A variant of Cowling–Price's and Miyachi's theorems for the Bessel–Struve transform on ℝd

ABSTRACT In this paper, we consider the Bessel–Struve transform on . We establish new versions of Cowling–Price's and Miyachi's theorems for the Bessel–Struve transform on . The techniques of the proofs are based on the properties of the Bessel–Struve kernel, the relation between the Bessel–Struve transform with the classical Fourier transform and on the positivity of the multi-variables Weyl and multi-variables Sonine transforms. The results of this paper are new, and they have novelty and generalize some results exist in the literature.


Introduction
An old Theorem of Hardy [1] proved way back in 1933 says that a function f and its Fourier transformf cannot both have arbitrary gaussian decay unless f is identically zero. Defining ϕ a (x) = e −ax 2 , we can state Hardy's Theorem more precisely as follows: if both f /ϕ a and f /ϕ b are in L ∞ (R) for some positive numbers a,b then f = 0 whenever ab > 1/4. Moreover, when ab = 1/4 the function f is a constant multiple of ϕ a and when ab < 1/4 there are infinitely many linearly independent functions satisfying both conditions. In 1983, Cowling and Price [2] generalized Hardy's Theorem by replacing the L ∞ estimates by L p estimates. They proved that if f /ϕ a ∈ L p (R) andf /ϕ b ∈ L q (R) for some p and q satisfying 1 < p, q < ∞, then f = 0 whenever ab > 1/4. The same conclusion holds even when ab = 1/4 provided either p or q is finite. In 1997, Miyachi [3] proved the following generalization of Hardy's Theorem. If f is a measurable function on R such that f /ϕ a ∈ L 1 (R) + L ∞ (R) and R log + |f (ξ )/ϕ 1/4a (ξ )| λ dξ < ∞ for some positive constants a and λ, then f is a constant multiple of ϕ a . The Cowling-Price's principle and their variants have been studied by many authors for various Fourier transforms, for examples (cf. [4][5][6][7]) and others.
In this paper we consider the differential operator γ , γ > 0, on R * given by This operator is called Bessel-Struve operator. It is connected with Dunkl theory [8,9]. In fact, Trimèche in [10] has introduced this operator and has built the harmonic analysis associated with this operator. In particular, the generalized Fourier transform associated with γ , called Bessel-Struve transform was studied. The purpose of the present paper is twofold. On one hand, we want to prove a variant of Cowling-Price's theorem for the Bessel-Struve transform on R d . We note that the analogue of the Cowling-Price's theorem for the Bessel-Struve transform on the real line was studied in [11]. Our version is different and it is motivated by the version of the Cowling-Price given by Ray and Sarkar in the cadre of symmetric spaces (cf. [12]). On the other hand, we want to prove a variant of Miyachi's theorem for the Bessel-Struve transform on R d . We note also that the analogue of Hardy's theorem was proved for the Bessel-Struve transform on R in [11], and the analogue of Beurling's theorem was proved for the Bessel-Struve transform on R in [13].
The structure of this paper is the following. In §2, we recall some basic facts about the harmonic analysis results related to the operator γ . §3 is devoted to give variants of Cowling-Price's theorem for the Bessel-Struve transform on R d . The last section of the paper aims to prove an analogue of Miyachi's theorem for the Bessel-Struve transform on R d .

Preliminaries
In order to confirm the basic and standard notations, we briefly overview the theory of Bessel-Struve operator and related harmonic analysis. Main reference is [10,11,13].

The Bessel-Struve kernel
The Bessel-Struve kernel is the function K γ ,d given by From the relations (1) and (2) we have This kernel has a holomorphic extension to C d × C d and possesses the following properties: (ii) For all ν ∈ N d , x ∈ R d and z ∈ C d , where D ν z = ∂ |ν| ∂z ν 1 1 · · · ∂z ν d d and |ν| = ν 1 + · · · + ν d .

The Bessel-Struve transform
In the following, we denote by The generalized heat kernel related to the Bessel-Struve transform N γ ,d (·, s), s > 0, is given by where C is a positive constant. Some basic properties of the Bessel-Struve transform are the following (cf. [10]) where F is the classical Fourier transform on R d and the operator t V γ ,d is multi-variable Weyl transform defined by Remark 2.1: By taking g ≡ 1 in (12) we can deduce that for all f ∈ L 1 (dμ γ ,d ), Using the relations (10) and (11), we obtain the following.

Cowling-Price's theorem for the Bessel-Struve transform
We shall prove a generalization of Cowling-Price's theorem for the Bessel-Struve transform F γ ,d on R d .

Theorem 3.1: Let f be a measurable function on
and  Proof: Clearly (15) implies that f belongs to L 1 (dμ γ ,d ) and thus, F γ ,d (f )(ξ ) exists for all ξ ∈ R d . Moreover, it has an entire holomorphic extension on C d satisfying for some s > 0, Actually, it follows from (7) and (5) that for all z = ξ + iη ∈ C d , Using Hölder's inequality and (15) we can obtain that

Hence it follows from (16) that
Here we use the following lemma.

Lemma 3.1: ([7]) Let h be an entire function on C d such that
|h(z)| ≤ C e a Re z 2 (1 + Im z ) l for some l > 0, a > 0 and

and Q ∈ P M (R d ). Then h is a polynomial with deg h ≤ min{l, (m − M − d)/q} and, furthermore if m ≤ q + M + d, then h is a constant.
Hence by this lemma g is a polynomial, we say P b , with deg P b ≤ min{n/p + (2γ If ab > 1 4 , then we can choose positive constants, a 1 , b 1 such that a > a 1 = 1/(4b 1 ) > 1/4b. Then f and F γ ,d (f ) also satisfy (15) and (16) with a and b replaced by a 1 and b 1 respectively. Therefore, it follows that F γ ,d (f )(ξ ) = P b 1 (x) e −b 1 ξ 2 . But then F γ ,d (f ) cannot satisfy (16) unless P b 1 ≡ 0, which implies f ≡ 0. This proves (i).

Miyachi's theorem for the Bessel-Struve transform Theorem 4.1: Let f be a measurable function on R d such that
and for some constants a, b, λ > 0 and 1 ≤ p, q ≤ ∞. (18) and (19).
To prove this result we need the following lemmas.
Proof: From the hypothesis it follows that e −a y 2 g belongs to L 1 γ ,d (R d ). Then by Proposition 2.1, we deduce that t V γ ,d (e −a y 2 g) is defined almost everywhere on R d . Here we consider two cases.
where r is the conjugate exponent of r. Since for t > 0 (cf. [4]), it follows from (13) that (ii) If r = ∞, then it follows from (21) that This completes the proof.

Lemma 4.3: Let p,q in [1, ∞] and f a measurable function on R d such that
for some a > 0. Then for all z ∈ C d , the integral Proof: The first assertion easily follows from (5) and Hölder's inequality. We shall prove (23). The relation (22) implies that f belongs to L 1 (dμ γ ,d ) and thus, t V γ ,d (f ) in L 1 (R d ) by (13). Hence by (10), for all ξ , η ∈ R d , it follows from Lemma 4.2 that Therefore, the desired result follows.

Proof of Theorem 4.1:
We will divide the proof in each case.
(i) ab > 1 4 . Let h be a function on C d defined by This function is entire on C d and by (23) we see that for all ξ ∈ R d and η ∈ R d . On the other hand, we note that Then it follows from (25) and (26) that h satisfies the assumptions in Lemma 4.1 and thus, h is a constant and Since ab > 1 4 , (19) holds whenever C = 0 and the injectivity of F γ ,d implies that f = 0 almost everywhere. (ii) ab = 1 4 . As in the previous case, it follows that (18) and (19) for all δ ∈ (b, 1/4a).
The following is an immediate consequence of Theorem 4.1.

Corollary 4.1: Let f be a measurable function on R d such that
and for some constants a, b > 0, 1 ≤ p, q ≤ ∞ and 0 < r ≤ ∞.

Corollary 4.2:
Let f be an integrable function on R d and p, q ∈ [1, ∞] such that e a · 2 f belongs to L p (dμ γ ,d ) + L q (dμ γ ,d ), for some positive a. Further assume that for some positive numbers b and c. If ab = 1 4 , then we get f (x) = A e −a x 2 , x ∈ R d and A is a positive constant. If ab < 1 4 , then there are infinitely many linearly independent functions meeting the hypotheses. Otherwise ab > 1 4 , f vanishes almost everywhere on R d .

Conclusions
In the present paper, we have successfully studied two qualitative uncertainty principles for the Bessel-Struve transform on R d . The obtained results have a novelty and contribution to the literature, and they improve and generalize the results of Hamem et al. [11] and Negzaoui [13]. It is our hope that this work motivate the researchers to study the quantitative uncertainty principles for the Bessel-Struve transform on R d .