Pseudo-Differential Operators associated with Bessel Type Operators - I

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1 Thai Journal of Mathematics Volume 8 (21 Number 1 : Online ISSN Pseudo-Differential Operators associated with Bessel Type Operators - I B. B. Waphare Abstract : In this paper Bessel type differential operator α,β, the pseudo differential type operators are defined and the symbol classes H m and H m are introduced. It is established that Pseudo-differential type operator associated with symbols belonging to these classes are continuous linear mappings of the Zemanian space H α,β into itself. Integral representation for Pseudo-differential type operator h α,β,a is obtained. Finally it is shown that Pseudo-differential type operators satisfy L 1 -norm inequality. Keywords : Pseudo-Differential type operator; Bessel type operator; symbol class; integral representation; Hankel type convolution; Hankel type transformation; Sobolev type space. 2 Mathematics Subject Classification : Primary 44A15; Secondary 46F12. 1 Introduction The Hankel type transformation, (h α,β φ(x = (xy α+β J α β (xyφ(ydy (1.1 is extended by Zemanian [7] to distributions belonging to H α,β, the dual of the function space H α,β which consists of all complex valued infinitely differentiable functions defined on I = (, satisfying, ρ α,β m,k for every m, k N. (ϕ = sup x m (x 1 D x k ( x 2β 1 φ(x < (1.2 x I Copyright c 21 by the Mathematical Association of Thailand. All rights reserved.

2 52 Thai J. Math. 8(1 (21/ B. B. Waphare Zaidman [5], [6] has used Schwartz s theory of the Fourier transform of distributions in I (R n in the study of Pseudo-differential operators. But Zemanian s theory of the Hankel type transform has not been used so far to develop a theory of Pseudo-differential operators associated with Bessel type operators as a special case. The purpose of the present paper is to change this Scenario (situation 2 Notations and Terminology We define the differential operators P α,β, Q α,β and S α,β as, P α,β = P α,β,x = x 2α D x x 2β 1 (2.1 Q α,β,x = x 2β 1 D x x 2α (2.2 α,β = α,β,x = Q α,β P α,β = x 2β 1 D x x 4α D x x 2β 1 = (2β 1(4α + 2β 2x 4(α+β 1 ( (2α + 2β 1 x 4α+4β 3 D x + x 2(2α+2β 1 D 2 x where, D x = d dx Following [7, p. 139] and [2, p. 948], we can establish the following relations for any φ H α,β h α,β,1 ( xφ = P α,β h α,β φ (2.4 h α,β,1 (P α,β φ = y h α,β φ (2.5 h α,β ( α,β φ = y 2 h α,β φ (2.6 ( x 1 D k ( x 2β 1 θφ = k ( ki ( x 1 i ( D x θ x 1 k i ( D x x 2β 1 φ (2.7 i= r α,β,x φ(x = b j x 2j+2α ( x 1 r+j ( D x x 2β 1 φ(x, (2.8 j= where b j are constants depending only on α β. We also need a lemma due to Haimo [1] for the Hankel type convolution transform. Lemma 2.1. Let (x, y, z be the area of a triangle with sides x, y, z if such a triangle exists for fixed (a b 1 2, set, D(x, y, z = 23(a b 1 (Γ(a b π Γ(a b (xyz 2(a b [ (x, y, z] 2(a b 1 (2.9

3 Pseudo-Differential Operators associated with if exists and zero otherwise. We note that D(x, y, z and that it is symmetric in x, y, z. Further we have the following basic formula: i(zt D(x, y, z d µ(z = i(xt i(yt (2.1 where, d µ(x = x 2(a b+1 2 a b dx (2.11 Γ(a b + 1 i(x = 2 a b Γ(a b + 1 x (a b J a b (x (2.12 Let f L 1 (, Then its associated function f(x, y is defined by, f(x, y = f(z D(x, y, z d µ(z < x, y < (2.13 Lemma 2.2. Let f and g be functions of L 1 (, and let, f # g(x = f(x, y g(y d µ(y < x < (2.14 Then the integral defining f#g(x converges for almost all x, < x <, and f # g L 1 f L 1 g L 1 ( The Pseudo-Differential type Operator h α,β,a Definition 3.1. Let a(x, y be a complex valued function belonging to the space C (I I, where I = (, and let its derivatives satisfy certain growth conditions such as (3.5. Then the Pseudo-differential type operator h α,β,a associated with the symbol a(x,y is defined by, where, (h α,β,a u(x = U α,β (y = (h α,β u (y = In case a(x, y = b(y, then clearly we have, (xy α+β J α β (xy a(x, y U α,β (y dy (3.1 (xy α+β J α β (xy u(x dx; (α β 1 2 (3.2 (h α,β,a u(x = h α,β [b(yu α,β (y]

4 54 Thai J. Math. 8(1 (21/ B. B. Waphare If a(x, y possesses a power series expansion in ( y 2 with variable coefficients depending on x, that is, a(x, y = N a k (x ( y 2 k (3.3 k= Then using formula (2.6 one can show that (h α,β,a u (x = N a k (x ( α,β k u(x (3.4 k= so that the Pseudo-differential type operator associated with the symbol (3.3 could be a finite order differential operator involving α,β. Definition 3.2. The function a(x, y : C (I I C belongs to class H m if and only if for every q N, i N, p N, there exists K p,i,m,q > such that, (1 + x q ( x 1 D x i ( y 1 D y p a (a, y Kp,i,m,q (1 + y m p (3.5 where D y = d dy and m is a fixed real number. If a(x, y satisfies (3.5 with q =, then the symbol class will be denoted by H m ; clearly H m H m. One can easily show that a(x, y = ( 1 + x 2 n ( 1 + y 2 m 2 n >, m R is an element of H m, but it does not belong to H m. Nevertheless a(x, y = e x2 (1 + y 2 m 2, m R belongs to H m. Theorem 3.3. Let the symbol a(x, y H m (or Hm. Then for (α β 1 2, the pseudo-differential type operator h α,β,a is a continuous linear mapping of H α,β into itself. Proof. Let φ(y = (h α,β,a u (y, u H α,β (I. Then using formulae (2.4, (2.5 and Zemanian s technique, [6, p. 141], we have, k C r r= (P α,β,k P α,β φ (y = y r+1 2 x 1 2 (y 1 D y r a(x, y ( x k r u(x J α β+k r (xy dx, where C r are certain positive real numbers. Set a r (x, y = (y 1 D y r a(x, y Now using formula (2.7 and induction, we obtain, = ( y n (P α,β,k P α,β φ (y k ( 1 k r C r y r+1 2 x α β+k+n r+1 r= n ( n i (x 1 D x i a r (x, y (x 1 D x n i ( x 2β 1 u(x J α β+k+n r (xy dx. i=

5 Pseudo-Differential Operators associated with Setting α β + k r = λ and using formula (2.1, we can obtain, ( 1 n y n (y 1 D y k ( y 2β 1 φ(y = k ( 1 k r C r x 2λ+n+1 n ( n i (x 1 D x i a r (x, y (x 1 D x n i ( x 2β 1 u(x J λ+n (xy (xy λ dx. i= Now setting n = t + s where s, t N, and s m, and n = t in turn, in the above expression and using (3.5 with q = and assumption (α β 1 2, we can estimate the above expression in absolute value as follows: There exists a constant K m,ν,r such that, r= (1 + y s y t (y 1 D y k y 2β 1 φ (y k 2 m (1 + y m (1 + y p r= t+s (1 + x 2λ+t+s+1 i= K m,v,r ( t+s i (x 1 D x t+s i ( x 2β 1 u(x t ( +(1 + x 2λ+t+1 K ti m,v,r (x 1 D x t i ( x 2β 1 u(x i= dx Thus, y t (y 1 D y k y 2β 1 φ(y k K r= 2 m (1 + y m (1 + y s 1 t+s ( (1 + x 2λ+t+s+1 t+s i (x 1 D x t+s i (x 2β 1 u(x i= t ( +(1 + x 2λ+t+1 ti (x 1 D x t i ( x 2β 1 u(x dx i= Now, we can choose a non-negative integer N such that N > 2(α β+k+t+s+3. Then using the fact that (1 + y m /(1 + y s 2 for y, s m, we obtain, [ t+s i= ( t+s i K y t (y 1 D y k y 2β 1 φ(y K k 2 m+1 (1 + x N 2 r= (x 1 D x t+s+i x 2β 1 u(x t ( + ti (x 1 D x t i x 2β 1 u(x ] dx k t+s i= i= ( t+s i i= N ( Nj t ρ α,β j,t+s i (u + ( ti N ( Nj P α,β j= i= j= j,t i (u.

6 56 Thai J. Math. 8(1 (21/ B. B. Waphare Therefore by using (1.2, we have, ρ α,β t,k (φ K k r= j= N ( Nj [ t+s ( t+s i ρ α,β i= j,t+s i (u + t i= ( ti ρ α,β j,t i (u ] (3.6 where K is a positive constant. The continuity of h α,β,a follows from (3.6 and hence the proof is complete. 4 Integral Representation for h α,β,a The function a η (y, associated with the symbol a(x, y and defined by, a η (y = (xy α+β J α β (xy [ (xη α+β J α β (xt a(x, η ] dx (4.1 will play a fundamental role in our investigation. An estimate for a η (y is given by the following lemma: Lemma 4.1. Let the symbol a(x, y H m. Then the function a η (y defined by (4.1 satisfies the inequity, a η (y A α,β,m,r,q (1 + η 2α+m+4r (1 + y 2α (1 + y 2r 1 where A α,β,m,r,q is a positive constant. Proof. For r N, using formula (2.6, we have, ( y 2 r a η (y = = (xy α+β J α β (xy ( α,β r [ (xη α+β J α β (xη a(x, η ] dx (xy α+β J α β (xy b j x 2j+2α (x 1 D x r+j x 2β 1 j= [ (xη α+β J α β (xη a(x, η ] dx. Using formula (2.7 we obtain, ( y 2 r a η (y = (xy α+β J α β (xy j= r+j b j x 2j+2α i= ( r+j i ( (x 1 D x i a(x, η (x 1 D x r+j i x (α β J α β (xη η α+β dx. Now using the formula, (x 1 D x m x (α β J α β (xη = ( η m x (α β m J α β+m (xη. We can obtain, ( y 2 r a η (y (xyα+β J α β (xy b j x 2j+2α j=

7 Pseudo-Differential Operators associated with r+j ( r+j i (x 1 D x i a(x, η η 2r+2j 2i+2α (xη (α β r j i J α β+r+j i (xη dx i= B α,β y 2α y 2α x 2α (xy (α β J α β (xy r+j ( b j x 2j+2α r+j i (x 1 D x i a(x, η η 2r+2j 2i+2α j= i= (xη (α β r j i J α β+r+j i (xη dx r+j i= i= ( r+j i B α,β y 2α D η 2r+2j 2i+2α (1 + η m x 2(α β+1+2j (1 + x q dx r+j ( r+j i D η 2r+2j 2i+2α (1 + η m j= for q > 2(α β + j + 1. i= B (2(α β + 2j + 2, q 2(α β 2j 2 ; Therefore there exists a constant A α,β,m,r,q such that, a η (y A α,β,m,r,q (1 + η m+4r+2α (1 + y 2α (1 + y 2r 1, for every r >. This completes the proof. Now we are ready to obtain an integral representation for the Pseudo-differential type operator h α,β,a as the following theorem: Theorem 4.2. For any symbol a(x, y H m, the associated operator h α,β,a can be represented by [ ] (h α,β,a u(x = (xy α+β J α β (xy a η (y U α,β (η dy, u H α,β (I, where U α,β (η = (h α,β u (η and all involved integrals are convergent. Proof. We have, a η (y = Now by inversion, formula, we have, (xy α+β J α β (xy [ (xη α+β J α β (xη a(x, η ] dx a η (y (xy α+β J α β (xy dy = (xη α+β J α β (xη a(x, η (4.2

8 58 Thai J. Math. 8(1 (21/ B. B. Waphare Therefore, (h α,β,a u(x = = = (xη α+β J α β (xη a(x, η U α,β (η dx U α,β (η d η (xy α+β J α β (xy dy a η (y (xy α+β J α β (xy dy U α,β (η a η (y d η. Using (4.1 for a η (y, the above change in the order of integration can be justified and the existence of the last integral can be proved. Note also that as U α,β (η H α,β (I, we have, U α,β (η C η 2α (1 + η l, for all l >. Thus, (h α,β,a u(x (xy 2α (xy (α β J α β (xy A α,β,m,r,q (1 + y 2α (1 + y 2r 1 (1 + η m+4r+2α C η 2α (1 + η l d η dy D α,β,m,r,q x 2α (1 + y 2(α β+1 (1 + y 2r 1 dy (1 + η 2(α β+m+4r l+1 d η Since (α β 1 2, and l and r can be chosen sufficiently large, the above integrals are convergent. This completes the proof. 5 An L 1 - Norm Inequality In this section we shall need the following Definition 5.1. (Sobolev type space The space G s (R, s R, is defined to be the set of all those elements u H α,β (I, which satisfy, u G s = η s+2β 1 h α,β (u L 1 <. (5.1 Lemma 5.2. For (α β 1 2 and r N,there exists a constant C r,m,q > such that, A η (y C r,m,q (1 + η m y 2α (1 + y 2r 1, (5.2 where, A η (y = h α,β ( x 2α a(x, η(y. (5.3

9 Pseudo-Differential Operators associated with Proof. Proceeding as in the proof of Lemma 4.1, we can obtain, ( y 2 r A η (y = = (xy α+β J α β (xy ( α,β r [ x 2α a(x, η ] dx (xy α+β J α β (xy j= b j x 2j+2α (x 1 D x r+j a(x, η dx. Therefore, ( y 2 r A η (y (xy α+β J α β (xy j= b j x 2j+2α D r+j,m,q (1 + η m (1 + x q dx y 2α x 2α (xy (α β J α β (xy b j x 2j+2α j= D r+j,m,q (1 + η m (1 + x q dx y 2α B j,m,q (1 + η m (1 + x 2(α β+2j+1 q dx. j= Choosing q > 2(α β + r + 1, we can obtain, A η (y C r,m,q (1 + η m (1 + y 2r 1 y 2α, where C r,m,q is a positive constant. Thus the proof is complete. Now we prove our main theorem. Theorem 5.3. Let (α β 1 2. Then for all m N, there exists C > such that, m h α,β,a (u G C u G l u H α,β (I. (5.4 Proof. From Theorem 4.2, equation (4.1 and the relation (2.1 we have, Hence A l= (xy α+β J α β (xy (h α,β,a u(x dx = η 2α U α,β (ηdη a η (y U α,β (η d η y 2β 1 (xy α+β J α β (xy (h α,β,a u(x dx = z 2α D(η, y, zdz x 2α a(x, η(zx α+β J α β (zxdx, (5.5

10 6 Thai J. Math. 8(1 (21/ B. B. Waphare 1 where, A = [2 α β Γ(α β + 1] 2 An application of the estimate (5.2 to (5.5 yields, y 2β 1 h α,β (h α,β,a u(y C r,m,q A m D α,β,m,r,q ( m l l= (1+η m η 2α U α,β (ηdη η l+2α U α,β (ηdη (Γ(α β dz. Set f(z = (1 + z 2r 1 L 1 (, ; for r >, and z 2(α β+1 (1+z 2r 1 D(η, y, zdz (5.6 (1+z 2r 1 D(η, y, zz 2(α β+1 2 (α β g(η = 2 α β Γ(α β+1η l+2β 1 U α,β (η L 1 (,, for all l =, 1, 2, m Thus according to (2.13 and (2.14 we obtain, f(η, y = and (f # g(y = f(z D(η, y, z z 2(α β+1 2 (α β (Γ(α β dz f(η, y g(η η 2(α β+1 2 (α β (Γ(α β d η Now if we apply (2.15 to (5.6, we get y 2β 1 h α,β (h α,β,a u(y L 1 m D α,β,r,m,q l= ( m l η l+2β 1 U α,β (η L 1 (1 + z 2r 1 L 1 C m ( m l η l+2β 1 h α,β u(η L 1 (5.7 l= Now inequality (5.4 follows from the inequality (5.7. This completes the proof. Conclusions

11 Pseudo-Differential Operators associated with If we take α = µ 2 and β = 1 4 µ 2 respectively, P µ = P µ,x = x µ+ 1 2 Dx x µ 1 2, in (2.1, (2.2 and (2.3, we obtain Q µ = Q µ,x = x µ 1 2 D x x µ+ 1 2 and µ = µ,x = Q µ P µ = x µ 1 2 D x x 2µ+1 x µ 1 2 = D 2 x + (1 4µ2 x 2 which are operators studied in Zeamnian [7] and later on in Pathak and Pandey [3] with P µ, Q µ, µ respectively replaced by N µ, M µ, S µ. 2. Author claims that results developed in this paper are stronger than Pathak and Pandey [3]. Remark : It is proposed to obtain more results on Pseudo-differential operators associated with Bessel type operators in future. Acknowledgement : I would like to thank the referees for their comments and suggestions on the manuscript. References [1] D. T. Haimo, Integral equations associated with Hankel convolutions, Trans. Amer. Math. Soc., 116 (1965, [2] E. L. Koh and A. H. Zemanian, The complex Hankel and I-transformations of Generalized functions, SIAM J. Appl. Math., 16 (1968, [3] R. S. Pathak and P. K. Pandey, A class of Pseudo-Differential operators Associated with Bessel operators, J. Math. Anal. Appl. 196 (1995, [4] L. Schwartz, Theorie des Distributions, Herman, Paris, [5] S. Zaidman, Distributions and Pseudo-Differential operators, Logman, Essex, England, [6] S. Zaidman, On some simple estimates for Pseudo-differential operators, J. Math. Anal. Appl. 39 (1972, [7] A. H. Zemanian, Generalized Integral Transformations, Interscience, Newyork, (Received 16 June 28 B. B. Waphare MAEER s MIT Arts, Commerce Science College, Alandi (D, Tal. Khed, Dist. Pune,

12 62 Thai J. Math. 8(1 (21/ B. B. Waphare Maharashtra 41215, INDIA. principal@mitacasc.com and bbwaphare@mitpune.com