A FAMILY OF POLYNOMIALS AND THEIR APPLICATION TO INEQUALITIES OF NORMS OF POISSON WICK PRODUCTS

TO INEQUALITIES OF NORMS OF POISSON WICK PRODUCTS

ALBERTO LANCONELLI and AUREL I. STAN

A family of polynomials{Pn(z)}n≥1, whose coefficients satisfy a system of linear equations, is introduced first. An inequality about the real part ofPn(z) is pre- sented next, for complex numberszwhose real part is between 0 and 1. Finally, using Stein Complex Interpolation Theorem, we prove some inequalities about the norms of Poisson Wick products.

AMS 2010 Subject Classification: 46B70, 60H40, 60H10.

Key words: Euler numbers, Poisson probability measure, Wick product, second quantization operator, Stein Complex Interpolation Theorem.

1. INTRODUCTION

The Wick product, at least in the Gaussian case, is intimately related to the definition of stochastic integral, see [3] (Chapter 13) and [2]. For this reason, we believe that it is important to find inequalities about the norms of Wick products. In order to bound the Wick products, we need to use second quantization operators. In this paper, we present a family of inequalities about the norms of Poisson Wick products.

The paper is structured as follows. In Section 2, we define a family of polynomials in terms of some systems of equations whose determinants are Euler numbers. In Section 3, we introduce the Poisson Wick product, while in Section 4 we present some inequalities about the norms of Poisson Wick products.

2. A FAMILY OF POLYNOMIALS

Let us consider the following sequence of polynomials:

f₀(z) := 1, (2.1)

f1(z) := 1−1 2z², (2.2)

and for all n≥2,

fn(z) := 1−c1z²+c2z⁴−c3z⁶+· · ·+ (−1)ⁿcnz²ⁿ, (2.3)

MATH. REPORTS15(65),4(2013), 443–457

wherec₁,c₂,· · ·,c_n are real numbers satisfying the system ofnequations and nunknowns:

S_n:







































−c₁ + c2 − c3 + · · · + (−1)ⁿcn = −¹₂ c1 − ⁴₂

c2 + ⁶₂

c3 − · · · + (−1)^{n−1 2n}₂

cn = 0 c₂ − ⁶₄

c₃ + · · · + (−1)^{n−2 2n}₄

c_n = 0 c3 − · · · + (−1)^{n−3 2}₆ⁿ

cn = 0

· · · · cn−1 − _2n−2²ⁿ

cn = 0 .

We prove now two propositions about the system Sn.

Proposition 2.1. Let D0 := 1, D1 := −1, and for all n ≥ 2, let Dn

denote the determinant of the system S_n. Then:

1. For all n≥1, we have:

2n 2n

D_n+ 2n

2n−2

Dn−1+· · ·+ 2n

0

D₀ = 0.

(2.4)

2. For all n ≥ 2, the determinant Dn of the system Sn is an odd integer.

In particular Dn6= 0, and thus, the systemSn has a unique solution (c1, c₂, . . ., c_n).

3. We have:

D0 = 1, D1 = −1, D₂ = 5, D₃ = −61, D4 = 1385, D5 = −50521,

· · · , D_n = E_n, (2.5)

· · · ,

where E1, E2, . . . are Euler numbers, i.e., for all n≥0:

E_n =

d²ⁿ

dx²ⁿ sechx

|x=0

, (2.6)

or equivalently, since the hyperbolic secant function is even, its Taylor expansion about x= 0 is:

sech(x) =

∞

X

n=0

E_n (2n)!x²ⁿ. (2.7)

Proof. 1. For alln≥2, the determinant of the systemS_n is:

Dn:=

−1 1 −1 . . . (−1)ⁿ⁻¹ (−1)ⁿ

2 2

− ⁴_{2 6}

2

· · · (−1)^{n−2 2n−2}₂

(−1)^{n−1 2}₂ⁿ 0 ⁴₄

− ⁶₄

· · · (−1)^{n−3 2n−2}₄

(−1)^{n−2 2n}₄

0 0 ⁶₆

· · · (−1)^{n−4 2n−2}₆

(−1)^{n−3 2n}₆

· · · · 0 0 0 · · · ²ⁿ⁻²_2n−2

− _2n−2²ⁿ .

Expanding this determinant after the last row, we obtain:

Dn = − 2n

2n−2

Dn−1−D_n⁽¹⁾, (2.8)

where:

D_n⁽¹⁾:=

−1 1 −1 . . . (−1)ⁿ⁻² (−1)ⁿ

2 2

− ⁴_{2 6}

2

· · · (−1)^{n−3 2n−4}₂

(−1)^{n−1 2}₂ⁿ 0 ⁴₄

− ⁶₄

· · · (−1)^{n−4 2n−4}₄

(−1)^{n−2 2n}₄

0 0 ⁶₆

· · · (−1)^{n−5 2n−4}₆

(−1)^{n−3 2n}₆

· · · ·

0 0 0 · · · ²ⁿ⁻⁴_{2n−4 2n}

2n−4

,

since:

2n−2 2n−2

= 1.

Expanding now D⁽¹⁾n after its last row, we obtain:

Dn = − 2n

2n−2

Dn−1− 2n

2n−4

Dn−2+D⁽²⁾_n , (2.9)

where:

D_n⁽²⁾:=

−1 1 −1 . . . (−1)ⁿ⁻³ (−1)ⁿ

2 2

− ⁴_{2 6}

2

· · · (−1)^{n−4 2n−6}₂

(−1)^{n−1 2n}₂ 0 ⁴₄

− ⁶₄

· · · (−1)^{n−5 2n−6}₄

(−1)^{n−2 2n}₄

0 0 ⁶₆

· · · (−1)^{n−6 2n−6}₆

(−1)^{n−3 2}₆ⁿ

· · · · 0 0 0 · · · ²ⁿ⁻⁶_2n−6

− _2n−6²ⁿ ,

since:

2n−4 2n−4

= 1.

We continue this sequence of reasoning, expanding nextD⁽²⁾n after its last row, and so on, obtaining in the end:

D_n := − 2n

2n−2

Dn−1− 2n

2n−4

Dn−2− · · · − 2n

0

D₀, (2.10)

which is equivalent to:

2n 2n

Dn+ 2n

2n−2

Dn−1+ 2n

2n−4

Dn−2+· · ·+ 2n

0

D0 = 0.

2. We prove now by induction on n, thatDn is an odd number.

For n= 0 and n= 1, since D₀ = 1 and D₁ =−1,D₀ and D₁ are odd.

For n≥2, assuming that D_k is odd, for all k≤n−1, if we pass to the congruence modulo 2 in the recursive relation (2.10), then we obtain:

D_n ≡ −

2n 2n−2

Dn−1+ 2n

2n−4

Dn−2+· · ·+ 2n

0

D₀

(mod 2)

≡ −

2n 2n−2

·1 + 2n

2n−4

·1 +· · ·+ 2n

0

·1

(mod 2)

≡ − 2²ⁿ⁻¹−1

(mod 2)

≡ 1 (mod 2).

Thus,D_n is an odd integer.

3. Let us consider the following sequence of identities:

0 0

D0 = 1 (2.11)

2 2

D1+ 2

0

D0 = 0 (2.12)

4 4

D₂+ 4

2

D₁+ 4

0

D₀ = 0 (2.13)

· · · .

If we multiply (2.11) by x⁰/0!, (2.12) by x²/2!, (2.13) by x⁴/4!, and so on, and sum up the resulting identities, where x is a real number in a small neighborhood of 0, then we obtain:

∞

X

n=0 n

X

k=0

2n 2k

D_k x²ⁿ

(2n)! = 1.

(2.14)

Canceling (2n)! and changing the order of summation, we obtain:

1 =

∞

X

k=0

D_k (2k)!x^2k

∞

X

n=k

x^2(n−k) [2(n−k)]!

=

∞

X

k=0

Dk

(2k)!x^2k

∞

X

m=0

x^2m (2m)!

=

∞

X

k=0

Dk

(2k)!x^2kcosh(x).

Dividing both sides by cosh(x), we obtain:

∞

X

k=0

D_k

(2k)!x^2k = sech(x).

(2.15)

Thus,D0,D1,D2,. . . are Euler numbers.

It is known that the even indexed Euler numbers are positive, while the odd indexed ones are negative (see [1]). Using Cramer rule, with a little bit of effort, it can be proven that, for alln≥1, all the components: c₁,c₂,. . .,c_n, of the unique solution of the system S_n, are positive numbers. We are listing

below the first four polynomials of the sequence{f_n}_n≥1: f₁(z) := 1−1

2z², (2.16)

f₂(z) := 1− 6

10z²+ 1 10z⁴, (2.17)

f3(z) := 1− 75

122z²+ 15

122z⁴− 1 122z⁶, (2.18)

f4(z) := 1−1708

2770z²+ 350

2770z⁴− 28

2770z⁶+ 1 2770z⁸. (2.19)

We have the following proposition:

Proposition 2.2. For all n≥1, we have:

1. The polynomial function z 7→ f_n(z) maps the line of purely imaginary numbers, iR, into the interval[1, ∞) of the set of real numbers.

2. The polynomial functionz7→fn(z) maps the stripS of all complex num- bers having the real part between 0 and 1 (including 0 and 1) into the half-plane of all complex numbers of real part greater than or equal to1/2.

Proof. Letn≥1 be fixed.

1. For all purely imaginary numbers z=iy, where y∈R, we have:

fn(z) = 1−c1(iy)²+c2(iy)⁴−c3(iy)⁶+· · ·+ (−1)ⁿcn(iy)²ⁿ

= 1 +c1y²+c2y⁴+c3y⁶+· · ·+cny²ⁿ

∈ [1,∞),

sincec₁,c₂,. . .,c_nare positive numbers, andy^2k≥0, for ally ∈Randk∈ {1, 2,. . .,n}. We observe that fn(0) = 1.

2. We will prove first that for all complex numberz, such thatRe(z) = 1, we haveRe(fn(z))≥1/2. Letz= 1+iy, wherey∈R. Using Newton binomial formula, we have:

f_n(1 +iy) = 1−c₁(1 +iy)²+c₂(1 +iy)⁴− · · ·+ (−1)ⁿc_n(1 +iy)²ⁿ

= 1−c1+c2−c3+· · ·+ (−1)ⁿcn

+ 2

2

c1− 4

2

c2+ 6

2

c3+· · ·+ (−1)ⁿ⁻¹cn

2n 2

cn

y²

+ 4

4

c4− 6

4

c3+· · ·+ (−1)ⁿ⁻²cn

2n 4

cn

y⁴

· · · +

2n−2 2n−2

cn−1−

2n 2n−2

c_n

y²ⁿ⁻²

+c_ny²ⁿ+iIm(f_n(z)).

Since (c₁, c₂, . . ., c_n) is the solution of the system S_n, we can see now that:

Re(fn(1 +iy)) = 1

2+cny²ⁿ

≥ 1 2, for all y∈R. We observe that Re(fn(1)) = 1.

To prove that for all z in S = {z ∈ C | 0 ≤ Re(z) ≤ 1} we have Re(fn(z))≥1/2, we proceed as follows.

Consider the analytic function gn:S →C, defined as:

g_n(z) := e^−fn(z). (2.20)

We show first that:

|z|→∞,z∈Slim |g_n(z)| = 0.

(2.21)

Indeed, ifz=x+iy, withxandyreal, then using again Newton binomial formula we have:

|g_n(z)|= exp [−Re(f_n(z))]

= exp

−c_ny²ⁿ+Qn−1(x)y²ⁿ⁻²+· · ·+Q₁(x)y²+Q₀(x) (2.22) ,

where Q₀, Q₁, . . ., Qn−1 are polynomials. Since [0, 1] is a compact interval, the polynomials Qi, 0 ≤ i≤ n−1 are bounded on [0, 1]. Thus, there exists M >0, such that for alli∈ {0, 1,. . .,n−1} and allx∈[0, 1], we have:

|Q_i(x)| ≤M.

(2.23)

It follows from (2.22) that, for all z=x+iy∈S, we have:

|g_n(z)| ≤exp

−c_ny²ⁿ+M y²ⁿ⁻²+· · ·+M y²+M . (2.24)

Since we have:

|y| ≥ |z| − |x| ≥ |z| −1,

it follows that if z ∈ S, and |z| → ∞, we have |y| → ∞, too. Because the even polynomial −c_ny²ⁿ+M y²ⁿ⁻² +· · ·+M y²+M has a negative leading coefficient −c_n, we have:

(2.25) lim

|y|→∞

−c_ny²ⁿ+M y²ⁿ⁻²+· · ·+M y²+M

=−∞.

Thus, we conclude from (2.24) that:

|z|→∞,z∈Slim |g_n(z)|= 0.

Since g_n is analytic on S, and sup_|y|=R|g_n(z)| →0, as R→ ∞, applying the Maximum Modulus Principle, on rectanglesL_R={z∈S | −R≤Im(z)≤ R}, forR >0, we obtain:

exp(−inf{Re(f_n(z))|z∈S})

= sup{|g_n(z)||z∈S}

= lim

R→∞sup{|g_n(z)||z∈LR}

= lim

R→∞max{sup

|y|≤R

|g_n(iy)|, sup

|y|≤R

g_n(1 +iy), sup

0≤x≤1

|g_n(x±iR)|}

= max{sup

y∈R

|g_n(iy)|,sup

y∈R

|g_n(1 +iy)|}

= max{exp(−1),exp(−1/2)}

= exp(−1/2).

Therefore, Re(f_n(z))≥1/2, for allz∈S.

3.POISSON WICK PRODUCT

Let abe a fixed positive number, and let W:={0, 1, 2,. . .} denote the set of non-negative integers. The Poisson probability measure with mean a, is the measure P defined on the set of all subsets B ofW, by the formula:

P(B) := X

n∈B

aⁿ n!e^−a, (3.1)

for all subsets B of W.

The Poisson probability measure P has finite moments of all orders, and so, every polynomial function belongs to the space L²(W, P). Thus, we can apply the Gram-Schmidt procedure to the monomial functions: 1, x, x², . . ., obtaining a sequence of orthogonal polynomials{C_n(x)}n≥0, called theCharlier polynomials. The polynomials Cn(x), n ≥ 0, are chosen to have the leading coefficient 1. They satisfy the recursive relation:

(3.2) xCn(x) =Cn+1(x) + (a+n)Cn(x) +anCn−1(x),

for all n ≥ 0, where C−1(x) := 0. This recursive relation implies that the Szeg¨o-Jacobi parameters of the Poisson probability measure with mean aare:

(3.3) αn=a+n

and

(3.4) ω_n=an,

for all n≥0. It is known that the square of theL² norm of C_n is:

(3.5) kCnk²₂=ω1ω2· · ·ωn=n!aⁿ,

for all n ≥ 0. Since {C_n}_n≥0 is an orthogonal basis of L²(W, P), for every functionf ∈L²(W,P) there exists a unique sequence{f_n}_n≥0 ⊂C, such that:

(3.6) f(x) =

∞

X

n=0

f_nC_n(x),

where the above series is convergent in the L² norm. The square of the L² norm of f is:

(3.7) kf k²₂=

∞

X

n=0

|f_n|²aⁿn!<∞.

The Poisson Wick product is defined first for any two Charlier polynomi- als as:

C_mC_n := C_m+n. (3.8)

This definition is extended in a bilinear way to linear combinations of Charlier polynomials, that means iff =P∞

n=0f_nC_n(x) andg=P∞

n=0g_nC_n(x), then:

(fg)(x) :=

∞

X

k=0

X

m+n=k

fmgnCk(x).

(3.9)

The series from the right-hand side of the above formula may not be convergent in L². Later, we will be able to make the series convergent using second quantization operators.

There is an important family of random variables that is closed with respect to the Wick product. For every complex number t, we define:

E_t(x) :=

∞

X

n=0

tⁿ

n!aⁿC_n(x), (3.10)

for all x ∈W. We callE_t an exponential function (see [4]). It is very easy to see that every exponential function belongs to L²(W, P), and for all s, t∈C, we have:

(3.11) E_s E_t=E_s+t.

We observe that the exponential functions have been defined in terms of the orthogonal Charlier polynomials. However, their point-wise formula is known to be (see [4]):

(3.12) E_t(x) =

1 + t

a x

e^−t,

for all t∈C and x∈W. From this point-wise formula it is easy to check, by a direct computation, that each exponential function belongs to every space L^p(W, P), for 1 ≤ p < ∞ (not true for p = ∞). Moreover, the complex vector space spanned by the exponential functions is dense in L^p(W, P), for all 1≤p <∞.

For every complex number c, we define thesecond quantization operator of c times the identity, and denote it by Γ(cI), in the following way. For every f =P

n=0f_nC_n, where{f_n}_n≥0⊂C, we define:

Γ(cI)f :=

∞

X

n=0

cⁿf_nC_n. (3.13)

From this definition, it is easy to see that for allc∈C, with|c| ≤1, Γ(cI) is a bounded linear operator, of operatorial norm equal to 1, from L²(W, P) toL²(W,P).

There are also point-wise formulas and probabilistic interpretations for the second quantization operator Γ(cI), from which it can be proved that, for all c ∈ [0, 1], Γ(cI) is a bounded operator from L^p(W, P) to L^p(W, P), of operatorial norm 1, for all 1≤p≤ ∞(even for p=∞) (see [5]).

The family of exponential functions is closed with respect to the second quantization operator ofcI, since for all c,t∈C, we have:

(3.14) Γ(cI)E_t=E_ct.

4. INEQUALITIES FOR NORMS OF POISSON WICK PRODUCTS

We denote by Exp the complex vector space spanned by {E_t}_t∈C, that meansf ∈Expif and only if there exist n∈N,c₁,c₂,. . .,c_n∈C, and t₁,t₂, . . .,t_n∈C, such that:

f = c₁E_t1 +c₂E_t2+· · ·+c_nE_tn. (4.1)

We have the following point-wise formula:

Lemma 4.1. For allα andβ in C,f andg in Exp, and n∈W, we have:

{[Γ(αI)f][Γ(βI)g]}(n)

=

∞

X

k=0

[a(1−α)]^k

k! e^−a(1−α)

∞

X

l=0

[a(1−β)]^l

l! e^−a(1−β)

× X

p+q+r=n

n p, q, r

α^pβ^qγ^rf(k+p)g(l+q), (4.2)

where

γ := 1−α−β.

(4.3)

This lemma can be checked directly for two exponential functionsf =E_s and g =E_t, for s and t complex numbers, see [5]. Based on the result of this lemma, we have:

Theorem 4.2. Let α and β be non-negative real numbers, such that:

(4.4) α+β≤1.

Then there exists a bounded linear operator Γ1(αI, βI) : L¹(W ×W, P ⊗P)→L¹(W, P), such that, for all F in L¹(W×W, P⊗P), we have:

kΓ₁(αI, βI)F k₁ ≤ |kF k|₁, (4.5)

and for all f and g in L¹(W, P):

(4.6) Γ1(αI, βI) (f ⊗g) = [Γ(αI)f][Γ(βI)g],

where P ⊗P denotes the product measure ofP and P,|k · |k₁ is the L¹ norm in the space L¹(W×W, P⊗P), and (f ⊗g)(x, y) := f(x)g(y), for all x and y in W.

In particular, for all f andg in L¹(W,P), [Γ(αI)f][Γ(βI)g]belongs to L¹(W, P), and:

k[Γ(αI)f][Γ(βI)g]k₁ ≤ kf k₁ · kgk₁. (4.7)

See [5] for a proof. The bounded linear operator Γ1(αI, βI), from the above lemma, is defined as:

(Γ₁(αI, βI)F) (n) =

∞

X

k=0

[a(1−α)]^k

k! e^−a(1−α)

∞

X

l=0

[a(1−β)]^l

l! e^−a(1−β)

× X

p+q+r=n

n p, q, r

α^pβ^qγ^rF(k+p, l+q), (4.8)

for all F ∈L¹(W×W,P⊗P) andn∈W.

Theorem 4.3. Let α and β be non-negative real numbers, such that:

(4.9) α+β≤1.

If we consider the restriction of the operator Γ1(αI, βI) to the space L^∞(W×W, P⊗P):

Γ∞(αI, βI) := Γ1(αI, βI)|_L^∞_(W×W,P⊗P), (4.10)

then for all F in L^∞(W×W, P ⊗P), we have:

kΓ∞(αI, βI)F k∞ ≤ |kF k|∞, (4.11)

and for all f and g in L^∞(W,P):

(4.12) Γ∞(αI, βI) (f ⊗g) = [Γ(αI)f][Γ(βI)g], where |k · |k∞ is theL^∞ norm in the space L^∞(W×W, P ⊗P).

In particular, for all f and g in L^∞(W, P), [Γ(αI)f][Γ(βI)g] belongs to L^∞(W, P), and:

k[Γ(α)f][Γ(β)g]k_∞ ≤ kf k_∞· kgk_∞. (4.13)

See [5] for a proof. Forp= 2, using the orthogonal basis{C_m⊗C_n}_m≥0,n≥0 of L²(W⊗W,P ⊗P), an operator Γ2(αI,βI) can be defined as:

Γ₂(αI, βI) X

(m,n)∈W×W

λ_m,nC_m(x)C_n(y) = X

k≥0

X

m+n=k

λ_m,nα^mβⁿC_k(x).

Using now Cauchy-Schwarz inequality, it is not hard to prove the follow- ing:

Theorem 4.4. Let α and β be complex numbers such that:

(4.14) |α|²+|β|²≤1.

Then there exists a bounded linear operator Γ₂(αI, βI) : L²(W ×W, P ⊗P)→L²(W, P), such that, for all F in L²(W×W, P⊗P), we have:

kΓ2(αI, βI)F k₂ ≤ |kF k|₂, (4.15)

and for all f and g in L²(W, P):

(4.16) Γ₂(αI, βI) (f ⊗g) = [Γ(αI)f][Γ(βI)g], where |k · |k₂ is the L² norm in the space L²(W×W, P⊗P).

In particular, for all f andg in L²(W,P), [Γ(αI)f][Γ(βI)g]belongs to L²(W, P), and:

k[Γ(α)f][Γ(β)g]k₂ ≤ kf k₂· kgk₂ . (4.17)

Moreover, if α and β are nonnegative real numbers such that α+β ≤1, then:

Γ2(αI, βI) := Γ1(αI, βI)|_L2(W×W,P⊗P). (4.18)

See [5] for a proof.

Since Γ₁(·,·) extends Γ₂(·,·), which in turn extends Γ_∞(·,·), we denote all of them simply by Γ(·,·).

Using the last three theorems, we can prove now the main result of this paper:

Theorem 4.5. For all n∈N, let

f_n(z) := 1−c₁z²+c₂z⁴−c₃z⁶+· · ·+ (−1)ⁿc_nz²ⁿ (4.19)

be the polynomial whose coefficients are obtained by solving the system S_n. Then

• If 1 ≤p ≤2, and q denotes the conjugate of p, i.e., (1/p) + (1/q) = 1, then for all positive real numbers α andβ, such that:

(4.20) α^1/fn(2/q)+β^1/fn(2/q) ≤1,

there exists a bounded linear operatorΓ_p(αI, βI) :L^p(W×W,P⊗P)→ L^p(W, P), such that, for all F in L^p(W×W, P ⊗P), we have:

kΓp(αI, βI)F k_p ≤ |kF |k_p, (4.21)

and for all f and g in L^p(W, P):

(4.22) Γp(αI, βI)(f ⊗g) = [Γ(αI)f]⊗[Γ(βI)g].

In particular, for all f and g in L^p(W, P), [Γ(αI)f][Γ(βI)g] belongs to L^p(W, P), and:

k[Γ(αI)f][Γ(βI)g]k_p ≤ kf k_p· kgk_p . (4.23)

• If 2≤p≤ ∞, then for all positive real numbers α and β, such that:

(4.24) α^1/fn(2/p)+β^1/fn(2/q) ≤1,

there exists a bounded linear operatorΓ_p(αI, βI) :L^p(W×W,P⊗P)→ L^p(W, P), such that, for all F in L^p(W×W, P ⊗P), we have:

kΓp(αI, βI)F k_p ≤ |kF |k_p, (4.25)

and for all f and g in L^p(W, P):

(4.26) Γp(αI, βI)(f ⊗g) = [Γ(αI)f]⊗[Γ(βI)g].

In particular, for all f and g in L^p(W, P), [Γ(αI)f][Γ(βI)g] belongs to L^p(W, P), and:

k[Γ(αI)f][Γ(βI)g]k_p ≤ kf k_p· kgk_p . (4.27)

Proof. Letc anddbe positive numbers such that:

(4.28) c+d≤1

For any two Banach spacesU andV, let us denote by B(U,V) the space of all bounded linear operators from U toV. Let

S := {x+iy|x∈R, y ∈R,0≤x≤1}.

Let us consider the function:

(4.29) ϕ:S→B L²(W×W, P ⊗P), L²(W, P) ,

defined by:

ϕ(z) := Γ

c^fn(z)I, d^fn(z)I

. (4.30)

Since we know from Proposition 2.2 that for all z ∈ S, the real part of fn(z) is at least 1/2, and c and d are positive numbers not exceeding 1, we have:

c^fn(z)

2

+ d^fn(z)

2

=c^2Re(fn(z))+d^2Re(fn(z))≤c¹+d¹ ≤1.

Theorem 4.4 implies now that, for allz∈S,ϕ(z)∈B(L²(W×W,P⊗P), L²(W,P)) and:

kϕ(z)k_L2(W×W,P⊗P),L²(W,P)≤1.

Hence,

(4.31) sup

Re(z)=1

kϕ(z)k_L2(W×W,P⊗P),L²(W,P)≤1.

We also know from Proposition 2.2 that for all z ∈ iR, fn(z) is a real number greater than or equal to 1. Thus,c^fn(z)andd^fn(z)are two real positive numbers, such that:

(4.32) c^fn(z)+d^fn(z) ≤1.

Thus, it follows from Theorem 4.2 that ϕ(z) ∈ B(L¹(W×W, P ⊗P), L¹(W,P)) and

(4.33) sup

Re(z)=0

kϕ(z)k_L1(W×W,P⊗P),L¹(W,P)≤1.

It is not hard to see that ϕis an analytic function.

It follows now from Stein Complex Interpolation Theorem, that for all t∈(0, 1), we have:

(4.34) kϕ(t)k_Lpt(W×W,P⊗P),L^pt(W,P) ≤1, where

(4.35) 1

p_t = 1−t 1 + t

2.

This condition is equivalent to (we drop the subscript t):

(4.36) t= 2

q.

If p∈[1, 2], andα and β are positive real numbers, such that:

(4.37) α^{1/[1−(2/q2)]}+β^{1/[1−(2/q2)]} ≤1, then we define:

c := α^{1/[1−(2/q2)]}

(4.38) and

d := β^{1/[1−(2/q2)]}, (4.39)

and obtain c+d≤1. Thus, we proved the first part of the theorem.

The second part of the theorem can be proven exactly in the same way, by interpolating betweenL^∞and L², instead of L¹ and L².

REFERENCES

[1] M. Abramowitz and I.A. Stegun (Editors), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York, 1970.

[2] P. Da Pelo, A. Lanconelli and A.I. Stan,An Itˆo formula for a family of stochastic integrals and related Wong-Zakai theorems. Stochastic Process. Appl. 123(2013), 3183–3200.

[3] H.-H. Kuo,White Noise Distribution Theory. CRC Press, Boca Raton, 1996.

[4] A. Lanconelli and L. Sportelli,A Connection Between the Poissonian Wick Product and the Discrete Convolution. COSA5(2011), 689–699.

[5] A. Lanconelli and A.I. Stan,A H¨older inequality for norms of Poissonian Wick products.

To appear in IDAQP (2013).

Received 13 August 2013 Universita’ degli Studi di Bari, Dipartimento di Matematica,

Via E. Orabona, 4, 70125 Bari - Italia lanconelli@dm.uniba.it The Ohio State University, Department of Mathematics, 1465 Mount Vernon Avenue, Marion, OH 43302, U.S.A.

stan.7@osu.edu