`Journal of Applied Mathematics and Stochastic AnalysisVolumeΒ 2008, Article IDΒ 254897, 22 pageshttp://dx.doi.org/10.1155/2008/254897`
Research Article

## HΓΆlder-Type Inequalities for Norms of Wick Products

1Dipartimento di Matematica, UniversitΓ  di Bari, Campus Universitario, Via E. Orabona 4, 70125 Bari, Italy
2Department of Mathematics, Ohio State University at Marion, 1465 Mount Vernon Avenue, Marion, OH 43302, USA

Received 3 November 2007; Revised 21 January 2008; Accepted 26 February 2008

Copyright Β© 2008 Alberto Lanconelli and Aurel I. Stan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Various upper bounds for the -norm of the Wick product of two measurable functions of a random variable , having finite moments of any order, together with a universal minimal condition, are proven. The inequalities involve the second quantization operator of a constant times the identity operator. Some conditions ensuring that the constants involved in the second quantization operators are optimal, and interesting examples satisfying these conditions are also included.

#### 1. Introduction

It was proven in [1] that for any positive numbers and , such that , any normally distributed random variable , and any and complex-valued Borel measurable functions, such that both random variables and are square integrable, the Wick product is square integrable and the following inequality holds: Here denotes the second quantization operator and the identity operator of the one-dimensional Hilbert space . The authors' motivation was to find a Hausdorff-Young-type inequality for the theory of Bosonian Fock spaces and they believed that (1.1) was indeed an inequality of this type, based on their feeling that the Wick product is an analogue concept of the convolution product from the theory of Fourier transform. After discussing with other mathematicians and thinking more about it, they have become convinced that the Wick product is in fact a simpler product, playing for the theory of Bosonian Fock spaces a role similar to the classic product of two series. Together this reconsideration and the condition strongly suggest that (1.1) is in fact a HΓΆlder-type inequality for the theory of Gaussian Hilbert spaces (Bosonian Fock spaces).

We will generalize inequality (1.1) to other types of random variables , and in some cases find the optimal constants and . Moreover, we will prove that no matter how we choose a nonconstant random variable , having finite moments of any order, the condition cannot be improved. In Section 2, we present a minimal background about the SzegΓΆ-Jacobi parameters of a random variable having finite moments of any order. We define a set of basic properties and prove some connections between these properties in Section 3. Section 4 is dedicated completely to proving a fundamental necessary condition that we call the universal minimal (unimprovable) condition. The main inequalities of the paper are proven in Section 5. Finally, in Section 6, we provide many examples in support of the results proven in the previous section. Some of these examples demonstrate that the estimates from Section 5 are optimal.

#### 2. Background

Let , , be a probability space and a random variable having finite moments of all orders. That means, for all , , where denotes the expectation with respect to . Since has finite moments of all orders, all the terms of the sequence, are square integrable, and thus we can apply the Gram-Schmidt orthogonalization procedure to obtain a sequence of orthogonal polynomial random variables , , . The inner product that we are using is for all , measurable, such that and . Also, are polynomial functions chosen, such that for all , if is not the null polynomial, then has the degree equal to and a leading coefficient of . In fact, if is a discrete random variable taking on only different values with positive probabilities, then is the null polynomial for all . If is not a discrete random variable, or is a discrete random variable taking on a countable set of values with positive probabilities, then for all , is a polynomial of degree with a leading coefficient equal to .

It is well known that there exist two sequences of real numbers and , such that for all ,When , (the null polynomial) and we can choose . Also, if is a discrete random variable taking on only different values with positive probabilities, then for , the equality (2.1) must be understood in the almost-sure sense, and we can choose and for all . The sequences and are called the SzegΓΆ-Jacobi parameters of . Moreover, are called the principal SzegΓΆ-Jacobi parameters of . It is well known that for all , (see, e.g., [2, 3]).

Let if is discrete and takes on only values with positive probabilities, and otherwise. We define the Hilbert space , where for all and . is in fact the closure of the space in , , . For many classic probability measures , , , where denotes the smallest sub-sigma-algebra of with respect to which is measurable. We denote by the space of all random variables of the form , where is a polynomial of degree at most , and define , that is, is the orthogonal complement of into for all . For convenience, we define and to be the null space. For all , and is called the homogenous chaos space of order generated by . We will also call the chaos space generated by .

For any , we define the Wick product of and , as . Observe that if is finite, then for all and , such that . It is not hard to see that is in fact the projection of the point-wise product on the space . We extend now the Wick product by bilinearity, defining formally for all and , Since it is not guaranteed that , may not belong to .

Definition 2.1. For any complex number , define the second quantization operator of , where denotes the identity operator of the one-dimensional Hilbert space , spanned by , as a densely defined operator on , defined bywhere for all .

A random variable belongs to the domain of if and only if .

#### 3. Wick-HΓΆlder Property

Definition 3.1. Let and be two fixed positive numbers. Let be a random variable, having finite moments of all orders, and let denote the chaos space generated by . is said to be (, )-Wick-HΓΆlderian, if, for all positive numbers and , such that , and for all and , such that and , there exists , and the following inequality holds:

Since for any fix the function , , , is non-decreasing, if is an -Wick-HΓΆlderian random variable, then is also -Wick-HΓΆlderian for all . By taking we conclude from (3.1) that .

Definition 3.2. Let be a fixed positive number. Let be a random variable, having finite moments of all orders. is said to be -Wick-HΓΆlderian if is -Wick-HΓΆlderian.

Again, if is a -Wick-HΓΆlderian random variable, then is also -Wick-HΓΆlderian for all .

Definition 3.3. Let be a random variable, having finite moments of all orders. is said to be Wick-HΓΆlderian if there exists a positive number , such that is -Wick-HΓΆlderian.

Proposition 3.4. If is a random variable, having finite moments of all orders, then the following two conditions are equivalent:
(1) is Wick-HΓΆlderian;(2) there exist two positive numbers and , such that is -Wick-HΓΆlderian.

Proof. This implication is obvious.
Let us assume that is , -Wick-HΓΆlderian for some and . Let .
Claim 1. is -Wick-HΓΆlderian.
Indeed, let and , such that . Let , and , such that and , where , , and represents the sequence of orthogonal polynomials, having a leading coefficient equal to , generated by . Let denote the -norm. Let and . We have and . Since , , and , where denotes the orthogonality relation, applying the Pythagorean theorem, we obtainBecause , we have and . Thus, and . However, implies . Similarly, we have . Since and is an -Wick-HΓΆlderian random variable, we haveThus, since and , we haveHence, is -Wick-HΓΆlderian.

Definition 3.5. Let be a family of random variables, having finite moments of all orders. The family is said to be uniformly Wick-HΓΆlderian if there exists a positive number , such that for all , is -Wick-HΓΆlderian.

It follows from the proof of the previous proposition that a family is uniformly Wick-HΓΆlderian if and only if there exists a pair of positive numbers, such that for all , is -Wick-HΓΆlderian.

From now on, to make the notation easier, we say that a random variable is of class -W-H, class -W-H, or class W-H if is -Wick-HΓΆlderian, -Wick-HΓΆlderian, or Wick-HΓΆlderian, respectively. We also say that a uniformly Wick-HΓΆlderian family , of random variables, is of class unif.-W-H.

#### 4. A Universal Minimal Condition

In this section we prove a very important condition about any two corresponding multipliers involved in a Wick product inequality.

Lemma 4.1. Let be a random variable, having finite moments of all orders, such that the support of contains at least two distinct points (that means is not almost surely constant). Let and be the first two orthogonal polynomials, with a leading coefficient equal to , generated by . If and are two positive numbers, such that for all , (i.e., and are polynomial functions of degree at most ), the following inequality holds: then one must have:

Proof. The fact that the support of contains at least two points guarantees that . Let be next orthogonal polynomial and let and be the first two principal SzegΓΆ-Jacobi parameters of . We have and (it is possible that , in which case ). As before, denotes the -norm. Let us apply inequality (4.1) to the random variables and , where and are arbitrary real numbers, such that . Sincethe inequality,means thatfor all and all . Subtracting first from both sides of this inequality, and then dividing both sides of the resulting inequality, by the strictly positive number , we conclude that the inequality,holds for all and . Passing to the limit, as , in the last inequality, we obtainfor all real numbers . Moving all terms to the right, we conclude that the quadratic trinomialmust be nonnegative for all real values of . Therefore, must be greater than one, and the discriminant must be less than or equal to zero. This is equivalent to , which in turn means . Dividing both sides of this inequality by the positive number , we conclude that

Corollary 4.2. If is a nonconstant random variable of class -W-H, then is at least .

We will call the condition the universal minimal (unimprovable) condition. We will also say that a nonconstant random variable of class -W-S satisfies the best Wick-HΓΆlder inequality. We know from [1] that every Gaussian random variable satisfies the best Wick-HΓΆlder inequality. However, there are many other random variables of class -W-H, and in the next section we will give some sufficient conditions that guarantee this property.

Let be a random variable, having finite moments of any order, and let and be the sequence of orthogonal polynomials and chaos space generated by , respectively. If is a sequence of complex numbers, then we denote by the densely defined linear operator on that maps for all . We call the sequence the multiplier of the operator . It is clear that for all , . The converse is also true, namely, if is a linear operator defined on the space of all polynomial functions of that leaves all homogenous chaos spaces generated by invariant, then there exists a sequence of complex numbers , such that . If , then , and we say that the operator (or the multiplier ) respects the vacuum space . Doing the same proof as in Lemma 4.1, we can prove the following result.

Lemma 4.3. Let be a random variable, having finite moments of all orders, such that the support of contains at least distinct points, where is fixed. Let , and be the orthogonal polynomial of degree , generated by . If and are two sequences of complex numbers, such that , and for all , , the following inequality holds: then one must have

We call inequality (4.11) the generalized universal minimal (unimprovable) condition. Even though we will not be using this generalized condition in this paper, we would like to reformulate it in words, so that some other mathematicians might use it in the future.

If the norm of the Wick product of and is always bounded above by the product of the norms of and , where and are two multipliers respecting the vacuum space, then the sum of the reciprocals of the square of the modulus of any two corresponding terms of the sequences, and , must be at most .

We extend now the universal minimal condition from the case to the case for . If we pay attention to the proof of the universal minimal condition, when dividing by and then passing to the limit as , then we can observe that, in fact, we were differentiating an inequality twice with respect to . Therefore, we will attack the case in the same manner, based on two very simple observations.

Observation 1. If and are two functions from to that are twice differentiable, such that for all , and there exists an , such that , then , and .

Proof. This can be seen intuitively by drawing a picture for the graphs of and , or of , and mathematically by using the formulaand a similar relation for .

We can formulate this observation in the following way: we can differentiate twice an inequality between two functions at the points where the functions are equal (touching each other), and the inequality is preserved.

Observation 2. If is a real number, then the function is differentiable if and only if , in which case .

We are generalizing now Lemma 4.1 to powers .

Lemma 4.4. Let be a random variable, having finite moments of all orders, such that the support of contains at least two distinct points. Let and be the first two orthogonal polynomials, with a leading coefficient equal to , generated by . Let be fixed. If and are two positive numbers, such that for all , , the following inequality holds: then one must have

Proof. Let and , where and are arbitrary real numbers. Let us consider the functions , and , . Since for all , and , according to Observation 1, we haveBecause has finite moments of all orders, and , a simple application of dominated convergence theorem shows that we can put the derivatives inside the expectations. Thus, applying Observation 2 twice and the product rule of differentiation, we get that for all , Setting , we get nowSince , , and , we have and . Thus, the becomes, after dividing both sides by the positive number for all . Now moving all terms to the right and writing the condition that the discriminant of the quadratic function in be nonpositive, we obtain as before that

Our technique for proving the universal minimal condition for is similar to the proof of Theorem 3 from [4].

Finally, we can also see that for multipliers, the inequalityfor a fix , also implies the condition thatfor all .

#### 5. Random Variables with (π,πΌ)-Subadditive Omega Parameters

We present first the following lemma.

Lemma 5.1. Let be a sequence of positive numbers, such that there exist a number , greater than or equal to , and a sequence , of nonnegative numbers, such that the series is convergent, and for all and natural numbers, with , it holds that Then, for all nonnegative numbers , it holds that where , , and for . If or , then and .

Proof. Since for all and , we conclude that the product is convergent. For , the inequality (5.2) is obvious since . We prove now by induction on that for all , we havefor all . For , we have nothing to prove, since , and is defined to be in this case.
Let us suppose now that (5.3) holds for and all and prove that it continues to hold for and all . Indeed, we may assume that since for and , (5.3) is trivial because . It follows from (5.2) thatUsing now this inequality, the induction hypothesis, and the classic Pascal identity , we getHence, for all , we haveIn order to make this inequality also true, for , we have to replace by .

We introduce now three definitions.

Definition 5.2. Given a number , greater than or equal to , and a sequence of nonnegative numbers , such that the series is convergent, then a sequence of nonnegative numbers is said to be (, )-subadditive if the following inequality,holds for all natural numbers and , such that .

Definition 5.3. Given a number , greater than or equal to , then a sequence of nonnegative numbers is said to be -subadditive, if it is , -subadditive, where denotes the identical zero sequence, that is, if the following inequality,holds for all natural numbers and . In particular, if , then a -subadditive sequence is called simply subadditive.

Definition 5.4. If is a number, greater than or equal to , and is a sequence of nonnegative numbers, then the sequence is said to be exp--subadditive iffor all and positive integers.

Observation 3. An exp--subadditive sequence of nonnegative numbers is also exp--subadditive for all .

Proof. Applying the inequality for all , , and , we getfor all and positive integers.

Observation 4. For all , any exp--subadditive sequence of nonnegative numbers is also -subadditive.

Proof. Since , the function , , is convex. Thus, for all , we have

We present now the main result of this paper.

Theorem 5.5. Let be a random variable, having finite moments of any order, and let be the principal SzegΓΆ-Jacobi parameters of . Let and be two numbers that are greater than or equal to 1. If the sequence satisfies the condition for all , where denotes the dimension of the chaos space generated by , then is of class -W-H.

Proof. Let and , such that . Let be the chaos space generated by . Let and , such that and . Let be the orthogonal sequence of polynomials, with a leading coefficient of , generated by . We distinguish between two cases.
Case 1. If , then there exist two unique sequences of complex numbers and , such thatWe haveUsing the triangle inequality, we obtainFrom the Cauchy-Bunyakovsky-Schwarz inequality, inequality (5.2), and Newton's binomial formula, we obtain
Case 2. If , then all the inequalities used in Case 1, remain the same or become strict inequalities, due to the fact that after a while, all 's become zero. Thus, for example, some complete sums like from Case 1 will become incomplete (that means some of the pais (, ), with , will be missing) in Case 2, and therefore instead of , we will have . Therefore, all the inequalities from Case 1 will also remain true in Case 2. Hence, is of class (, )-W-H in this case too.

Corollary 5.6. Let be a number greater than or equal to , and a sequence of nonnegative numbers producing a convergent series . Let be a random variable, having finite moments of any order, and let be the principal SzegΓΆ-Jacobi parameters of . If the sequence is -subadditive, then is of class -W-H, where .

Corollary 5.7. If is a random variable, having finite moments of all orders, such that its principal SzegΓΆ-Jacobi sequence is -subadditive, then is of class -W-H.

Corollary 5.8. If is a random variable, having finite moments of all orders, such that its principal SzegΓΆ-Jacobi sequence is exp--subadditive, then is of class -W-H. In particular, by taking , it is concluded that if denotes the chaos space generated by , and if , , such that and , then and the following inequality holds:

Observation 5. If , then to prove that the inequality,
holds whenever both expectations from the right-hand side are finite, we do not need the condition that each -binomial coefficient is less than or equal to , but it is enough to assume that for each , the sum of all binomial coefficients, having on the top, is less than or equal to times the sum of all classic binomial coefficients having on the top, that meansfor all .

Proof. The proof of this observation follows line by line the proof of Theorem 5.5, and it uses the fact that for all pairs (, ) such that is equal to a fixed number since , the product is always the same (independent of the pair) and equal to .

The following proposition will be useful in showing later that is optimal for some particular random variables .

Proposition 5.9. Let be a random variable having finite moments of all orders. We assume that the probability distribution of has an infinite support. Let be the chaos space generated by and the principal SzegΓΆ-Jacobi sequence of . We define for all , and . Then,
(1) if there exists a positive number such that the inequality,holds for all and , such that and , then ;(2) if satisfies (5.20), then is optimal (i.e., the smallest among all positive 's satisfying this inequality);(3) if for all , then is optimal.

Proof. (1) Let such that (5.20) holds whenever both expectations from its right-hand side are finite. For every , let us choose and . The inequality is equivalent to . This implies that . Summing up from to , we obtain for all . Taking the th root in both sides of this inequality and then passing to the superior limit as , we get . Thus, .
(2) It follows from 1.(3) It follows from letting and in Observation 5.

#### 6. Examples

Example 6.1. Let be a fixed real number and let be a random variable, having finite moments of all orders, and the principal SzegΓΆ-Jacobi parameters for all . It is clear that is an exp--subadditive (in fact exp--additive) sequence. Therefore, is also -subadditive. Thus, is of class -W-H. Moreover, the following lemma holds.

Lemma 6.2. For all , it holds that Formula (6.2) holds even for .

Proof. If , then we have Since , the function , , is convex, and thus Hence,Since , as , we conclude that exists and is equal to .
If , then in a similar way, we can show thatand it follows again that .

From formula (6.2), inequality (6.1), Corollary 5.8, and Proposition 5.9, we obtain the following proposition.

Proposition 6.3. Let be a random variable, having finite moments of any order whose principal SzegΓΆ-Jacobi parameters are for all , where is a fixed real number, such that . Let be the chaos space generated by .
(1) If and are positive numbers, such that , then for all and , such that and , there exists and the following inequality holds:(2) is the smallest among all positive numbers , for which the inequality,holds for all and , such that and .

Example 6.4. This is a modification of the previous example. Let be a fixed real number, such that . Let be a random variable whose principal SzegΓΆ-Jacobi parameters are for all , where . The condition follows from the inequality , and ensures that for all . The reason why we have chosen will become more transparent later.

Claim 1. If , then the sequence is -subadditive.

Indeed, for all and natural numbers, we haveMultiplying the second inequality by and adding the resulting inequality to the first one, we get

Claim 2. If , then the sequence is not -subadditive.

Indeed, if we define , then for all , we have

Claim 3. If , then there exists a sequence of positive numbers, such that is convergent, and the sequence is (, )-subadditive.

As before, let , . Let be a fixed natural number. Let us find a positive number , such thatfor all . This inequality is equivalent tofor all . Since , if we choose such thatthen the last inequality holds, for all . In fact, for even, if we choose (that means ), then we can see that cannot be chosen smaller than the value of the solution of (6.14). Solving (6.14) for , we getSince and , we can see that for all . Moreover, since we have chosen , the series has the same nature as the series , which is convergent.

Therefore, according to Theorem 5.8 for , is of class -W-H, while for , is of class -W-H for some ( depends on ). Moreover, the family is not uniformly Wick-HΓΆlderian (that means we cannot find the same for all ). This follows from the observation that , while . If we assume the existence of two positive numbers and , such that the inequality: holds for all since , where and , are the orthogonal polynomials of degree 1 and 2, respectively, generated by , then we would conclude thatThis inequality reduces to for all , which is impossible since the left-hand side converges to a positive number, while the right-hand side tends to zero as goes to .

Example 6.5. Let us take now , and consider the family of random variables whose principal SzegΓΆ-Jacobi parameters are for all , where . If we take the other SzegΓΆ-Jacobi parameters to be for all , where is a fixed real number, then we can see that are exactly the centered and rescaled Meixner random variables.

We can see exactly as in Example 6.5 that for , is -subadditive, while for , it is not -suadditive. If , unfortunately, for , the series is not convergent. Moreover, as we saw in the previous example, since (6.14) cannot be avoided for even, we can see that the sequence is not (, )-subadditive, for any nonnegative sequence , such that .

Claim 1. If , then for all , we haveSince and , we may assume that .

We fix , and prove by induction on that for all , .

Let us prove first this inequality for . Since for all