom R= BX1(a1; t) ∩ (X0∩X1)then we have

that

[

λ≥0

λ(dom S−dom R) =X0∩X1. (29)

The conjugate function ϕ₀∗of ϕ0is defined on X0∗and is given by

ϕ₀∗(z) = 1 p0₀kzk p0₀ X∗ 0, ∀z∈X ∗ 0.

The conjugate function S∗ of S can be obtained from Lemma 5 where z is replaced by−z. We have that

S∗(z) = ( 1 p0 0kzk p0₀ X∗₀ +h−z, a0i if z∈X ∗ 0; +∞ if z∈ X₀∗+X₁∗
\X∗₀. It is clear that S∗(−z) =  ϕ∗₀(−z) +hz, a0i if z∈ X0∗; +∞ if z∈ X∗₀+X∗₁
\X₀∗. (30) The conjugate ϕ∗₁of ϕ1is defined on X1∗and is given by

ϕ₁∗(z) = sup x∈B_X1(0;t) hz, xi =tkzk_X∗ 1 , ∀z∈X ∗ 1.

Recall that, from Lemma 4, that R∗(z) =  _tkzk X∗₁ +hz, a1i if z∈X∗1; +∞ if z∈ X₀∗+X∗₁
\X₁∗. It is clear that R∗(−z) =  ϕ∗₁(z) +h−z, a1i if z∈ X1∗; +∞ if z∈ X∗₀+X∗₁
\X₁∗. (31) By (29), (30) and (31), we conclude from Theorem 2.1 that Ep0 is

subdifferen-tiable. It is clear that the functions ϕ0and ϕ1are convex and proper. Therefore

we conclude by Key Lemma that the decomposition x = x0,opt+x1,opt, where

x1,opt ∈ BX1(0, t), is optimal for the Ep0- functional if and only if there exists

y∗∈X₀∗∩X∗₁such that:    1 p0 x_0,opt p0 X₀ = hy∗, x0,opti − 1 p0 0 ky∗k p0₀ X₀∗; hy∗, x1,opti =tky∗k_X∗ 1.

3.1.2 Case: p0=1

For the case p0=1 we have the followimg E– functional

E(t, x; X0, X1) = inf kx1k_X1≤t kx−x1kX0. (32) In this case E(t, x; X0, X1) = (ϕ0⊕ϕ1) (x), where ϕ0(u) =  _k uk_X 0 if u∈X0; +∞ if u∈ (X0+X1) \X0.

and ϕ1is the function defined on the sum X0+X1as in (27). The functions

ϕ0: X0−→R∪ {+∞}and ϕ1: X1−→R∪ {+∞}are defined by

ϕ0(u) =kukX0 and ϕ1(u) =



0 if u∈ B_X1(0; t);

+∞ if u∈X1\BX₁(0; t).

The decomposition x=x0,opt+x1,opt, where x0,opt ∈X0and x1,opt ∈ BX1(0, t)

is optimal for the E– functional (32) if E(t, x; X0, X1) =

x−x1,opt

_X

0, x1,opt∈ BX1(0, t), x0,opt =x−x1,opt.

Theorem 3.2. The decomposition x = x0,opt+x1,opt, where x1,opt ∈ BX₁(0, t), is

optimal for the E– functional (32) if and only if there exists y∗ ∈ X₀∗∩X∗₁ such that

ky∗k_X∗ 0 ≤1 and: ( x0,opt _X 0 = hy∗, x0,opti; hy∗, x1,opti =tky∗k_X∗ 1 .

Proof. For given a0∈X0and a1∈X1such that x=a0+a1, the E– functional

can be defined as follows E(t, x; X0, X1) = inf

y∈X0∩X1

{S(y) +R(y)},

where S, R : X0∩X1−→R∪ {+∞}are functions defined by

S(y) =ka0+ykX0 and R(u) =



0 if u∈ BX1(a1; t) ∩ (X0∩X1);

+∞ otherwise.

It is clear from Lemma 2, that the function S is convex, lower semicontinuous and proper. The function R is the same as in the previous case and by the same reason, we have

[

λ≥0

The conjugate function ϕ₀∗of ϕ0is defined on X0∗and is given by ϕ₀∗(z) = sup u∈X0  hz, ui − kuk_X0=  _{0 if z∈ B} X∗₀(0; 1); +∞ if z∈X₀∗\B_X∗ 0(0; 1).

The conjugate functions S∗ of S can be obtained from Lemma 5 where z is replaced by−z. We have that

S∗(z) = ( h−z, a0i if z∈ BX∗ 0(0; 1); +∞ if z∈ X₀∗+X₁∗
\BX∗₀(0; 1). It is clear that S∗(−z) =  ϕ∗₀(−z) +hz, a0i if z∈ X0∗; +∞ if z∈ X∗₀+X∗₁
\X₀∗. (34) and as in the previous case

R∗(−z) =



ϕ∗₁(z) +h−z, a1i if z∈ X1∗;

+∞ if z∈ X∗₀+X∗₁

\X₁∗. (35) By (33), (34) and (35), we conclude from Theorem 2.1 that the E- functional is subdifferentiable. Since the functions ϕ0 and ϕ1 are convex and proper,

then we conclude by Key Lemma the decomposition x = x0,opt+x1,opt, where

x1,opt ∈ BX1(0, t), is optimal for the E– functional (32) if and only if there

exists y∗∈ X₀∗∩X∗₁such thatky∗k_X∗

0 ≤1 and: ( x_0,opt X0 = hy∗, x0,opti; hy∗, x1,opti =tky∗k_X∗ 1.

3.2

Optimal decomposition for the L– functional

Let x∈X0+X1and let t>0 be a fixed parameter. We consider the following

L– functional Lp0,p1(t, x; X0, X1) =_x=xinf 0+x1  1 p0 kx0kpX00+ t p1 kx1kpX11  , (36) where 1≤p0, p1<∞.

We are interested in characterization of optimal decomposition for this L– functional, i.e., characterization of x0,opt ∈ X0and x1,opt ∈ X1such that x =

x0,opt+x1,optand

Lp0,p1(t, x; X0, X1) =  1 p0 x_0,opt p0 X0+ t p1 x_1,opt p1 X1  . It can be written as the infimal convolution

where functions ϕ0and ϕ1are both defined on the sum X0+X1as follows ϕ0(u) = ( ₁ p0 kuk p0 X0 if u∈X0; +∞ if u∈ (X0+X1) \X0. (37) and ϕ1(u) = ( _t p1 kuk p1 X1 if u∈X1; +∞ if u∈ (X0+X1) \X1. (38) In this case the functions ϕ0 : X0−→ R∪ {+∞}and ϕ1 : X1 −→R∪ {+∞}

are defined by ϕ0(u) = 1 p0 kukp0 X0 and ϕ1(u) = t p1 kukp1 X1.

The following theorem gives a characterization of optimal decomposition for L– functional.

Theorem 3.3(Optimal decomposition for Lp0,p1– functional). Let 1<p0, p1<

∞. Then the decomposition x=x0,opt+x1,optis optimal for the L– functional (36) if

and only if there exists an element y∗∈X∗₀∩X₁∗such that:

   1 p0 x_0,opt p0 X0 = hy∗, x0,opti − 1 p0₀ ky∗k p0₀ X₀∗; t p1 x1,opt p1 X₁ = hy∗, x1,opti − t p₁0 y∗ t p0₁ X₁∗.

Proof. For given a0∈X0and a1∈X1such that x=a0+a1, we can define the

Lp0,p1- functional as follows

Ep0(t, x; X0, X1) =_y∈Xinf 0∩X1

{S(y) +R(y)},

where functions S and R are defined as in (18). Moreover Lp0,p1(t, x; X0, X1) = (ϕ0⊕ϕ1) (x),

where functions ϕ0 and ϕ1 are defined by (37) and (38) respectively. From

Lemma 2 and Lemma 3, the functions S and R are convex, proper and lower semicontinuous and since dom S=dom R=X0∩X1, we have that

[

λ≥0

λ(dom S−dom R) =X0∩X1. (39)

The respective conjugate functions S∗and R∗of S and R are given in Lemma 5. We have that S∗(z) = ( 1 p0₀kzk p0₀ X∗₀ +hz, a0i if z∈X ∗ 0; +∞ if z∈ X∗₀+X∗₁
\X₀∗. and R∗(z) = ( t p0 1 z_t p0₁ X∗₁ +h−z, a1i if z∈X ∗ 1; +∞ if z∈ X₀∗+X₁∗
\X∗₁.

The conjugate functions ϕ∗₀ of ϕ0 and ϕ1∗ of ϕ1 are defined on X0∗ and X1∗

respectively and are given by

ϕ₀∗(z) = 1 p0₀kzk p0 0 X∗₀, ∀z∈X ∗ 0, and ϕ₁∗(z) = t p0₁ z t p0₁ X₁∗, ∀z∈X ∗ 1. It is clear that S∗(z) =  ϕ∗₀(−z) +hz, a0i if z∈X0∗; +∞ if z∈ X₀∗+X₁∗
\X∗₀. (40) and R∗(z) =  ϕ∗₁(z) +h−z, a1i if z∈X1∗; +∞ if z∈ X₀∗+X₁∗
\X∗₁. (41) By (39), (40) and (41), we conclude from Theorem 2.1 that L– functional is sub-differentiable. Since the functions ϕ0and ϕ1are convex and proper, then we

conclude by Key Lemma that the decomposition x = x0,opt+x1,opt is optimal

for the L– functional (36) if and only if there exists an element y∗ ∈ X∗₀∩X₁∗

such that:    1 p0 x_0,opt p0 X0 = hy∗, x0,opti − 1 p0₀ ky∗k p0₀ X₀∗; t p1 x1,opt p1 X1 = hy∗, x1,opti − t p0 1 y∗ t p0₁ X₁∗.

Some important particular cases will be discussed in the next section.

3.2.1 Important particular cases 3.2.2 Case1<p0< +∞ and p1=1

Consider the L– functional Lp0,1(t, x; X0, X1) =_x=xinf 0+x1  1 p0 kx0kp_X0₀+tkx1kX1  , (42)

where 1<p0< +∞. Let ϕ0and ϕ1be functions defined on the sum X0+X1

as follows ϕ0(u) = ( ₁ p0 kuk p0 X0 if u∈X0; +∞ if u∈ (X0+X1) \X0. (43)

and ϕ1(u) =  tkuk_X 1 if u∈X1; +∞ if u∈ (X0+X1) \X1. (44)

Then the L– functional (42) can be written as the following infimal convolution Lp0,1(t, x; X0, X1) = (ϕ0⊕ϕ1) (x).

In this case the functions ϕ0 : X0−→ R∪ {+∞}and ϕ1 : X1 −→R∪ {+∞}

are defined by

ϕ0(u) = 1

p0

kukp0

X0 and ϕ1(u) =tkukX1.

Theorem 3.4 (Optimal decomposition for Lp0,1– functional). Let 1 < p0 <

+∞. Then the decomposition x= x0,opt+x1,opt is optimal for the Lp0,1– functional

if and only if there exists y∗ ∈X₀∗∩X∗₁such thatky∗k_X∗

1 ≤t and    1 p0 x_0,opt p0 X0 = hy∗, x0,opti − 1 p₀0 ky∗k p0₀ X∗₀; t x_1,opt X1 = hy∗, x1,opti.

Proof. Given a0∈X0and a1∈ X1such that x =a0+a1, the Lp0,1– functional

can be defined as

Lp0,1(t, x; X0, X1) =_y∈Xinf 0∩X1

{S(y) +R(y)}, where functions S and R are defined on X0∩X1by

S(y) = 1

p0

ka0−ykp_X0₀ and R(y) =tky+a1kX1.

Moreover

Lp0,1(t, x; X0, X1) = (ϕ0⊕ϕ1) (x),

where functions ϕ0 and ϕ1 are defined by (43) and (44) respectively. From

Lemma 2 and Lemma 3, the functions S and R are convex, proper and lower semicontinuous and are such that

[

λ≥0

λ(dom S−dom R) =X0∩X1. (45)

The respective conjugate functions S∗and R∗of S and R are given in Lemma 5. We have that S∗(z) = ( 1 p0₀kzk p0₀ X∗ 0 +hz, a0i if z∈X ∗ 0; +∞ if z∈ X∗₀+X∗₁
\X₀∗. and R∗(z) = ( h−z, a1i if kzkX₁∗≤t, z∈ BX∗₁(0; t); +∞ if z∈ X₀∗+X₁∗
\B_X∗ 1(0; t).

The conjugate functions ϕ∗₀of ϕ0and ϕ∗1of ϕ1are defined on X∗0 and X∗1and are equal to ϕ₀∗(z) = 1 p0₀kzk p0 0 X∗₀, ∀z∈X ∗ 0, and ϕ₁∗(z) =  _{0 if z∈ B} X∗ 1(0; t); +∞ if z∈X₁∗\BX∗ 1(0; t). It is clear that S∗(z) =  ϕ∗₀(−z) +hz, a0i if z∈X0∗; +∞ if z∈ X₀∗+X₁∗
\X∗₀. (46) and R∗(z) =  ϕ∗₁(z) +h−z, a1i if z∈X1∗; +∞ if z∈ X₀∗+X₁∗
\X∗₁. (47) By (45), (46) and (47), we conclude from Theorem 2.1 that Lp0,1- functional is

subdifferentiable. Since the functions ϕ0and ϕ1are convex and proper, then

we conclude by Key Lemma the decomposition x = x0,opt+x1,opt is optimal

for the Lp0,1- functional if and only if there exists y∗ ∈ X

∗ 0 ∩X1∗ such that ky∗k_X∗ 1 ≤t and    1 p0 x0,opt p0 X0 = hy∗, x0,opti − 1 p0₀ ky∗k p0₀ X₀∗; t x1,opt _X 1 = hy∗, x1,opti. 3.2.3 Case p0=1 and 1<p1< +∞

Consider the L– functional L1,p1(t, x; X0, X1) =_x=xinf 0+x1  kx0kX0+ t p1 kx1kXp1₁  , (48)

where 1<p1< +∞. Let ϕ0and ϕ1be functions defined on the sum X0+X1

as follows ϕ0(u) =  _k uk_X 0 if u∈X0; +∞ if u∈ (X0+X1) \X0. (49) and ϕ1(u) = ( _t p1 kuk p1 X1 if u∈X1; +∞ if u∈ (X0+X1) \X1. (50)

Then the L– functional (48) can be written as the following infimal convolution L1,p1(t, x; X0, X1) = (ϕ0⊕ϕ1) (x).

In this case the functions ϕ0 : X0−→ R∪ {+∞}and ϕ1 : X1 −→R∪ {+∞}

are defined by ϕ0(u) =kukX0 and ϕ1(u) = t p1 kukp1 X1.

Theorem 3.5 (Optimal decomposition for L1,p1-functional). Let 1 < p1 <

+∞. Then the decomposition x= x0,opt+x1,opt is optimal for the L1,p1– functional

if and only if there exists y∗ ∈X₀∗∩X∗₁such thatky∗k_X∗

0 ≤1 and    x0,opt _X 0 = hy∗, x0,opti; t p1 x1,opt p₁ X1 = hy∗, x1,opti − t p₀0 z_t p0₁ X∗ 1.

Proof. Given a0∈X0and a1∈ X1such that x =a0+a1, the L1,p1– functional

can be defined by

L1,p1(t, x; X0, X1) =_y∈Xinf 0∩X1

{S(y) +R(y)}, where functions S and R are defined on X0∩X1by

S(y) =ka0−ykX0 and R(y) = t p1 ky+a1kp_X1₁, Moreover Lp0,1(t, x; X0, X1) = (ϕ0⊕ϕ1) (x),

where functions ϕ0 and ϕ1 are defined by (49) and (50) respectively. From

Lemma 2 and Lemma 3, the functions S and R are convex, proper and lower semicontinuous and are such that

[

λ≥0

λ(dom S−dom R) =X0∩X1. (51)

The respective conjugate functions S∗and R∗of S and R are given in Lemma 5. We have that S∗(z) = ( hz, a0i if kzkX∗₀ ≤1, z∈ BX∗ 0(0; 1); +∞ if z∈ X₀∗+X₁∗
\B_X∗ 0 (0; 1). and R∗(z) = ( t p0 1 z_t p0₁ X∗₁ +h−z, a1i if z∈X ∗ 1; +∞ if z∈ X₀∗+X₁∗
\X∗₁.

The conjugate functions ϕ∗₀ of ϕ0 and ϕ1∗ of ϕ1 are defined on X0∗ and X1∗

respectively and are equal to

ϕ₀∗(z) =

 _{0 if z∈ B}

X∗₀(0; 1);

+∞ if z∈X₀∗\BX∗ 0(0; 1).

and ϕ₁∗(z) = t p0₁ z t p0₁ X₁∗, ∀z∈X ∗ 1. It is clear that S∗(z) =  ϕ∗₀(−z) +hz, a0i if z∈X0∗; +∞ if z∈ X₀∗+X₁∗
\X∗₀. (52) and R∗(z) =  ϕ∗₁(z) +h−z, a1i if z∈X1∗; +∞ if z∈ X₀∗+X₁∗
\X∗₁. (53) By (51), (52) and (53), we conclude from Theorem 2.1 that L1,p1– functional

is subdifferentiable. Since the functions ϕ0 and ϕ1 are convex and proper,

then we conclude by Key Lemma that the decomposition x = x0,opt+x1,opt is

optimal for the L1,p1- functional if and only if there exists y∗ ∈X

∗ 0∩X∗1 such thatky∗k_X∗ 0 ≤1 and    x_0,opt X0 = hy∗, x0,opti; t p1 x1,opt p₁ X1 = hy∗, x1,opti − t p0₀ z_t p0₁ X∗ 1.

3.2.4 Optimal decomposition for the K– functional

Consider the K– functional K(t, x; X0, X1) = inf x=x0+x1  kx0kX0+tkx1kX1  . It can be expressed as the following infimal convolution

K(t, x; X0, X1) = (ϕ0⊕ϕ1) (x),

where functions ϕ0and ϕ1are defined on the sum X0+X1as follows

ϕ0(u) =  _k uk_X 0 if u∈X0; +∞ if u∈ (X0+X1) \X0. (54) and ϕ1(u) =  tkuk_X 1 if u∈X1; +∞ if u∈ (X0+X1) \X1. (55)

In this case the functions ϕ0 : X0−→ R∪ {+∞}and ϕ1 : X1 −→R∪ {+∞}

are defined by

ϕ0(u) =kukX0 and ϕ1(u) =tkukX1.

The decomposition x = x0,opt+x1,opt, where x0,opt ∈ X0 and x1,opt ∈ X1 is

optimal for K– functional if K(t, x; X0, X1) = x_0,opt X0+t x_1,opt X1.

Theorem 3.6. The decomposition x=x0,opt+x1,optis optimal for the K– functional

if and only if there exists y∗ ∈X₀∗∩X∗₁such thatky∗k_X∗

0 ≤1,ky∗kX∗1 ≤t and ( x_{0,opt X} 0 = hy∗, x0,opti; t x_1,opt X1 = hy∗, x1,opti.

Proof. For given a0∈X0and a1∈X1such that a0+a1=x, the K– functional

is equal to

K(t, x; X0, X1) = inf y∈X0∩X1

{S(y) +R(y)}, where functions S and R are defined on X0∩X1by

S(y) =ka0−ykX0 and R(y) =tky+a1kX1,

Moreover

K(t, x; X0, X1) = (ϕ0⊕ϕ1) (x),

where functions ϕ0 and ϕ1 are defined by (54) and (55) respectively. From

Lemma 2 and Lemma 3, the functions S and R are convex, proper and lower semicontinuous and are such that

[

λ≥0

λ(dom S−dom R) =X0∩X1. (56)

The respective conjugate functions S∗and R∗of S and R are given in Lemma 5. We have that S∗(z) = ( hz, a0i if kzkX₀∗ ≤1, z∈ BX∗₀(0; 1); +∞ if z∈ X₀∗+X₁∗
\B_X∗ 0 (0; 1). and R∗(z) = ( h−z, a1i if kzkX₁∗≤t, z∈ BX∗₁(0; t); +∞ if z∈ X₀∗+X₁∗
\B_X∗ 1(0; t).

The conjugate functions ϕ∗₀of ϕ0and ϕ∗1of ϕ1are defined on X∗0 and X∗1and

are equal to ϕ₀∗(z) =  _{0 if z∈ B} X∗₀(0; 1); +∞ if z∈X₀∗\B_X∗ 0(0; 1). and ϕ₁∗(z) =  _{0 if z∈ B} X∗₁(0; t); +∞ if z∈X₁∗\B_X∗ 1(0; t). It is clear that S∗(z) =  ϕ∗₀(−z) +hz, a0i if z∈X0∗; +∞ if z∈ X₀∗+X₁∗
\X∗₀. (57)

and R∗(z) =  ϕ∗₁(z) +h−z, a1i if z∈X1∗; +∞ if z∈ X₀∗+X₁∗
\X∗₁. (58) By (56), (57) and (58), we conclude from Theorem 2.1 that K– functional is subdifferentiable. Since the functions ϕ0and ϕ1are convex and proper, then

we conclude by Key Lemma the decomposition x=x0,opt+x1,optis optimal for

the K- functional if and only if there exists y∗ ∈X₀∗∩X∗₁such thatky∗k_X∗ 0 ≤1, ky∗k_X∗ 1 ≤t and ( x_0,opt X0 = hy∗, x0,opti; t x1,opt _X 1 = hy∗, x1,opti.

4

Geometry of optimal decomposition for the

Cou-ple

(`

, X

)

on

R

4.1

Introduction and Formulation of the problem

We are interested in an extension of a theorem that we have establshed in (Niyobuhungiro, 2013a) for the couple(`2_{, X)onR}n_{. We consider the couple}

(`p, X), 1< p < +∞, in finite dimensional case. More precisely, we consider

the spaceRn_{, with the normkxk}

`p and some Banach space X onRn equipped

with norm k·k_X. We will denote by X∗ the dual space of X defined by the dual norm

kyk_X∗=sup{hy, xi: x∈X; kxk_X ≤1},

where h· , ·iis the dual product between X∗and X. We will consider the L– functional Lp,1(t, x;`p, X) = inf x=x0+x1  1 p kx0k p `p+tkx1kX  . (59)

In subsection 4.2 we present a result (Theorem 4.1) which gives the character-ization of optimal decomposition for the L– functional (59). Subsection (4.3) presents the particular case where p = 2 in order to illustrate that, indeed Theorem 4.1 generalizes our theorem in (Niyobuhungiro, 2013a). Finally sub-section 4.4 presents a geometrical illustration of Theorem 4.1 for the case p=3 in the plane.

4.2

Optimal decomposition for L

(

t, x;

`

, X

)

– functional

In this subsection we use Key Lemma to characterize the optimal decomposi-tion for the L– funcdecomposi-tional (59). In terms of infimal convoludecomposi-tion, the L-

func-tional (59) can be expressed as Lp,1(t, x;`p, X) = (F0⊕F1) (x), where F0(u) = 1 pkuk p `p and F1(u) =tkukX. (60)

We recall that 1 < p < ∞. Thus we have that ∂F₀(u) consists of the single vector

∂F0(u) = ∇F0(u) =

n

|u|p−1sgn(u)o. (61) Let us denote by tB_X∗ the ball of X∗of radius t with center at the origin:

tBX∗={y∈Rn: kyk_X∗ ≤t},

and let p0 be the conjugate of p. i.e., 1_p+ _p10 = 1. Then we can obtain the

conjugate functions of F0and F1from Lemma 5 as follows:

F₀∗(y) = 1 p0kyk p0 `p0 and F ∗ 1(y) =  0 if y∈tBX∗; +∞ else. (62)

Theorem 4.1. Let F0and F1be defined onRn by (60). Let us define the setΩ by

Ω={v∈Rn : ∇F0(v) ∈tBX∗}, (63)

where tBX∗ is the ball of X∗of radius t with center at the origin. Then for x∈ Rn

we have that

(1) If x ∈ Ω then the optimal decomposition for Lp,1– functional is given by

x0,opt=x and x1,opt=0.

(2) If x /∈Ω then x=x0,opt+x1,optis optimal decomposition for Lp,1– functional

if and only if (a) ∇F₀ x_0,opt
X∗ =t (b) x1,opt,∇F0 x0,opt
= x_1,opt X ∇F₀ x_0,opt
X∗ =t x_1,opt X.

Proof. For(1), since Lp,1– functional is convex and dom Lp,1(t, x;`p, X) =Rn,

then it is subdifferentiable and we can apply Key Lemma to check that if x∈Ω then the decomposition x0,opt=x and x1,opt =0 is optimal. It will be enough

to prove that this decomposition satisfies the characterization (14). Indeed, if x ∈ Ω, then y = ∂F0(x) = ∇F0(x) ∈ tBX∗. It follows from Proposition 2.1

that

F0(x) =hy, xi −F0∗(y). (64)

As y∈tBX∗ therefore from (62) we have

It then follows trivially from definition of F1that

F1(0) =hy, 0i −F1∗(y). (65)

Considering both (64) and (65) we conclude that the decomposition x0,opt=x

and x1,opt = 0 satisfies the sufficient conditions for optimality (14) and is

therefore optimal for Lp,1-functional.

For (2), let x /∈ Ω and x = x0,opt+x1,opt be an optimal decomposition and

show that(a)and (b) hold. In this case we must have x1,opt 6= 0. Indeed if

x1,opt =0 then x0,opt=x and from Key Lemma there exists y∗satisfying



F0(x) = hy∗, xi −F₀∗(y∗);

F₁∗(y∗) =0. (66)

By Proposition 2.1, the first equation of (66) is equivalent to

y∗∈∂F0(x) = ∇F0(x), (67)

and the second equation of (66) means, from conjugate of F1, that

y∗∈tBX∗. (68)

Considering (67) and (68), we have that

∇F0(x) ∈tBX∗,

which means, from (63), that x∈Ω.

If x0,opt=0 then we must have x1,opt =X and

(

F₀∗(y∗) = _p10ky∗kp

0

`p0 =0;

y∗∈∂F1(x).

It means that y∗ = 0 and x1,opt = 0. This case would be a degenerate one

where the given x=0.

Now let us consider the nondegenerate case when x /∈Ω and both x0,optand

x1,opt are different from zero. In this case, by Key Lemma, there exists element

y∗which satisfies



F0 x0,opt
= hy∗, x0,opti −F0∗(y∗);

F1 x1,opt
= hy∗, x1,opti −F₁∗(y∗). (69)

The first equation of (69) is equivalent (by Proposition 2.1 and (61)) to y∗∈∂F0 x0,opt
. Since ∂F0 x0,opt
= ∇F0 x0,opt
=n x0,opt   p−1 sgn(x0,opt) o , then y∗= ∇F0 x0,opt
.

The second equation of (69) can be rewritten as t

x1,opt

_X =y∗, x1,opt , with kyx,FkX∗≤t.

Therefore we obtain that t x1,opt X = ∇F0 x0,opt
, x1,opt  and ∇F₀ x_0,opt
X∗ ≤t. (70)

It follows then by definition of the dual norm that t x1,opt X ≤ ∇F₀ x_0,opt
X∗ x1,opt X. (71)

Since x1,opt 6=0 then (71) is equivalent to

t≤

∇F₀ x_0,opt

X∗. (72)

Combining (70) and (72), we conclude that

∇F₀ x_0,opt

X∗ =t. (73)

Which proves(a). Combining (70) and (73) we conclude by Cauchy-Schwarz inequality, that t x1,opt X = ∇F0 x0,opt
, x1,opt  ≤ ∇F₀ x_0,opt
X∗ x1,opt X =t x_1,opt X. Which proves(b).

Next let us assume that x /∈Ω and that a decomposition x=x˜0+x˜1satisfies

(a)and(b)and show that x=x˜0+x˜1is optimal. It follows from(a)that the

element y= ∇F0(x˜0) =∂F0(x˜0)satisfies

F0(x˜0) =hy, ˜x0i −F0∗(y). (74)

From(b), we conclude that

tkx˜1kX=h∇F0(x˜0), ˜x1i. (75)

Since kyk_X∗ = k∇F0(x˜0)kX∗ = t then it follows from expression of F₁∗ that

F₁∗(∇F0(x˜0)) =0. Therefore (75) can be written as

F1(x˜1) =tkx˜1kX = h∇F0(x˜0), ˜x1i −F1∗(∇F0(x˜0)). (76)

Considering (74) and (76), we conclude from Key Lemma, that the decomposi-tion x=x˜0+x˜1is optimal for the Lp,1-functional.

Proposition 4.1(Geometrical interpretation of Theorem 4.1 when x /∈Ω). If

x /∈Ω then Theorem 4.1 means that x1,optis orthogonal to the supporting hyperplane

to tBX∗which goes through the point∇F₀ x_0,opt
.

Proof. Suppose that condition(b)in Theorem 4.1 is satisfied. i.e., x1,opt,∇F0 x0,opt
= x_1,opt X ∇F₀ x_0,opt
X∗ =t x_1,opt X. (77)

By definition of the dual norm we have that t x_1,opt X =t sup kyk_X∗≤1 y, x1,opt=t sup ktyk_X∗≤t y, x1,opt= sup ktyk_X∗≤t ty, x1,opt .

Putting z=ty, we have t x1,opt X = sup kzk_X∗≤t z, x1,opt  = sup z∈tB_X∗ z, x1,opt . By (77), we then have t x_1,opt X = sup z∈tB_X∗

z, x1,opt=∇F0 x0,opt
, x1,opt .

Let us denote by Pvu= hu,vi kvk2_`2

v, the vector projection of u onto v. As x1,opt6=0

we have that sup_z∈tB X∗z, x1,opt  x_1,opt X = sup z∈tB_X∗ Px1,optz _{`2</sub> = Px1,opt∇F0 x0,opt
<sub>`2}, and therefore sup z∈tB_X∗ Px1,optz _{`</sub>2 = Px1,opt∇F0 x0,opt
<sub>`}2.

Therefore, among all elements from tBX∗, the element∇F₀ x_0,opt
gives the

maximum length of vector projection onto x1,opt.

We illustrate Proposition 4.1 in Figure 1.

The Figure 1 illustrate the fact that for p6= 2, the ball tBX∗ and the setΩ are

Figure 1: Geometry of optimal decomposition for Lp,1 t, x;`p, X
where x /∈

Ω.

4.3

Special case: p

=

2

In the case p = 2, we are concerned with the optimal decomposition for the L-functional L2,1  t, x;`2, X= inf x=x0+x1  1 2kx0k 2 `2+tkx1kX  . (78)

The function F0is therefore

F0(u) = 1

2kuk

2

`2, (79)

and F1remains unchanged. i.e.,

F1(u) =tkukX. (80)

It is clear that

∂F0(u) = ∇F0(u) ={|u|sgn(u)} =u,

and the setsΩ and tBX∗coincide:

Ω=tBX∗.

Corollary 1. Let F0and F1be defined onRnby (79) and (80). Let us define the set

Ω by

Ω=tBX∗,

where tBX∗ is the ball of X∗of radius t with center at the origin. Then for x∈ Rn

we have that

(1) If x ∈ Ω then the optimal decomposition for (78) is given by x0,opt = x and

x1,opt=0.

(2) If x /∈ Ω then x=x0,opt+x1,optis optimal decomposition for (78) if and only

if (a) x_0,opt X∗=t (b) x1,opt, x0,opt= x1,opt X x0,opt _X∗=t x1,opt X.

In this case, the geometrical meaning of condition (2)in Corollary 1 means that x1,opt is orthogonal to the supporting hyperplane to tBX∗ which goes

through the point x0,opt. It is equivalent to the fact that x0,opt is the nearest

element (in the metric of`2_{) of tB}

X∗to the point x : inf kx0kX∗≤t kx−x0k`2 = x−x0,opt <sub>`2</sub>. We illustrate Corollary 1 in Figure 2.

4.4

Geometrical illustration

In this section we give a concrete illustration of the geometry of optimal de-composition based on Theorem 4.1. We consider couple(`p, X)on the plane for p=3 and where the unit ball of X is the rotated ball of`1by the rotation matrix Rθ =  cos θ sin θ −sin θ cos θ  , for θ=30 degrees. i.e.,

R_θ = √ 3/2 1/2 −1/2 √3/2  .

We have that the norm in X of x= [x1, x2]T is given by

kxk_X = R −1 θ x _`1 =      √ 3 2 x1− 1 2x2      +      1 2x1+ √ 3 2 x2      . In this case we have that

∇F0(u) =

h

|u1|2sgn(u1),|u2|2sgn(u2)

i . So the setΩ can be written as

Ω=  v∈R2: h |v1|2sgn(v1),|v2|2sgn(v2) iT _X∗ ≤t  , where the norm in X∗of y= [y1, y2]T is given by

kyk_X∗= R −1 θ y _`∞=max (     √ 3 2 y1− 1 2y2      ,      1 2y1+ √ 3 2 y2      ) .

The case (1)of Theorem 4.1 says that if x ∈ Ω then the decomposition x =

x+0 is optimal for the L3,1- functional, i.e.,

L3,1  t, x;`3, X= inf x=x0+x1  1 3kx0k 3 `3+tkx1kX  = 1 3kxk 3 `3.

The case(2) of Theorem 4.1 is illustrated in Figure 3. So we see that in this situation the setΩ is not convex. The unit ball of X∗is R_θ(B_{`</sub>∞), whereB<sub>`}∞is the unit ball of`∞. If x belongs to the blue area of Figure 3 (above), then x0,opt

is the corresponding corner point of Ω and y∗ is the corresponding corner

point of tBX∗ (see Figure 3 (below)). The same holds for areas 1, 3 and 4. In

other situations, x0,opt belongs to the boundary of Ω such that the direction

of x1,opt is the direction perpendicular to the tangent line to tBX∗ which goes

−5 −4 −3 −2 −1 0 1 2 3 4 5 −4 −3 −2 −1 0 1 2 3 4 x

Ω

tB

Figure 3: Geometry of Optimal Decomposition for the Couple (`p, X) for p=3, X=Rθ `1
and θ=30◦, t=2.

5

Attouch–Brezis theorem and existence of

opti-mal decomposition for infiopti-mal convolution

We recall that for a regular couple(X0, X1), the dual spaces X0∗ and X∗1 also

form a compatible couple X∗₀, X∗₁
(see for example the book (Bennett and Sharpley, 1988)). In this situation, both X₀∗ and X∗₁ are linearly and continu-ously embedded in(X0∩X1)∗ and we will call X0∗, X∗1
conjugate couple. In

this section we will prove that the optimal decomposition always exists for the conjugate couple.

5.1

Existence of optimal decomposition for the K– functional

Let y be an element from X₀∗+X₁∗and a parameter t>0 be given. We consider the K– functional on the conjugate couple X₀∗, X₁∗
.

K(t, y; X0∗, X∗1) =_y=yinf 0+y1  ky0kX∗₀ +tky1kX∗₁  ,

where the infimum extends over all representations of y=y0+y1for y0∈ X0∗

and y1∈ X1∗. Define the functions ψ0, ψ1: X0∗+X1∗−→R∪ {+∞}as follows

ψ0(y) =  _kyk X∗ 0 if y∈X ∗ 0; +∞ if y∈ X₀∗+X₁∗
\X∗₀. and ψ1(y) =  _tkyk X₁∗ if y∈X∗1; +∞ if y∈ X₀∗+X₁∗
\X₁∗.

In this way, we can write the K– functional in terms of infimal convolution of

ψ0and ψ1.

K(t, y; X₀∗, X∗1) = (ψ0⊕ψ1) (y) =_y=yinf

0+y1

(ψ0(y0) +ψ1(y1)),

where the infimum extends over all representations of y=y0+y1for y0and

y1in X0∗+X1∗. Let us define the functions ϕ0, ϕ1 : X0∩X1−→R∪ {+∞}as

follows ϕ0(x) =  _{0 if kxk} X0 ≤1; +∞ if kxk_X 0 >1. and ϕ1(x) =  _{0 if kxk} X1 ≤t; +∞ if kxk_X 1 >t.

By Lemma 5, it is clear that the functions ϕ0and ϕ1have for conjugates, the

Theorem 5.1. Let y be an element from X∗₀+X₁∗and a parameter t > 0 be given. Then there exists an optimal decomposition for the K– functional

K(t, y; X0∗, X∗1) =_y=yinf 0+y1  ky0kX∗₀ +tky1kX∗₁  .

Proof. It is clear that the functions ϕ0 and ϕ1 satisfy conditions in Attouch–

Brezis Theorem. In fact it clear that they are convex, proper and lower semi-continuous. Moreover, since dom ϕ0 = BX0(0; 1) ∩ (X0∩X1)and dom ϕ1 =

BX1(0; t) ∩ (X0∩X1)then we have that

∪λ≥0λ(dom ϕ0−dom ϕ1) =X0∩X1.

Therefore, by Attouch–Brezis Theorem, we have that

K(t, y; X0∗, X∗1) = (ψ0⊕ψ1) (y) = (ϕ₀∗⊕ϕ∗₁) (y) = (ϕ0+ϕ1)∗(y),

and there exists y0∈X∗0and y1∈X∗1with y=y0+y1such that

K(t, y; X0∗, X∗1) =ϕ0∗(y0) +ϕ∗₁(y1)

Equivalently

K(t, y; X0∗, X∗1) =ky0kX∗₀+tky1kX∗₁.

5.2

Existence of optimal decomposition for the L– functional

As in the previous section, we let y be an element from X₀∗+X₁∗and a param-eter t > 0 be given. We consider the L– functional on the conjugate couple

X∗₀, X∗₁
. Lp0,p1(t, y; X ∗ 0, X∗1) =_y=yinf 0+y1  1 p0 ky0k_Xp0∗ 0 + t p1 ky1kp_X1∗ 1  ,

where the infimum extends over all representations of y=y0+y1for y0∈ X0∗

and y1 ∈ X1∗, and where 1 < p0, p1 < +∞. Define the functions ψ0, ψ1 :

X₀∗+X₁∗−→R∪ {+∞}as follows ψ0(y) = ( ₁ p0 kyk p0 X∗ 0 if y∈X ∗ 0; +∞ if y∈ X∗₀+X∗₁
\X₀∗. and ψ1(y) = ( _t p1 kyk p1 X₁∗ if y∈X ∗ 1; +∞ if y∈ X∗₀+X∗₁
\X₁∗.

In this way, we can write the L– functional in terms of infimal convolution of

ψ0and ψ1. Lp0,p1(t, y; X ∗ 0, X∗1) = (ψ0⊕ψ1) (y) = inf y=y0+y1 (ψ0(y0) +ψ1(y1)),

where the infimum extends over all representations of y=y0+y1for y0and

y1in X0∗+X1∗. Let us define the functions ϕ0, ϕ1 : X0∩X1−→R∪ {+∞}as

follows ϕ0(x) = 1 p0₀kxk p0₀ X0 and ϕ1(x) = t p0₁ x t p0₁ X1 , where p0_j is such that _p1

i +

1

p0_i = 1, j = 0, 1. By Lemma 5, it is clear that the

functions ϕ0and ϕ1have for conjugates, the functions ψ0and ψ1respectively. Theorem 5.2. Let y be an element from X∗₀+X₁∗and a parameter t > 0 be given. Then there exists an optimal decomposition for the L– functional

Lp0,p1(t, y; X ∗ 0, X∗1) =_y=yinf 0+y1  1 p0 ky0k_Xp0∗ 0 + t p1 ky1k p₁ X∗ 1  ,

Proof. It is clear that the functions ϕ0 and ϕ1 satisfy conditions in Attouch–

Brezis Theorem. In fact it clear that they are convex, proper and lower semi-continuous. Moreover, since dom ϕ0=dom ϕ1=X0∩X1then we have that

∪λ≥0λ(dom ϕ0−dom ϕ1) =X0∩X1.

Therefore, by Attouch–Brezis Theorem, we have that Lp0,p1(t, y; X

∗

0, X∗1) = (ψ0⊕ψ1) (y) = (ϕ∗₀⊕ϕ∗₁) (y) = (ϕ0+ϕ1)∗(y),

and there exists y0∈X∗0and y1∈X∗1with y=y0+y1such that

Lp0,p1(t, y; X ∗ 0, X∗1) =ϕ∗0(y0) +ϕ∗₁(y1) Equivalently Lp0,p1(t, y; X ∗ 0, X∗1) = 1 p0 ky0kXp0∗ 0+ t p1 ky1kpX1∗ 1.

5.3

Existence of optimal decomposition for the E– functional

Let y be an element from X₀∗+X₁∗and a parameter t>0 be given. We consider the E– functional on the conjugate couple X₀∗, X₁∗
.

Ep0(t, y; X ∗ 0, X1∗) = inf y1∈X∗1 ky1kX₁∗≤t 1 p0 ky−y1k_Xp0∗ 0, 1< p< +∞.

In this case we define the functions ψ0, ψ1: X0∗+X1∗−→R∪ {+∞}as follows

ψ0(y) = ( ₁ p0 kyk p0 X₀∗ if y∈X ∗ 0; +∞ if y∈ X∗₀+X∗₁
\X₀∗.

and ψ1(y) = ( 0 if y∈tBX∗ 1, kykX1∗≤t; +∞ if y∈ X₀∗+X₁∗
\tB_X∗ 1.

In this way, we can write the E– functional in terms of infimal convolution of

ψ0and ψ1. Ep0(t, y; X ∗ 0, X1∗) = (ψ0⊕ψ1) (y) =_y=yinf 0+y1 (ψ0(y0) +ψ1(y1)),

where the infimum extends over all representations of y=y0+y1for y0and

y1in X0∗+X1∗. Let us define the functions ϕ0, ϕ1 : X0∩X1−→R∪ {+∞}as

follows

ϕ0(x) = 1

p0₀kxk

p0₀

X0 and ϕ1(x) =tkxkX1.

By Lemma 5, it is clear that the functions ϕ0and ϕ1have for conjugates, the

functions ψ0and ψ1respectively.

Theorem 5.3. Let y be an element from X∗₀+X₁∗and a parameter t > 0 be given. Then there exists an optimal decomposition for the E– functional

Ep0(t, y; X ∗ 0, X1∗) = inf y1∈X∗1 ky1kX₁∗≤t 1 p0 ky−y1kXp0∗ 0, 1≤ p< +∞.

Proof. It is clear that the functions ϕ0 and ϕ1 satisfy conditions in Attouch–

Brezis Theorem. In fact it clear that they are convex, proper and lower semi-continuous. Moreover, since dom ϕ0=dom ϕ1=X0∩X1then we have that

∪λ≥0λ(dom ϕ0−dom ϕ1) =X0∩X1.

Therefore, by Attouch–Brezis Theorem, we have that Ep0(t, y; X

∗

0, X1∗) = (ψ0⊕ψ1) (y) = (ϕ₀∗⊕ϕ∗₁) (y) = (ϕ0+ϕ1)∗(y),

and there exists y0∈X∗0and y1∈tBX∗₁ with y=y0+y1such that

Ep₀(t, y; X0∗, X1∗) =ϕ∗0(y0) +ϕ₁∗(y1) Equivalently Ep0(t, y; X ∗ 0, X1∗) = 1 p0 ky−y1kp_X0∗ 0.

6

Final remarks and discussions

In this paper, we have used results developped in (Niyobuhungiro, 2013b) to give mathematical characterization of optimal decomposition for K–, L– and E– functionals. Remember that for the regular Banach couple (X0, X1),

there exist two specific convex, lower semicontinuous and proper functions

ϕ0: X0−→R∪ {+∞}and ϕ1: X1−→R∪ {+∞}for each of the K–, L– and

E– functionals such that they can be written as a function F : X0+X1 −→ R∪ {+∞}defined by

F(x) = (ϕ0⊕ϕ1) (x) = inf x=x0+x1

(ϕ0(x0) +ϕ1(x1)),

where the infimum extends over all representations x = x0+x1of x with x0

and x1 in X0+X1 and where ϕ0 : X0+X1 −→ R∪ {+∞} and ϕ1 : X0+

X1 −→ R∪ {+∞}are respective extensions of ϕ0 and ϕ1 on X0+X1 in the

following way ϕ0(u) =  ϕ0(u) if u∈X0; +∞ if u∈ (X0+X1) \X0. and ϕ1(u) =  ϕ1(u) if u∈X1; +∞ if u∈ (X0+X1) \X1.

However, it is important to notice that the extended functions ϕ0and ϕ1could

stop to be lower semicontinuous even if ϕ0 and ϕ1are. The reformulation of

the infimal convolution F(x) = (ϕ0⊕ϕ1) (x)as:

F(x) = (ϕ0⊕ϕ1) (x) = inf y∈X₀∩X1

(S(y) +R(y)),

where S and R are functions defined on X0∩X1with values inR∪ {+∞}by

S(y) =ϕ0(a0−y) and R(y) =ϕ1(a1+y),

was useful as it helps overcome some technical difficulties appearing because two different Banach spaces are involved.

We have been able to use the characterization obtained in order to give a geometrical interpretation of optimal decomposition for Lp,1– functional for

the couple(`p, X)in finite dimensional case, which generalizes our previous result. (Attouch and Brezis, 1986) theorem was also used to show the existence of optimal decomposition for the K–, L– and E– functionals for the conjugate couple.

References

H. Attouch and H. Brezis. Duality for the sum of convex functions in general banach spaces. In: Aspects of Mathematics and its Applications. J.A. Barroso ed., North-Holland, Amsterdam, pages 125–133, 1986.

C. Bennett and R. Sharpley. Interpolation of Operators. Pure and Applied Math-ematics, Vol. 129, Academic Press, Orlando, 1988.

J. Bergh and J. Löfström. Interpolation Spaces. An Introduction. Springer, Berlin, 1976.

Y. A. Brudnyi and N. Y. Krugljak. Interpolation Functors and Interpolation Spaces, Volume 1. North-Holland, Amsterdam, 1991.

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