Generalized Hankel-Schwartz Type Transformations on L p,ν Spaces of Distributions

Transcription

1 Int. Journal of Math. Analysis, Vol. 4, 21, no. 51, Generalized Hankel-Schwartz Type Transformations on L p,ν Spaces of Distributions B. B. Waphare MAEER s MIT Arts, Commerce and Science College Alandi (D), Pune Maharashtra, India bbwaphare@mitpune.com, principal@mitacsc.com Abstract In this paper we have defined two Hankel-Schwartz type transformations. The Hankel-Schwartz type transformations are defined on L p,ν and L p,ν spaces. Further it is shown that transformations defined by (1) and (2) are bounded linear operators of L p,ν. Finally we have established the bounded linearness of distributional generalized Hankel- Schwartz type transformation on L p,ν spaces. Keywords: Hankel-Schwartz transformation, bounded linear operator, generalized transformation, distributions, Bessel function. 1 Introduction In Betancor and Negrin [2], two Hankel-Schwartz transformations are defined by B μ,1 (f)(y) = x 2μ+1 b μ (xy) f(x) dx (1) B μ,2 (f)(y) =y 2μ+1 b μ (xy) f(x) dx (2) where b μ (t) = t μ J μ (t) and J μ is the Bessel function of first kind and order μ. These transformations have been studied by Altenbung[1], Schuitman [14], Lee [6], Sanchez [13], Mendez [7] and many others.

2 2516 B. B. Waphare Inspired by Betancor and Negrin [2], we define Hankel-Schwartz type transformations by H α,β,1 (f) (y) = x 4α C α,β (xy) f(x) dx (3) where H α,β,2 (f) (y) =y 4α C α,β (xy) f(x) dx, (4) C α,β (xy) =(xy) (α β) J α β (xy). In this paper we study the behaviour of transformations (3) and (4) on the L p,ν spaces introduced by Rooney [1]. We prove that H α,β,1 - transformation is a bounded linear operator of L p,ν into L p,2p(3α+β) ν, provided that 1 <p<, (α β) > 1 and 2 (2α + β) < ν < 2(3α + β). Further if 1 <p< 2 p, (α β) > 1and 2β < ν < 1 then H 2 p α,β,2 is a bounded linear operator of L p,ν into L p, ν 2(α β) p. Iff L q, (ν 4α)q ν and g L p,ν then mixed Parseval s equation can be stated as f(x) H α,β,1 (g) (x) dx = H α,β,2 (f)(y) g (y) dy (5) provided that 1 <p<, 1 p + 1 q =1, (α β) > 1 2 and 2 (2α + β) < ν p < 2(3α + β). The relation (5) suggests definitions of the generalized Hankel-Schwartz type transformations. More precisely we define the distributional generalized H α,β,1 - transformation on L q,ν+(1 ν)q as the adjoint of the classical H α,β,2 transform, so that H α,β,1 f,φ = f, H α,β,2 φ (6) for every f L q, ν+(1 ν)q and for every φ L q, (ν 4α) q ν provided that ) 1 <q<, (α β) (1 1 and 2 (2α + β) <ν 1 < 2(3α + β). 2 q The distributional generalized H α,β,2 transformation on L p,2(3α+β)p ν as the adjoint of the H α,β,1 - transformation, through is defined H α,β,2 f, φ = f, H α,β,1 φ, (7)

3 Generalized Hankel-Schwartz type transformations 2517 for all f L p, 2(3α+β)p ν, for all φ L p,ν provided that 1 <p<, (α β) > 1 and 2 (2α + β) < ν < 2(3α + β). 2 p Note that the relations (6) and (7) appear to be extensions of the mixed Parseval equation (5). 2 Preliminaries In this section we briefly recall some definitions and properties of L p,ν spaces (see Rooney [1]). Let 1 p<,νis real. We define L p,ν as the collection of functions f, measurable on (, ) and which satisfy f p,ν = [ x ν 1 f (x) p dx ] 1/p <. The space D (, ) consists of all smooth complex valued function having compact support contained in (, ). We will need the following in the sequel. Lemma 2.1 D (, ) is dense in L p,ν for any ν and any p satisfying 1 p<. Forγ>,Rea>, Reb>, let (I γ,a,η f)(x) = ν (a) 1 (1 u γ ) a 1 u νη 1 f (ux) du (8) (J γ,b,ξ f)(x) = ν (a) 1 (u γ 1) a 1 u ν(b+ξ 1) 1 f (xu) du, (9) where η and ξ are complex numbers. I ν,a,ξ and J ν,b,ξ are generalizations of the Riemann-Liouville and Weyl fractional integrals, respectively. There are vast literatures of these fractional integrals, particularly for ν =1and ν =2(see Kober [5] for an excellent summary, and Erdelyi [4] for many applications). γ Lemma 2.2 If < Reη, I pν ν,a,η is a bounded linear operator of L p,γ into γ itself and if > Re ξ, then J pν ν,b,ξ is a bounded linear operator of L p,γ into itself. The following behaviours near the origin and the infinity of the function C α,β (z) can be deduced of the correspondent ones of the Bessel function J α β (z) and they will be used in the sequel. C α,β (z) =O (1),asz (1)

4 2518 B. B. Waphare C α,β (z) z 2α,asz. (11) 3 The Hankel-Schwartz type transformations on L p,ν spaces In this section we study the behaviours of the Hankel-Schwartz type transformations on the L p,ν spaces. Theorem 3.1 The H α,β,1 - transformation is a bounded linear operator of L p,ν into L p,2p(3α+β) ν, provided that 1 p <, (α β) > 1 and 2 2(2α + β) < ν < 2(3α + β) p Proof By definition of H α,β,1 - transform, we have H α,β,1 (φ) ( ) 1 y By using (1) and (11), we obtain ( ) x = C α,β x 4α φ (x) dx y = y 2(3α+β) C α,β (u) u 4α φ (uy) du. y 2(3α+β) H α,β,1 (φ) ( ) [ 1 1 A u 4α φ (uy) du + y 1 ] u 2α φ (uy) du, (12) where φ L p,ν and A is some positive constant. By (8) and (9), we can write (I 1, 1, 2(3α + β) φ) (y) = (J 1, 1, 2(5α +4β) φ) (y) = 1 u 4α φ (uy) du u 2α φ (uy) du. Now by Lemma 2.2, I 1, 1, 2(3α + β) and J 1, 1, 2(2α + β) are bounded linear operators of L p,ν into itself provided that 2 (2α + β) < ν < 2(3α + β), 1 < p p< and (α β) > 1. 2 Then, from (12) if 2 (2α + β) < ν p < 2(3α + β), 1 <p< and (α β) > 1 2,

5 Generalized Hankel-Schwartz type transformations 2519 y 2(3α+β) H α,β,1 (φ) where A 1 is a positive constant. Thus we have ( 1 y ) ν,p A 1 φ ν,p, under above conditions. H α,β,1 (φ) (y) 2p (3α+β) ν,p A 1 φ ν,p This shows that H α,β,1 is a bounded linear operator of L p,ν into L p, 2p (3α+β) ν under the imposed hypotheses. Corollary 3.2 If 1 <p<, (α β) > 1 2 and 2β < ν p < 1, then H α,β,2 is a bounded linear operator from L p,ν into L p, ν 2(α β)p. Proof As H α,β,2 (φ) (y) =y 4α H α,β,1 (x 4α φ)(y), proof follows from Theorem The distributional generalized Hankel-Schwartz type transformations on L p,ν In this section we define the distributional generalized Hankel-Schwartz type transformations on L p,ν. Now we prove generalizations of the mixed Parseval s equation which can be understood from definitions of said transformations. 1 Lemma 4.1 If f L p,ν,g L q, (ν 4α)q ν, 1 < p <, + 1 = 1, p q (α β) > 1 and 2(2α + β) < ν < 2(3α + β), then 2 p g(x) H α,β,1 (f) (x) dx = H α,β,2 (g) (y) f (y) dy. (13) Proof Let f,g D (, ). Then by Fubinis theorem we have

6 252 B. B. Waphare g (x) H α,β,1 (f) (x) dx = = = g (x) y 4α C α,β (xy) f (y) dy dx f(y) y 4α C α,β (xy) g (x) dx dy H α,β,2 (g) (y) f (y) dy. Thus (13) is true if f D (, ) and g D (, ), and hence, since D (, ) is dense in L p,ν, the general result will be true if we show that both sides of (13) represent bounded bilinear functionals on L p,ν L q, 2q(3α+β) ν. As 1 <p<, = 1, by using Holder s inequality we have p q g (x) H α,β,1 f (x) dx H α,β,1 (f) 2p(3α+β) ν,p g (ν 4α)q ν,q. < 2(3α + β), H α,β,1 is a bounded linear opera- and we can deduce that Moreover as 2 (2α + β) < ν p tor of L p,ν into L p,2p(3α+β) ν g (x) H α,β,1 (f) (x) dx A f ν,p g (ν 4α)q ν,q, where A is a certain positive constant. Left hand side of (13) is a bounded bilinear function on L p,ν L q,2q(3α+β) ν as the right hand side of (13) by a similar calculation and the result follows. Definition 4.2 The distributional generalized H α,β,1 - transformation on L q,ν+(1 ν)q as the adjoint of the classical H α,β,2 - transform through H α,β,1 f, φ = f, H α,β,2 φ,f L q, ν+(1 ν)q,φ L q, (ν 4α)q ν. (14) ) Theorem 4.3 If 1 <q<, (α β) > (1 1 and 2(2α + β) <ν 1 2 q < 2(3α + β), then the distributional generalized H α,β,1 - transformation is a bounded linear operator of L q,ν+(1 ν)q into L q,(ν 4α)q ν. Proof Proof follows from Corollary 3.2. Remarks

7 Generalized Hankel-Schwartz type transformations 2521 (i) (4.2) represents an extension of the mixed Parseval equation (13). (ii) If f L p,ν, then by involving again Holder s inequality f(x) φ (x) dx f ν,p φ ν+(1 ν)q,q, where =1.Thusf generates a regular distribution in p q L q,ν+(1 ν)q, and in this sense, L p,v L q, ν+(1 ν)q. If f L p,ν, then we can define the distributional generalized H α,β,1 transformation H α,β,1 f of f and the classical H α,β,1 - transformation in the sense of equality in L q, (ν 4α)q ν. In effect, if φ L q, (ν 4α)q ν then by using (13), H α,β,1 f, φ = (H α,β,1 f)(x) φ (x) dx = f (x)(h α,β,2 f)(x) dx = f, H α,β,2 f = ( H α,β,1 f, φ). Thus the classical H α,β,1 - transformation is a special case of disttibutional generalized H α,β,1 - transform. In a similar way we can define the H α,β,2 - transformation. More precisely if f L p, 2p (3α+β) ν the H α,β,2 - transformation H α,β,2 f of f is defined by H α,β,2 f, φ = f, H α,β,1 φ, for every φ L p,ν. Theorem 4.4 If 1 <p<, (α β) > 1 and 2(2α + β) < ν < 2 p 2(3α + β), then the distributional generalized H α,β,2 - transformation is a bounded linear operator of L p, 2p(3α+β) ν into L p,ν. (iii) the classical H α,β,2 - transformation is a special case of the distributional generalized H α,β,2 - transformation. References [1] G. Altenburg, Bessel transformation in Raumen Von Grund functionen uber dem interval Q =(, ) und derem dual raumen, Math. Nachr, 18 (1982), [2] J. J. Betancor and E. R. Negrin, The generalized Bessel transformations on the spaces L p,v of distributions, J. Korean Math. Soc., 27, No. 1 (199),

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9 Generalized Hankel-Schwartz type transformations 2523 [19] A. H. Zemanian, Distribution theory and transform analysis, McGraw Hill Book Co., New York, [2] A. H. Zemanian, Generalized integral transformations, Inder-Science Pubishers, New York, Received: September, 21