On Pseudo-Differential Operator Associated with Bessel Operator

Transcript

1 Int. J. Contemp. Math. Sciences, Vol. 6, 2, no. 25, On Pseudo-Differential Operator Associated with Bessel Operator Akhilesh Prasad and Vishal Kumar Singh Department of Applied Mathematics Indian School of Mines Dhanbad-8264, India apr vks Abstract In this paper pseudo-differential operator(p.d.o) P (x, D) in terms of a symbol is defined and inverse Hankel transform of this symbol is also defined. It is shown that the p.d.o is bounded in certain Sobolev type space associated with Hankel transform. A special case is discussed. Mathematics Subject Classification: 46F2; 46F5; 47G3 Keywords: Pseudo-differential operator; Hankel transform; Hankel convolution Introduction The Hankel-type transformation of φ L (I), I (, ) is defined by (H μ φ)(x) (xy) μ J μ (xy)φ(y)y 2μ+ dy, x I (.) where (xy) μ J μ (xy)y 2μ+ represents the kernel of this transformation, as usual, J μ is the Bessel function of the first kind and order μ. We shall assume that through out this paper that μ /2. Since x μ J μ (x) is bounded on I, the Hankel-type transformation H μ (φ)(x) is bounded on I, provided x 2μ+ φ(x) dx < (.2)

2 238 A. Prasad and V. K. Singh clearly (H μ φ)() 2 μ Γ(μ +) The inversion formula for (.) is given by φ(x) φ(y)y 2μ+ dy. (.3) (xy) μ J μ (xy)h μ φ(y)y 2μ+ dy, x I. (.4) The above transformation has been used [2, 6]. Altenburg [] introduced the space H consisting of all infinitely- differentiable functions φ defined on I (, ), such that for all m, k N the quantites γ m,k (φ) sup( + x 2 ) m (x d/dx) k φ(x) <. (.5) x I Zaidman [8] studied a class of pseudo-differential operators (p.d.o s) using Swartz s. theory of Fourier transformation. Pseudo -differential operators associated to a numerical valued symbol a(x, y) were discussed by Pathak and Prasad [6]. Moreover the Hankel transformation (.) finds wide applications in the Hankel convolution theory [2, 4, 5]. Therefore, it is natural to develop a theory of pseudo-differential operators depending on the transformation (.). In the investigation of the pseudo-differential operator P (x, D) depending on the transformation H μ, it assume that the symbol a(x, y) posses derivatives which satisfy certain growth conditions. One formula for such an operator appears as follows: where (H μ,a φ)(x) (H μ φ)(x) (xy) μ J μ (xy)a(x, y)h μ φ(y)y 2μ+ dy, x I (.6) (xy) μ J μ (xy)φ(y)y 2μ+ dy, x I. (.7) From [7] the symbol a(x, y) is defined to be the complex valued infinitely differentiable function on I I which satisfy (x d/dx) α (y d/dy) β a(x, y) C α+β+ α!β!( + y) m β (.8) α, β N where m is a fixed real number. The class of all such symbols is defined by H m. From [6] we know that for any φ, ψ H: (x d/dx) k (φ, ψ) k ( k υ )(x d/dx) υ φ(x d/dx) k υ ψ. (.9) υ The theory of Hankel convolution studied by Belhadj and Betancor [2]. In this paper we have used the Hankel transformation defined by (.) to develop a theory of pseudo-differential operator associated with Bessel operator corresponding to [5].

3 Pseudo-differential operator associated with Bessel operator The Hankel Convolution From Zemanian [9] we recall that the following results on Hankel convolution in the sequel Δ(x, y, z) be the area of the triangle with sides x, y, z if such a triangle exists. For μ>, set D(x, y, z) 2 3μ /2 (π) /2 [Γ(μ + )] 2 (Γ(μ +/2)) (xyz) 2μ [ Δ(x, y, z) 2μ ] (2.) if Δ exists and zero otherwise. We note that D(x, y, z) and that D(x, y, z) is symmetric in x, y, z and we have where and From [5] we know that j(zt)d(x, y, z)dμ(z) j(xt)j(yt) (2.2) dμ(z) [2 μ Γ(μ + )] z 2μ+ dz (2.3) j(x) 2 μ Γ(μ +)x μ J μ (x). (2.4) J μ (xξ)j μ (xλ) (xλξ) μ z μ J 2 μ μ (zx)d(ξ, λ, z)dμ(z). (2.5) Γ(μ +) Next we define the space L p μ (I), p< as the space of all real measurable function on satisfying [ /p f p f(x) dμ(x)] p < (2.6) Lemma 2. Let f L μ (I) then the associated function f(x, y) is defined by f(x, y) f(z)d(x, y, z)dμ(z), < x,y <. (2.7) Lemma 2.2 Let f and g be functions of L μ (I) and the Hankel convolution of f and g be defined by (f#g)(x) f(x, y)g(y)dμ(y), <x<. (2.8) Then the integral defining (f#g)(x) converses for all x, < x <, and and (f#g)(x) (g#f)(x) almost everywhere. (f#g)(x) f g (2.9)

4 24 A. Prasad and V. K. Singh 3 Pseudo-Differential Operator P (x, D) Definition 3. Let us define the pseudo-differential operator P (x, D) by (P (x, D)φ)(x) (xξ) μ J μ (xξ)a(x, ξ)h μ φ(ξ)ξ 2μ+ dξ, x I (3.) where φ H(I),I (, ),μ /2 and we assume that the symbol a(x, ξ) is defined as the Hankel-type transformation: a(x, ξ) with condition that for all λ I, ξ I and (xλ) μ J μ (xλ)(h μ a)(λ, ξ)λ 2μ+ dλ, x I (3.2) (H μ a)(λ, ξ) k(λ), λ I,ξ I (3.3) where k(λ) L μ(i), μ /2. Now we prove a boundedness result P (x, D) for which we need the following Sobolev type space. Definition 3.2 For s, μ R and p<, the space G s μ,p set of all those elements φ H (I), which satisfy is defined to the φ G s µ,p η s H μ φ p (3.4) we usually call G s μ,p the Sobolev type space. Theorem 3. Let μ /2, then P (x, D)φ G µ, k φ G µ,, φ H(I) (3.5) Proof: We have (P (x, D)φ)(x) where a(x, ξ) (xξ) μ J μ (xξ)a(x, ξ)h μ φ(ξ)ξ 2μ+ dξ, x I (3.6) (xλ) μ J μ (xλ)(h μ a)(λ, ξ)λ 2μ+ dλ, x I. (3.7)

5 Pseudo-differential operator associated with Bessel operator 24 Therefore by changing the order of integration using Fubini s theorem, we have (P (x, D)φ)(x) (xξ) μ J μ (xξ) (xλ) μ J μ (xλ)(h μ a)(λ, ξ)λ 2μ+ dλ H μ φ(ξ)ξ 2μ+ dξ, x I 2 μ Γ(μ +) (xξ) μ (xλ) μ (H μ a)(λ, ξ)(λξ) 2μ+ H μ φ(ξ) (xλξ) μ z μ J μ (zx)d(ξ, λ, z)dμ(z)dλdξ (xξ) μ (xλ) μ (H μ a)(λ, ξ)(λ, ξ) 2μ+ H μ φ(ξ) (xλξ) μ z μ J 2 μ μ (zx)d(ξ, λ, z) Γ(μ +) 2 μ Γ(μ +) z2μ+ dzdλdξ (xz) μ J (2 μ Γ(μ + )) 2 μ (zx) [ ] (λξ) 2μ+ (H μ a)(λ, ξ)h μ φ(ξ)d(ξ, λ, z) z 2μ+ dzdλdξ. (3.8) An application of the inverse Hankel transform yields (xz) μ J μ (xz)(p (x, D)φ)(x)x 2μ+ dx In other wards, we have (2 μ Γ(μ + )) 2 ξ 2μ+ H μ φ(ξ)λ 2μ+ (H μ a)(λ, ξ)d(ξ, λ, z)dλdξ. (3.9) H μ (P (x, D)φ(x))(z) using the inequality (3.3) we have H μ (P (x, D)φ(x))(z) (H μ a)(λ, ξ)d(ξ, λ, z) (H μ φ)(ξ)dμ(λ)dμ(ξ), (3.) k(λ)d(ξ, λ, z) (H μ φ)(ξ)dμ(λ)dμ(ξ) (3.) (k#h μ φ)(z). (3.2)

6 242 A. Prasad and V. K. Singh Hence H μ (P (x, D)φ(x))(z) dμ(z) (k#h μ φ)(z)dμ(z). (3.3) Now applying the definition (3.4) and (2.9) P (x, D)φ G k µ, φ G,φ H(I). (3.4) µ, 4 Property of Symbol Let us now consider the special case when symbol a(x, ξ) is separable in the form a(x, ξ) a(x)c(ξ) (4.) where H μ (a(x))(λ) L μ(i) and c(ξ) is a bounded measurable function on I (c(ξ)) M, for all ξ I. Since a(x) and [H μ a(x)] (λ) L μ (I), therefore (xλ) μ J μ (xλ)[(h μ a(x))] (λ)λ 2μ+ dλ, x I (4.2) a(x, ξ) (xλ) μ J μ (xλ)(h μ a(x, ξ))(λ)λ 2μ+ dλ, (xλ) μ J μ (xλ)(h μ a(x))(λ)c(ξ)λ 2μ+ dλ x I (xλ) μ J μ (xλ)(h μ a(x))(λ)λ 2μ+ dλc(ξ). (4.3) Thus (H μ a(x, ξ))(λ) H μ (a(x))(λ)c(ξ), which is measurable function on I I for all ξ I, since c(ξ) M and (H μ a)(λ, ξ) MH μ (a(x))(λ) L μ (I). Thus Therefore, by the preceding theorem k(λ) H μ (a(x))(λ). (4.4) P (x, D)φ G µ, k φ G µ,,φ H(I). (4.5) Acknowledgement: This work is supported by University Grants Commission, Govt. of India, under grant no.f.no.34-45/28(sr).

7 Pseudo-differential operator associated with Bessel operator 243 References [] G. Altenburg, Bessel Transformation in Roümen von Grund funktionen uber dem intervall Ω (, ) and derem Dual-raumen, Math Nachr. 8(982): [2] M. Belhadj and J. J. Betancor., Hankel transformation and Hankel convolution of tempered Beurling distributions, Rocky Mountain J. Math., 3(4)(2): [3] J. J. Betancor and I. Merrero., Some properties of Hankel convolution operators, Canad. Math. Bull., 36(4),(993): [4] J. N. Pandey., An extension of Haimo s form Hankel convolution, Pacific J. Math, 96(969): [5] R. S. Pathak and S. Pathak., Certain pseudo-differential operators associated with Bessel operator, Indian. J. pure. appl. Math, 3(2): [6] R. S. Pathak and A.Prasad., Continuity of pseudo-differential operators associated with Bessel operator in some Gevrey spaces, appl. anal., 8(3)(22): [7] L. Rodino., Linear Partial Differential Operators in Gevrey spaces, World Scientific, Singapore (993) [8] S. Zaidman., Distributions and Pseudo-differential Operators, Longmann Esex, England (99). [9] A. H. Zemanian., Generalised Integral Transformations, Interscience, New York (962). Received: December, 2