Boundedness of Some Pseudodifferential Operators on Bessel-Sobolev Space 1

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1 M a t h e m a t i c a B a l k a n i c a New Series Vol. 2, 26, Fasc. 3-4 Boundedness of Some Pseudodifferential Operators on Bessel-Sobolev Space 1 Miloud Assal a, Douadi Drihem b, Madani Moussai b Presented by V. Kiryakova We study the continuity of generalized pseudodifferential operator B,σ on Sobolev- Bessel space, with > 1/2 and σ in the class of symbols. Also, we give the analogous result related to the commutator [B,σ, I ϕ ] where I ϕ = F 1 B ϕfb )) and ϕ is being a suitable function. AMS Subj. Classification: 35 E 45 Key Words: Bessel operator, Commutator, Pseudodifferential operator 1. Introduction The continuity of the pseudodifferential operators on Sobolev space has been introduced in literature by using the classical Fourier transform on R n. Throughout this paper we fix > 1/2, the weight function v x) = 1 2 Γ+1) x2+1 and we define the generalized pseudodifferential operator B,σ, on [, + [, by the formula B,σ f x) = j xξ) σ x, ξ) F B f ξ) v ξ) dξ, for all f S R), where: S R) is the Schwartz s subspace of even functions. j are the normalized Bessel functions of first kind and order given by j λ) = 2Γ+1) π 1/2 Γ+1/2) 1 1 t 2 ) 1/2 cosλt)dt. 1 Research of the third author was supported partially by Acc. Prog. 2 MDU 543.

2 36 M. Assal, D. Drihem, M. Moussai See for example [4]). σ belongs to S 1,m ; the class of even symbols with respect to the second variable and satisfying, for all, β, γ N, N 1,m,β,γ σ) = sup x,ξ 1+x 2 ) 1+ξ 2 ) m γ 1 x β 1 x) ξ F B is the Fourier-Bessel transform given by F B f λ) = ξ ) γ σ x, ξ) <. j λx) f x) v x) dx, f S R), λ R). It has been proved in [5] that F B is an isomorphism from S R) into itself and its inverse is F 1 B = F B. In this work we shall be interested in the continuity of B,σ and of the commutator [B,σ, I ϕ ] on the Bessel-Sobolev space E, where I ϕ = F 1 B ϕf B )) with ϕ is a differentiable even function adeuately chosen. We recall here that is the set of even distributions f on R satisfying E f E = 2) s FB f ξ) 1/p p v ξ) dξ) <. For more details, we refer to the works of M. Assal and M. Nessibi [1]. See also Pathak and Pandey [2], [3]. Our results are the following: Theorem 1. Let s, r R, 1 p, < and σ S 1,m. If one of the following assertions holds ) i) r < +1 1, s > m + + 1) 1 p and p > 1, ) ii) s r > m + + 1) 1 1 p and p 1, iii) s r > m + +1 and p = 1, then B,σ is a bounded operator from E to E r,. Theorem 2. Let s, r, λ R, 1 p, < and σ S 1,m. Let ϕ λ C 1R) such that ϕ λ ξ) 1 + ξ ) λ. If one of the following assertions holds i) r < , s > m λ ) 1 1 p ) and p > 1,

3 Boundedness of Some ψ.d.o. on Bessel-Sobolev Space 361 ) ii) s r > m λ ) 1 1 p and + 1 p 1, iii) s r > m λ and p = 1, then [B,σ, I ϕλ ] is a bounded operator from E to E r,. This paper is organized as follows. In Section 2 we collect some harmonic analysis results related to the Bessel operator. Section 3 is devoted to the proof of Theorems 1 and 2. Some remarks concerning the continuity from E to itself are also given in this section. 2. Preparations In this section we recall some basic results in harmonic analysis related to the Bessel operators see [5]). All functions and spaces are defined on R. For a Banach space E let E denotes its norm. We set C k R) = C k, L p R) = L p, etc... C k denotes the space of even functions of class k... The spaces S, S 1,m and will be as defined above. We denote L p [, + [) the space of all functions E f defined on [, + [ such that f L p <, where f L p = ess sup x [,+ [ f x) p v x) dx) 1/p if 1 p <, f x) if p =. We recall that the Bessel operator is given by L = 1 satisfies the following properties 1) L ) k j λx)) = λ 2k j λx), x 2+1 d dx x 2+1 d dx) and 2) L ) k = 2k i=1 ) C i x i 1 d i, x dx where C i R and i N with i i. For instance ) L ) 2 1 d 2 ) 1 d 3 ) 1 = 4 + 2) + 1) ) + x 4 d 4. x dx x dx x dx Noting that, we can obtain 2) by induction on k. Next, we shall need the generalized translation operator Tx defined, for all x, y [, [ and suitable function f by π ) Tx f y) = f x 2 + y 2 + 2xy cos θ sin θ) 2 dθ; Γ+1) π 1/2 Γ+1/2)

4 362 M. Assal, D. Drihem, M. Moussai The above translation operator satisfies the following properties see [5]) T x j λ )) y) = j λx) j λy), T x f L p f L p for 1 p, T x f y) = f t) W x, y, t) t 2+1 dt, where t W x, y, t) is supported on [ x y, x + y] and W x, y, t) t 2+1 dt = 1. As usual C denotes the constant with many vary from line to line. If 1 p then its conjugate is given by p = p p Proofs 3.1. Some estimates The following propositions are useful: Proposition 1. Let 1 p. Then there exists a constant C >, such that for all function f defined on [, + [ [, + [ with f, y) L p and f x, ) L, one has 3) ) 1 f, y) L p v y) dy C ) 1 f x, ) p L v p x) dx. P r o o f. Consider the operator T { x f x, ) L }) { y f, y) L p }. Then we obtain 3) by interpolation on T. Then it suffices, and is not difficult, to prove 3) for p = 1 and for p =. Proposition 2. Let σ S 1,m. Then, for all k N, there exists a constant C = C m,k > such that 4) F B [σ, ξ)] t) C 1 + t 2) k 2 ) m.

5 Boundedness of Some ψ.d.o. on Bessel-Sobolev Space 363 t 1, P r o o f. Using 1) and 2) together with j xt) 1, we obtain, for all F B [σ, ξ)] t) Ct 2k 2k i=1 C i C t 2k 2) m 2k x i 1 x i=1 C t 2k 2) m 2k i=1 x) i σ x, ξ) v x) dx 1 + x 2 ) i N 1,m [ i ] + +1,i, σ). 1 x x ) i σ x, ξ) dx 1+x 2 Here [x] + denotes the greatest integer less than or eual to x. On the other hand, for all < t < 1, we have F B [σ, ξ)] t) 2 k 1 + t 2) k σ x, ξ) v x) dx ) 1 C 2 k N 1,m [+ 2] σ) + t 2 ) k 2 ) m, ,, which ends the proof Proof of Theorem 1. The case i). Using Fubini s theorem and the properties of the translation operators we obtain = = σ x, ξ) j xξ) j xη) v x) dx ) σ x, ξ) W ξ, η, t) j xt) t 2+1 dt v x) dx W ξ, η, t) A σ t, ξ) t 2+1 dt, where A σ t, ξ) = F B [σ, ξ)] t). This leads to 5) F B B,σ f) η) = Therefore, 5) and Hölder s ineuality yield F B B,a f) η) f E + F B f ξ) [ T ξ A σ, ξ) η) ] v ξ) dξ. 2 ) sp Tξ A σ, ξ) η) ) 1/p p v ξ) dξ.

6 364 M. Assal, D. Drihem, M. Moussai Applying 4), we obtain 6) Tξ A σ, ξ) η) C 2) ξ+η m ξ η = C 2) m T ξ g 1 η), 1 + t 2 ) k W ξ, η, t) t 2+1 dt where g 1 t) = 1 + t 2) k. Since T x g 1 L C g 1 L, then B,σ f E r, C f E 2 ) ) 1/ r η 2+1 dη 2 ) m s)p ξ 2+1 dξ ) 1/p The case ii). Using 5) together with Hölder s ineuality, we obtain. B,σ f E r, [ + f E 2 ) sp 2) rp T ξ A σ, ξ) η) /p p v ξ) dξ] v η) dη. Combining 6) and Peetre s ineuality, we get 7) 2 ) r T ξ A σ, ξ) η) C2 r 2) m+r η) 2) r T ξ g 1 η). We take into account that W ξ, η,.) is supported on [ ξ η, ξ + η], then ξ η t, implies the lift-hand side of 7) is bounded by C2 r 2) m+r T ξ g 2 η), where g 2 t) = 1 + t 2) r k, with k is at our disposal. Taking k > r + +1 Proposition 2 leads to then [ g 2 L 2 ) m s+r)p Tξ g 2 η) ) ] /p p ξ 2+1 dξ η 2+1 dη 2 ) m s+r)p ) /p Tξ g 2 p L ξ 2+1 dξ 2 ) ) /p m s+r)p ξ 2+1 dξ.

7 Boundedness of Some ψ.d.o. on Bessel-Sobolev Space 365 Hence we have the desired result. The case iii). We shall proceed as above. So, using Peetre s ineuality, we obtain [ B,σ f E 2 ) r r,p FB f ξ) T ξ A σ, ξ) η) v ξ) dξ] v η) dη 2 s [ ) 2 s 2 ) s r η) 2) s FB f ξ) T ξ A σ, ξ) η) v ξ) dξ] v η) dη Now, as in 6) and since W ξ, η,.) is supported on [ ξ η, ξ + η], we obtain by Peetre s ineuality η) 2) s T ξ A σ., ξ) η) C 2) ξ+η m ξ η C2 m 2) m T ξ g 3 η), 1 + t 2 ) s k W ξ, η, t) t 2+1 dt where g 3 t) = 1 + t 2) s k+m. Hence for k > s + m, it holds B,σ f) E r, C g 3 L f E s,1 2 ) r s+m) η 2+1 dη. R e m a r k 1. Under the hypotheses of Theorem 1, with m < + 1). B,σ becomes a bounded operator on E. and 3.3. Proof of Theorem 2. We shall give the proof first for the case λ = 1. The case i). The use of 5) gives F B B σ, I ϕλ f) η) = + F B I ϕλ B σ, f) η) = ϕ λ η) Then, it holds ϕ λ ξ) F B f ξ) [ T ξ A σ, ξ) η) ] v ξ) dξ + F B f ξ) [ T ξ A σ, ξ) η) ] v ξ) dξ. F B [B σ,, I ϕλ ] f) η) =

8 366 M. Assal, D. Drihem, M. Moussai + ξ η) 1 The elementary estimate ϕ λ t ξ η) + η) dtf Bf ξ) [ T ξ A σ, ξ) η) ] v ξ) dξ. 1 + η + t ξ η) ) η ) t ξ η ) for t 1, ξ, η ) together with Hölder s ineuality imply that F B [B σ,, I ϕλ ] f) η) is bounded by 8) C f E { 1 + ξ η p 1+ξ 2 ) sp 1 + ξ η ) p Tξ A σ, ξ) η) } 1/p p ξ 2+1 dξ. Next, the same argument given in the proof of Theorem 1 yields 9) ξ η 1 + ξ η ) T ξ A σ., ξ) η) C 2 ) m T ξ g 4 η), where g 4 t) = t 1 + t) 1 + t 2) k. The contraction property of the translation operator leads to estimate 8) by the desired term, i.e. C f E ) 1 2 ) ) 1/p m s)p ξ 2+1 dξ. The case ii). Combining relations 8) and 9), we obtain F B [B σ,, I ϕ ] f E r, CV f E, where V = + [ + 1+ξ 2 ) m s)p 1+η 2 ) rp 1+η) p T ξ g 4 η) p ξ 2+1 dξ ] /p η 2+1 dη. On the other hand, Peetrs s ineuality and the estimate ) 1 ) ξ η ) lead to 2 ) r ) 1 T ξ g 4 η) C 2 ) r ) 1 T ξ g 5 η),

9 Boundedness of Some ψ.d.o. on Bessel-Sobolev Space 367 where g 5 t) = t 1 + t) t 2) r k. Now, Proposition 1 and the choice of k > r give the correct bound, i.e. { V 2 ) m s+r)p 1 T ) p ξ g 5 η) η 2+1 dη [ C g 5 L 2 ) ] /p m s+r)p ) p ξ 2+1 dξ. ) p / The case iii). It is easy to obtain the following estimates F B [B σ,, I ϕ ]) f η) C ) 1 [ + C 2 m s 2 ) m s ) C g 6 L f E s,1 ξ 2+1 dξ 2 ) s FB f ξ) 2) ] m s T ξ g 4 η) ξ 2+1 dξ + } /p ) 2 s η) 2) F m s B f) ξ) T ξ g 4 η) ξ 2+1 dξ 2 ) m s ) 1, where g 6 t) = t 1 + t) 1 + t 2) m s k. The choice of k > 2 + m s together with the conditions in iii) give the desired result. R e m a r k 2. The proof of the case λ 1 is similar to the above one with adeuately changes by using the following ineuality We omit the details. 1 + η + t ξ η) ) λ 2 λ 1 + η ) λ 1 + t ξ η ) λ. R e m a r k 3. Let λ <. If ϕ λ satisfies the property ϕ λ ξ) ϕ λ η) A ξ η λ, then Theorem 2 is also valid. For example ϕ λ belongs to the Lipschitz space Λ λ. R e m a r k 4. Under the hypotheses of Theorem 2. Define a = λ 2 1, b = λ 3 2 and c = λ 2 + 1) p.

10 368 M. Assal, D. Drihem, M. Moussai If one of the following holds: i) λ 1, m < a and p 1, ii) λ < 1, m < b and p 1, iii) λ < 1, b < m < c, p = 1 and p 2, iv) λ < 1, m > max b, c) and p = 1, then [B,σ, I ϕλ ] is a bounded operator on E. References [1] M. A s s a l and M. N e s s i b i. Bessel-Sobolev Type Spaces. Mathematica Balkanica, 18, 24, [2] R. S. P a t h a k and P. K. P a n d e y. A class of pseudodifferential operators associated with Bessel operators. J. Math. Anal., 213, 1995, [3] R. S. P a t h a k and P. K. P a n d e y. Sobolev type space associated with Bessel operators. J. Math. Anal., 215, 1997, [4] E. M. S t e i n. Harmonic Analysis, real-variable methods, orthogonality and oscillatory integrals, Princeton Univ. Press, Princeton New Jersey [5] K. T r i m e c h e. Generalized Harmonic Analysis and Wavelet Packets. Gordon and Breach Science Publisher, Taylor & Francis, Australia 21. a Department of Mathematics, Received Campus Universitaire, MRAZKA, IPEIN, Nabeul, 8, Tunisia Miloud.Assal@fst.rnu.tn b Department of Mathematics, M Sila University, P.O. Box 166, M Sila 28, Algeria douadidr@yahoo.fr mmoussai@yahoo.fr