A duality approach for optimal decomposition in real ... - DiVA

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Department of Mathematics

A duality approach for optimal decomposition in real interpolationer Japhet Niyobuhungiro LiTH-MAT-R–2013/08–SE

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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 characterization 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 characterize 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 ) on Rn . We will also apply the Attouch–Brezis theorem 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., X0 and X1 are 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 X0 and 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 k x − wk X0 + t kwk X1 . w ∈ X1

The K– functional is at the center of the so–called K– method of real interpolation that is basically concerned with the construction of suitable families of ∗ Department of Mathematics, Linköping University, SE–581 83 Linköping, Sweden. E-mail: [email protected] † Department of Applied Mathematics, National University of Rwanda, P.O. Box: 56 Butare Rwanda. E-mail: [email protected]

1

real interpolation spaces between X0 and X1 . The K– functional is a particular case of the more general L– functional which is defined by   p p (1) L p0 ,p1 (t, x; X0 , X1 ) = inf k x − wk X00 + t kwk X11 , w ∈ X1

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 X ≤t

k x − w k X0 .

1

Definition 1.1 (Exact and near minimizers). We say that the element (which depends on x and t) wt ∈ X1 is a near minimizer for the functional (1) if there exists C > 0 independent of x and t such that p

p

k x − wt k X00 + t kwt k X11 ≤ CL p0 ,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   1 t p0 p1 L p0 ,p1 (t, x; X0 , X1 ) = inf , (3) k x 0 k X0 + kx k x = x0 + x1 p0 p 1 1 X1 for 1 ≤ p0 , p1 < ∞. Problem. Suppose that the optimal decomposition x = x0,opt + x1,opt for K– functional (respectively for L– and E– functionals) corresponding to the element x exists. Give a characterization of this decomposition. In other words, what are the mathematical properties of x0,opt and x1,opt . For example, in the case of the L– functional (3), we want the mathematical properties of the decomposition x = x0,opt + x1,opt such that L p0 ,p1 (t, x; X0 , X1 ) =



1

x0,opt p0 + t x1,opt p1 . X X1 0 p0 p1

In Section 2, we will provide some results which will be used in the subsequent sections. In Section 3, we state and prove theorems on dual characterization of optimal decomposition for the E–, L– and K– functionals. In Section 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 establshed in (Niyobuhungiro, 2013a) for the couple (`2 , X ) on Rn . 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

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 + X1 as follows  ϕ0 (u) if u ∈ X0 ; (4) ϕ0 ( u ) = +∞ if u ∈ ( X0 + X1 ) \ X0 . 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 − x i , ∀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 F 6= ∅. Proposition 2.1. Any points y ∈ E∗ and x ∈ E satisfy the inequality, well–known as Young–Fenchel inequality F ( x ) + F ∗ (y) ≥ hy, x i ,

(6)

where F ∗ is the conjugate of F. i.e., the function defined from E∗ into R ∪ {+∞} by F ∗ (y) = sup {hy, x i − F ( x )} , for y ∈ E∗ . x ∈dom E

Equality in (6) holds if and only if y ∈ ∂F ( x ): y ∈ ∂F ( x ) ⇐⇒ F ( x ) + F ∗ (y) = hy, x i . The following theorem establishes sufficient conditions for the infimal convolution ( 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 ) = ϕ 1 ( a 1 + y ) , y ∈ ( X0 ∩ X1 ) ,

(7)

be convex, lower semicontinuous and proper. Let ϕ0∗ and ϕ1∗ be the respective conjugate functions of ϕ0 and ϕ1 . Suppose that 3

(1) the sets dom S and dom R satisfy [

λ (dom S − dom R) = X0 ∩ X1

(8)

λ ≥0

(2) The conjugate function S∗ of S is given by  ∗ ϕ0 (−z) + hz, a0 i if z ∈ X0∗ ;
S∗ (z) = +∞ if z ∈ X0∗ + X1∗ \ X0∗ . (3) The conjugate function R∗ of R is given by  ∗ ϕ1 (z) + h−z, a1 i if z ∈ X1∗ ;
R∗ (z) = +∞ if z ∈ X0∗ + X1∗ \ X1∗ .

(9)

(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) theorem to prove Theorem 2.1. The theorem by H. Attouch and H. Brezis provides a sufficient condition for the conjugate of the sum of two convex lower semicontinuous 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 (dom ϕ − dom ψ) is a closed vector space.

(11)

λ ≥0

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 convolution. 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 ) =

inf

x = x0 + x1

{ F0 ( x0 ) + F1 ( x1 )}

(13)

is subdifferentiable at the point x ∈ dom ( F0 ⊕ F1 ). Then the decomposition x = x0,opt + x1,opt is optimal for (13) if and only if there exists y∗ ∈ E∗ that is dual to both x0,opt and x1,opt with respect to F0 and F1 respectively. i.e.,
 F0 x0,opt
= hy∗ , x0,opt i − F0∗ (y∗ ) ; (14) F1 x1,opt = hy∗ , x1,opt i − F1∗ (y∗ ) . 4

Lemma 2. Let 1 ≤ p0 < +∞ and let a0 ∈ X0 be given. Then the function S : X0 ∩ X1 −→ R ∪ {+∞} defined by S (y) =

1 p k a0 − yk X00 , p0

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 u j j=1 ∈ X0 ∩ X1 converges to y in the norm of X0 ∩ X1 :

lim u j − y X ∩X = 0. (15) j→+∞

0

1

From definition of the function S, we can write S (y) =

1 1 p

a 0 − y + u j − u j p0 . k a0 − yk X00 = X0 p0 p0

It follows from triangle inequality that S (y) ≤

 p0

1 

a0 − u j + u j − y . X0 X0 p0

(16)

Since by definition

u j − y

X0 ∩ X1



 = max u j − y X , u j − y X ,

it follows that



u j − y ≤ u j − y X X 0

0

0 ∩ X1

1

.

Putting this into (16) we have S (y) ≤

 p0

1 

a0 − u j + u j − y . X0 X0 ∩ X1 p0

This is equivalent to S (y) ≤

 p0

1/p0 1  p0 S ( u j ) + u j − y X ∩X . 0 1 p0

It follows that

 1/p0 + u j − y X [ p0 S (y)]1/p0 ≤ p0 S(u j )

0 ∩ X1

Taking limit, it follows from (15), that
S (y) ≤ lim inf S u j , j→+∞

which establishes lower semicontinuity of S at y. 5

.

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 p k a + yk X11 , p1 1

(17)

is convex, proper and lower semicontinuous. Lemma 4. Let B X1 ( a1 ; t) denote the ball of X1 centered at a1 ∈ X1 and of radius t and let R : X0 ∩ X1 −→ R ∪ {+∞} be the function defined by  0 if u ∈ B X1 ( a1 ; t) ∩ ( X0 ∩ X1 ) ; R (u) = +∞ otherwise. Then its conjugate function R∗ is equal to  t kzk X ∗ + hz, a1 i if z ∈ X1∗ ; 1
R∗ (z) = +∞ if z ∈ X0∗ + X1∗ \ X1∗ . For the proof, we will need two claims. Claim 2.1. Let us assume that z ∈ X1∗ . Then sup sup hz, yi = hz, yi y ∈ X0 ∩ X1 y ∈ X1 k a 1 − y k X1 ≤ t k a 1 − y k X1 ≤ t Proof of Claim 2.1. It is clear that sup sup hz, yi ≤ hz, yi . y ∈ X0 ∩ X1 y ∈ X1 k a 1 − y k X1 ≤ t k a 1 − y k X1 ≤ t We only need to show that sup sup hz, yi ≥ hz, yi . y ∈ X0 ∩ X1 y ∈ X1 k a 1 − y k X1 ≤ t k a 1 − y k X1 ≤ t In this end, it is enough to prove that for y ∈ X1 such that k a1 − yk X1 ≤ t, we have that sup hz, yi ≥ hz, yi . y ∈ X0 ∩ X1 k a 1 − y k X1 ≤ t   Put a1 − y = u and take the an element 1 − n1 u in X1 . Then

     

1 − 1 u = 1 − 1 kuk ≤ 1 − 1 t. X1

n n n X1 6

By denseness of X0 ∩ X1 in X1 , take un ∈ X0 ∩ X1 and a1,n ∈ X0 ∩ X1 such that

 

un − 1 − 1 u ≤ t and k a1 − a1,n k ≤ t . X1

n 2n 2n X1 Let us pick yn = a1,n − un . We must check that lim ky − yn k X1 = 0 and k a1 − yn k X1 ≤ t.

n→+∞

Indeed,

ky − yn k X1 = k a1 − u − ( a1,n − un )k X1 ≤ k a1 − a1,n k X1 + kun − uk X1

   

1 1

u − 1 − = k a1 − a1,n k X1 + u + 1 − u − u

n n n X1



  



1 1

≤ k a1 − a1,n k X1 + un − 1 − u + 1 − u − u

n n X1 X1 t t t 2t ≤ + + = . 2n 2n n n Therefore lim ky − yn k X1 = 0.

n→+∞

We also have that

k a1 − yn k X1 = k a1 − ( a1,n − un )k X1



≤ k a1 − a1,n k X1 + un − 1 −



u − 1− ≤ k a1 − a1,n k X1 + n



1 u+ 1− u

n X1

  

1 1

1− + u u

n n X1 X1 1 n



  t t 1 ≤ + + 1− t = t. 2n 2n n Therefore

k a1 − yn k X1 ≤ t. Then we conclude that sup hz, yi ≥ lim hz, yn i = hz, yi . n→+∞ y ∈ X0 ∩ X1 k a 1 − y k X1 ≤ t

7




Claim 2.2. Suppose that z ∈ X0∗ + X1∗ \ X1∗ . Then sup hz, yi = +∞ y ∈ X0 ∩ X1 k a 1 − y k X1 ≤ t
Proof of Claim 2.2. The fact that z ∈ X0∗ + X1∗ \ X1∗ means z is defined on X0 ∩ X1 and is unbounded on the set B X1 (0; 1) ∩ ( X0 ∩ X1 ). In fact, if it were bounded on B X1 (0; 1) ∩ ( X0 ∩ X1 ), then it would nbe possible to extend it as a o bounded linear functional on X1 . Put the set Ω = y ∈ X0 ∩ X1 : k a1 − yk X1 ≤ t . We need to prove that sup hz, yi = +∞. y∈Ω


Notice that hz, yi is well defined because z ∈ ( X0 ∩ X1 )∗ = X0∗ + X1∗ . Assume by contradiction that sup hz, yi = C < +∞. y∈Ω

Choose a1,ε ∈ X0 ∩ X1 such that k a1 − a1,ε k X1 <   t Ω t = u ∈ X0 ∩ X1 : ku − a1,ε k X1 ≤ . 2 2

t 2

and put the set

We have in that case, that

k a1 − uk X1 = k a1 − a1,ε + a1,ε − uk X1 ≤ k a1 − a1,ε k X1 + k a1,ε − uk X1 t t ≤ + = t, 2 2 and that Ω t ⊂ Ω. Condider the set B X1 (0; 4t ) ∩ ( X0 ∩ X1 ) and take an element 2

v ∈ B X1 (0; 4t ) ∩ ( X0 ∩ X1 ). Note that u = (2v + a1,ε ) ∈ Ω t . In fact 2

t t k2v + a1,ε − a1,ε k X1 = 2 kvk X1 ≤ 2 = . 4 2 For such an element v we have that D a E 1 1 hz, vi = z, − 1,ε + hz, 2v + a1,ε i ≤ C0 + C = C1 < +∞, 2 2 2 where C0 and C1 are some constants. We conclude that sup v∈B X1 (0;1)∩( X0 ∩ X1 )

hz, vi ≤ 4

C1 < +∞. t

This is a contradiction of the fact that z is unbounded on B X1 (0; 1) ∩ ( X0 ∩ X1 ). Therefore sup hz, yi = +∞. y∈Ω

8

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

sup hz, yi . y ∈ X0 ∩ X1 k a 1 − y k X1 ≤ t

From Claim 1, we have that R∗ (z) =

sup sup hz, y − a1 i + hz, a1 i . hz, yi = y ∈ X1 y ∈ X1 k a 1 − y k X1 ≤ t k a 1 − y k X1 ≤ t

Therefore R∗ (z) = t kzk X ∗ + hz, a1 i . 1

If z ∈

X0∗

+ X1∗




\ X1∗ ,

then it follows from Claim 2 that

sup hz, yi = +∞. y ∈ X0 ∩ X1 k a 1 − y k X1 ≤ t This completes the proof of Lemma 4. Lemma 5. Let a0 ∈ X0 and a1 ∈ X1 be given. Let functions S and R be defined on X0 ∩ X1 by S (y) =

1 p k a0 − yk X00 p0

and let pi0 , i = 0, 1 be such that

R (y) =

and 1 pi

+

1 pi0

t p ky + a1 k X11 , p1

(18)

= 1. Then

(1) In the case 1 < p0 , p1 < +∞, we have that ( p00 1 if z ∈ X0∗ ; 0 k z k X ∗ + h z, a0 i ∗ p S (z) = 0 0
+∞ if z ∈ X0∗ + X1∗ \ X0∗ .

(19)

and ( ∗

R (z) =

t p10

z p0

1∗ + h−z, a1 i t X 1

if z ∈ X1∗ ;


+∞ if z ∈ X0∗ + X1∗ \ X1∗ .

(2) In the case p0 = p1 = 1, we have that ( hz, a0 i if kzk X ∗ ≤ 1, z ∈ B X0∗ (0; 1) ; ∗ 0
S (z) = +∞ if z ∈ X0∗ + X1∗ \B X0∗ (0; 1) .

(20)

(21)

and ( ∗

R (z) =

h−z, a1 i if kzk X ∗ ≤ t, z ∈ B X1∗ (0; t) ; 1
+∞ if z ∈ X0∗ + X1∗ \B X1∗ (0; t) . 9

(22)

Proof. We will prove (19) and (21) only. The case of (20) and (22) is similar. (1) Case 1 < p0 < +∞. Assume that z ∈ X0∗ . Let us define the function ϕ : R −→ R+ in the following way 1 | t | p0 . p0

t 7−→ ϕ(t) =

It is clear that this function is convex, lower semicontinuous, proper and even. The dual ϕ∗ of ϕ is by definition   1 p0 ∗ ∗ ∗ . ϕ (t ) = sup t · t − |t| p0 t ∈R 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 p kyk X00 . p0

Let us calculate its conjugate. n o T ∗ (z) = sup hz, yi − ϕ(kyk X0 ) . y ∈ X0

This can be rewritten as follows T ∗ (z) = sup sup {hz, yi − ϕ(t)} t ≥0 k y k X = t 0

= sup t ≥0

  

t sup hz, yi − ϕ(t) k y k X =1

  

0

n o = sup t kzk X ∗ − ϕ(t) . t ≥0

Since ϕ is an even function then n o T ∗ (z) = sup t kzk X ∗ − ϕ(t) = ϕ∗ (kzk X ∗ ). 0

t ∈R

0

Hence T ∗ (z) = ϕ∗ (kzk X ∗ ). 0

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

1 p0 kzk X0∗ . 0 0 p0 10

0

Let us now calculate S∗ . We have that 

∗

S (z) =

{hz, yi − S(y)} =

sup

sup y ∈ X0 ∩ X1

y ∈ X0 ∩ X1

1 p hz, yi − ky − a0 k X00 p0



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 ∩ X1 is dense in both X0 and X1 then as T is a continuous function with respect to the norm of X0 , we have that S∗ (z) = sup {hz, y − a0 i − T (y − a0 )} + hz, a0 i . y ∈ X0

This is equal to S∗ (z) = T ∗ (z) + hz, a0 i .

(24)

Whence S∗ (z) =

1 p0 kzk X0∗ + hz, a0 i . 0 0 p0


Now let us assume that z ∈ X0∗ + X1∗ \ X0∗ . In this case z is unbounded on B X0 (0; 1) ∩ ( X0 ∩ X1 ). Therefore we must have that supy∈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 ) = k y k X0 . Its conjugate function is by definition n o T ∗ (z) = sup hz, yi − kyk X0 . y ∈ X0

By the same reasoning as above, we have that     T ∗ (z) = sup sup {hz, yi − t} = sup t sup hz, yi − t  t ≥0 k y k = t t ≥0  k y k =1 X0

X0

Equivalently n o T ∗ (z) = sup t kzk X ∗ − 1 t ≥0

0

11

.

It is clear that ( T ∗ (z) =

0 +∞

if kzk X ∗ ≤ 1, z ∈ B X0∗ (0; 1) ; 0
if z ∈ X0∗ + X1∗ \B X0∗ (0; 1) .

It follows from (24), that ( hz, a0 i if kzk X ∗ ≤ 1, z ∈ B X0∗ (0; 1) ; ∗ 0
S (z) = +∞ if z ∈ X0∗ + X1∗ \B X0∗ (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 E p0 (t, x; X0 , X1 ) =

inf

k x1 k X ≤ t 1

3.1.1

1 p k x − x1 k X00 . p0

(25)

Case: p0 > 1

For p0 > 1, we can express the E– functional (25) as the infimal convolution E p0 (t, x; X0 , X1 ) = ( ϕ0 ⊕ ϕ1 ) ( x ) , where ϕ0 and ϕ1 are functions defined on the sum X0 + X1 by ( p0 1 p0 k u k X0 if u ∈ X0 ; ϕ0 ( u ) = +∞ if u ∈ ( X0 + X1 ) \ X0 . 12

(26)

and ϕ1 is defined by  0 ϕ1 ( u ) = +∞

if u ∈ B X1 (0; t) ; if u ∈ ( X0 + X1 )\B X1 (0; t) .

(27)

where B X1 (0; t) is the ball of X1 of radius t and centered at 0: n o B X1 (0; t) = x ∈ X0 + X1 : k x k X1 ≤ t . In this case the functions ϕ0 : X0 −→ R ∪ {+∞} and ϕ1 : X1 −→ R ∪ {+∞} are defined by  1 0 if u ∈ B X1 (0; t) ; p0 ϕ0 ( u ) = kuk X0 and ϕ1 (u) = + ∞ if u ∈ X1 \B X1 (0; t) . p0 Let us fix a0 ∈ X0 and a1 ∈ X1 such that x = a0 + a1 . Then we can rewrite the E– functional (25) as follows E p0 (t, x; X0 , X1 ) =

inf

k a1 − y k X ≤ t 1

1 p k a0 + yk X00 , y ∈ X0 ∩ X1 . p0

Equivalently, E p0 (t, x; X0 , X1 ) =

inf

y ∈ X0 ∩ X1

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

where S, R : X0 ∩ X1 −→ R ∪ {+∞} are functions defined by  1 0 if y ∈ B X1 ( a1 ; t) ∩ ( X0 ∩ X1 ) ; p S (y) = k a0 + yk X00 and R (y) = +∞ otherwise. p0 (28) The following theorem gives the dual characterization of optimal decomposition for the E p0 – functional. i.e., the decomposition x = x0,opt + x1,opt , where x0,opt ∈ X0 and x1,opt ∈ B X1 (0, t) such that E p0 (t, x; X0 , X1 ) =

1

x − x1,opt p0 , x1,opt ∈ B X (0, t) , x0,opt = x − x1,opt . 1 X0 p0

Theorem 3.1. Let 1 < p0 < ∞. Then the decomposition x = x0,opt + x1,opt , where x1,opt ∈ B X1 (0, t), is optimal for the E p0 – functional if and only if there exists y∗ ∈ X0∗ ∩ X1∗ such that: 

p0 p00  1 1

p0 x0,opt X0 = h y∗ , x0,opt i − p00 k y∗ k X0∗ ;  hy∗ , x1,opt i = t ky∗ k ∗ . X1

Proof. We have shown that for fixed a0 ∈ X0 and a1 ∈ X1 such that x = a0 + a1 , the E p0 – functional can be defined by E p0 (t, x; X0 , X1 ) =

inf

y ∈ X0 ∩ X1

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

13

where functions S and R are defined by (28). Moreover E p0 (t, x; X0 , X1 ) = ( ϕ0 ⊕ ϕ1 ) ( x ) , where functions ϕ0 and ϕ1 are defined by (26) and (27) respectively. It follows 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 set B X1 ( a1 ; t) ∩ ( X0 ∩ X1 ). It is proper because B X1 ( a1 ; t) ∩ ( X0 ∩ X1 ) 6= ∅ because the space X0 ∩ X1 is dense in X1 . Moreover, since dom S = X0 ∩ X1 and dom R = B X1 ( a1 ; t) ∩ ( X0 ∩ X1 ) then we have that [

λ (dom S − dom R) = X0 ∩ X1 .

(29)

λ ≥0

The conjugate function ϕ0∗ of ϕ0 is defined on X0∗ and is given by ϕ0∗ (z) =

1 p0 kzk X0∗ , ∀z ∈ X0∗ . 0 0 p0

The conjugate function S∗ of S can be obtained from Lemma 5 where z is replaced by −z. We have that ( p00 1 if z ∈ X0∗ ; z k k ∗ + h− z, a0 i 0 ∗ X p0 S (z) = 0
+∞ if z ∈ X0∗ + X1∗ \ X0∗ . It is clear that 

∗

S (−z) =

ϕ0∗ (−z) + hz, a0 i +∞

if z ∈ X0∗ ;
if z ∈ X0∗ + X1∗ \ X0∗ .

(30)

The conjugate ϕ1∗ of ϕ1 is defined on X1∗ and is given by ϕ1∗ (z) =

sup x ∈B X1 (0;t)

hz, x i = t kzk X ∗ , ∀z ∈ X1∗ .

Recall that, from Lemma 4, that  t kzk X ∗ + hz, a1 i ∗ 1 R (z) = +∞

1

if z ∈ X1∗ ;
if z ∈ X0∗ + X1∗ \ X1∗ .

It is clear that R∗ (−z) =



ϕ1∗ (z) + h−z, a1 i +∞

if z ∈ X1∗ ;
if z ∈ X0∗ + X1∗ \ X1∗ .

(31)

By (29), (30) and (31), we conclude from Theorem 2.1 that E p0 is subdifferentiable. It is clear that the functions ϕ0 and ϕ1 are convex and proper. Therefore we conclude by Key Lemma that the decomposition x = x0,opt + x1,opt , where x1,opt ∈ B X1 (0, t), is optimal for the E p0 - functional if and only if there exists y∗ ∈ X0∗ ∩ X1∗ such that: 

p0 p00  1 1

p0 x0,opt X0 = h y∗ , x0,opt i − p00 k y∗ k X0∗ ;  hy∗ , x1,opt i = t ky∗ k ∗ . X1

14

Case: p0 = 1

3.1.2

For the case p0 = 1 we have the followimg E– functional E (t, x; X0 , X1 ) =

inf

k x1 k X ≤ t

k x − x 1 k X0 .

(32)

1

In this case E (t, x; X0 , X1 ) = ( ϕ0 ⊕ ϕ1 ) ( x ) , where  ϕ0 ( u ) =

kuk X0 if u ∈ X0 ; +∞ if u ∈ ( X0 + X1 ) \ X0 .

and ϕ1 is the function defined on the sum X0 + X1 as in (27). The functions ϕ0 : X0 −→ R ∪ {+∞} and ϕ1 : X1 −→ R ∪ {+∞} are defined by  0 if u ∈ B X1 (0; t) ; ϕ0 (u) = kuk X0 and ϕ1 (u) = +∞ if u ∈ X1 \B X1 (0; t) . The decomposition x = x0,opt + x1,opt , where x0,opt ∈ X0 and x1,opt ∈ B X1 (0, t) is optimal for the E– functional (32) if

E (t, x; X0 , X1 ) = x − x1,opt X , x1,opt ∈ B X1 (0, t) , x0,opt = x − x1,opt . 0

Theorem 3.2. The decomposition x = x0,opt + x1,opt , where x1,opt ∈ B X1 (0, t), is optimal for the E– functional (32) if and only if there exists y∗ ∈ X0∗ ∩ X1∗ such that ky∗ k X ∗ ≤ 1 and: 0

(

x0,opt

= hy∗ , x0,opt i; hy∗ , x1,opt i = t ky∗ k X ∗ . X0

1

Proof. For given a0 ∈ X0 and a1 ∈ X1 such 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  0 if u ∈ B X1 ( a1 ; t) ∩ ( X0 ∩ X1 ) ; S (y) = k a0 + yk X0 and R (u) = +∞ 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 (dom S − dom R) = X0 ∩ X1 .

λ ≥0

15

(33)

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

The conjugate functions S∗ of S can be obtained from Lemma 5 where z is replaced by −z. We have that ( h−z, a0 i if z ∈ B X0∗ (0; 1);
∗ S (z) = +∞ if z ∈ X0∗ + X1∗ \B X0∗ (0; 1). It is clear that ϕ0∗ (−z) + hz, a0 i +∞

if z ∈ X0∗ ;
if z ∈ X0∗ + X1∗ \ X0∗ .

(34)

and as in the previous case  ∗ ϕ1 (z) + h−z, a1 i R∗ (−z) = +∞

if z ∈ X1∗ ;
if z ∈ X0∗ + X1∗ \ X1∗ .

(35)

S∗ (−z) =



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 ∈ B X1 (0, t), is optimal for the E– functional (32) if and only if there exists y∗ ∈ X0∗ ∩ X1∗ such that ky∗ k X ∗ ≤ 1 and: 0

(

x0,opt

= hy∗ , x0,opt i; X0 hy∗ , x1,opt i = t ky∗ k X ∗ . 1

3.2

Optimal decomposition for the L– functional

Let x ∈ X0 + X1 and let t > 0 be a fixed parameter. We consider the following L– functional   1 t p p L p0 ,p1 (t, x; X0 , X1 ) = inf (36) k x0 k X00 + k x1 k X11 , x = x0 + x1 p0 p1 where 1 ≤ p0 , p1 < ∞. We are interested in characterization of optimal decomposition for this L– functional, i.e., characterization of x0,opt ∈ X0 and x1,opt ∈ X1 such that x = x0,opt + x1,opt and  

1

x0,opt p0 + t x1,opt p1 . L p0 ,p1 (t, x; X0 , X1 ) = X0 X1 p0 p1 It can be written as the infimal convolution L p0 ,p1 (t, x; X0 , X1 ) = ( ϕ0 ⊕ ϕ1 ) ( x ) , 16

where functions ϕ0 and ϕ1 are both defined on the sum X0 + X1 as follows ( p0 1 p0 k u k X0 if u ∈ X0 ; ϕ0 ( u ) = (37) +∞ if u ∈ ( X0 + X1 ) \ X0 . and ( ϕ1 ( u ) =

t p1

p

kuk X11 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 t p p kuk X00 and ϕ1 (u) = kuk X11 . p0 p1

The following theorem gives a characterization of optimal decomposition for L– functional. Theorem 3.3 (Optimal decomposition for L p0 ,p1 – functional). Let 1 < p0 , p1 < ∞. Then the decomposition x = x0,opt + x1,opt is optimal for the L– functional (36) if and only if there exists an element y∗ ∈ X0∗ ∩ X1∗ such that:  0

 1 x0,opt p0 = hy∗ , x0,opt i − 10 ky∗ k p0∗ ; p0 X0 p0 X0

p1

0 t y ∗ p1  t x

1,opt X = h y∗ , x1,opt i − p0 p t X∗ . 1

1

1

1

Proof. For given a0 ∈ X0 and a1 ∈ X1 such that x = a0 + a1 , we can define the L p0 ,p1 - functional as follows E p0 (t, x; X0 , X1 ) =

inf

y ∈ X0 ∩ X1

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

where functions S and R are defined as in (18). Moreover L p0 ,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 (dom S − dom R) = X0 ∩ X1 .

(39)

λ ≥0

The respective conjugate functions S∗ and R∗ of S and R are given in Lemma 5. We have that ( p00 1 if z ∈ X0∗ ; 0 k z k X ∗ + h z, a0 i ∗ p S (z) = 0 0
+∞ if z ∈ X0∗ + X1∗ \ X0∗ . and ( ∗

R (z) =

t p10

z p0

1∗ + h−z, a1 i t X 1

if z ∈ X1∗ ;


+∞ if z ∈ X0∗ + X1∗ \ X1∗ . 17

The conjugate functions ϕ0∗ of ϕ0 and ϕ1∗ of ϕ1 are defined on X0∗ and X1∗ respectively and are given by ϕ0∗ (z) =

1 p0 kzk X0∗ , ∀z ∈ X0∗ , 0 0 p0

ϕ1∗ (z) =

0 t

z p1

, ∀z ∈ X1∗ . p10 t X1∗

and

It is clear that S∗ (z) =



ϕ0∗ (−z) + hz, a0 i +∞

if z ∈ X0∗ ;
if z ∈ X0∗ + X1∗ \ X0∗ .

(40)

ϕ1∗ (z) + h−z, a1 i +∞

if z ∈ X1∗ ;
if z ∈ X0∗ + X1∗ \ X1∗ .

(41)

and 

∗

R (z) =

By (39), (40) and (41), we conclude from Theorem 2.1 that L– 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 L– functional (36) if and only if there exists an element y∗ ∈ X0∗ ∩ X1∗ such that:  0

 1 x0,opt p0 = hy∗ , x0,opt i − 10 ky∗ k p0∗ ; p0 X p X 0

p10

00 t y ∗ p1  t x

= h y , x i − ∗ 1,opt 1,opt X p t X∗ . p0 1

1

1

1

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

3.2.1

Important particular cases

3.2.2

Case 1 < p0 < +∞ and p1 = 1

Consider the L– functional  L p0 ,1 (t, x; X0 , X1 ) =

inf

x = x0 + x1

1 p k x0 k X00 + t k x1 k X1 p0

 ,

(42)

where 1 < p0 < +∞. Let ϕ0 and ϕ1 be functions defined on the sum X0 + X1 as follows ( p0 1 p0 k u k X0 if u ∈ X0 ; ϕ0 ( u ) = (43) +∞ if u ∈ ( X0 + X1 ) \ X0 .

18

and  ϕ1 ( u ) =

t k u k X1 +∞

if u ∈ X1 ; if u ∈ ( X0 + X1 ) \ X1 .

(44)

Then the L– functional (42) can be written as the following infimal convolution L p0 ,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 p kuk X00 and ϕ1 (u) = t kuk X1 . p0

Theorem 3.4 (Optimal decomposition for L p0 ,1 – functional). Let 1 < p0 < +∞. Then the decomposition x = x0,opt + x1,opt is optimal for the L p0 ,1 – functional if and only if there exists y∗ ∈ X0∗ ∩ X1∗ such that ky∗ k X ∗ ≤ t and 1

 

1 p0



0

x0,opt p0 = hy∗ , x0,opt i − 10 ky∗ k p0∗ ; X p X0 0 0

t x1,opt = hy∗ , x1,opt i.

X1

Proof. Given a0 ∈ X0 and a1 ∈ X1 such that x = a0 + a1 , the L p0 ,1 – functional can be defined as L p0 ,1 (t, x; X0 , X1 ) =

inf

y ∈ X0 ∩ X1

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

where functions S and R are defined on X0 ∩ X1 by S (y) =

1 p k a0 − yk X00 p0

R ( y ) = t k y + a 1 k X1 .

and

Moreover L p0 ,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 (dom S − dom R) = X0 ∩ X1 .

(45)

λ ≥0

The respective conjugate functions S∗ and R∗ of S and R are given in Lemma 5. We have that ( p00 1 z if z ∈ X0∗ ; k k ∗ + h z, a0 i 0 ∗ X p0 S (z) = 0
+∞ if z ∈ X0∗ + X1∗ \ X0∗ . and ( ∗

R (z) =

h−z, a1 i if kzk X ∗ ≤ t, z ∈ B X1∗ (0; t) ; 1
+∞ if z ∈ X0∗ + X1∗ \B X1∗ (0; t) . 19

The conjugate functions ϕ0∗ of ϕ0 and ϕ1∗ of ϕ1 are defined on X0∗ and X1∗ and are equal to ϕ0∗ (z) =

1 p0 kzk X0∗ , ∀z ∈ X0∗ , 0 0 p0

ϕ1∗ (z) =



0 +∞



ϕ0∗ (−z) + hz, a0 i +∞

if z ∈ X0∗ ;
if z ∈ X0∗ + X1∗ \ X0∗ .

(46)

ϕ1∗ (z) + h−z, a1 i +∞

if z ∈ X1∗ ;
if z ∈ X0∗ + X1∗ \ X1∗ .

(47)

and if z ∈ B X1∗ (0; t) ; if z ∈ X1∗ \B X1∗ (0; t) .

It is clear that ∗

S (z) = and R∗ (z) =



By (45), (46) and (47), we conclude from Theorem 2.1 that L p0 ,1 - 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 is optimal for the L p0 ,1 - functional if and only if there exists y∗ ∈ X0∗ ∩ X1∗ such that ky∗ k X ∗ ≤ t and 1

  

3.2.3

1 p0

0

x0,opt p0 = hy∗ , x0,opt i − 10 ky∗ k p0∗ ; X p0 X0 0

t x1,opt = hy∗ , x1,opt i.

X1

Case p0 = 1 and 1 < p1 < +∞

Consider the L– functional  L1,p1 (t, x; X0 , X1 ) =

inf

x = x0 + x1

k x 0 k X0 +

t p kx k 1 p 1 1 X1

 ,

(48)

where 1 < p1 < +∞. Let ϕ0 and ϕ1 be functions defined on the sum X0 + X1 as follows  kuk X0 if u ∈ X0 ; ϕ0 ( u ) = (49) +∞ if u ∈ ( X0 + X1 ) \ X0 . and ( ϕ1 ( u ) =

t p1

p

kuk X11 if u ∈ X1 ; +∞ if u ∈ ( X0 + X1 ) \ X1 .

20

(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) = kuk X0 and ϕ1 (u) =

t p kuk X11 . p1

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∗ ∩ X1∗ such that ky∗ k X ∗ ≤ 1 and 0 

x0,opt = hy∗ , x0,opt i;  X0

p0

 pt x1,opt p1 = hy∗ , x1,opt i − t0 zt X1∗ . p X 1

1

1

0

Proof. Given a0 ∈ X0 and a1 ∈ X1 such that x = a0 + a1 , the L1,p1 – functional can be defined by L1,p1 (t, x; X0 , X1 ) =

inf

y ∈ X0 ∩ X1

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

where functions S and R are defined on X0 ∩ X1 by S ( y ) = k a 0 − y k X0

R (y) =

and

t p ky + a1 k X11 , p1

Moreover L p0 ,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 (dom S − dom R) = X0 ∩ X1 .

(51)

λ ≥0

The respective conjugate functions S∗ and R∗ of S and R are given in Lemma 5. We have that ( hz, a0 i if kzk X ∗ ≤ 1, z ∈ B X0∗ (0; 1) ; ∗ 0
S (z) = +∞ if z ∈ X0∗ + X1∗ \B X0∗ (0; 1) . and ( ∗

R (z) =

t p10

z p0

1∗ + h−z, a1 i t X 1

if z ∈ X1∗ ;


+∞ if z ∈ X0∗ + X1∗ \ X1∗ .

The conjugate functions ϕ0∗ of ϕ0 and ϕ1∗ of ϕ1 are defined on X0∗ and X1∗ respectively and are equal to  0 if z ∈ B X0∗ (0; 1) ; ϕ0∗ (z) = +∞ if z ∈ X0∗ \B X0∗ (0; 1) . 21

and ϕ1∗ (z) =

0 t

z p1

, ∀z ∈ X1∗ . p10 t X1∗

It is clear that 

∗

S (z) =

ϕ0∗ (−z) + hz, a0 i +∞

if z ∈ X0∗ ;
if z ∈ X0∗ + X1∗ \ X0∗ .

(52)

ϕ1∗ (z) + h−z, a1 i +∞

if z ∈ X1∗ ;
if z ∈ X0∗ + X1∗ \ X1∗ .

(53)

and 

∗

R (z) =

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∗ ∈ X0∗ ∩ X1∗ such that ky∗ k X ∗ ≤ 1 and 0 

x0,opt = hy∗ , x0,opt i;  X0

p0  pt x1,opt p1 = hy∗ , x1,opt i − t0 zt X1∗ . p X 1

3.2.4

1

0

1

Optimal decomposition for the K– functional

Consider the K– functional K (t, x; X0 , X1 ) =

inf

x = x0 + x1



 k x 0 k X0 + t k x 1 k X1 .

It can be expressed as the following infimal convolution K (t, x; X0 , X1 ) = ( ϕ0 ⊕ ϕ1 ) ( x ) , where functions ϕ0 and ϕ1 are defined on the sum X0 + X1 as follows  kuk X0 if u ∈ X0 ; ϕ0 ( u ) = +∞ if u ∈ ( X0 + X1 ) \ X0 .

(54)

and  ϕ1 ( u ) =

t k u k X1 +∞

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) = kuk X0 and ϕ1 (u) = t kuk X1 . 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 ) = x0,opt X + t x1,opt X . 0

1

22

Theorem 3.6. The decomposition x = x0,opt + x1,opt is optimal for the K– functional if and only if there exists y∗ ∈ X0∗ ∩ X1∗ such that ky∗ k X ∗ ≤ 1, ky∗ k X ∗ ≤ t and 0

(

1



x0,opt = hy∗ , x0,opt i;

X0

t x1,opt X = hy∗ , x1,opt i. 1

Proof. For given a0 ∈ X0 and a1 ∈ X1 such 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 ∩ X1 by S ( y ) = k a 0 − y k X0

R ( y ) = t k y + a 1 k X1 ,

and

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 (dom S − dom R) = X0 ∩ X1 .

(56)

λ ≥0

The respective conjugate functions S∗ and R∗ of S and R are given in Lemma 5. We have that ( hz, a0 i if kzk X ∗ ≤ 1, z ∈ B X0∗ (0; 1) ; ∗ 0
S (z) = +∞ if z ∈ X0∗ + X1∗ \B X0∗ (0; 1) . and ( ∗

R (z) =

h−z, a1 i if kzk X ∗ ≤ t, z ∈ B X1∗ (0; t) ; 1
+∞ if z ∈ X0∗ + X1∗ \B X1∗ (0; t) .

The conjugate functions ϕ0∗ of ϕ0 and ϕ1∗ of ϕ1 are defined on X0∗ and X1∗ and are equal to  0 if z ∈ B X0∗ (0; 1) ; ϕ0∗ (z) = +∞ if z ∈ X0∗ \B X0∗ (0; 1) . and ϕ1∗



0 if z ∈ B X1∗ (0; t) ; +∞ if z ∈ X1∗ \B X1∗ (0; t) .



ϕ0∗ (−z) + hz, a0 i +∞

(z) =

It is clear that ∗

S (z) =

if z ∈ X0∗ ;
if z ∈ X0∗ + X1∗ \ X0∗ . 23

(57)

and R∗ (z) =



ϕ1∗ (z) + h−z, a1 i +∞

if z ∈ X1∗ ;
if z ∈ X0∗ + X1∗ \ X1∗ .

(58)

By (56), (57) and (58), we conclude from Theorem 2.1 that K– 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 is optimal for the K- functional if and only if there exists y∗ ∈ X0∗ ∩ X1∗ such that ky∗ k X ∗ ≤ 1, 0 ky∗ k X ∗ ≤ t and 1 (

x0,opt = hy∗ , x0,opt i;

X0

t x1,opt X = hy∗ , x1,opt i. 1

4

4.1

Geometry of optimal decomposition for the Couple (` p , X ) on Rn 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 ) on Rn . We consider the couple (` p , X ), 1 < p < +∞, in finite dimensional case. More precisely, we consider the space Rn , with the norm k x k` p and some Banach space X on Rn 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, x i : x ∈ X; k x k X ≤ 1} , where h · , · i is the dual product between X ∗ and X. We will consider the L– functional   1 p p L p,1 (t, x; ` , X ) = inf (59) k x0 k ` p + t k x1 k X . x = x0 + x1 p In subsection 4.2 we present a result (Theorem 4.1) which gives the characterization 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 subsection 4.4 presents a geometrical illustration of Theorem 4.1 for the case p = 3 in the plane.

4.2

Optimal decomposition for L p,1 (t, x; ` p , X )– functional

In this subsection we use Key Lemma to characterize the optimal decomposition for the L– functional (59). In terms of infimal convolution, the L- func24

tional (59) can be expressed as L p,1 (t, x; ` p , X ) = ( F0 ⊕ F1 ) ( x ) , where F0 (u) =

1 p kuk` p p

and

F1 (u) = t kuk X .

(60)

We recall that 1 < p < ∞. Thus we have that ∂F0 (u) consists of the single vector n o ∂F0 (u) = ∇ F0 (u) = |u| p−1 sgn (u) . (61) Let us denote by tB X ∗ the ball of X ∗ of radius t with center at the origin: t B X ∗ = { y ∈ Rn : k y k X ∗ ≤ t } , and let p0 be the conjugate of p. i.e., 1p + p10 = 1. Then we can obtain the conjugate functions of F0 and F1 from Lemma 5 as follows:  1 0 if y ∈ tB X ∗ ; p0 F0∗ (y) = 0 kyk p0 and F1∗ (y) = (62) ` +∞ else. p Theorem 4.1. Let F0 and F1 be defined on Rn by (60). Let us define the set Ω by Ω = {v ∈ Rn : ∇ F0 (v) ∈ tB X ∗ } ,

(63)

where tB X ∗ 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 L p,1 – functional is given by x0,opt = x and x1,opt = 0. (2) If x ∈ / Ω then x = x0,opt + x1,opt is optimal decomposition for L p,1 – functional if and only if


(a) ∇ F0 x0,opt X ∗ = t





(b) x1,opt , ∇ F0 x0,opt = x1,opt X ∇ F0 x0,opt X ∗ = t x1,opt X . Proof. For (1), since L p,1 – functional is convex and dom L p,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 ) ∈ tB X ∗ . It follows from Proposition 2.1 that F0 ( x ) = hy, x i − F0∗ (y) .

(64)

As y ∈ tB X ∗ therefore from (62) we have F1∗ (y) = 0. 25

It then follows trivially from definition of F1 that 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 L p,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∗ , x i − 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∗ ∈ tB X ∗ .

(68)

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

∇ F0 ( x ) ∈ tB X ∗ , which means, from (63), that x ∈ Ω. If x0,opt = 0 then we must have x1,opt = X and (

F0∗ (y∗ ) =

1 p0

p0

ky∗ k 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,opt and x1,opt are different from zero. In this case, by Key Lemma, there exists element y∗ which satisfies
 F0 x0,opt
= hy∗ , x0,opt i − F0∗ (y∗ ) ; (69) F1 x1,opt = hy∗ , x1,opt i − F1∗ (y∗ ) . The first equation of (69) is equivalent (by Proposition 2.1 and (61)) to
y∗ ∈ ∂F0 x0,opt . Since o p −1

n ∂F0 x0,opt = ∇ F0 x0,opt = x0,opt sgn( x0,opt ) , then
y∗ = ∇ F0 x0,opt . 26

The second equation of (69) can be rewritten as



 t x1,opt X = y∗ , x1,opt , with ky x,F k X ∗ ≤ t. Therefore we obtain that




t x1,opt X = ∇ F0 x0,opt , x1,opt

and




∇ F0 x0,opt ∗ ≤ t. X

(70)

It follows then by definition of the dual norm that




t x1,opt X ≤ ∇ F0 x0,opt X ∗ x1,opt X .

(71)

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


t ≤ ∇ F0 x0,opt X ∗ .

(72)

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




∇ F0 x0,opt ∗ = t. X

(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 x0,opt X ∗ x1,opt X

= t x1,opt . X

Which proves (b). Next let us assume that x ∈ / Ω and that a decomposition x = x˜0 + x˜1 satisfies ( a) and (b) and show that x = x˜0 + x˜1 is optimal. It follows from ( a) that the element y = ∇ F0 ( x˜0 ) = ∂F0 ( x˜0 ) satisfies F0 ( x˜0 ) = hy, x˜0 i − F0∗ (y) .

(74)

From (b), we conclude that t k x˜1 k X = h∇ F0 ( x˜0 ) , x˜1 i .

(75)

Since kyk X ∗ = k∇ F0 ( x˜0 )k X ∗ = t then it follows from expression of F1∗ that F1∗ (∇ F0 ( x˜0 )) = 0. Therefore (75) can be written as F1 ( x˜1 ) = t k x˜1 k X = h∇ F0 ( x˜0 ) , x˜1 i − F1∗ (∇ F0 ( x˜0 )) .

(76)

Considering (74) and (76), we conclude from Key Lemma, that the decomposition x = x˜0 + x˜1 is optimal for the L p,1 -functional. Proposition 4.1 (Geometrical interpretation of Theorem 4.1 when x ∈ / Ω). If x∈ / Ω then Theorem 4.1 means that x1,opt is orthogonal to the supporting hyperplane
to tB X ∗ which goes through the point ∇ F0 x0,opt . Proof. Suppose that condition (b) in Theorem 4.1 is satisfied. i.e.,







x1,opt , ∇ F0 x0,opt = x1,opt X ∇ F0 x0,opt X ∗ = t x1,opt X . 27

(77)

By definition of the dual norm we have that



 t x1,opt X = t sup y, x1,opt = t sup k y k X ∗ ≤1



 y, x1,opt =

ktyk X ∗ ≤t

sup



 ty, x1,opt .

ktyk X ∗ ≤t

Putting z = ty, we have





 t x1,opt X = sup z, x1,opt = sup z, x1,opt . z∈tB X ∗

kzk X ∗ ≤t

By (77), we then have






 t x1,opt X = sup z, x1,opt = ∇ F0 x0,opt , x1,opt . z∈tB X ∗

Let us denote by Pv u = we have that

hu,vi v, kvk2`2

the vector projection of u onto v. As x1,opt 6= 0



supz∈tB ∗ z, x1,opt




X

= sup z = ∇ F x

P

P

2, x x 0 0,opt 1,opt 1,opt 2

x1,opt ` ` z∈tB X∗

X

and therefore



sup P x1,opt z z∈tB X ∗

`2




= P x1,opt ∇ F0 x0,opt 2 . `


Therefore, among all elements from tB X ∗ , the element ∇ F0 x0,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 p 6= 2, the ball tB X ∗ and the set Ω are two different sets. This makes the situation more complicated.

28


Figure 1: Geometry of optimal decomposition for L p,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     1 2 2 L2,1 t, x; ` , X = inf (78) k x 0 k `2 + t k x 1 k X . x = x0 + x1 2 The function F0 is therefore F0 (u) =

1 kuk2`2 , 2

(79)

and F1 remains unchanged. i.e., F1 (u) = t kuk X .

(80)

It is clear that ∂F0 (u) = ∇ F0 (u) = {|u| sgn (u)} = u, and the sets Ω and tB X ∗ coincide: Ω = tB X ∗ . We obtain the following corollary to Theorem 4.1

29

Corollary 1. Let F0 and F1 be defined on Rn by (79) and (80). Let us define the set Ω by Ω = tB X ∗ , where tB X ∗ 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,opt is optimal decomposition for (78) if and only if

(a) x0,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 tB X ∗ 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 k x − x0 k`2 = x − x0,opt `2 . k x0 k X ∗ ≤ t

We illustrate Corollary 1 in Figure 2.

Figure 2: Illustration of Corollary 1.

30

4.4

Geometrical illustration

In this section we give a concrete illustration of the geometry of optimal decomposition 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 `1 by the rotation matrix   cos θ sin θ Rθ = , − sin θ cos θ for θ = 30 degrees. i.e.,  √ 3/2 √1/2 . Rθ = 3/2 −1/2 We have that the norm in X of x = [ x1 , x2 ] T is given by √ √

3 1 1 3

−1 x − x + x + x2 . k x k X = Rθ x 1 = 2 1 2 2 2 1 2 ` In this case we have that i h ∇ F0 (u) = |u1 |2 sgn(u1 ), |u2 |2 sgn(u2 ) . So the set Ω can be written as

h  iT

2 2 2

Ω = v ∈ R : |v1 | sgn(v1 ), |v2 | sgn(v2 )

 ≤t , X∗

where the norm in X ∗ of y = [y1 , y2 ] T is given by ) ( √ √

3 1 1 3

−1 y − y , y + y2 . kyk X ∗ = Rθ y ∞ = max 2 1 2 2 2 1 2 ` 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.,     1 1 3 x + t x L3,1 t, x; `3 , X = inf k 0 k `3 k 1 k X = k x k3`3 . x = x0 + x1 3 3 The case (2) of Theorem 4.1 is illustrated in Figure 3. So we see that in this situation the set Ω is not convex. The unit ball of X ∗ is Rθ (B`∞ ), where B`∞ 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 tB X ∗ (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 tB X ∗ which goes
through y∗ = ∇ F0 x0,opt . This is illustrated by the two bold parallel lines.

31

4

x

area 3

3

x1opt 2

1

x

0

O

0opt

Ω

−1

area 4 −2

area 1

−3

−4 −5 5

−4

−3

−2

−1

0

1

2

3

4

5

area 3

4

3

2

y∗ 1

O

0

tBX*

area 4

−1

−2

−3

area 1

−4

−5

−6

−4

−2

0

2

4

6

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

32

5

Attouch–Brezis theorem and existence of optimal decomposition for infimal convolution

We recall that for a regular couple
( X0 , X1 ), the dual spaces X0∗ and X1∗ also form a compatible couple X0∗ , X1∗ (see for example the book (Bennett and Sharpley, 1988)). In this situation, both X0∗ and X1∗ are linearly and continu
ously embedded in ( X0 ∩ X1 )∗ and we will call X0∗ , X1∗ 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∗ , X1∗ ) = inf k y0 k X ∗ + t k y1 k X ∗ , y = y0 + y1

0

1

where the infimum extends over all representations of y = y0 + y1 for y0 ∈ X0∗ and y1 ∈ X1∗ . Define the functions ψ0 , ψ1 : X0∗ + X1∗ −→ R ∪ {+∞} as follows 

kyk X ∗ if y ∈ X0∗ ; 0
+∞ if y ∈ X0∗ + X1∗ \ X0∗ .



t kyk X ∗ 1 +∞

ψ0 (y) = and ψ1 (y) =

if y ∈ X1∗ ;
if y ∈ X0∗ + X1∗ \ X1∗ .

In this way, we can write the K– functional in terms of infimal convolution of ψ0 and ψ1 . K (t, y; X0∗ , X1∗ ) = (ψ0 ⊕ ψ1 ) (y) =

inf

y = y0 + y1

(ψ0 (y0 ) + ψ1 (y1 )) ,

where the infimum extends over all representations of y = y0 + y1 for y0 and y1 in X0∗ + X1∗ . Let us define the functions ϕ0 , ϕ1 : X0 ∩ X1 −→ R ∪ {+∞} as follows  0 if k x k X0 ≤ 1; ϕ0 ( x ) = +∞ if k x k X0 > 1. and  ϕ1 ( x ) =

0 if k x k X1 ≤ t; +∞ if k x k X1 > t.

By Lemma 5, it is clear that the functions ϕ0 and ϕ1 have for conjugates, the functions ψ0 and ψ1 respectively. 33

Theorem 5.1. Let y be an element from X0∗ + X1∗ and a parameter t > 0 be given. Then there exists an optimal decomposition for the K– functional   K (t, y; X0∗ , X1∗ ) = inf k y0 k X ∗ + t k y1 k X ∗ . y = y0 + y1

0

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 semicontinuous. Moreover, since dom ϕ0 = B X0 (0; 1) ∩ ( X0 ∩ X1 ) and dom ϕ1 = B X1 (0; t) ∩ ( X0 ∩ X1 ) then we have that

∪λ≥0 λ (dom ϕ0 − dom ϕ1 ) = X0 ∩ X1 . Therefore, by Attouch–Brezis Theorem, we have that K (t, y; X0∗ , X1∗ ) = (ψ0 ⊕ ψ1 ) (y) = ( ϕ0∗ ⊕ ϕ1∗ ) (y) = ( ϕ0 + ϕ1 )∗ (y) , and there exists y0 ∈ X0∗ and y1 ∈ X1∗ with y = y0 + y1 such that K (t, y; X0∗ , X1∗ ) = ϕ0∗ (y0 ) + ϕ1∗ (y1 ) Equivalently K (t, y; X0∗ , X1∗ ) = ky0 k X ∗ + t ky1 k X ∗ . 0

5.2

1

Existence of optimal decomposition for the L– functional

As in the previous section, we let y be an element from X0∗ + X1∗ and a parameter t >
0 be given. We consider the L– functional on the conjugate couple X0∗ , X1∗ .   1 t p0 p1 ∗ ∗ L p0 ,p1 (t, y; X0 , X1 ) = inf k y0 k X ∗ + ky k ∗ , y = y0 + y1 0 p0 p 1 1 X1 where the infimum extends over all representations of y = y0 + y1 for y0 ∈ X0∗ and y1 ∈ X1∗ , and where 1 < p0 , p1 < +∞. Define the functions ψ0 , ψ1 : X0∗ + X1∗ −→ R ∪ {+∞} as follows ( p0 1 ∗ p0 k y k X0∗ if y ∈ X0 ;
ψ0 (y) = +∞ if y ∈ X0∗ + X1∗ \ X0∗ . and ( ψ1 (y) =

t p1

p

kyk X1∗ if y ∈ X1∗ ; 1
+∞ if y ∈ X0∗ + X1∗ \ X1∗ .

In this way, we can write the L– functional in terms of infimal convolution of ψ0 and ψ1 . L p0 ,p1 (t, y; X0∗ , X1∗ ) = (ψ0 ⊕ ψ1 ) (y) =

34

inf

y = y0 + y1

(ψ0 (y0 ) + ψ1 (y1 )) ,

where the infimum extends over all representations of y = y0 + y1 for y0 and y1 in X0∗ + X1∗ . Let us define the functions ϕ0 , ϕ1 : X0 ∩ X1 −→ R ∪ {+∞} as follows

0 t 1 p0

x p1 ϕ0 ( x ) = 0 k x k X00 and ϕ1 ( x ) = 0 , p0 p 1 t X1 where p0j is such that

1 pi

+ p10 = 1, j = 0, 1. By Lemma 5, it is clear that the i functions ϕ0 and ϕ1 have for conjugates, the functions ψ0 and ψ1 respectively. Theorem 5.2. Let y be an element from X0∗ + X1∗ and a parameter t > 0 be given. Then there exists an optimal decomposition for the L– functional   t 1 p p L p0 ,p1 (t, y; X0∗ , X1∗ ) = inf ky1 k X1∗ , ky0 k X0∗ + y = y0 + y1 0 1 p0 p1 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 semicontinuous. Moreover, since dom ϕ0 = dom ϕ1 = X0 ∩ X1 then we have that

∪λ≥0 λ (dom ϕ0 − dom ϕ1 ) = X0 ∩ X1 . Therefore, by Attouch–Brezis Theorem, we have that L p0 ,p1 (t, y; X0∗ , X1∗ ) = (ψ0 ⊕ ψ1 ) (y) = ( ϕ0∗ ⊕ ϕ1∗ ) (y) = ( ϕ0 + ϕ1 )∗ (y) , and there exists y0 ∈ X0∗ and y1 ∈ X1∗ with y = y0 + y1 such that L p0 ,p1 (t, y; X0∗ , X1∗ ) = ϕ0∗ (y0 ) + ϕ1∗ (y1 ) Equivalently L p0 ,p1 (t, y; X0∗ , X1∗ ) =

5.3

1 t p p ky0 k X0∗ + ky k 1∗ . 0 p0 p 1 1 X1

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∗ . E p0 (t, y; X0∗ , X1∗ ) =

1 p inf ky − y1 k X0∗ , 1 < p < +∞. 0 ∗ p y 1 ∈ X1 0 k y1 k X ∗ ≤ t 1

In this case we define the functions ψ0 , ψ1 : X0∗ + X1∗ −→ R ∪ {+∞} as follows ( p0 1 ∗ p0 k y k X0∗ if y ∈ X0 ;
ψ0 (y) = +∞ if y ∈ X0∗ + X1∗ \ X0∗ . 35

and ( ψ1 (y) =

0 +∞

if y ∈ tB X1∗ , kyk X ∗ ≤ t;
1 if y ∈ X0∗ + X1∗ \tB X1∗ .

In this way, we can write the E– functional in terms of infimal convolution of ψ0 and ψ1 . E p0 (t, y; X0∗ , X1∗ ) = (ψ0 ⊕ ψ1 ) (y) =

inf

y = y0 + y1

(ψ0 (y0 ) + ψ1 (y1 )) ,

where the infimum extends over all representations of y = y0 + y1 for y0 and y1 in X0∗ + X1∗ . Let us define the functions ϕ0 , ϕ1 : X0 ∩ X1 −→ R ∪ {+∞} as follows ϕ0 ( x ) =

1 p0 k x k X00 and ϕ1 ( x ) = t k x k X1 . p00

By Lemma 5, it is clear that the functions ϕ0 and ϕ1 have for conjugates, the functions ψ0 and ψ1 respectively. Theorem 5.3. Let y be an element from X0∗ + X1∗ and a parameter t > 0 be given. Then there exists an optimal decomposition for the E– functional E p0 (t, y; X0∗ , X1∗ ) =

1 p inf ky − y1 k X0∗ , 1 ≤ p < +∞. 0 ∗ p y 1 ∈ X1 0 k y1 k X ∗ ≤ t 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 semicontinuous. Moreover, since dom ϕ0 = dom ϕ1 = X0 ∩ X1 then we have that

∪λ≥0 λ (dom ϕ0 − dom ϕ1 ) = X0 ∩ X1 . Therefore, by Attouch–Brezis Theorem, we have that E p0 (t, y; X0∗ , X1∗ ) = (ψ0 ⊕ ψ1 ) (y) = ( ϕ0∗ ⊕ ϕ1∗ ) (y) = ( ϕ0 + ϕ1 )∗ (y) , and there exists y0 ∈ X0∗ and y1 ∈ tB X1∗ with y = y0 + y1 such that E p0 (t, y; X0∗ , X1∗ ) = ϕ0∗ (y0 ) + ϕ1∗ (y1 ) Equivalently E p0 (t, y; X0∗ , X1∗ ) =

1 p ky − y1 k X0∗ . 0 p0

36

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 + x1 of 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) if u ∈ X0 ; ϕ0 ( u ) = +∞ 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 ϕ0 and ϕ1 could stop to be lower semicontinuous even if ϕ0 and ϕ1 are. 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 ∩ X1 with values in R ∪ {+∞} 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 L p,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.

37

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 Mathematics, 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. J. Niyobuhungiro, 2013a. Geometry of Optimal Decomposition for the Couple
`2 , X on Rn . LiTH-MAT-R–2013/06–SE, Linköping University. J. Niyobuhungiro, 2013b. Optimal decomposition for infimal convolution on Banach Couples. LiTH-MAT-R–2013/07–SE, Linköping University.

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