© Hindawi Publishing Corp. THE PSEUDODIFFERENTIAL OPERATOR A(x,D)

The pseudodifferential operator (p.d.o.) A(x,D), associated with the Bessel operator d2/dx2


Introduction. The theory of the Hankel transformation
where 0 < y < ∞ and J µ is the Bessel function of the first kind and of order µ, has been extended by Zemanian [14] to distributions belonging to H µ , the dual of the test function space H µ , consisting of all complex-valued smooth functions φ defined on I = (0, ∞) and satisfying for each pair of nonnegative integers m and k.Pseudodifferential operators associated with a numerical valued symbol a(x, y) were investigated by R. S. Pathak and Pandey [8,9] and by R. S. Pathak and S. Pathak [10].We will use the notation and terminology of [4,7,11,12,14].The differential operators N µ , M µ , and S µ are defined by ) (1.5) From [14, page 139] and [5, page 948] we know the following relations for any φ ∈ H µ : h µ+1 N µ φ = −yh µ φ, (1.6) h µ S µ φ = −y 2 h µ φ, (1.7) where the b j are constants depending only on µ.
Next, we define the space L p σ (I), 1 ≤ p < ∞, as the space of those real-valued measurable functions on I for which where Let (x,y,z) denote the area of a triangle with sides x, y, and z, if such a triangle exists.For fixed µ ≥ −1/2, set (1.12) if exists, and zero otherwise.Then D(x, y, z) ≥ 0 and that D(x, y, z) is symmetric in x, y, z.From [13, page 411], we know that where From (1.13) it follows that and the Hankel convolution of f and g is defined by (1.17) From [9, page 102] we know that if f (x) ∈ L q σ (I), g(x) ∈ L p σ (I) with p > 1, q > 1, and 1/p + 1/q > 1, then f #g r ≤ f q g p , (1.18) where Many algebraic and topological properties of the generalized Hankel convolution have been given by Betancor and his associates [1,2,3,6].
We also note the differentiation formula In this paper, a general class H m ρ,δ of symbols associated with the differentiation formula (1.20) and similar to the Hörmander class S m ρ,δ is introduced.For ρ = 1/2 and δ = 0, the class H m ρ,δ reduces to the symbol class H m 0 studied by R. S. Pathak and Pandey [8].An integral representation for A(x, D) is given when the symbol belongs to the class H m ρ,δ .It is shown that A(x, D) is a continuous linear mapping of Zemanian's space H µ into itself.The space L p σ ,α is defined and an L p σ ,α -boundedness result is also obtained.

The pseudodifferential operator A(x, D)
Definition 2.1.Let a(x, ξ) be a complex-valued smooth function belonging to the space C ∞ (I × I), where I = (0, ∞), and let its derivatives satisfy certain growth conditions such as (2.3).The pseudodifferential operator (p.d.o.) A(x, D), associated with the symbol a(x, ξ), is defined by where A few examples of elements of H m ρ,δ are given as (i) We give a proof for example (i).We have Then using definitions (1.3) and (2.1), we have By induction, we get (2.7) Using formula (1.8), we get (2.8) Next, applying (1.20), we have (2.9) Using formula (1.6), we get (2.10) By induction, we have (2.11) Therefore, which in view of formula (2.7) can be written as (2.13) (2.14) Now an application of formula (1.8) yields (2.15) Then using inequality (2.3) and the boundedness property of the Bessel function, we find that the right-hand side of (2.15) is bounded by (2.16) Let p be a positive integer greater than or equal to µ +k+m/2+n/2+1/2, then we can write (2.17) where D λ,n,k,m is a positive constant.From (2.18), the continuity of A(x, D) follows.
3. An integral representation.The function a x (ξ), associated with the symbol a(x, ξ) and defined by will play a fundamental role in our investigation.An estimate for a x (ξ) is given by the following lemma.
Proof.Using property (1.7), we can write In view of (1.9), we have Again applying formula (1.20), we have Therefore, Then, using inequality (2.3) and the boundedness property of the Bessel function, we have for m < −µ − 3/2 − t.Therefore there exists a constant E µ,m,t such that Theorem 3.2.For any symbol a(x, ξ) ∈ H m ρ,δ , the associated operator A(x, D) can be represented by where all the involved integrals are convergent for µ (3.10) Therefore, Since (h µ f )(η) ∈ H µ (I), we have Now using the above estimate and (3.2), we obtain (3.14) The above integrals are convergent since µ ≥ −1/2, and t can be chosen greater than 1/2 and l sufficiently large.