New Poisson inequality for the Radon transform of infinitely differentiable functions

Poisson inequality for the Radon transform is a key tool in signal analysis and processing. An analogue of the Hardy–Littlewood–Poisson inequality for the Radon transform of infinitely differentiable functions is proved. The result is related to a paper of Luan and Vieira (J. Inequal. Appl. 2017:12, 2017) and to a previous paper by Yang and Ren (Proc. Indian Acad. Sci. Math. Sci. 124(2):175-178, 2014).


Introduction
The Radon transform PI, which is defined as the Cauchy principal value of the following singular integral for any x ∈ R, has been widely used in physics, engineering, and mathematics. The following Poisson inequality PI(hg) ≤ hPIg (1.1) was first studied in [1][2][3]5]. It was proved that (1.1) holds if h, g ∈ L 2 (R) satisfy that suppf ⊆ R + (R + = [0, ∞)) and suppĝ ⊆ R + in [21]. In 2014, Yang and Ren also obtained more general sufficient conditions by weakening the above condition in [24]. Recently, Luan and Vieria established the first necessary and sufficient condition in the time domain and a parallel result in the frequency domain for the Poisson inequality in [16].
It is natural that there have been attempts to define the complex signal and prove the Poisson inequality in a multidimensional case. Definition 1. 1 The partial Radon transform PI j of a function h ∈ L p (R n ) (1 ≤ p < ∞) is given by The total Radon transform PI of a function h ∈ L p (R n ) (1 ≤ p < ∞) is defined as follows: dy.
The existence of the singular integral above and its boundedness property were proved in [10,19]. The iterative nature of the Radon transform in L p (R n ) (p > 1) was shown in [6]. It was shown that The operations PI i and PI j commute with each other, where i, j = 1, 2, . . . , n.
It is known that the Fourier transformĥ of h ∈ L 1 (R n ) is defined as follows (see [7]): where x ∈ R n . Let D(R n ) be the space of infinitely differentiable functions in R n with a compact support and D (R n ) be the space of distributions, that is, the dual of D(R n ) (see [15,23]). This definition is consistent with the ordinary one when T is a continuous function. Put We denote by D D + (R n ), D D -(R n ) and D D 0 (R n ) the set of functions in D(R n ) that are supported on D + , D -, and D 0 , respectively.
The Fourier transformφ is a linear homeomorphism from S(R n ) onto itself. Meanwhile, the following identity holds: The Fourier transform F : S (R n ) → S (R n ) defined as for any ϕ ∈ S(R n ) is a linear isomorphism from S (R n ) onto itself. For the detailed properties of S(R n ) and S (R n ), we refer the readers to [18,20]. For ν ∈ S (R n ) and ϕ ∈ S(R n ), it is obvious that andν is defined as follows: So we obtain that ν =ν in the distributional sense. Following the definition in [16], a function ϕ belongs to the space where |k| = k 1 + k 2 + · · · + k n and k = (k 1 , k 2 , . . . , k n ). In the sequel, we denote by D L p (R n ) the dual of the corresponding spaces As a consequence, we have Then the Radon transform of h is defined by (see [8]) In [16], Luan and Vieira proved that the total Radon transform is a linear homeomor- So the following inequality holds: in the distributional sense. Let be a nonempty subset of R, define (see [16]) where t is a nonzero real number. Hence we have

Main lemmas
In this section, we shall introduce some lemmas.
dy for x ∈ R n from the total Radon transform.
It is clear that the Poisson inequality is satisfied if and only if We use W k,p (R) to denote the Sobolev space where the derivative D m f is understood in the distributional sense.

Lemma 2.2
Suppose that 1 < p ≤ 2. Then, for fixed x ∈ R, the function xy for any y ∈ R and μ ∈ W 1,p (R) is in L p (R) and Proof Since μ ∈ W 1,p (R), we have Now we prove that ν ∈ L p (R). We observe that for fixed x ∈ R by using the generalized Minkowski inequality, which involves that ν ∈ L p (R). Since (see [9]) it follows that ∇ν = σ (u)∇u = ku 2 -1 1/2 ∇u, which yields that Thus we have (see [11,22]) for each ϕ ∈ C 1 0 (R n ). On the other hand, we have R n from the definition of W 1,p (R), which is the desired result.

Poisson inequality for W 1,p (R) functions
In this section, we develop a characterization of W 1,p (R) functions which satisfy the Poisson inequality PI(hg) ≤ hPIg.

Thus (3.2) holds if and only if
which yields thatǧ(w) =ĝ(-w). It is known that the above equation is equivalent to Replacing t by 1 y , we obtain that (see [14]) Set (see [13]) respectively. It follows that for all r > 0 sufficiently large, which yields that (see [17]) for all r 0. Put a(s) g(PI -1 (w ζ )) On the other hand, we have for t 0, where χ [0,S(ζ )] stands for the characteristic function of [0, S(ζ )], which yields that (see [12]) but this is impossible. Consider the following problem (see [15]): As a consequence, we get Now we give an application of Theorem 3.1.
Proof By condition (3.5), we obtain that (see [4]) for any t ∈ I -, which is equivalent to for any t ∈ I -.
By the embedding theorem and Hölder's inequality, we obtain Let = δ(ρ)/ρ. We estimate the first term on the right-hand side of (3.6) as follows: It follows that from the previous inequality and Lemma 2.2. Since we obtain h(u)k j+1 dx ≤ γ (1 -) -γ 2 jγ ρ -1 k -1 n y Since we obtain that and (3.14) Combining (3.13) and (3.14), we have

Conclusions
This paper was mainly devoted to studying a new Poisson inequality for the Radon transform of infinitely differentiable functions. An application of it was also given.