ON THE NORM CONTINUITY OF THE HK-FOURIER TRANSFORM

. In this work we study the Cosine Transform operator and the Sine Transform operator in the setting of Henstock-Kurzweil integration theory. We show that these related transformation operators have a very diﬀerent behavior in the context of Henstock-Kurzweil functions. In fact, while one of them is a bounded operator, the other one is not. This is a generalization of a result of E. Liﬂyand in the setting of Lebesgue integration.


Introduction
If f belongs to the space of real valued Lebesgue integrable functions, L 1 (R), the Fourier transform is defined for every real number s as where the integral is taken in the Lebesgue sense. When f is in L 2 (R), the Fourier transform of f can be defined as where the limit is taken in the norm topology of L 2 (R) and (f n ) n≥1 is a sequence in L 1 (R) ∩ L 2 (R) such that f n − f 2 → 0, as n → ∞.
While the operator F 1 has an integral representation on its domain, the operator F 2 shares this property only on a dense subspace of its domain. This happens also for the Fourier transform operator F p defined on L p (R) for 1 < p < 2. Recently, it was shown in [13] that having an integral representation implies additional properties for the Fourier transform operator. Pointwise continuity and the Riemann-Lebesgue lemma were shown to be valid on a larger subspace of the domain of each of the operators F p for 1 < p ≤ 2. The proof relies on the Henstock-Kurzweil integral which has the remarkable property that every Lebesgue integrable function is also integrable in the setting of the Henstock-Kurzweil theory with values of both integrals coinciding.
If f ∈ L 2 (R), then e −isx f (x) is not necessarily Lebesgue integrable. However, the HK-Fourier transform operator defined by the same formula (1.1) is well defined as a Henstock-Kurzweil integral for each s = 0 and any function of bounded variation vanishing at infinity [11,10]. See definition below. We say "HK-Fourier transform" in order to emphasize the use of Henstock-Kurzweil integral [17]. Furthermore, it was shown in [13] that F p and the HK-Fourier transform operator coincide in the intersections of their domains.
In this paper we look at norm continuity of the Henstock-Kurzweil Fourier transform operator. There is also a pending question concerning pointwise continuity of a HK-Fourier transform function F HK f (s) at the origin. We have not answered this question but we show below that there is a type of smoothness even at s = 0 in the case of the "real part" of the HK-Fourier transform operator, namely the Cosine transform operator.

Henstock-Kurzweil Fourier transform
where the supreme is taken over all (finite) partitions P of I. If I = R, then f is of bounded variation if and only if exists in R. We will denote the set of bounded variation functions over an interval I ⊆ R as BV (I). If I ⊆ R is an unbounded interval, we define BV o (I) as the subspace of BV (I) consisting of the functions which have limit zero at ±∞: In [17] the Henstock-Kurzweil integral was employed to study the Fourier transform. Later, in [11,10], it was proved that (1.1) makes sense as a Henstock-Kurzweil integral over the space BV o (R). The norm in BV (I) is taken as Note that over BV o (R) the norms · BV (R) and · BV (R) are equivalent: Definition 2.2. Let 0 < p < ∞ and X ⊂ R. For any Lebesgue measurable function f : X → R we define The real vector space of functions f such that f p < ∞ is denoted by L p (X) and W p denotes the subspace of function on which · p vanishes.
For real numbers p ≥ 1, · p is a seminorm on L p (X) and induces a norm in the quotient space L p (X)/W p . We will denote the completion of this space with respect to its norm by L p (X). Similarly, for p ≥ 1 we define L p (X, C) and L p (X, C) by considering functions f : X → C. For p = ∞ and f : X → R, we define f ∞ to be the essential supremum of |f |, and L ∞ (R) denotes the vector space of all Lebesgue measurable functions f for which f ∞ < ∞.
If A X is a Lebesgue measurable set and m denotes the Lebesgue measure, then given a Lebesgue measurable function f defined on A such that m(X A) = 0, we will denote by the same symbol f the trivial extension of f to a (measurable) function on X. That is, we extend the function as zero on X A. Furthermore, for a function f ∈ L p (X), or f ∈ L p (X, C), we will denote by the same symbol f the (unique) element that defines the function in L p (X) or in L p (X, C), respectively.
In order to introduce the definition of the Henstock-Kurzweil integral, we consider the system of extended real numbers R := R ∪ {±∞}, and for each interval of [a, b] is called γ-fine according to the following cases: For a ∈ R and b = ∞, For a = −∞ and b ∈ R, For a = −∞ and b = ∞,

The number A is the integral of f over [a,b] and it is denoted by
Using the convention 0 · (±∞) = 0, an extra condition for f is f (±∞) = 0 [1].
The space of Henstock-Kurzweil integrable functions defined on an interval I R will be denoted by HK(I). Two fundamental theorems over the Henstock-Kurzweil integral are the following, the cases on [−∞, ∞] and [−∞, b] are analogous, see [1].
The second integral on the right side of the equation is a Riemann-Stieljes integral and  The space HK(I) is a seminormed space with the Alexiewicz seminorm, which is defined as The quotient space HK(I)/W(I) will be denoted by HK(I), where W(I) is the subspace of HK(I) on which the Alexiewicz seminorm vanishes [3]. By HK(R, C) will be denoted the space . The completion of the spaces HK(R) and HK(R, C) with respective given norms will be denoted by HK(R) and HK(R, C).
Let us consider S(R), the Schwartz space of real valued functions defined on R. We know that the Fourier transform operators F 1 and F 2 are well defined on S(R) and L 1 (R)∩L 2 (R) and have an integral representation given by (1.1) valid for every s ∈ R. Because of their density in L 2 (R), both spaces are used to extend the Fourier transform over L 2 (R), see [4] and [16]. We also know that HK(R) ∩ BV (R) is a dense subspace of L 2 (R). An important point is and for f ∈ BV o (R) the integral in (1.1) is well defined as a Henstock-Kurzweil integral for each s = 0. This means that on a dense subspace of L 2 (R), not contained in L 1 (R), the Fourier transform operator F 2 is represented by an integral. Furthermore, a similar asseveration holds true for the Fourier transform operator with domain L 2 (R, C). See [11] and [13].
For any unbounded subset X ⊂ R, we denote by C ∞ (X) the space of complex valued continuous functions on X vanishing at infinity [14].
Definition 2.7. The HK-Fourier transform exists for every s = 0 and is defined by where the integral is a Henstock-Kurzweil integral.
We define the norm The next proposition is a corollary of [12, Theorem 1].
Proposition 1. The HK-Fourier Transform operator with domain HK(R)∩BV (R) and codomain L 2 (R, C) is a bounded operator.
Proof. From the Plancherel Theorem and [12, Theorem 1] we get The Henstock-Kurzweil Fourier Sine Transform is given by We have that f ∈ HK(R) ∩ BV (R) and for s > 0, it follows that Note that this function is not an element of HK(R). Therefore, the image of the space HK(R) ∩ BV (R) under the action of F s HK is not contained in HK(R). The previous example shows that the HK-Fourier Sine transform cannot be defined as a bounded operator from BV o (R) into HK(R). However, for the HK-Fourier Cosine transform a different situation occurs. The integrability of the Fourier Cosine and Sine transforms of functions in BV o (R) is a problem that has been attacked in different ways. The aim is to obtain a wide variety of subspaces of BV o (R) where the transforms are integrable. In [9], Liflyand studied the integrability of these transforms in the sense of Lebesgue. Among others, he showed that when f ∈ BV o (R) is locally absolutely continuous with its derivative in a space W, then the Fourier transform of f belongs to L 1 (R). Here W being the subspace of functions g ∈ L 1 (R) such that In analogy with the above we take the space which is not empty because S(R) ⊂ Λ. Let AC loc (R) be the space of locally absolutely continuous functions on R. In this setting, we provide the next proposition.
Proof. The proof for the HK-Fourier Sine transform is obtained from the Multiplier Theorem and the equality A similar formula for (F c HK g)(s) is valid, which proves the proposition.
This shows that taking into account the Henstock-Kurzweil integration theory, the subspace of BV o (R) where the HK-Fourier transform of each of its elements is integrable, it is larger than the one considered by Liflyand. For might not exist, so that (F c HK f )(s) is not well defined at the point s = 0. Without resorting to the condition over the space Λ, we prove in theorem 1 below that F c HK can be extended to a bounded linear transformation from BV o (R) into HK(R). We will need some lemmas.
We set R + = [0, ∞) and Proof. This follows from elementary properties of the integral b a f (x) dx and consideration of the cases a · b ≥ 0 or a · b < 0.
Remark 1. The Sine Integral function, see [6] and [15], is given by sin y y dy.
It has the following properties: Let us consider the set of functions Ω : . For given 0 ≤ u ≤ v and 0 < t, we make the change of variable y = tx. It follows that Therefore, because h t is an even function, we have that Ω is a bounded set in HK(R) and Proof. We have as a consequence of the Multiplier Theorem: Therefore, applying (2.7) we obtain the proof.
As we already mentioned, (F c HK f )(s) might not be well defined at the point s = 0. However, it does have certain regularity even there. This regularity implies the Bounded Linear Transformation theorem for the HK-Fourier Cosine Transform operator. Proof. We know that By Fubini's Theorem, Therefore, by Lemma 2.9, we get that Moreover, since F c HK (f ) is an even function, (2.9) with a = 0 and (2.10) yield: Therefore, the norm of F c HK (f ) is finite. To show that F c HK (f ) belongs to HK[0, 1], we prove the existence of the limit Given ε > 0 take R > 0 great enough such that (2.14) if b, b < δ for some positive δ. Here C is a constant depending only on f and R. By using (2.8), and the two previous estimations one proves the existence of (2.12).
Similarly, we prove that F c HK f ∈ HK(R) by showing existence of We get as before the same estimation in (2.13), for any b, b > 0 and R > 0 great enough. Now to show that the integral in (2.14) is small, we estimate the integral The main argument is to show that each of these integrals can be viewed as convergent alternating series. First we take f continuous and nonincreasing in [0, R] such that f (R) > 0. Note that for a given y > 0 andỹ = y + π, theñ It follows that the first two integrals on the right side of (2.16) tends to zero for b ≥ b 1 great enough. The last integral on the right side of (2.16) can be written as Note that |f (y/b)−f (0)| and |f (y/b )−f (0)| are arbitrarily small whenever |y| ≤ b δ and δ > 0 is small enough under the hypothesis that f is continuous. Because f is nonincreasing these two integrals define alternating series with respective initial terms See [5], [18]. Summing up, the previous arguments applied to f 1 and f 2 give (2.15), which proves the theorem.
This theorem has its implications to interpolation theory for the classical Fourier Transform on the space L p (R Also, we consider the spaces L p (R) ∩ BV o (R) and L q (R) ∩ HK(R) with given norms .
Similarly for f L q ∩HK . In [7, Theorem 6.3.1] it is proved that the space of bounded variation functions defined on a compact interval [a, b] is a Banach space. With a few changes over that proof it is possible to show that the space BV 0 (R) is a Banach space. Therefore, L p (R) ∩ BV 0 (R) is a Banach space of real valued functions defined on R, whereas elements in the Banach space L q (R) ∩ HK(R) are classes of functions.
Proof. The density of D( ) in L 2 (R, C) follows since: S(R, C) ⊂ D( ), and it is a dense subspace of L 2 (R, C). Moreover, F 2 restricted to S(R, C) is a bijection onto S(R, C) ⊂ HK(R, C). In order to prove that is a closed operator, we take a sequence (f n ) in D( ) such that f n → f in L 2 -norm and f n → Υ in HK-norm.
Both together must imply f ∈ D( ) and f = Υ.
Note that Υ might belong to the completion of HK(R, C). Since F 2 is an unitary operator on L 2 (R, C), one has F 2 f n ∈ L 2 ([s, t] , C) ∩ L 1 ([s, t] , C) for every s, t ∈ R. Therefore, One can use the Cauchy-Bunyakovsky-Schwarz inequality to prove that the last equality holds true for every [s, t] ⊂ R. This shows that F 2 f = Υ ∈ HK(R, C), proving that f ∈ D( ).