New convolutions for an oscillatory integral operator on the half-line

The main purpose of this work is to present three new convolutions for oscillatory integral operators defined on the positive half-line and in the framework of L1 Lebesgue spaces. Therefore, such new functions introduced by the new convolutions will have factorization properties when considering the oscillatory integral operators under consideration. Moreover, some fundamental and operational properties of the mentioned integral operator are also studied in the first part of the paper. INTRODUCTION Convolutions are very important operations in Functional Analysis, Integral Operators and Integral Transforms and in their applications. In fact, probably the most important property of a convolution – besides generating new functions upon previously given two or more functions – is to satisfy a factorization property which is typically intrinsically associated with one or more than one integral operators. In most of the cases, such factorization property is fundamental to solve consequent integral equations which can be characterized by those convolutions. The properties of the convolutions may also be very diverse up to the concrete definition of each convolution (cf. [1, 2, 7, 10, 15, 23, 24, 25]). The integral operators which are associated with each one of the know convolutions may have also a very distinct nature (see, e.g., [3, 4, 5, 6, 8, 9, 11, 12, 13, 14, 21, 22]). Moreover, the range of possible applications of convolutions to other sciences is also wide (cf. [16, 17, 18, 19, 20, 26]). Here, we will introduce some new convolutions associated with oscillatory integral operators on the positive half-line. We start by defining those operators. Let L(R+) denote the Lebesgue space of all absolutely integrable complex-valued functions on R+. For any function f ∈ L(R+), we define the integral operator S ±,± by (S ±,± f )(x) = 1 √ 2π ∫ +∞ 0 (± cos(xy) ± sin(xy)) f (y)dy, x ∈ R+, (1) respectively. The main aim of this work is to introduce new convolutions for the operators just presented. That is to say, in the present case, to introduce new functions, generated by two previously given functions, which will exhibit certain factorization identities when considering the above operators applied to all those functions. This will achieved in the final section of this work. FUNDAMENTAL PROPERTIES OF THE INTEGRAL OPERATOR S +,+ In the present section, we will be concentrated in studying some fundamental properties of the operator S +,+: this will be the case of some mapping properties. The Integral Operator S +,+ in the Framework of the Schwartz Space Let us denote by S(R+) the Schwartz space on R+, and by C∞ 0 (R+) the Banach space of all continuous and infinitely differentiable functions on R+ that vanish at infinity. In this subsection, the operator S +,+ is studied in the framework of the Schwartz space. We prove that if f ∈ S(R+) then, S +,+ f ∈ S(R+). The following lemma shows some basic properties of the integral operator S +,+ in the Schwartz space S(R+). Lemma 1 Suppose that f ∈ S(R+). Then, (i) S +,+ f is continuous; (ii) ( dk dxk S +,+ f ) (x) =  ik+3S +,−(y f )(x), if k is odd ikS +,+(y f )(x), if k is even , k ∈ N; (iii) S +,+ ( dk dyk (y α f ) ) (x) =  ik+1xkS +,−(y f )(x), if k is odd ik xkS +,+(y f )(x), if k is even , k ∈ N, α ∈ Z; (iv) ||S +,+ f ||∞ ≤ 1 π || f ||1. Proof. (i) For any x ∈ R+, we have lim h→0 (S +,+ f )(x + h) = lim h→0 1 √ 2π ∫ ∞ 0 [cos((x + h)y) + sin((x + h)y)] f (y)dy = 1 √ 2π ∫ ∞ 0 f (y) lim h→0 [cos((x + h)y) + sin((x + h)y)]dy = (S +,+ f )(x). (ii) Let us compute the first derivative of S +,+ f . We have ( d dx S +,+ f ) (x) = 1 √ 2π d dx ∫ ∞ 0 (cos(xy) + sin(xy)) f (y)dy = 1 √ 2π ∫ ∞ 0 f (y) d dx (cos(xy) + sin(xy)) dy = 1 √ 2π ∫ ∞ 0 y f (y) (cos(xy) − sin(xy)) dy = S +,−(y f )(x), where the interchange of the derivative and the integral is valid since f ∈ S(R+). Applying the same procedure, we obtain that d2 dx2 (S +,+ f )(x) = −[S +,+(y f )](x), d3 dx3 (S +,+ f )(x) = −[S +,−(y f )](x), d4 dx4 (S +,+ f )(x) = [S +,+(y f )](x), and the result follows by induction. (iii) Integrating by parts and having into consideration that f ∈ S(R+), we obtain S +,+ ( d dy (y f ) ) (x) = 1 √ 2π ∫ ∞ 0 d dy (y f (y)) (cos(xy) + sin(xy))dy = 1 √ 2π y f (y)(cos(xy) + sin(xy))|0 − 1 √ 2π ∫ ∞ 0 y f (y)(−x sin(xy) + x cos(xy))dy = − 1 √ 2π x ∫ ∞ 0 y f (y)(cos(xy) − sin(xy))dy = −xS +,− (y f ) (x). With same procedure, we deduce that S +,+ ( d2 dy2 (y f ) ) (x) = −x2S +,+ (y f ) (x) S +,+ ( d3 dy3 (y f ) ) (x) = x3S +,− (y f ) (x) S +,+ ( d4 dy4 (y f ) ) (x) = −x4S +,+ (y f ) (x), and the result follows by induction. (iv) From a direct computation, it follows ||S +,+ f ||∞ = sup x∈R+ ∣∣∣∣∣ 1 2π ∫ +∞ 0 (cos(xy) + sin(xy)) f (y)dy ∣∣∣∣∣ ≤ 1 √ 2π sup x∈R+ ∫ +∞ 0 |(cos(xy) + sin(xy))| | f (y)|dy ≤ 1 √ 2π sup x∈R+ ∫ +∞ 0 √ 2| f (y)|dy