On Bi-dimensional Second Variation

Jurancy Ere´u^∗, Jos´e Gim´enez, Nelson Merentes

On Bi-dimensional Second Variation

Abstract. In this paper we present the concept of bounded second variation of a real valued function defined on a rectangle in R². We use Hardy-Vitali type technics in the plane in order to extend the classical notion of function of bounded second variation on intervals of R. We introduce the class BV²(I_a^b), of all functions of bounded second variation on a rectangle I_a^b ⊂ R², and show that this class can be equipped with a norm with respect to which it is a Banach space. Finally, we present two results that show that integrals of functions of first bounded variation (on I_a^b) are in BV²(I^b_a).

2000 Mathematics Subject Classification: 26B30, 26B35.

Key words and phrases: Functions of Bounded Second Variation, Functions of Bounded Variation .

1. Introduction. In 1881, C. Jordan ([6]) introduces the concept of function of bounded variation after a rigorous study of the proof given by Dirichlet ([3]) on his work on the convergence of the Fourier series of a function and showed that a function is of bounded variation if and only if it is the difference of two monotone functions. The notion was extended to functions defined on the plane in 1905 by Hardy and Vitali ([1], [4], [15]). In 1908, De La Vall´ee Poussin ([14]), introduced the notion of second variation of a function showing that a function is of second bounded variation if and only if it is the difference of two convex functions. The subsequent denomination of this class as functions of bounded convexity, apparently, is due to A. W. Roberts and D. E. Varberg ([10] and [11]). A few years later, in 1911, F.

Riesz ([9]) proved that a function F is of bounded second variation on an interval [a, b] if and only if it is the indefinite Lebesgue integral of a function f of bounded variation.

In this work we present an extension of the classical notion of function of se- cond bounded variation, defined on a closed real interval, to functions defined on a

∗This research has been partly supported by the Central Bank of Venezuela. We want to give thanks to the library staff of B.C.V. for compiling the references.

rectangle Υ in R² and introduce the Banach space, BV²([a, b], Υ), of all functions of bounded second variation on Υ. We show that there is a polynomial which is a universal mayorant for functions in the unit ball of BV²([a, b], Υ). In the last part, we present two results that show that integrals of functions of first bounded variation are in the class introduced. This last contributions complement some of the work done by F.A. Talalyan in [13] and by M. Wróbel in [16].

2. Preliminaries. Given an interval [a, b] ⊂ R we will use the notation Π([a, b]) to denote the set of all partitions of [a, b], whereas Π3([a, b]) will denote the subset of Π([a, b]) consisting of partitions of [a, b] with at least three points.

Recall that a function u : [a, b] −→ R is said to be of bounded variation (in the sense of Jordan) if

V (u; [a, b]) := sup

ξ

Xn j=1

|u(t^j) − u(tj−1)| < ∞,

where the supremum is taken over the set of all partitions ξ = {a = t^o< t1<· · · <

tn= b} ∈ Π([a, b]).

The notion of bounded second variation in the sense of De La Vall´ee Poussin is defined as follows:

A function u : [a, b] → R is of bounded second variation if and only if

V²(u; [a, b]) := sup

π∈Π3([a,b]) mX−2

i=0

|u[tⁱ⁺¹, ti+2] − u[tⁱ, ti+1]| < ∞,

where

(1) u[ti+1, ti+2] := u(ti+2) − u(tⁱ⁺¹)

ti+2− tⁱ⁺¹ , i = 0, ..., m− 2.

The class of all the functions of bounded second variation (on [a, b]), in the sense of De La Vall´ee Poussin, is denoted by BV²([a, b]).

The following are known properties of functions in BV²([a, b]) ([8], [10] and [12]).

Proposition 2.1 Let u ∈ BV²([a, b]).

(1) If v ∈ BV²([a, b])and λ is any real constant, then V²(λu; [a, b]) = |λ|V²(u; [a, b])

and

V²(u + v; [a, b]) ¬ V²(u; [a, b]) + V²(v; [a, b]).

(2) (Monotonicity) If a < c < d < b, then V²(u; [c, d]) ¬ V²(u; [a, b]).

(3) (Semi-additivity) If a < c < b, then u ∈ BV²([a, c]), u ∈ BV²([c, b]) and V²(u; [a, b]) ­ V²(u; [a, c]) + V²(u; [c, b]).

(4) u[y0, y1]is bounded for all y0, y1∈ [a, b].

(5) u is Lipschitz and therefore absolutely continuous on [a, b].

(6) u ∈ BV²([a, b])if and only if u = u1− u²,where u1, u2 are convex functions.

(7) A necessary and sufficient condition for a function F to be the integral of a function f ∈ BV ([a, b]) is that F ∈ BV²([a, b]).

(8) If u is twice differentiable with u⁰⁰ integrable on [a, b] then u ∈ BV²([a, b])and V²(u; [a, b]) =Rb

a |u⁰⁰(t)|dt.

2.1. Bi-dimensional variation.

Let a = (a1, a2), b = (b1, b2) ∈ R², such that a1< b1and a2< b2, In the sequel, we will use the symbol I_a^b to denote the basic rectangle [a1, b1] × [a², b2].

For ξ := {tⁱ}ⁿi=0 ∈ Π([a¹, b1]) and η := {s^j}^mj=0 ∈ Π([a², b2]) we will use the following notation:

(i) ∆10u(ti, s) := u(ti, s)− u(tⁱ−1, s) for s∈ [a², b2] fixed.

(ii) ∆01u(t, sj) := u(t, sj) − u(t, sj−1) for t ∈ [a¹, b1] fixed.

(iii) ∆11u(ti, sj) := u(ti−1, sj−1) + u(tisj) − u(tⁱ−1, sj) − u(tⁱ, sj−1).

Definition 2.2 Let u : I_a^b→ R.

(A) If s ∈ [a², b2] is fixed, define the variation in the sense of Jordan of u in [a1, b1] × {s} by

V_[a1,b1](u(·, s)) := sup

ξ

Xn i=1

|∆¹⁰u(ti, s)| ,

where the supremum is taken over the set of all partitions ξ ∈ Π([a¹, b1]).

(B) Similarly for t ∈ [a¹, b1], definethe variation in the sense of Jordan of u in {t} × [a², b2] by

V[a2,b2](u(t, ·)) := sup

η

Xm j=1

|∆⁰¹u(t, sj)| ,

where the supremum is taken over the set of all partitions η ∈ Π([a², b2]).

(C) Define the variation of u, in the sense of Hardy-Vitali as

V (u, I_a^b) := sup

(ξ,η)

Xn i=1

Xm j=1

|∆¹¹u(tisj)| ,

where the supremum is taken over the set of all partitions (ξ, η) ∈ Π([a¹, b1])×

Π([a2, b2]).

(D) The total variation of u on I_a^b is defined as

T V (u, I_a^b) := V_[a1,b1](u(·, s⁰)) + V_[a₂,b₂](u(t0,·)) + V (u, Ia^b), where (t0, s0) is any point in I_a^b.

(E) u is said to be of total bounded variation if T V (u, I_a^b) < ∞. The class of all functions u : I_a^b→ R of total bounded variation is denoted as BV (Ia^b).

For the definition of the total variation of u in (D ) it is irrelevant the choice of the point (t0, s0) ∈ Ia^b. This fact is one of the consequences of the next proposition, which proof follows in a straightforward manner from the definitions and will be omitted.

Proposition 2.3 Suppose V (f, I_a^b) < ∞. Then

V_[a2,b2](u(t2,·)) − V[a2,b2](u(t1,·))_{¬ V (f; [t}1, t2] × [a2, b2]) ¬ V (f; Ia^b)

for all a1¬ t¹< t2¬ b¹.

A similar estimate holds for V_[a1,b1](u(·, s)).

Theorem 2.4 ([2]) The space BV (I_a^b) is a Banach algebra with respect to the norm

kuk := |u(a)| + T V (u, Ia^b).

For further information about properties of functions in BV (I_a^b) the reader is referred to [5] (c.f. also [1, 2]).

3. Bi-dimensional Second Variation. Assume u : I_a^b→ R. Let^{ξ:= {ti}n}i=0∈ Π₃([a₁, b₁])andη:= {sj}^m_j=0∈ Π3([a₂, b₂]). We will use the following notations:

(i) For each s ∈ [a2, b2] fixed, set

u[ti+1, ti+2; s] := u(ti+2, s)− u(tⁱ⁺¹, s) ti+2− tⁱ⁺¹ ,

∆10u[ti+1, ti+2; s] := u[ti+1, ti+2; s] − u[tⁱ, ti+1; s] and

V_[a1,b1]² (u(·, s)) := sup

ξ nX−2

i=0

|∆¹⁰u[ti+1, ti+2; s]| . (ii) Similarly, for each fixed t ∈ [a¹, b1]

u[t; sj+1, sj+2] := u(t, sj+2) − u(t, s^j+1) sj+2− s^j+1 ,

∆01u[t; sj+1, sj+2] := u[t; sj+1, sj+2] − u[t; s^j, sj+1] and (2)

V_[a2,b2]² (u(t, ·)) := sup

η mX−2

j=0

|∆⁰¹u[t; sj+1, sj+2]| .

Definition 3.1 Let u : I_a^b→ R.

The second variation of u on I_a^bin the sense of De La Vall´ee Poussin, is defined as

V²(u, I_a^b) := sup

(ξ,η) V²

Iba (u, ξ × η), where V²

Iba (u, ξ × η) :=ⁿP⁻²

i=0 mP−2

j=0

∆²₁₁u(ti, sj) with

∆²₁₁u(ti, sj) : = ∆01u[ti+2; sj+1, sj+2] − ∆01u[ti+1; sj+1, sj+2] t_i+2− ti+1

−

∆01u[ti+1; sj+1, sj+2] − ∆01u[ti; sj+1, sj+2] t_i+1− ti



,

the supremum being taken over the set of all partitions (ξ, η) ∈ Π³([a1, b1]) × Π3([a2, b2]).

The total bi-dimensional second variation of u in the sense of De La Vall´ee Poussin, is defined by

T V²(u, I_a^b) := V²(u, I_a^b) (3)

+ V²

[a1,b1](u(·, a²)) + V²

[a1,b1](u(·, b²)) + V_[a2,b2]² (u(a1,·)) + V_[a2,b2]² (u(b1,·))

and a function u : I_a^b → R, is said to be of bounded (bi-dimensional) second variation if

T V²(u, I_a^b) < ∞.

The class of all functions u ∈ R^Iab of bounded second variation is denoted by V²(I_a^b); that is,

V²(I_a^b) :=

u : I_a^b→ R/ T V²(u, I_a^b) < ∞ .

Remark 3.2 In order to define ∆²₁₁we only considered combinations of expressions of type ∆01 (see (2)). Actually, ∆²₁₁ could have been also defined in the following way:

For eachj∈ {0, 2, . . . , m − 2} fixed, define the function∆²₀₁[u]_j : [a1, b1] → Ras the (second order) difference quotient variation

(4) ∆²₀₁[u]_j(t) := ∆₀₁u[t; sj, sj+1].

Then, it is readily seen that

(5) ∆²₁₁u(ti, sj) = ∆²₀₁[u]_j+1[ti+1, ti+2] − ∆²01[u]_j+1[ti, ti+1]

(we recall that the notation f[a, b] stands for the difference quotientf[a, b] := (f(b) − f(a))/(b − a)(see (1))).

If, instead of proceeding as set forth above, we were chosen to consider the subse- quent difference quotients variations of expressions of type ∆10, then, an analogous reasoning would lead us to differences of type

(6) ∆²₁₀[u]_i+1[sj+1, sj+2] − ∆²10[u]_i+1[sj, sj+1]

but, as it is easy to verify after expanding and regrouping, the difference (5) is precisely ∆²₁₁u(ti, sj). Thus our definition ofV²(u, I_a^b)is independent of the choice of either ∆₀₁ or ∆₁₀, to perform the subsequent difference quotients variations.

Example 3.3 Let u : I_a^b→ R such that u(x, y) = (x + y)². Then u ∈ V²(I_a^b).

Indeed: let ξ := {tⁱ}ⁿi=0∈ Π³([a1, b1]) and η := {s^j}^mj=0∈ Π³([a2, b2]).

1. To estimate V_[a2,b2]² (u) notice that for all t ∈ [a¹, b1]

sup

ξ m−2X

j=0

|∆⁰¹[t, sj+1, sj+2]| = Z b2

a2

 ∂²

∂s²u(t, s) 

 ds = 2(b²− a²).

2. Similarly, for s ∈ [a², b2],

sup

ξ nX−2

i=0

|∆¹⁰u[ti+1, ti+2; s]| = 2(b¹− a¹).

3. Finally, a straightforward computation yields

∆01u[t; sj+1, sj+2] = sj+2− s^j for all t ∈ [a¹, b1].

It follows that ∆²₁₁u(ti, sj) = 0 and T V²(u, I_a^b) = 4(b₁− a1+ b₂− a2).

The proof of the following lemma follows in a straightforward manner from the definition.

Lemma 3.4 If u and v belong to V²(I_a^b), and λ is any real constant, then T V²(λu, I_a^b) = |λ|T V²(u, I_a^b)

and

T V²(u + v, I_a^b) ¬ T V²(u, I_a^b) + T V²(v, I_a^b).

Lemma 3.5 Let u ∈ V²(I_a^b)

(Monotonicity) If c = (c1, c2) and d = (d1, d2) with a1 < c1 < d1 < b1 and a2< c2< d2< b2 then

T V²(u, I_c^d) ¬ T V²(u, I_a^b).

(Semi-additivity)If a1< x < b1 and a2< y < b2 then V²

u, I_(a^(x,b_1,a²₂⁾₎ + V²

u, I_(x,a^(b1,b₂₎²⁾

¬ V²(u, I_a^b).

Proof For the sake of brevity we will omit the proof of part (a) which is similar (but much longer) to the correspondent one for functions in BV (I_a^b)) taking into account, of course, the specific bi-dimensional setting.

To prove (b), let ξ := {tⁱ}ⁿi=0 ∈ Π³([a1, x]), ξe:= {e^ti}ⁿi=0 ∈ Π³([x, b1]) and η :=

{s^j}^mj=0∈ Π³([a2, b2]). Then

n−2

X

i=0 mX−2

j=0

∆²₁₁u(ti, sj)+

n−2

X

i=0 mX−2

j=0

∆²₁₁u(e^ti, sj)

¬

n−2

X

i=0 mX−2

j=0

∆²₁₁u(ti, sj)+

mX−2 j=0

^∆01u[e^t1; s_j+1, s_j+2] − ∆01u[e^t0; s_j+1, s_j+2]

e^t1−e^t0

− h_{∆01u[tn; sj+1, sj+2] − ∆01u[tn}

−1; s_j+1, s_j+2] tn− tn−1

i

+

n−2

X

i=0 mX−2

j=0

∆²₁₁u(e^ti, sj)¬ V²(u, I_a^b).

Therefore V²

u, I_(a^(x,b_1,a²₂⁾₎ + V²

u, I_(x,a^(b1,b₂₎²⁾

¬ V²(u, I_a^b). _

Definition 3.6 By lemma 3.4, V²(I_a^b) is a linear space. In the sequel it will be denoted as BV²(I_a^b).

Remark 3.7 In the one dimensional case the notion of second variation plays a role similar to the one played by second derivatives in several senses; for instance, if [a, b] ⊂ R and f : [a, b] → R satisfies V²(f, [a, b]) = 0, then f must be an affine (linear + constant) function. It is readily seen that if I_a^b ⊂ R² is a rectangle and u :R² → R is any affine function then the coefficients of u depend linearly of the values of u over the vertices of I_a^band

(7) ∆11u[a, b] := u(b1, b2) − u(a¹, b2) − u(b¹, a2) + u(a1, a2) = 0.

The following proposition generalizes this fact for functions of bounded second va- riation on a rectangle I_a^b.

Theorem 3.8 A function u : I_a^b → R satisfies the conditions ∆¹¹u[a, b] = 0 and T V²(u, I_a^b) = 0if and only if there are constants A, B, C such that u(x, y) = Ax + By + C.

Proof If T V²(u, I_a^b) = 0 then, for all (x, y) ∈ (a¹, b1) × (a², b2) the following implications hold:

V_[a²_2,b2](u(a1,·)) = 0 =⇒ u(a1, b2) − u(a1, y)

b2− y =u(a1, y) − u(a1, a2) y− a2 ; (8)

V_[a²_2,b2](u(b1.·)) = 0 =⇒ u(b₁, b₂) − u(b1, y)

b2− y =u(b₁, y) − u(b1, a₂) y− a2 ; (9)

V²

[a1,b1](u(·, a2)) = 0 =⇒ u(b1, a2) − u(x, a2)

b1− x =u(x, a2) − u(a1, a2) x− a1 ; (10)

V²

[a1,b1](u(·, b2)) = 0 =⇒ u(b1, b2) − u(x, b2)

b1− x = u(x, b2) − u(a1, b2) x− a1

(11)

while V²(u, I_a^b) = 0 implies 1

b1− x

u(b1, b2) − u(b¹, y)

b2− y −u(b1, y)− u(b¹, a2) y− a²

−

u(x, b2) − u(x, y)

b2− y −u(x, y)− u(x, a²) y− a2



= 1

x− a¹

u(x, b2) − u(x, y)

b2− y −u(x, y)− u(x, a²) y− a²

−

u(a1, b2) − u(a¹, y)

b2− y −u(a1, y)− u(a¹, a2) y− a²



.

The first and fourth summands of this last equality vanish, by virtue of (8) and (9). Hence, we have

− 1

b1− x

h_{u(x, b2}) − u(x, y)

b2− y −u(x, y) − u(x, a2) y− a2

i=

1 x− a1

h_{u(x, b2}) − u(x, y)

b2− y −u(x, y) − u(x, a2) y− a2

i

and solving this equation for u(x, y) we get

u(x, y) =

y− a2

b2− a²



u(x, b2) +

b2− y b2− a²



u(x, a2).

(12)

On the other hand, making use of (10) and (11) we obtain the relations

u(x, a2) =

x− a¹ b1− a¹



u(b1, a2) +

b1− x b1− a¹



u(a1, a2) and

u(x, b2) =

x− a¹ b1− a¹



u(b1, b2) +

b1− x b1− a¹



u(a1, b2), which, combined with (12) yields

u(x, y) = _{y− a2}

b₂− a2

 _{x− a1}

b₁− a1

u(b₁, b₂) +_{y− a2}

b₂− a2

 _{b1− x}

b₁− a1

u(a₁, b₂) (13)

+_{b2− y}

b₂− a2

 _{x− a1}

b₁− a1

u(b1, a2) +_{b2− y}

b₂− a2

 _{b1− x}

b₁− a1

u(a1, a2).

Now, although (13) was established assuming that (x, y) ∈ (a¹, b1) × (a², b2), a straightforward computation shows that if we replace in (13), (x, y) by any point in the boundary of I_a^b, then we get an identity. Thus (13) actually holds for all (x, y) ∈ Ia^b. Finally, notice that after expanding and regrouping terms in the right hand side of (13), the coefficient of the product xy is

∆11u[a, b]

(b2− a²)(a1− b¹)

which is zero, by hypothesis. Thus u must be an affine function.

Conversely if u = Ax + By + C then V²(u, I_a^b) = 0 and, by Proposition 2.1,

V²(u(., a2)[a1, b1]) =Z b₁ a1

  ∂²

∂x²(Ax + Ba2+ C)   dx = 0

and

V²(u(., b₂)[a₁, b1]) =Z b₁ a1

  ∂²

∂x²(Ax + Bb₂+ C)   dx = 0 analogously

V²(u(a₁, .)[a2, b2]) =Z b2

a2

  ∂²

∂x²(Aa₁+ By + C)   dy = 0 and

V²(u(b1, .)[a2, b2]) = Z b2

a2

  ∂²

∂x²(Ab1+ By + C)   dy = 0.

Therefore, by (3),

T V²(u, I_a^b) = 0. _

Based on Theorem 3.8, and in the discussion previous to it, the following defi- nition is now natural.

Definition 3.9 For any u ∈ BV²(I_a^b) define

kuk := Σ |u|[a, b] + T V²(u, I_a^b), (14)

where Σ |u|[a, b] := |u(a¹, a2)| + |u(b¹, b2)| + |u(a¹, b2)| + |u(b¹, a2)|.

Corollary 3.10 k · k is a norm on BV²(I_a^b).

Proof Let u ∈ BV²(I_a^b). By definition kuk ­ 0 and clearly u = 0 implies kuk = 0.

On the other hand, if kuk = 0, then T V²(u, I_a^b) = 0 and |∆¹¹u[a, b]| ¬ Σ |u|[a, b] = 0. It follows, by (13), that u ≡ 0.

On the other hand, the properties:

(P2) ∀ α ∈ R : kαuk = |α|kuk and

(P3) ku + vk ¬ kuk + kvk, (u, v ∈ BV²(I_a^b))

follow readily from the definition and the properties of the functionals of modulus

(| |) and supremum (sup). 

In the following proposition we present a polynomial mayorant for the functions in the unit ball of (BV²(I_a^b), k · k). This property will be fundamental in proving that BV²(I_a^b) is a Banach space.

Proposition 3.11 There is a fourth degree polynomial p : R² → R such that for all u ∈ BV²(I_a^b)

|u(x, y)| ¬ p(x, y) kuk for all (x, y) ∈ Ia^b.

In particular, BV²(I_a^b) is a subspace of B(Ia^b), the Banach space of all bounded functions on I_a^b with the sup norm.

Proof Let u be in BV²(I_a^b). Put δ1:= b1− a¹and δ2:= b2− a². Then, by (3) and definition (14) for all (x, y) ∈ (a¹, b1) × (a², b2), we have the following inequalities

^{u(a1, b}_b²₂^{) − u(a}_{− y} ^{1, y)−u(a1, y}_y^{) − u(a}_{− a2} ^{1, a2)¬ kuk;}

(15)

^{u(b1, b2}_b2^{) − u(b}_{− y} ^{1, y)−u(b1, y}_y^{) − u(b}_{− a2} ^{1, a2)¬ kuk;}

(16)

^{u(b1, a2}_b1^{) − u(x, a}_{− x} ^{2)−u(x, a2}_x^{) − u(a}_{− a1} ^{1, a2)¬ kuk;}

(17)

^{u(b1, b2}_b1^{) − u(x, b}_{− x} ^{2)−u(x, b2}_x^{) − u(a}_{− a1} ^{1, b2)¬ kuk}

(18)

whereas V²(u, I_a^b) ¬ kuk implies 

_b1¹_{− x}h_{u(b1, b2) − u(b1, y)}

b2− y −u(b1, y) − u(b1, a2) y− a2

−_{u(x, b2}) − u(x, y)

b2− y −u(x, y) − u(x, a2) y− a2

i

− 1

x− a1

h_{u(x, b2}) − u(x, y)

b2− y −u(x, y) − u(x, a2) y− a2

−_{u(a1, b2) − u(a1, y)}

b2− y −u(a₁, y) − u(a1, a₂) y− a2

i^{¬ kuk.}

(19)

Now, (17), in turn, implies

(20) _{(b1− x)(x − a}^δ1 ₁₎^{u(x, a2)¬ kuk +u(b}_b1¹_{− x}^{, a2)+u(a}_{x− a}^{1, a}₁²⁾ or

|u(x, a2)| ¬(b1− x)(x − a1)kuk

δ₁ +^{(x − a1)u(b}_δ1 ^{1, a2)+(b1− x)u(a}_δ1 ^{1, a2).} Similarly, (18) implies

(21) |u(x, b2)| ¬(b₁− x)(x − a1)kuk

δ1 +^{(x − a1)u(b}_δ1 ^{1, b2)+(b1− x)u(a}_δ1 ^{1, b2).} On the other hand, from inequality (19),

_{(b1− x)(x − a}^δ1δ2₁^{u(x, y)}_{)(b2− y)(y − a2)}^{¬ kuk +} 

^u(a_{(x − a}^{1, b2) − u(a}_{1)(b2− y)}^{1, y)−u(a}_{(x − a}^{1, y) − u(a}_{1)(y − a}^{1, a}₂₎^2)+ 

^u(b_(b¹₁^{, b}_{− x)(b}^{2) − u(b}_{2− y)}^{1, y)−u(b}_(b¹₁^{, y}_{− x)(y − a}^{) − u(b1, a}₂₎^2)+ 

_{(b1− x)(x − a}^{δ1u(x, b}₁²_)(b⁾ _{2− y)}⁺_{(b1− x)(x − a}^{δ1u(x, a}₁²_{)(y − a}⁾ ₂₎^, and from this, by (15) and (16), we get

_{(b1− x)(x − a}^δ1δ2₁^{u(x, y)}_{)(b2− y)(y − a2)} ^{¬ kuk +}_x^kuk_{− a1} ^{+ k}_b1^uk_{− x}⁺ 

_{(b1− x)(x − a}^{δ1u(x, b}₁²_)(b⁾ _{2− y)}⁺ 

_{(b1− x)(x − a}^{δ1u(x, a}₁²_{)(y − a}⁾ ₂₎ or equivalently

|u(x, y)| ¬ (b₁− x)(x − a1)(b₂− y)(y − a2)

δ1δ2 kuk +(b₁− x)(b2− y)(y − a2)

δ1δ2 kuk

+(x − a1)(b2− y)(y − a2) δ₁δ₂ kuk +

^{(y − a2}_δ^{)u(x, b}₂ ²⁾

+^{(b2− y)u(x, a}_δ2 ^2).

Taking into account (20), (21)and the fact thatΣ |u|[a, b] ¬ kuk, we obtain

|u(x, y)| ¬ h_{(b1− x)(x − a1)(b2− y)(y − a2)}

δ1δ2 +(b₁− x)(b2− y)(y − a2) δ1δ2

(22)

+(x − a1)(b₂− y)(y − a2)

δ1δ2 +(y − a2)(b₁− x)(x − a1) δ2δ1

+(y − a2)(x − a1)

δ₂δ₁ +(y − a2)(b1− x)

δ₁δ₂ +(b2− y)(x − a1) δ₂δ₁ +(b2− y)(b1− x)(x − a1)

δ₂δ₁ +(b2− y)(b1− x) δ₂δ₁

i

kuk.

Finally, by regrouping the right hand side of this inequality we may define

p(x, y) := 1 + δ⁻¹₁ δ⁻¹₂ (b1− x)(x − a1)(b2− y)(y − a2)

+ δ⁻¹₁ (b1− x)(x − a1) + δ₂⁻¹(b2− y)(y − a2). _