INTERPOLATION THEORY FOR THE HK-FOURIER TRANSFORM

. We use the Henstock–Kurzweil integral and interpolation theory to extend the Fourier cosine transform operator, broadening some classical properties such as the Riemann–Lebesgue lemma. Furthermore, we show that a qualitative diﬀerence between the cosine and sine transform is preserved on diﬀerentiable functions.


Introduction
We shall deal with real Banach spaces denoted by X and with their complexification given by X + iX. Also, given two Banach spaces X and Y , we denote by L(X, Y ) the Banach space of all bounded linear operators T : X → Y with the operator norm given by T L(X,Y ) = sup { T (x) Y : x X ≤ 1}. For any T ∈ L(X, Y ) we defineT (x + iy) := T (x) + iT (y) (x, y ∈ X).
It follows that T L(X,Y ) = T L(X+iX,Y +iY ) . This procedure has been used by several authors [24,2,17]. We recall that for any p ∈ [1, ∞) and X ⊂ R, the symbol L 1 (X) denotes the space of all Lebesgue measurable functions f : X → R with Moreover, we denote by W p = {f : R → R | f (x) = 0 a.e.} ≡ the subspace of L p (X) on which · L p vanishes. It is known that · L p is a seminorm for all p ∈ [1, ∞) and induces a norm on the quotient space L p (X)/W p , under which it is complete. We will denote this space with respect to its norm by L p (X), [27]. Similarly, for p ∈ [1, ∞) we define L p (X, C) and L p (X, C) by considering functions f : X → C.

JUAN H. ARREDONDO AND ALFREDO REYES
For p = ∞ and f : X → R, we define f ∞ as the essential supremum of |f |, and L ∞ (X) is the vector space of all Lebesgue measurable functions f for which f ∞ < ∞. Similarly, we define L ∞ (X), L ∞ (X, C) and L ∞ (X, C). 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. Furthermore, for a function f ∈ L p (X) or f ∈ L p (X, C), we will call by the same symbol f the (unique) element that defines this function in L p (X) or in L p (X, C), respectively. Also, the characteristic function of a set E is given by χ E (x) = 1 if x ∈ E and zero otherwise.
If f belongs to L 1 (R) ∩ L p (R), the Fourier transform is defined for every real number s as where the integral is taken in the Lebesgue sense. F c p and F s p are called Fourier cosine and Fourier sine transforms, respectively. Furthermore, by interpolation theory, the operator F p (f ) is extended to L p (R) for p ∈ [1, 2] as a bounded operator where 1/p + 1/q = 1 and The value of γ p is given by the Hausdorff-Young inequality [25], the sharp Hausdorff-Young inequality [5,29], [15,Theorem 5.7] and [3]. For any unbounded subset X ⊂ R, the space C ∞ (X) denotes the complex valued continuous functions on X vanishing at infinity [25]. We denote the space of bounded variation functions by BV (R) and by BV 0 (R) the subspace of functions vanishing at infinity, [12,4,31]. Also BV 0 (R, C) is the corresponding complexification of BV 0 (R).
In [30] the Henstock-Kurzweil integral was employed to study the Fourier transform. In [20,22] it was proved that (1.1) makes sense as a Henstock-Kurzweil integral on BV 0 (R). In fact, we have the following statement in [23]. INTERPOLATION THEORY FOR THE HK-FOURIER TRANSFORM   403 Definition 1.1. The HK-Fourier transform exists for every s = 0, and is defined by where the integral is in the Henstock-Kurzweil sense. Analogously, we define the HK-Fourier cosine transform F c HK and the HK-sine Fourier transform F s HK as in (1.1).
We say "HK-Fourier transform" in order to emphasize the use of the Henstock-Kurzweil integral [30]. Moreover, F HK (f )(s) is pointwise defined and is continuous except at zero; see example 3(d) in [30]. Note that F HK is well defined because the Henstock-Kurzweil integral contains the Lebesgue integral, [14,19]. F 1 can be seen as an extension of the HK-Fourier transform restricted to BV 0 (R), Moreover, F p is an extension of F 1 , so that F p is an extension of F HK .
The relation between F p and F HK was first studied in [23], while the operator F c HK was studied in [3]. This work builds on these references.

Henstock-Kurzweil Fourier transform
The space of Henstock-Kurzweil integrable functions defined on an interval I is denoted by HK(I). This space is a seminormed space with the Alexiewicz seminorm, defined as The quotient space HK/W(I) will be denoted by HK (I), where W(I) is the subspace of HK (I) for which the Alexiewicz seminorm vanishes [7]. The completion will be denoted by HK (I) and its complexification will be written as HK (R, C).
We study the HK-Fourier cosine transform defined by Notice that for s = 0 and f ∈ BV 0 (R), F c HK (f )(0) might not be defined. Also, we have that for all f ∈ L 1 (R) ∩ BV 0 (R) and s ∈ R. However, a partial result about the question of continuity at s = 0 was proved in [3,Theorem 1]. In fact, F c HK is bounded while F s HK is not. Actually, Theorem 1 and Proposition 3 in [3] imply the following statement. (i) The HK-Fourier cosine transform is a bounded linear operator from BV 0 (R) into HK (R). (ii) The Fourier transform is a densely defined closed operator from L 2 (R) into HK (R).
We shall show that differences and similitudes between the Fourier cosine and Fourier sine transforms also hold on the classical Sobolev space W 1,1 (R). It is expected that these transforms are bounded operators with the same domain and codomain for functions with enough smoothness, for example as in the Schwartz space [25]. See also [16].

Interpolation theory
We consider a couple (X, Y ) of complex Banach spaces such that X and Y are continuously embedded in a Hausdorff topological vector space V , i.e., X ⊂ V and Y ⊂ V with continuous inclusion. This couple is called a complex interpolation couple. In this case the intersection X ∩ Y is a linear subspace of V , and it is a Banach space under the norm Remark 3.1. It follows from [18] that the space X +Y is isometric to the quotient Throughout this section we shall consider S = {z ∈ C : 0 ≤ Re(z) ≤ 1} and we shall use the complex space X +Y and the space F(X, Y ) of functions f : S → X +Y holomorphic on the interior of the strip S and continuous up to its boundary, such that the maps t → f (it) and t → f (1 + it) are continuous from the real line into X and Y , respectively. Therefore, F(X, Y ) is a Banach space with the norm given by These facts can be consulted in [ consisting of the functions vanishing at z = θ. Moreover, N θ is closed (see [6,18] In order to construct the interpolation space of L 1 (R) and BV 0 (R) we consider the space This yields uniform convergence of the sequence to f . Similarly, there exists [f ] ∈ L 1 (R) such that f n −f L 1 → 0 (n → ∞). It follows that there exists a subsequence (f n k ) k≥1 of (f n ) n≥1 converging pointwise a.e. to f ; see [27,8]. From the fact that (f n ) n≥1 converges uniformly to f , we get that f (x) =f (x) a.e., yielding f ∈ L 1 (R) and Therefore, if a ∈ L 1 (R) + BV 0 (R), then it is an equivalence class given by a = (f, g) + D. Nevertheless, we shall write a = f + g to simplify notation. Also, we define a L 1 +BV0 := inf This is a norm, by standard arguments. Then we consider the completion of the space L 1 (R) + BV 0 (R), denoted by L 1 (R) + BV 0 (R). In addition, on the product 406 JUAN H. ARREDONDO AND ALFREDO REYES Thus the sum space L 1 (R)+BV 0 (R) is a Banach space with the quotient norm [26]. Its elements are equivalence classes of the formā = f +g = ([f ], g)+D ; however, we will just writeā = f + g. We have the following characterization. .
Therefore, we have characterized the real space L 1 (R) + BV 0 (R). The complexification space of this space is given by Similarly, we define the real space L ∞ (R) + HK (R) and its complexification L ∞ (R, C) + HK (R, C). We will consider complex spaces and omit the symbol (R, C) to simplify notation. Furthermore, for the complex interpolation couples Formula (3.2) is well defined on L 1 (R) + BV 0 (R). By interpolation theory, F c 1 is extended to L 1 (R) + BV 0 (R) for each s = 0. Thus, from Theorem 2.1 and Theorem 3.6 we conclude that The following estimate for its norm is valid: for every θ ∈ (0, 1), where C = 4π Si(π) and Si(x) := 2 π x 0 sin(y) y dy.
holds true pointwise almost everywhere and the Riemann-Lebesgue lemma is satis- Proof.

Corollary 3.11.
For u ∈ W 1,1 (R, C), F c 1 (u) belongs to HK (R, C). The proof of Corollary 3.11 follows from the fact that W 1,1 (R) ⊂ BV 0 (R), and then by Theorem 2.1, F c HK W 1,1 (R) ⊂ HK (R). Therefore, the range of the Sobolev space W 1,1 (R, C) under the HK-Fourier cosine transform is contained in HK (R, C). Explicitly, The Fourier cosine and sine transforms are continuous operators on L 2 (R, C), while their qualitative differences appear even on the space of functions W 1,1 (R, C) that have a degree of regularity. In the following example we show this difference.

Example 3.12. Let us define
For each x ∈ (0, 1), we have We extend h over R as an odd map. Also, we consider an even function ϕ ∈ C ∞ c (R) such that 0 ≤ ϕ(x) ≤ 1, with ϕ(x) = 1 for |x| ≤ 1/2 and vanishing for |x| ≥ 1. We define f (x) := h(x)ϕ(x), for all x ∈ R. Thus, f is an odd map belonging to We analyze the convergence of the Henstock-Kurzweil integral: Thus, for 0 < b < ∞, we get from Lebesgue's dominated convergence theorem, Fubini's theorem and Hake's theorem [4]: In fact,

JUAN H. ARREDONDO AND ALFREDO REYES
Integrating by parts, By Lebesgue's dominated convergence theorem we conclude that Therefore the limit of I 2 is zero. Writing explicitly the integrand of the integral I 1 we get Therefore, the integral in (3.6) does not exist and F s HK (f ) does not belong to HK (R). In conclusion, Then, the HK-Fourier sine transform remains unbounded on W 1,1 (R), in contrast with relation (3.5). Also, in [3,Example 1] it was established that The function given in Example 3.12 is a slight variation of one considered in [11]. We can proceed in the same way to define the space L 2 (R, C) + BV 0 (R, C). Therefore, we have the continuous inclusions for 0 < θ < 1. Now for the extended operators F c 2 and F c HK on L 2 (R, C) and on BV 0 (R, C) respectively, we define the map    INTERPOLATION THEORY FOR THE HK-FOURIER TRANSFORM   411 for every θ ∈ (0, 1) and C given by (3.3). Similarly, for the operators F c p and F c HK on L p (R, C) and on BV 0 (R, C) respectively, we define where f = f p + g with 1/p + 1/q = 1. This operator is a generalization of the map considered in [3,Corollary 1]. For the couples given by X 1 = L p (R, C) and X 2 = L q (R, C), with 1 ≤ p ≤ 2, and Y 1 = BV 0 (R, C) and Y 2 = HK (R, C) we have from Theorem 3.6 that for every θ ∈ (0, 1), and the following estimate for the norm: where C is given in (3.3).
For 1 < p < 2, the relation between F c p , F c 1 and F c 2 is given by the decomposition of L p (R) in [23], which implies that for each f p + g ∈ L p (R) + BV 0 (R) there exist Proof. This follows by taking a sequence (f n ) n≥1 on L p (R) such that f n → f with f = f 1 + f 2 ∈ L p (R), and using that the sequence f n = f 1,n + f 2,n has the property f i,n → f i in the norm of L i (R), i = 1, 2; see [23].
As a consequence of Proposition 3.15, the range of W 1,p (R), for 1 < p ≤ 2, under the action of the L p -Fourier transform operator is contained in L 1 (R). Explicitly, This relation contrasts with (3.8).