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- JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2014, Vol. 59, No. 7, pp. 9-20 This paper is available online at http://stdb.hnue.edu.vn REVERSE INEQUALITIES FOR THE FOURIER COSINE CONVOLUTION AND APPLICATIONS TO INVERSE HEAT SOURCE PROBLEMS Nguyen Xuan Thao and Bui Minh Khoi School of Applied Mathematics and Informatics, Hanoi University of Science and Technology Abstract. Convolution inequalities and inverse problems play an important role in mathematical analysis. This paper studies inverse inequalities for the Fourier cosine convolution and the backward heat problem. We give new reverse inequalities for the Fourier cosine convolution and their applications to inverse heat source problems: ut = uxx + f (x, t), 0 < x < ∞, t > 0, our purpose is to evaluate the stability of non-negative heat source f (x, t) from any observations u(x, t0 ), where 0 < t0 is a constant, t0 ≤ T + δ, T > 0, δ > 0, 0 < x < X, X > 0 or u(x0 , t), where 0 < x0 is a constant, 0 < t < T . Applying a new inverse inequality allow us to evaluate heat source through some initial observations with space or time variables. Keywords: Reverse inequalities, Fourier cosine convolution, one-dimensional inverse, heat source. 1. Introduction Inverse problems have developed robustly and attracted the attention of many mathematicians in recent decades. Two examples are inverse problems for partial differential equations [2, 4-7] and inverse problems for heat equations [1, 3, 8, 9]. Of these, the problem in [9] is recent research on the reverse heat source problem using reverse inequality for Laplace convolution. We show the following multidimensional heat source equation (see [9]): ∂t u(x, t) = ∆u(x, t) + f (t)φ(x), x ∈ Rn , t > 0, u(x, 0) = 0, x ∈ Rn , Received March 20, 2014. Accepted September 30, 2014. Contact Nguyen Xuan Thao, e-mail address: thao.nguyenxuan@hust.edu.vn 9
- Nguyen Xuan Thao and Bui Minh Khoi for φ is a given function and satisfies φ ≥ 0 in Rn , φ has compact support, { φ ∈ C ∞ (Rn ), n ≥ 4 φ ∈ L2 (Rn ), n ≤ 3. We then estimate the stabilization of heat source f (t), 0 < t < T , from the observation u(x0 , t), 0 < t < T, where x0 ∈ / suppφ. We have the following theorem: Theorem 1.1. ([9]) Let φ satisfy as above, and x0 ∈ / supp φ. We set { 4 4−n , n ≤ 3, p> 1, n ≥ 4. Then for an arbitrary δ > 0, there exists a constant C = C(x0 , φ, T, p, δ, U ) > 0 such that 1/pN ||f ||Lp (0,T ) ≤ C||u(x0 , .)||L1 (0,T +δ) for any f ∈ U, U = {f ∈ C [0, T ]; ||f ||C [0,T ] ≤ M, f changes the signs at most N-times}, M = const > 0, N ∈ N. This theorem was proven using reverse inequality for Laplace convolution (see [9]) and heat source conditions that separate variables to f (t)φ(x), x ∈ Rn , the authors estimated the stabilization of f (t) according to time variable, t, 0 < t < T . In this paper, we study one-dimensional inverse heat source problem with heat source f (x, t), x ∈ R+ , which does not contain separate variables, as follows: ut = uxx + f (x, t), 0 < x < ∞, t > 0, (1.1) under the marginal condition ux (0, t) = 0, ∀t > 0, (1.2) ux (x, t) → 0 when x → ∞, (1.3) u(x, t) → 0 when x → ∞, (1.4) and the initial condition u(x, 0) = 0, (1.5) Here we need to estimate the stability of f (x, t) from any observations u(x, t0 ), where 0 < t0 = const ≤ T + δ, T > 0, δ > 0, 0 < x < X, X > 0 or u(x0 , t), where 0 < x0 = const, 0 < t < T . The main finding of this paper proves the reverse inequality for Fourier cosine convolution (Section 3) and we then apply the new received result for one-dimensional inverse heat source problem (1.1) - (1.5). 10
- Reverse inequalities for the Fourier cosine convolution and applications to inverse heat source... 2. Some known results We present some convolutions and convolution inequalities used in this article. First of all, The Fourier cosine convolution is defined by [10]: ∫+∞ 1 (f ∗ g)(x) = √ f (y){g(x + y) + g(|x − y|)}dy, x ∈ R+ , (2.1) Fc 2π 0 for which the factorization property holds Fc (f ∗ g)(y) = (Fc f )(y)(Fc g)(y), y ∈ R+ , f, g ∈ L1 (R+ ), (2.2) Fc where the Fourier cosine transformation is √ ∫+∞ 2 (Fc f )(y) = f (x)cos(xy)dx. (2.3) π 0 The Laplace convolution is defined by [10]: ∫x (f ∗ g)(x) = f (t)g(x − t)dt, x > 0, (2.4) L 0 for which the factorization property holds L(f ∗ g)(p) = (Lf )(p)(Lg)(p), p ∈ C, Re p > α, L f, g are functions of exponential order, where the Laplace transform is of the form ∫+∞ (Lf )(p) = e−xp f (x)dx. 0 Next, the norm of function f on Lp (A × B), where A, B ⊆ R+ is defined by 1/p ∫ ∫ ||f (x, t)||Lp (A×B) = |f (x, t)|p dxdt B A for p > 0. Moreover, the following reverse convolution inequality holds (see [9]). Proposition 2.1. ([9]) Let p ≥ 1, δ > 0, 0 ≤ α < T, and f, g ∈ L∞ (0, T + δ) satisfy 0 ≤ f, g ≤ M < ∞, 0 < t < T + δ. 11
- Nguyen Xuan Thao and Bui Minh Khoi Then T +δ t 1/p ∫ ∫ ||f ||Lp (α,T ) ||g||Lp (0,δ) ≤ M (2p−2)/p f (s)g(t − s)dsdt α α In particular, for ∫t (f ∗ g)(t) = f (t − s)g(s)ds, 0 < t < T + δ L 0 and α= 0, we have ||f ||Lp (0,T ) ||g||Lp (0,δ) ≤ M (2p−2)/p ||f ∗ g||1/p L (0,T +δ) . L 1 Although it has an important role the study of inverse problems, not many reverse inequalities for convolutions have been studied. In next section, we study a new reverse inequality for the Fourier cosine convolution and apply it to a certain inverse heat problem. 3. A new reverse inequality for the Fourier cosine convolution In this section, we establish a new reverse inequality for the Fourier cosine convolution of two functions f, g in two-dimensional space. Theorem 3.1. Let p > 1, δ > 0, T > 0 and f, g ∈ L1 (D) ∩ Lp (D), satisfy 0 ≤ f, g ≤ M < ∞, (x, t) ∈ D, D = {(x, t) : 0 < x < ∞, 0 < t ≤ T + δ }. Then we have 1/p ||f (x, t)||Lp (R+ ×(0,T )) ||g(x, t)||Lp (R+ ×(0,δ)) 6 M (2p−2)/p ∥u(x, t)∥L1 (R+ ×(0,T +δ)) , here, ( ) √ ∫ t u(x, t) = 2π f (τ, ξ) ∗ g(τ, t − ξ) (x)dξ. Fc 0 Proof. Since 0 ≤ f, g ≤ M < ∞ for 0 < x < ∞, 0 < t ≤ T + δ and formula (2.1), we have ∫ t ∫+∞ f p (τ, ξ)g p (|x − τ |, t − ξ)dτ dξ 0 0 ∫ t ∫+∞ = f p−1 (τ, ξ)g p−1 (|x − τ |, t − ξ)f (τ, ξ)g(|x − τ |, t − ξ)dτ dξ 0 0 12
- Reverse inequalities for the Fourier cosine convolution and applications to inverse heat source... ∫ t ∫+∞ ≤M 2p−2 f (τ, ξ)g(|x − τ |, t − ξ)dτ dξ 0 0 ∫ t ∫+∞ ≤ M 2p−2 f (τ, ξ){g(x + τ, t − ξ)+ g(|x − τ |, t − ξ)}dτ dξ 0 0 √ ∫t ( ) ≤M 2p−2 2π f (τ, ξ) ∗ g(τ, t − ξ) (x)dξ Fc 0 ≤ M 2p−2 u(x, t). Hence +∞ ∫T +δ ∫ +∞ ∫ t ∫ ∫ T +δ ∫ +∞ f (τ, ξ)g (|x − τ |, t − ξ)dτ dξ dxdt ≤ M p p 2p−2 u(x, t) dxdt 0 0 0 0 0 0 (3.1) On the other hand, by using Fubini’s theorem and changing the variables in integrals, we have +∞ ∫ T +δ ∫ +∞ ∫ t ∫ f p (τ, ξ)g p (|x − τ |, t − ξ)dτ dξ dxdt 0 0 0 0 ∫+∞ ∫ T +δ ∫ T +δ ∫ +∞ = f p (τ, ξ)g p (|x − τ |, t − ξ)dτ dtdξdx 0 0 ξ 0 +∞ T +δ ∫+∞ ∫ T +δ ∫ ∫ = f p (τ, ξ)g p (|x − τ |, t − ξ)dtdτ dξdx 0 0 0 ξ +∞ T +δ ∫+∞ ∫ T +δ ∫ ∫ = f p (τ, ξ) g p (|x − τ |, t − ξ)dt dτ dξdx 0 0 0 ξ +∞ T +δ−ξ ∫+∞ ∫ T +δ ∫ ∫ = f p (τ, ξ) g p (|x − τ |, y)dy dτ dξdx 0 0 0 0 +∞ δ ∫+∞∫T ∫ ∫ ≥ f p (τ, ξ) g p (|x − τ |, y)dy dτ dξdx 0 0 0 0 +∞ δ +∞ ∫T ∫ ∫ ∫ ≥ f p (τ, ξ) g p (|x − τ |, y)dxdy dτ dξ 0 0 0 0 13
- Nguyen Xuan Thao and Bui Minh Khoi +∞ δ +∞ ∫T ∫ ∫ ∫ ≥ f p (τ, ξ) g p (x − τ, y)dxdy dτ dξ 0 0 0 τ +∞ δ +∞ ∫T ∫ ∫ ∫ ≥ f p (τ, ξ) g p (z, y)dzdy dτ dξ 0 0 0 0 ∫T ∫+∞ ∫δ ∫+∞ ≥ p f (τ, ξ)dτ dξ g p (z, y)dzdy. (3.2) 0 0 0 0 From (3.1), (3.2) we obtain ∫T ∫+∞ ∫δ ∫+∞ ∫ T +δ ∫ +∞ f p (τ, ξ)dτ dξ g p (z, y)dzdy ≤ M 2p−2 u(x, t) dxdt. 0 0 0 0 0 0 Thus 1/p ||f ||Lp (R+ ×(0,T )) ||g||Lp (R+ ×(0,δ)) 6 M (2p−2)/p ∥u(x, t)∥L1 (R+ ×(0,T +δ)) . The proof of the theorem is complete. Remark 3.1. For fixed 0 < t = t0 ≤ T + δ, t0 ( ) √ ∫ u(x, t0 ) = 2π f (τ, ξ) ∗ g(τ, t0 − ξ) (x)dξ, Fc 0 and if g(τ, t0 − ξ) ≥ g(τ, t0 − ξ0 ) for 0 ≤ ξ ≤ β, β = const, β < t0 , as in the proof Theorem 3.1, we have 1/p ||f ||Lp ((0,X)×(0,β)) ||g(x, t0 − ξ0 )||Lp (0,γ) 6 M (2p−2)/p ∥u(x, t0 )∥L1 (0,X+γ) . Indeed, since 0 ≤ f, g ≤ M < ∞ for 0 < x < ∞, 0 < t ≤ T + δ and formula (2.1), therefore ∫t0 ∫+∞ f p (τ, ξ)g p (|x − τ |, t0 − ξ)dτ dξ ≤ M 2p−2 u(x, t0 ). 0 0 Hence t +∞ ∫ X+γ ∫0 ∫ ∫ X+γ f (τ, ξ)g (|x − τ |, t0 − ξ)dτ dξ dx ≤ M p p 2p−2 u(x, t0 ) dx, (3.3) 0 0 0 0 14
- Reverse inequalities for the Fourier cosine convolution and applications to inverse heat source... for X > 0, γ > 0. On the other hand, by using Fubini’s theorem, changing the variables in integrals and since g(τ, t0 − ξ) ≥ g(τ, t0 − ξ0 ) for 0 ≤ ξ ≤ β, β = const, β < t0 , we have t +∞ ∫ X+γ ∫0 ∫ f p (τ, ξ)g p (|x − τ |, t0 − ξ)dτ dξ dx 0 0 0 ∫t0 ∫+∞ X+γ ∫ ≥ f p (τ, ξ)g p (|x − τ |, t0 − ξ) dxdτ dξ 0 0 0 ∫ ∫ ∫ t 0 X X+γ ≥ f p (τ, ξ)g p (x − τ, t0 − ξ) dxdτ dξ 0 0 τ ∫0 ∫ ∫ t X X+γ−τ ≥ f p (τ, ξ)g p (z, t0 − ξ) dzdτ dξ 0 0 0 ∫β ∫ ∫ X γ ≥ f p (τ, ξ)g p (z, t0 − ξ) dzdτ dξ 0 0 0 ∫β ∫X ∫γ ≥ f p (τ, ξ)g p (z, t0 − ξ0 ) dzdτ dξ 0 0 0 ∫β ∫X ∫γ ≥ p f (τ, ξ)dτ dξ g p (z, t0 − ξ0 ) dz. (3.4) 0 0 0 From (3.3), (3.4), we obtain 1/p ||f ||Lp ((0,X)×(0,β)) ||g(x, t0 − ξ0 )||Lp (0,γ) 6 M (2p−2)/p ∥u(x, t0 )∥L1 (0,X+γ) . Remark 3.2. For fixed x = x0 > 0 then ∫ t ∫+∞ u(x0 , t) = f (τ, ξ){g(x0 + τ, t − ξ)+g(|x0 − τ |, t − ξ)}dτ dξ, 0 0 and if g(x0 + τ, ξ) ≥ g(x0 + τ0 , ξ) for 0 ≤ τ ≤ α, α = const, as in the proof Theorem 3.1, we have 1/p ||f ||Lp ((0,α)×(0,T )) ||g(x0 + τ0 , t)||Lp (0,δ) 6 M (2p−2)/p ∥u(x0 , t)∥L1 (0,T +δ) . 15
- Nguyen Xuan Thao and Bui Minh Khoi Indeed, since 0 ≤ f, g ≤ M < ∞ for 0 < x < ∞, 0 < t ≤ T + δ, therefore ∫ t ∫+∞ f p (τ, ξ)g p (x0 + τ, t − ξ)dτ dξ ≤ M 2p−2 u(x0 , t). 0 0 Hence +∞ ∫ T +δ ∫ t ∫ ∫ T +δ f p (τ, ξ)g p (x0 + τ, t − ξ)dτ dξ dt ≤ M 2p−2 u(x0 , t)dt. (3.5) 0 0 0 0 On the other hand, by using Fubini’s theorem, changing the variables in integrals and since g(x0 + τ, ξ) ≥ g(x0 + τ0 , ξ) for 0 ≤ τ ≤ α we have +∞ T +δ ∫ t ∫ ∫ f p (τ, ξ)g p (x0 + τ, t − ξ)dτ dξ dt 0 0 0 +∞ T +δ ∫ T +δ ∫ ∫ = f p (τ, ξ)g p (x0 + τ, t − ξ)dtdτ dξ 0 0 ξ +∞ T +δ ∫ T +δ ∫ ∫ = f p (τ, ξ) g p (x0 + τ, t − ξ)dt dτ dξ 0 0 ξ +∞ T +δ−ξ ∫ T +δ ∫ ∫ = f p (τ, ξ) g p (x0 + τ, y)dy dτ dξ 0 0 0 +∞ δ ∫T ∫ ∫ ≥ f p (τ, ξ) g p (x0 + τ, y)dy dτ dξ 0 0 0 α δ ∫T ∫ ∫ ≥ f p (τ, ξ) g p (x0 + τ, y)dy dτ dξ 0 0 0 α δ ∫T ∫ ∫ ≥ f p (τ, ξ) g p (x0 + τ0 , y)dy dτ dξ 0 0 0 ∫T ∫α ∫δ ≥ p f (τ, ξ)dτ dξ g p (x0 + τ0 , y)dy. (3.6) 0 0 0 From (3.5), (3.6), we obtain 1/p ||f ||Lp ((0,α)×(0,T )) ||g(x0 + τ0 , t)||Lp (0,δ) 6 M (2p−2)/p ∥u(x0 , t)∥L1 (0,T +δ) . 16
- Reverse inequalities for the Fourier cosine convolution and applications to inverse heat source... Remark 3.3. In the reverse inequality for Laplace convolution (Proposition 2.1), the functions f , g are considered in one-dimensional space. As in Theorem 3.1, our reverse inequality for Fourier cosine convolution has a more complicated kernel (see (2.1), (2.4)) with the functions f , g in two-dimensional space. 4. Applications to inverse heat source problems In this section, we get the result of inverse problem (1.1) - (1.5) when applying Theorem 3.1 with to be specific, an evaluation of the stabilization of the heat source f (x, t) according to space or time. Theorem 4.1. We consider the heat equation (1.1) with marginal conditions (1.2) - (1.4) and an initial condition (1.5) where f ∈ L1 (D) ∩ Lp (D); 0 ≤ f ≤ M < ∞, for every (x, t) ∈ D and M is a constant, D = {(x, t) : 0 < x < ∞, 0 < t ≤ T + δ }, δ > 0, T > 0, p > 1. Then we have (i) For 0 < t0 = const ≤ T + δ, T > 0, δ > 0, 0 < x < X, X > 0, we evaluate the stability of heat source f (x, t) from any observations u(x, t0 ): 1/p ||f ||Lp ((0,X)×(0,t0 −γ 2 /2)) 6 C1 ∥u(x, t0 )∥L1 (0,X+γ) , for C1 is constant. (ii) For 0 < x0 = const, 0 < t < T , 1 < p < 2, α ≥ 0 we evaluate the stability of heat source f (x, t) from any observations u(x0 , t): 1/p ||f ||Lp ((0,α)×(0,T )) 6 C2 ∥u(x0 , t)∥L1 (0,T +δ) , for C2 is constant. Proof. (i) Considering t as a parameter, by applying Fourier cosine transformation (2.3) on both sides of (1.1), therefore: ∂ (Fc u(x, t))(y) = −y 2 (Fc u(x, t))(y) + (Fc f (x, t))(y), (4.1) ∂t for the condition (Fc u(x, 0))(y) = 0. Equation (4.1) has root as ∫t −y 2 t −y 2 t 2 (Fc u(x, t))(y) = C(y)e +e (Fc f (x, ξ))(y)ey ξ dξ, 0 since (Fc u(x, 0))(y) = 0 thus C(y) = 0. 17
- Nguyen Xuan Thao and Bui Minh Khoi Then, by using formula (2.2) we have ∫t ∫t −y 2 t y2 ξ (Fc f (x, ξ))(y)e−y 2 (t−ξ) (Fc u(x, t))(y) = e (Fc f (x, ξ))(y)e dξ = dξ 0 0 ∫t ( ) e−τ /4(t−ξ) 2 = (Fc f (x, ξ))(y)Fc √ (y)dξ t−ξ 0 ∫t ( ) e−τ /4(t−ξ) 2 = Fc f (τ, ξ) ∗ √ (y)dξ Fc t−ξ 0 t( ) ∫ e−τ 2 /4(t−ξ) = Fc f (τ, ξ) ∗ √ (x) dξ (y). Fc t−ξ 0 Therefore ∫t ( ) e−τ /4(t−ξ) 2 u(x, t) = f (τ, ξ) ∗ √ (x)dξ Fc t−ξ 0 ∫ t ∫+∞ ( ) e−(x−τ ) /4(t−ξ) e−(x+τ ) /4(t−ξ) 2 2 1 =√ f (τ, ξ) √ + √ dτ dξ. 2π t−ξ t−ξ 0 0 Set e−τ /4ξ 2 g(τ, ξ) = √ , ξ then there exists a constant N such that 0 ≤ g(τ, ξ) ≤ N < ∞ (0 < τ < ∞, 0 < ξ ≤ T + δ). Set M1 = max {M , N }. We have function g(τ, t0 − ξ) which does not decrease in 0 ≤ ξ ≤ β = t0 − γ 2 /2, 0 < γ 2 /2 ≤ t0 , thus, by applying Remark 3.1 we evaluate the stability of function f (x, t) from any observations u(x, t0 ), where 0 < t0 = const ≤ T + δ, T > 0, δ > 0, 0 < x < X, X > 0: e−x2 /4t0 ||f ||Lp ((0,X)×(0,t0 −γ 2 /2)) √ (2p−2)/p 6 M1 (2π)1/2p ∥u(x, t0 )∥1/p . t0 L1 (0,X+γ) Lp (0,γ) Hence 1/p ||f ||Lp ((0,X)×(0,t0 −γ 2 /2)) 6 C1 ∥u(x, t0 )∥L1 (0,X+γ) , for e−x2 /4t0 −1 (2π)1/2p √ (2p−2)/p C1 = M1 . t0 Lp (0,γ) 18
- Reverse inequalities for the Fourier cosine convolution and applications to inverse heat source... (ii) On the other hand, the function g(x0 +τ, ξ) does not increase in 0 ≤ τ ≤ α, thus, by applying Remark 3.2 we evaluate the stability of function f (x, t) from any observations u(x0 , t), where 0 < x0 = const, 0 < t < T , 1 < p < 2, α is an arbitrary non-negative constant: e−(x0 +α)2 /4t ||f ||Lp ((0,α)×(0,T )) √ 6 M1 (2p−2)/p (2π)1/2p ∥u(x0 , t)∥1/p . t L1 (0,T +δ) Lp (0,δ) Hence 1/p ||f ||Lp ((0,α)×(0,T )) 6 C2 ∥u(x0 , t)∥L1 (0,T +δ) , for e−(x0 +α)2 /4t −1 (2p−2)/p C2 = M1 (2π)1/2p √ . t Lp (0,δ) The proof of the theorem is complete. Remark 4.1. For fixed x = x0 > 0, 0 < t = t0 ≤ T + δ then ∫t0 ∫+∞ 1 u(x0 , t0 ) = √ f (τ, ξ){g(x0 + τ, t0 − ξ)+ g(|x0 − τ |, t0 − ξ)}dτ dξ, 2π 0 0 for t0 ≥ (x0 + α)2 /2, p > 1, as in the proof of Theorem 4.1, we obtain: ||f ||Lp ((0,α)×(0,t0 −(x0 +α)2 /2)) 6 C (u(x0 , t0 ))1/p , for ( )−1 e−(x0 +α) 2 /4t 0 C = M1 (2p−2)/p (2π)1/2p √ . t0 In [9], the authors evaluated the stabilization of f (t) according to time variable t (Theorem (1.1)) by using a reverse inequality for the Laplace convolution (Proposition 2.1) and heat source conditions that separate variables to f (t)φ(x), x ∈ Rn . Here, we evaluate the stabilization of heat source f (x, t), x ∈ R+ , which does not separate variables, according to space x or time t from some initial observations (Theorem 4.1), or in both variables from an initial observation (Remark 4.1) using the reverse inequality for the Fourier cosine convolution which was obtained in Theorem 3.1. Here is no any numerical example to illustrate the validity/effectiveness of the main result. In the future, we can apply the above results to give a numerical solution for a specific heat source problem and evaluate the advantages of the new method compared to old results. 19
- Nguyen Xuan Thao and Bui Minh Khoi REFERENCES [1] A. Boumenir and V. K. Tuan, 2010. An inverse problem for the heat equation. Proc. Amer. Math. Soc. 138, pp. 3911-3921. [2] L. Baudonin, A. Mercado, 2008. An inverse problem for Schrodinger equations with discontinuous main coefficient. Appl. Anal, 87, No. 10-11, pp. 1145-1165. [3] J. R. Cannon and S. P. Esteva, 1986. An inverse problem for the heat equation. Inverse Problems, 2, pp. 395-403. [4] V. Isakov, 1998. Inverse Problems for Partial Differential Equations. Springer-Verlag, Berlin. [5] A. Mercado, A. Osses and L. Rosier, 2008. Inverse problem for the schrodinger equation via Carleman inequalities with degenerate weight. Inverse problems, 24, No. 18, pp. 15-17. [6] S. Saitoh, 1984. A fundamental inequality in the convolution of L2 functions on the half line. Proc. Amer. Math. Soc., 91, pp. 285-286. [7] S. Saitoh, 2000. Weighted Lp -norm inequalities in convolutions, Survey on Classical Inequalities. Kluwer Academic Publishers, the Netherlands, pp. 225-234. [8] S. Saitoh, V. K. Tuan and M. Yamamoto, 2000. Reverse weighted Lp -norm inequalities in convolutions and stability in inverse problems. J. of Inequal. Pure and Appl. Math., 1(1), Article 7. [9] S. Saitoh, V. K. Tuan and M. Yamamoto, 2002. Reverse convolution inequalities and applications to inverse heat source problems. J. of Inequal. Pure and Appl. Math., 3(5), Article 80. [10] I. N. Sneddon, 1972. The Use of Integral Transforms. Mc Gray-Hill, New York. [11] Xiao-Hua, L., 1990. On the inverse of H¨older inequality. Math. Practice and Theory, 1, pp. 84-88. 20
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