On the boundedness of square function generated by the Bessel di erential operator in weighted Lebesque L p , α spaces

associated with the Bessel di erential operator Bt = d 2 dt2 + (2α+1) t d dt , α > −1/2, t > 0 on the half-line R+ = [0,∞). The aim of this paper is to obtain the boundedness of this function in Lp,α, p > 1. Firstly, we proved L2,α-boundedness by means of the Bessel-Plancherel theorem. Then, its weak-type (1, 1) and Lp,α, p > 1 boundedness are proved by taking into account vector-valued functions.

The Bessel translation operator is one of the most important generalized translation operators on the half-line R + = [ , ∞), [24,32]. It is used while studying various problems connected with Bessel operators (see, [22], [27] and bibliography therein).
In this paper, the square function associated with the Bessel di erential operator B t is introduced on the half-line R + = [ , ∞) and its L ,α − boundedness by means of the Bessel-Plancherel theorem is proved. Then, ( , ) weak-type and L p,α , < p < ∞ boundedness of this function are obtained by taking into account vector-valued functions. For this, some necessary de nitions and auxiliary facts are given in Section 2. The main results of the paper are formulated and proved in Section 3.

Preliminaries
Let R + = [ , ∞), C(R + ) be the set of continuous functions on R + , C (k) (R + ), the set of even k-times di erentiable functions on R + and S(R) be the Schwartz space consisting of in nitely di erentiable and rapidly decreasing functions on R and S + (R + ) be the subspace of even functions on S(R). For a xed parameter α > − , let L p,α = L p,α (R + ) be the space of measurable functions f de ned on R + and the norm is nite. In the case p = ∞, we identify L ∞ with C , the corresponding space of continuous functions vanishing at in nity. Denoted by T s , s ∈ R + the Bessel translation operator acts according to the law where and the following relations are known [25] : It is not di cult to see the following inequality For this, we de ne a measure on the [ , π] by dµ (ϕ) = c α (sin ϕ) α dϕ, where c α is de ned by (3). By using (2) and the Hölder inequality, we have Further, by using (5) and (4) we obtain It is known that the function u(t, s) = T s f (t), f ∈ C (R + ) is the solution the following Cauchy problem, (see [8,25]): The Bessel transform of order α > − of a function f ∈ L ,α is de ned by and the inverse Bessel transform is given by the formula is the normalized Bessel function and J α (z) is the Bessel function of the rst kind. From the following integral presentation for j α (t) (see [13], Eq. 8.411 (8)) and the equality takes place only at t = . We also note that, by using (8) and the Riemann-Lebesgue Lemma, we have lim Moreover, from (9) we have The asymptotic formula for J α (r) is as follows ( [28]): Then, the following asymptotic formula for j α (r) is obtained easily: The following Lemmas will be needed in proving the main results containing important properties of Bessel transform.
The generalized convolution generated by the Bessel translation operator for f , g ∈ L ,α is de ned by The convolution operation makes sense if the integral on the right-hand side of (13) is de ned; in particular, if f , g ∈ S + (R + ), then the convolution f ⊗ g also belongs to S + (R + ). Now, we list some properties of generalized convolution as follows: (see details in [25]) ), Further, by using (6) and the Hölder inequality it is not di cult to prove the corresponding Young inequality

Main results and proofs
In this part, the L ,α boundedness of the square function generated by the Bessel di erential operator is proved by Bessel-Plancherel formula, then its ( , ) weak-type and L p,α , < p < ∞ boundedness is obtained by using vector-valued functions.

De nition 3.1. Let
The square function associated with the Bessel di erential operator is de ned by An important trend in mathematical analysis and applications is to investigate convolution-type operators. Convolution type square functions have a very direct connection with L -estimates by the Plancherel theorem. For this reason, we have proved L ,α -boundedness of the square function (15), associated with the Bessel di erential operator by using Bessel-Plancherel formula (12) in the following.

Theorem 3.2.
Let the square function S f be de ned as (15). If f ∈ L ,α then there is c > such that Proof. Firstly, let f ∈ S + (R + ). By making use of the Fubini theorem and Bessel-Plancherel formula, we have Taking into account (14) and then using Fubini theorem, we get Since Φ t (x) = t − α− Φ x t , then using Lemma 1, we have By taking this into account in the formula (16) and using (12) we have where Firstly, let us estimate I . Since and taking into account (8) for the normalized Bessel function j α (t) we get Therefore, Now we estimate I . For this, we need the following asymptotic formula for j α (r), (cf. (11)): For arbitrary f ∈ L ,α , we will take into account that the Schwartz space S + (R + ) is dense in L ,α . Namely, let (f n ) be a sequence of functions in S + (R + ), which converges to f in L ,α -norm. From the "triangle inequality"

Hence, by (3.17) we get
This shows that the sequence (Sf n ) converges to (Sf ) in L ,α −norm. Thus and the proof is complete. Now, taking into account vector-valued functions spaces, we will obtain L p,α (R + ), < p < ∞ boundedness of the square function associated with the Bessel di erential operator. For this, necessary de nitions and theorems are given below. The rst theorem is well known as the Marcikiewicz interpolation theorem for the vector-valued functions. The other theorem is the extension of Benedek-Calderon-Panzone principle.
Let H be a seperable Hilbert space. We say that a function f de ned on R + = [ , ∞) and with values in H is measurable if the scalar valued function (f (x), h) is measurable for every h in H, where (, ) denotes the inner product of H and h denotes an arbitrary vector of H.
Now let H = R + and H = L ,α (R + , dt t ), α > − be the Hilbert space of square integrable functions on the half-line with respect to the measure dt t and the norm Since Φ ∈ S + (R + ) and ∞ ∫ Φ(x)x α+ dx = then we de ne K(x) to be the H -valued function given by So, the square function associated with the Bessel di erential operator (S f )(x) is the linear operator (Af )(x) = (f ⊗ K) (x) and Af takes its values in H . Thus, the condition (18) is equivalent to the following inequality x≥ y Now let us calculate (19). For this, since Φ ∈ S + (R + ), we take and for < < min {θ, q} by using Hölder inequality we have Finally, by using Theorem 3.4, we see that the square function associated with the Bessel di erential operator S f is of weak-type ( , ) and since we have already veri ed the L ,α (R + )-boundedness then by the Marcinkiewicz interpolation theorem for the vector-valued functions, (Theorem 3.3) Sf is also of type (p, p), < p < and consequently, by a simple duality argument S f is of type (p, p), < p < ∞.