Stability estimates in stationary inverse transport

Stability estimates in stationary inverse transport Guillaume Bal and Alexandre Jollivet ∗ April 8, 2008 Abstract We study the stability of the reconstruction of the scattering and absorption coefficients in a stationary linear transport equation from knowledge of the full albedo operator in dimension n ≥ 3. The albedo operator is defined as the mapping from the incoming boundary conditions to the outgoing transport solution at the boundary of a compact and convex domain. The uniqueness of the reconstruction was proved in [3, 4] and partial stability estimates were obtained in [13] for spatially independent scattering coefficients. We generalize these results and prove an L1 -stability estimate for spatially dependent scattering coefficients. 1 Introduction Let the spatial domain X ⊂ Rn , n ≥ 2, be a convex, open bounded subset with C 1 boundary ∂X, and let the velocity domain V be Sn−1 or an open subset of Rn which satisfies inf v∈V |v| > 0. Let Γ± = {(x, v) ∈ ∂X ×V ; ±n(x)v > 0} where n(x) denotes the outward normal vector to ∂X at x ∈ ∂X. The set Γ− is the set of incoming boundary condition while Γ+ is the set where we measure the outgoing solution to the following stationary linear Boltzmann transport equation in X × V : Z k(x, v ′ , v)f (x, v ′ )dv ′ = 0 in X × V, (1.1) v∇x f (x, v) + σ(x, v)f (x, v) − V f|Γ− = f− . Here, f (x, v) models the density of particles at position x ∈ X with velocity v ∈ V . The albedo operator A is then defined by A : f− 7→ f|Γ+ , (1.2) Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA; [email protected] and [email protected] ∗ 1 where f (x, v) is the solution to (1.1). The inverse transport problem consists of reconstructing the absorption coefficient σ(x, v) and the scattering coefficient k(x, v ′ , v) from knowledge of A. Stability estimates aim at controlling the variations in the reconstructed coefficients σ(x, v) and k(x, v ′ , v) from variations in A in suitable metrics. The forward transport equation has been analyzed in e.g. [5, 6, 8]. The inverse transport problem has been addressed in e.g. [3, 4, 9, 10] with stability estimates obtained in [9, 13]. For the two-dimensional case, in which proofs of uniqueness of the scattering coefficient are available only when it is sufficiently small or independent of the spatial variable, we refer the reader to e.g. [1, 11, 12]. To obtain our stability estimates, we follow a methodology based on the decomposition of the albedo operator into singular components [3, 4] and the use of appropriate functions on Γ± with decreasing support [13]. In dimensions n ≥ 3 the contribution due to single scattering is more singular than the contribution due to higher orders of scattering. As a consequence, the single scattering in a direction v ′ generated by a delta function f− = δx0 (x)δ(v − v0 ) is a one-dimensional curve on ∂X. In order to obtain general stability estimates for the scattering coefficient, one way to proceed is to construct test functions whose support converges to that specific curve. It turns out that it is simpler to work in a geometry in which this curve becomes a straight line. We now briefly introduce that geometry and refer the reader to section 2 below for a formal presentation. Let R be a positive real constant such that X is included in the ball B(R) of radius R centered at x = 0. On (B(R)\X) × V , the absorption and scattering coefficients vanish and we may solve the equation v∇x f = 0. This allows us to map back the incoming conditions f− on Γ− as incoming conditions, which we shall still denote by f− , on F− and map forward the outgoing solution f|Γ+ to an outgoing solution f+ on F+ , where we have defined F± := {(x ± Rv̂, v) ∈ Rn × V for (x, v) ∈ Rn × V s.t. vx = 0, |x| < R}. (1.3) In other words, F± is the union for each v ∈ V of the spatial points on a disc of radius R in a plane orthogonal to v and tangent to the sphere of radius R. The incoming boundary condition is thus now defined on F− while measurements occur on F+ and we may define the albedo operator still called A as an operator mapping f− defined on F− to the outgoing solution f+ on F+ . We may now verify that the single scattering in a direction v ′ generated by a delta function f− = δ(x − x0 )δ(v − v0 ) for (x0 , v0 ) ∈ F− is a one-dimensional segment in F+ ; see Fig. 1. Note also that the geometry we consider here may be more practical than the geometry based on Γ± . Indeed, we assume that the incoming conditions are generated on a plane for each direction of incidence, and, more importantly, that our measurements are acquired on a plane for each outgoing direction. This is how the collimators used in Computerized Tomography [7] are currently set up. Under appropriate assumptions on the coefficients, we aim to show that A is a well posed operator from L1 (F− ) to L1 (F+ ). We shall then obtain a stability esti2 v0' x0'+Rv0' P2 P3 x0'-sv0' v x0'-sv0'+t(s)v x0'-Rv0' P1 Figure 1: Geometry of the ballistic and single scattering components in dimension n = 3. The source term is non-zero in the vicinity (in F− ) of x′0 − Rv0′ in P1 = −Rv0′ + Πv0′ (R). The ballistic part is non-zero in the vicinity (in F+ ) of x′0 + Rv0′ in P2 = Rv0′ + Πv0′ (R). The thick “line” represents the support of the single scattering contribution in the vicinity (in F+ ) of the segment {x′0 − sv0′ + t(s)v; s ∈ (−R, R)} ⊂ P3 = Rv + Πv (R). See text for the notation. mate for the reconstruction σ(x) (or σ(x, |v|)) and k(x, v, v ′ ) with respect to the norm L(L1 (F− ), L1 (F+ )) of A. The rest of the paper is structured as follows. Because our geometry is not standard, we present a detailed analysis of the linear transport equation and of the singular decomposition of the albedo operator in section 2. Most of the material in that section is similar to that in [4]. One of the main physical constraints in the existence of solutions to (1.1) is that the system be “subcritical”, in the sense that the “production” of particles by the scattering term involving the scattering coefficient k(x, v, v ′ ) has to be compensated by the absorption of particles and the leakage of particles at the domain’s boundary. Although this may be seen implicitly in [4], we state explicitly that the decomposition of the albedo operator used in the stability estimates holds as soon as the forward transport problem is well-posed in a reasonable way. The stability results are stated in section 3. Under additional continuity assumptions on the absorption and scattering coefficients, we obtain that (i) the exponential of line integrals of the absorption coefficient and (ii) the scattering coefficient multiplied by the exponential of the integral of the absorption coefficient on a broken line are both stably determined by A in L(L1 (F− ), L1 (F+ )); see Theorem 3.2. Under additional regularity hypotheses on the absorption coefficient, we obtain a stability result for the absorption 3 coefficient in some Sobolev space H s and for the scattering coefficient in the L1 norm. The stability results in the geometry of (1.1) are presented in section 4. The proof of the stability results and the construction of the appropriate test functions are presented in section 5. Several proofs on the decomposition of the albedo operator and the uniqueness of the transport equation have been postponed to sections 6 and 7, respectively. 2 Transport equation and albedo operator We now state our main results on the stationary linear transport equation and the corresponding albedo operator. Let R be a positive real constant and let n ∈ N, n ≥ 2. Let V be Sn−1 or an open v subset of Rn which satisfies v0 := inf v∈V |v| > 0. For v ∈ V , we define v̂ := |v| . Then, n we consider the open subset O of R × V defined by O := {(x, v) ∈ Rn × V | |xv̂| < R, |x − (xv̂)v̂| < R}, (2.1) and let F be the set F := {(x, v) ∈ Rn × V | x ∈ Πv (R)}, (2.2) Πv (R) := {x ∈ Rn | xv = 0, |x| < R}, (2.3) where for all v ∈ V . For all v ∈ V we also consider Rv̂ + Πv (R) := {Rv̂ + x | x ∈ Πv (R)}. (2.4) When V = Sn−1 , then F is an open subset of T Sn−1 := {(x, v) ∈ Rn ×Sn−1 | vx = 0}, the tangent space to the unit sphere. When V is an open subset of Rn (which satisfies v0 = inf v∈V |v| > 0) then F is an open subset of the 2n − 1 dimensional manifold {(x, v) ∈ Rn × V | vx = 0}. We also define F± by F± := {(x ± Rv̂, v) ∈ Rn × V |(x, v) ∈ F }, (2.5) and recall that F− is the set of incoming conditions for the transport equation while F+ is the set in which measurements are performed. We consider the space L1 (O) with the usual norm Z |f (x, v)|dxdv, for f ∈ L1 (O). (2.6) kf kO := O We also consider the space L1 (F ) defined as the completed Banach space of the vector space of compactly supported continuous functions on F for the norm Z Z |v||f (x, v)|dxdv, f ∈ L1 (F ), (2.7) kf kF := V Πv (R) 4 and similarly the spaces L1 (F± ) defined as the completed Banach space of the vector space of compactly supported continuous functions on F± for the norm Z Z |v||f± (x ± Rv̂, v)|dxdv, f± ∈ L1 (F± ). (2.8) kf± kF± := V Πv (R) We assume that: 0 ≤ σ ∈ L∞ (Rn × V ), k(x, v ′ , v) is a measurable function on Rn × V × V, σ(x, v) = k(x, v ′ , v) = 0 for (x, v ′ , v) ∈ Rn × V × V, |x| > R, 0 ≤ k(x, v ′ ,R.) ∈ L1 (V ) for a.e. (x, v ′ ) ∈ Rn × V and σp (x, v ′ ) = V k(x, v ′ , v)dv belongs to L∞ (Rn × V ). (2.9) Under these conditions, we consider the stationary linear Boltzmann transport equation Z v∇x f (x, v) + σ(x, v)f (x, v) − k(x, v ′ , v)f (x, v ′ )dv ′ = 0 in O, (2.10) V f|F− = f− . Throughout the paper, for m ∈ N and for any subset U of Rm we denote by χU the characteristic function defined by χU (x) = 0 if x 6∈ U and χU (x) = 1 if x ∈ U . We now analyze the well-posedness of (2.10). The following change of variables is useful. Lemma 2.1. For f ∈ L1 (O), we have: Z Z Z f (x, v)dxdv = Πv (R) V O Proof. First using (2.1) we have Z Z Z f (x, v)dxdv = O V Z R f (y ± tv̂, v)dtdydv. (2.11) −R χO (x, v)f (x, v)dxdv. Rn Then for a.e. v ∈ V using the change of variables {x ∈ Rn | |v̂.x| < R, |x − (v̂x)v̂| < R} → Πv (R) × (−R, R), x 7→ (x − (v̂x)v̂, ±v̂x), we obtain (2.11). We introduce the following notation: T0 f = −v∇x f in the distributional sense, A1 f = −σf, (2.12) Z A2 f = k(x, v ′ , v)f (x, v ′ )dv ′ , T1 = T0 + A1 , T = T0 + A1 + A2 = T1 + A2 , V and the Banach spaces W := {f ∈ L1 (O, |v|dxdv); T0 f ∈ L1 (O)}, kf kW = kT0 f kO + k|v|f kO , W̃ := {f ∈ L1 (O); T0 f ∈ L1 (O)}, kf kW̃ = kT0 f kO + kf kO . 5 We consider the space L(F± ) defined as the completed Banach space of the vector space of compactly supported continuous functions on F± for the norm Z Z |f± (x′ ± Rv̂, v)|dx′ dv, for f± ∈ L(F± ). (2.13) kf± kL(F± ) := V Πv (R) Note that W ⊆ W̃ and L1 (F± ) ⊆ L(F± ). The spaces W̃ and L(F± ) are used only to define the unbounded operators T and T1 below. We obtain the following trace result. Lemma 2.2. We have kf|F± kF± ≤ Ckf kW , (2.14) for f ∈ W, where C = max((2R)−1 , 1) and kf|F± kL(F± ) ≤ C ′ kf kW̃ , (2.15) for f ∈ W̃, where C ′ = max((2R)−1 , v0−1 ). Proof. Let f be a C 1 function in Rn × V with compact support. Then from (2.11), it follows that Z Z Z R d (|v|| f (x′ ± tv̂, v)| + |v||f (x′ ± tv̂, v)|)dtdx′ dv. kf kW = (2.16) dt V Πv (R) −R ′ n ′ ′ Let R t vd ∈ V′ and x ∈ R such that vx = 0. Note that f′ (x ∓ Rv̂, v) = f (x′ ± tv̂, v) − f (x ± sv̂, v)ds for all t ∈ (−R, R). Hence |f (x ∓ Rv̂, v)| ≤ |f (x ± tv̂, v)| + ds R−R R d | f (x′ ± sv̂, v)|ds. Upon integrating the latter equality, we obtain −R ds Z R Z R 1 d ′ ′ |f (x ∓ Rv̂, v)| ≤ | f (x′ ± sv̂, v)|ds. |f (x ± tv̂, v)|dt + (2.17) 2R −R −R ds Combining (2.16) et (2.17), we obtain (2.14). The proof of (2.15) is similar. For a continuous function f− on F− , we define the following extension of f− in O: Jf− (x, v) = e−|v| −1 R R+xv̂ 0 σ(x−sv̂,v)ds f− (x − (xv̂ + R)v̂, v), (x, v) ∈ O. (2.18) Lemma 2.3. For f− ∈ L1 (F− ) with C = 2R(1 + v0−1 kσk∞ ), we have: kJf− kW ≤ Ckf− kF− . (2.19) Proof. Let f− be a compactly supported continous function on F− . From (2.11) and (2.18) it follows that Z Z Z R R −1 t ′ ′ k|v|Jf− kO = |v||f− (x − Rv̂, v)| e−|v| −R σ(x +sv̂,v)ds dtdx′ dv ≤ 2Rkf− kF− . (2.20) V Πv (R) −R One can check that Jf− satisfies T0 Jf− = −A1 Jf− in the distributional sense. Therefore using also (2.20) we obtain kT0 Jf− kO + k|v|Jf− kO ≤ (1 + kA1 |v|−1 k) ×k|v|Jf− kO = 2R(1 + v0−1 kσk∞ )kf− kF− , which proves the lemma. 6 2.1 Existence theory for the albedo operator We consider the following unbounded operators: T1 f = T1 f, Tf = T f, D(T1 ) = D(T) = {f ∈ W̃ ; f|F− = 0}. (2.21) The operator T1 : D(T1 ) → L1 (O) is close, one-to-one, onto, and its inverse T1 −1 is given for all f ∈ L1 (O) by Z R+xv̂ R −1 t −1 −1 T1 f (x, v) = −|v| e−|v| 0 σ(x−sv̂,v)ds f (x − tv̂, v)dt, (x, v) ∈ O. (2.22) 0 Lemma 2.4. The following statements hold: i. The bounded operator |v|T1 −1 in L1 (O) has norm less or equal to 2R and the bounded operator A2 |v|−1 in L1 (O) has norm less than k|v|−1 σp (x, v)kL∞ (O) . ii. Under the hypothesis σ − σp ≥ 0, (2.23) −2Rv0−1 kσp k∞ the bounded operator A2 T1 −1 in L1 (O) has norm less than 1 − e . iii. Assume either condition (2.23) or 2Rk|v|−1 σp (x, v)kL∞ (O) < 1. (2.24) Then I + A2 T1 −1 is invertible in L1 (O). Lemma 2.4 is proved in section 7. We denote by K the bounded operator in L (O, |v|dxdv) defined by K = T1 −1 A2 : Z R+xv̂ R −1 t −1 Kf (x, v) = −|v| e−|v| 0 σ(x−sv̂,v)ds (A2 f )(x − tv̂, v)dt, (x, v) ∈ X × V, 1 0 1 for all f ∈ L (O, |v|dxdv). The operator K also defines a bounded operator in L1 (O). This allows us to recast the stationary linear Boltzmann transport equation as the following integral equation: (I + K)f = Jf− . (2.25) The existence theory for the above integral equation is addressed in the following result. Proposition 2.5. The following statements hold: i. The conditions (2.26) and (2.27) below are equivalent. The bounded operator I + K in L1 (O) admits a bounded inverse in L1 (O). (2.26) −1 1 The bounded operator I + A2 T1 in L (O) admits a bounded inverse in L1 (O). (2.27) 7 ii. Assume either (2.23) or (2.24). Then condition (2.26) is satisfied. iii. If (2.26) is satisfied then the bounded operator I + K in L1 (O, |v|dxdv) admits a bounded inverse in L1 (O, |v|dxdv). (2.28) Proposition 2.5 is proved in Section 7. The following proposition deals with the existence of the albedo operator. Proposition 2.6. Assume (2.28). Then i. the integral equation (2.25) is uniquely solvable for all f− ∈ L1 (F− ) and f ∈ W ; ii. the albedo operator A : f− 7→ f+ = f|F+ is a bounded operator A : L1 (F− ) → L1 (F+ ). Proposition 2.6 is proved in Section 7. 2.2 Singular decomposition of the albedo operator We assume that condition (2.28) is satisfied. Let us consider the operator R : L1 (O, |v|dxdv) → L1 (F+ ), defined by
ψ 7→ Rψ := K 2 ψ |F+ , (2.29) for ψ ∈ L1 (O, |v|dxdv). Using the equality (in the distributional sense) T0 Kf = −A1 f − A2 f for f ∈ L1 (O, |v|dxdv) and the boundedness of the operators A1 and A2 from L1 (O, |v|dxdv) to L1 (O, dxdv) and using (2.14), we obtain that R is a well defined and bounded operator from L1 (O, |v|dxdv) to L1 (F+ ). We shall use the following lemma for the kernel distribution of R. Lemma 2.7. We have the following decomposition: Z β(x, v, x′ , v ′ )ψ(x′ , v ′ )dx′ dv ′ , Rψ(x, v) = (2.30) O for a.e. (x, v) ∈ F+ and for any ψ ∈ L1 (O, |v|dxdv), where 0 ≤ |v ′ |−1 β ∈ L∞ (O, L1 (F+ )). (2.31) In addition if k ∈ L∞ (Rn × V × V ), then for any ε′ > 0, δ > 0, and any 1 < p < 1 1 + n−1 there exists some nonnegative constant C(ε′ , δ, p) such that Z Z φ(x, v)β(x, v, x′ , v ′ )|v|dxdv V Rv̂+Πv (R) ≤ C(ε′ , δ, p) L∞ (Ox′ ,v′ ) Z Z V ′ |φ(x, v)|p dxdv Rv̂+Πv (R) 8  1′ p + ε′ kφkL∞ (F+ ) , (2.32) for any continuous compactly supported function φ on F+ such that suppφ ⊂ {(x, v) ∈ F+ | |v| < δ −1 }, and where p′ −1 + p−1 = 1. Lemma 2.7 is proved in Section 6. The last inequality shows that the kernel of the second scattering operator R is more regular than is indicated in (2.31). When V is bounded, then we can choose ε′ = 0 in (2.32), in which case we obtain that n |v ′ |−1 β ∈ L∞ (O, Lp (F+ )) for 1 < p < n−1 . This regularity is sufficient (while that described in (2.31) is not) to show that multiple scattering contributions do not interfere with our stability estimates. Taking account of Lemma 2.7, we have the following decomposition for the albedo operator. Lemma 2.8. Under condition (2.28), the following equality in the distributional sense is valid Z Z α(x, v, x′ , v ′ )φ− (x′ − Rv̂ ′ , v ′ )dx′ dv ′ Aφ− (x, v) = ZV Πv′ (R) (2.33) β(x, v, x′ , v ′ )((I + K)−1 Jφ− )(x′ , v ′ )dx′ dv ′ , + O for a.e. (x, v) ∈ F+ and for any C 1 compactly supported function φ− on F− , where α(x, v, x′ , v ′ ) = α1 (x, v, x′ , v ′ ) + α2 (x, v, x′ , v ′ ), −|v|−1 α1 (x, v, x′ , v ′ ) = e α2 (x, v, x′ , v ′ ) = |v|−1 R 2R 0 Z σ(x−sv̂,v)ds 2R e−|v| 0 −1 Rt 0 (2.34) δv (v ′ )δx−(xvˆ′ )vˆ′ (x′ ), σ(x−sv̂,v)ds−|v ′ |−1 R R+(x−tv̂)vˆ′ 0 (2.35) σ(x−tv̂−svˆ′ ,v ′ )ds ×k(x − tv̂, v ′ , v)δx−tv̂−((x−tv̂)vˆ′ )vˆ′ (x′ )dt, (2.36) for a.e. (x, v) ∈ F+ and (x′ , v ′ ) ∈ F , and where β is given by (2.30). Lemma 2.8 is proved in Section 6. The above decomposition is similar to that obtained in [3, 4] except that the multiple scattering contribution is written here in terms of the distribution kernel of R rather than that of R(I + K)−1 J. 3 Stability estimates In this section, we give stability estimates for the reconstruction of the absorption and scattering coefficient from the albedo operator following the approach in [13]. We assume that conditions (2.9) and (2.28) are satisfied and that there exists a convex open subset X of Rn with C 1 boundary ∂X such that X̄ ⊂ B(0, R) := {x ∈ Rn | |x| < R} and the function 0 ≤ σ|X×V is continous and bounded in X × V, the function 0 ≤ k|X×V ×V is continous and bounded in X × V × V, σ(x, v) = k(x, v, v ′ ) = 0 for x 6∈ X̄, (v, v ′ ) ∈ V × V. 9 (3.1) Let (σ̃, k̃) be a pair of absorption and scattering coefficients that also satisfy (2.9), (2.28), and (3.1). We denote by a superscript˜any object (such as the albedo operator à or the distribution kernels α̃i , i = 1, 2) associated to (σ̃, k̃). Let (x′0 , v0′ ) ∈ F such that the intersection of X and the straight line {tv0′ +x′0 | t ∈ R} is not empty. The point (x′0 − Rvˆ0′ , v0′ ) ∈ F− models the incoming condition and is fixed in the analysis that follows. For ε > 0 let fε ∈ C0∞ (F− ) such that kfε kF− = 1, fε ≥ 0 and suppfε ⊂ {(x′ − Rv̂ ′ , v ′ ) ∈ F− | |v ′ − v0′ | + |x′ − x′0 | < ε}. Hence |v ′ |fε is a smooth approximation of the delta function on F− at (x′0 − Rvˆ0′ , v0′ ) as ε → 0+ and is thus an admissible incoming condition in L1 (F− ). The support of fε is represented in Fig. 1. For a.e. (x, v) ∈ F , t ∈ R and v ′ ∈ V let E(x, t, v, v ′ ) ≥ 0 be defined by E(x, t, v, v ′ ) := e−|v| −1 Rt −R σ(x−sv̂,v)ds−|v ′ |−1 R R+(x−tv̂)v̂′ 0 σ(x−tv̂−svˆ′ ,v ′ )ds . (3.2) Replacing σ by σ̃ in (3.2) we also define Ẽ(x, t, v, v ′ ) for a.e. (x, v) ∈ F , t ∈ R and v′ ∈ V . Let δ > 0 and let φ be any compactly supported continuous function on F+ such that kφk∞ ≤ 1 and suppφ ⊆ {(x, v) ∈ F+ | |v| < δ −1 }. (3.3) Then using (2.33) and (3.2) we obtain for ε > 0 that Z Z |v|φ(x, v)(A − Ã)fε (x, v)dxdv = I1 (φ, ε) + I2 (φ, ε) + I3 (φ, ε), V (3.4) Rv̂+Πv (R) where Z Z  −1 RR |v|φ(x + Rv̂, v) e−|v| −R σ(x−sv̂,v)ds V Πv (R)  R −1 R −e−|v| −R σ̃(x−sv̂,v)ds fε (x − Rv̂, v)dxdv Z R Z Z (k(x − tv̂, v ′ , v)E(x, t, v, v ′ ) φ(x + Rv̂, v) I2 (φ, ε) = −R V ×V Πv (R)  −k̃(x − tv̂, v ′ , v)Ẽ(x, t, v, v ′ ) I1 (φ, ε) = fε (x − tv̂ − (x − tv̂)v̂ ′ − Rv̂ ′ , v ′ )dtdxdvdv ′ , I3 (φ, ε) = I31 (φ, ε) − I32 (φ, ε), (3.5) (3.6) (3.7) and where I31 (φ, ε) = Z Z V I32 (φ, ε) = Rv̂+Πv (R) Z Z V |v|φ(x, v) |v|φ(x, v) Rv̂+Πv (R) Z β(x, v, x′ , v ′ )((I + K)−1 Jfε )(x′ , v ′ )dx′ dv ′ dxdv, (3.8) Z ˜ ε )(x′ , v ′ )dx′ dv ′ dxdv. β̃(x, v, x′ , v ′ )((I + K̃)−1 Jf (3.9) O O 10 In addition using the estimate kφk∞ ≤ 1, item ii of Proposition 2.6 and the definition of fε , we obtain Z Z |v|φ(x, v)(A − Ã)fε (x, v)dxdv ≤ k(A − Ã)fε kF+ V Rv̂+Πv (R) ≤ kA − ÃkL(L1 (F− ),L1 (F+ )) kfε kF− = kA − ÃkL(L1 (F− ),L1 (F+ )) . 3.1 (3.10) First stability estimate We now prove a stability estimate under conditions (2.9), (2.28), and (3.1). Taking (3.5)–(3.7) into account, we obtain the following preparatory lemma: Lemma 3.1. Assume that (σ, k) and (σ̃, k̃) satisfy conditions (2.9), (2.28), and (3.1). Then the following limits and estimate hold: I1 (φ, ε) −→+ φ(x′0 + Rvˆ0′ , v0′ ) ε→0   RR RR −|v0′ |−1 −R σ(x′0 −svˆ0′ ,v0′ )ds −|v0′ |−1 −R σ̃(x′0 −svˆ0′ ,v0′ )ds × e −e , I2 (φ, ε) −→+ I21 (φ) + I22 (φ), ε→0 for any compactly supported continuous function φ on F+ , where Z Z R 1 1 I2 (φ) = (k − k̃)(x′0 + t′ vˆ0′ , v0′ , v) ′ |v0 | V −R × [φ(x + Rv̂, v)E(x, t, v, v0′ )] t=t(x′0 ,v0′ ,t′ ,v) dt′ dv, (3.11) (3.12) (3.13) ′ ,t′ ,v) x=x(x′0 ,v0 I22 (φ) Z Z R 1 k̃(x′0 + t′ vˆ0′ , v0′ , v) = ′ |v0 | V −R h i × φ(x + Rv̂, v)(E − Ẽ)(x, t, v, v0′ ) t=t(x′ ,v′ ,t′ ,v) dt′ dv, (3.14) 0 0 ′ ,t′ ,v) x=x(x′0 ,v0 where E and Ẽ are defined by (3.2) and   (t(x′0 , v0′ , t′ , v), x(x′0 , v0′ , t′ , v)) = −(x′0 + t′ vˆ0′ )v̂, x′0 + t′ vˆ0′ − ((x′0 + t′ vˆ0′ )v̂)v̂ , for t′ ∈ R. In addition, for all ε′ > 0, δ > 0 and for all 1 < p < 1 + some nonnegative real valued constant C(ε′ , δ, p) such that sup |I3 (φ, ε)| ≤ C ε>0 C(ε′ , δ, p) Z Z V  1′ p χsuppφ (x + Rv̂, v)dxdv Πv (R) 11 1 n−1 + ε′ (3.15) there exists ! , (3.16) for any compactly supported continuous function φ on F+ , which satisfies kφk∞ ≤ 1 and (3.3) for δ > 0, where p′ −1 + p−1 = 1 and C := 2Rk(I + K)−1 kL(L1 (O,|v|dxdv)) + 2Rk(I + K̃)−1 kL(L1 (O,|v|dxdv)) . (3.17) Lemma 3.1 is proved in Section 5. Taking account of Lemma 3.1 and (3.10), and choosing an appropriate sequence of functions “φ”, we obtain the main result of the paper: Theorem 3.2. Assume that n ≥ 3 and that (σ, k) and (σ̃, k̃) satisfy conditions (2.9), (2.28), and (3.1). Then the following estimates are valid: ′ −1 e−|v0 | RR −R σ(x′0 −svˆ0′ ,v0′ )ds |v0′ |−1 Z Z V R −R ′ −1 − e−|v0 | RR −R σ̃(x′0 −svˆ0′ ,v0′ )ds ≤ kA − ÃkL(L1 (F− ),L1 (F+ )) ; (3.18) (k − k̃)(x′0 + t′ vˆ0′ , v0′ , v) [E(x, t, v, v0′ )] t=t(x′0 ,v0′ ,t′ ,v) dt′ dv ′ ,t′ ,v) x=x(x′0 ,v0 ≤ 2R|v0′ |−1 kσ̃p (x′0 + t′ vˆ0′ , v0′ )kL∞ (Rt′ ) sup (E − Ẽ)(x, t, v, v0′ ) (x,v)∈F t∈R +kA − ÃkL(L1 (F− ),L1 (F+ )) , (3.19) where E and Ẽ are defined by (3.2), and where (t(x′0 , v0′ , t′ , v), x(x′0 , v0′ , t′ , v)) is defined by (3.15) for t′ ∈ R and v ∈ V . Theorem 3.2 is proved in Section 5. Remark 3.3. Under condition (2.26), we can obtain similar estimates to those in Theorem 3.2 for the albedo operator defined on L(F− ) with values in L(F+ ). Note that L(F± ) = L1 (F± ) when V is bounded. 3.2 Stability results under additional regularity assumptions The second inequality in Theorem 3.2 provides an L1 stability result for k(x, v ′ , v) provided that σ(x, v) is known. The first inequality in Theorem 3.2 shows that the Radon transform of σ(x, v) is stably determined by the albedo operator. Because the inverse Radon transform is an unbounded operation, additional constraints, including regularity constraints, on σ are necessary to obtain a stable reconstruction. We assume here that v | v ∈ V = Sn−1 , V0 := sup |v| < ∞, (3.20) |v| v∈V 12 and that the absorption coefficient σ does not depend on the velocity variable, i.e. σ(x, v) = σ(x), x ∈ Rn ; see also remark 3.5 below. Then let  M := (σ(x), k(x, v ′ , v)) ∈ L∞ (Rn ) × L∞ (Rn × V × V ) | (σ, k) satisfies (3.1), n (2.9) and (2.28), and σ|X ∈ H 2 +r̃ (X), kσkH n2 +r̃ (X) ≤ M, kσp k∞ ≤ M , (3.21) for some r̃ > 0 and M > 0. Using Theorem 3.2 for any (x′0 , v0′ ) ∈ F such that the intersection of X and the straight line {tv0′ + x′0 | t ∈ R} is not empty, we obtain the following theorem. Theorem 3.4. Assume that n ≥ 3. Under condition (3.20), for any (σ, k) ∈ M and (σ̃, k̃) ∈ M the following stability estimates are valid: kσ − σ̃kH s (X) ≤ C1 kA − ÃkθL(L1 (F− ),L1 (F+ )) , where − 12 ≤ s < n 2 + r̃, θ = Z Z n+2(r̃−s) , n+1+2r̃ (3.22) and C1 = C1 (R, X, v0 , V0 , M, s, r̃); R k(x′0 + t′ vˆ0′ , v0′ , v) − k̃(x′0 + t′ vˆ0′ , v0′ , v) dt′ dv V −R   1−θ θ ≤ C2 kA − ÃkL(L1 (F− ),L1 (F+ )) 1 + kA − ÃkL(L1 (F− ),L1 (F+ )) , for (x′0 , v0′ ) ∈ F such that x′0 + t′ v0′ ∈ X for some t′ ∈ R, and where θ = 0 < r < r̃, and C2 = C2 (R, X, v0 , V0 , M, r, r̃); in addition, (3.23) 2(r̃−r) , n+1+2r̃ kk − k̃kL1 (Rn ×V ×V )   ≤ C3 kA − ÃkθL(L1 (F− ),L1 (F+ )) 1 + kA − Ãk1−θ 1 1 L(L (F− ),L (F+ )) , where θ = 2(r̃−r) , n+1+2r̃ (3.24) 0 < r < r̃, and C3 = C3 (R, X, v0 , V0 , M, r, r̃). Theorem 3.4 is proved in Section 5. Remark 3.5. Theorem 3.4 can be extended to the case σ = σ(x, |v|) and V = {v ∈ Rn | 0 < λ1 ≤ |v| ≤ λ2 < ∞}. In this case the class M is replaced by the class  N := (σ(x, |v|), k(x, v ′ , v)) ∈ L∞ (Rn × V ) × L∞ (Rn × V × V ) | (σ, k) satisfies (3.1), (2.9) and (2.28), kσp k∞ ≤ M and for any λ ∈ (λ1 , λ2 ), n σ|X (., λ) ∈ H 2 +r̃ (X), sup kσ(., λ)kH n2 +r̃ (X) ≤ M . (3.25) λ∈(λ1 ,λ2 ) Then the left-hand side of (3.22) is replaced by supλ∈(λ1 ,λ2 ) kσ(., λ)kH n2 +r̃ (X) whereas the right-hand side of (3.22) and estimates (3.23)–(3.24) remain unchanged (see the proof of Theorem 3.4). 13 4 Stability in Γ± We now come back to the original geometry in (1.1) and present a similar stability result (Theorem 4.3 below) to Theorem 3.4. The case of a scattering coefficient k(x, v ′ , v) = k(v ′ , v) that does not depend of the space variable x was studied in [13]. We now introduce the notation we need to state our stability result. Recall that X ⊂ Rn , n ≥ 2, is an open bounded subset with C 1 boundary ∂X, and that V is Sn−1 or an open subset of Rn which satisfies v0 := inf v∈V |v| > 0, and that the linear stationary Boltzmann transport equation in X × V takes the form Z v∇x f (x, v) + σ(x, v)f (x, v) − k(x, v ′ , v)f (x, v ′ )dv ′ = 0 in X × V, (4.1) V f|Γ− = f− . We assume that (σ, k) is admissible if 0 ≤ σ ∈ L∞ (X × V ), k(x, v ′ , v) is a measurable function on X × V × V, and 0 ≤ k(x, v ′ ,R.) ∈ L1 (V ) for a.e. (x, v ′ ) ∈ X × V σp (x, v ′ ) = V k(x, v ′ , v)dv belongs to L∞ (X × V ). (4.2) For (x, v) ∈ (X × V ) ∪ Γ∓ , let τ± (x, v) be the real number defined by τ± (x, v) = sup{t > 0 |x ± sv ∈ X for all s ∈ (0, t)}. For (x, v) ∈ X × V , let τ (x, v) be defined by τ (x, v) = τ+ (x, v) + τ− (x, v). For (x, v) ∈ Γ∓ , we put τ (x, v) = τ± (x, v). We consider the measure dξ(x, v) = |n(x)v|dµ(x)dv on Γ± . We still use the notation T0 , T1 , T , A1 , and A2 as in (2.12) and introduce the following Banach space W := {f ∈ L1 (X × V ); T0 f ∈ L1 (X × V ), τ −1 f ∈ L1 (X × V )}, kf kW = kT0 f kL1 (X×V ) + kτ −1 f kL1 (X×V ) . We recall the following trace formula (see Theorem 2.1 of [4]) kf|Γ± kL1 (Γ± ,dξ) ≤ kf kW , for f ∈ W. (4.3) Estimate (4.3) is the analog of the estimate (2.14) in the previous measurement setting. For a continuous function f− on Γ− , we define J f− as the extension of f− in X × V given by : J f− (x, v) = e− R τ− (x,v) 0 σ(x−sv,v)ds f− (x − τ− (x, v)v, v), (x, v) ∈ X × V. (4.4) Note that J has the following trace property (see Proposition 2.1 of [4]): kJ f− kW ≤ Ckf− kL1 (Γ− ,dξ) , (4.5) for f− ∈ L1 (Γ− , dξ), where C = 1 + diam(X)v0−1 kσk∞ and where diam(X) := supx,y∈X |x − y|. Estimate (4.5) is the analog of estimate (2.19) in the previous measurement setting. 14 4.1 Existence theory for the albedo operator We denote by K the bounded operator of L1 (X × V, τ −1 dxdv) defined by Kf (x, v) = − Z τ− (x,v) − e 0 Rt 0 σ(x−sv,v)ds Z k(x, v ′ , v)f (x − tv, v ′ )dv ′ dt, (x, v) ∈ X × V. V for all f ∈ L1 (X ×V, τ −1 dxdv). We transform the stationary linear Boltzmann transport equation (4.1) into the following integral equation (I + K)f = J f− . (4.6) We have the following proposition, which is the analog of Proposition 2.6. Proposition 4.1. Assume that the bounded operator I + K in L1 (X × V, τ −1 dxdv) admits a bounded inverse in L1 (X × V, τ −1 dxdv). (4.7) Then i. the integral equation (4.6) is uniquely solvable for all f− ∈ L1 (Γ− , dξ), and f ∈ W, ii. the operator A : L1 (Γ− , dξ) → L1 (Γ+ , dξ), f− → f|Γ+ , is a bounded operator. This operator is called the albedo operator A for (4.1). The above proposition can be proved by slightly modifying the proofs of Propositions 2.3 and 2.4 of [4]. Remark 4.2. i. Assume that X is also convex. Let f ∈ L1 (F± ) be such that suppf ⊆ {(x, v) ∈ F± | x + tv ∈ X for some t ∈ R}, where F± is defined by (2.5) and R > diam(X). Then we obtain that: Z Z Z f (x ± Rv̂, v)|v|dxdv = f (γ± (x, v), v)dξ(x, v), (4.8) V Πv (R) Γ± where γ± (x, v) = x−(xv̂)v̂±Rv̂ for any (x, v) ∈ Γ± . Therefore, considering results on existence of the albedo operator A obtained in [4] and our assumptions (2.9), equality (4.8) leads us to define the albedo operator A from L1 (F− ) to L1 (F+ ). ii. The condition (4.7) is satisfied under either of the following constraints: kτ σp k∞ < 1, σ − σp ≥ 0. 15 (4.9) (4.10) iii. Assume that the bounded operator I + K in L1 (X × V ) admits a bounded inverse in L1 (X × V ). (4.11) ˜ to L1 (Γ+ , dξ) ˜ where dξ˜ = Then we can define the albedo operator from L1 (Γ− , dξ) min(τ, λ)dξ and where λ is a positive constant. To prove this latter statement, we need trace results for the functions f ∈ W̃ := {f ∈ L1 (X ×V ) | T0 f ∈ L1 (X ×V )}. iv. Under (4.9) and the condition kτ σk < ∞, the existence of the albedo operator A : L1 (Γ− , dξ) → L1 (Γ+ , dξ) is proved in [4] (Proposition 2.3) when V is an open subset of Rn (the condition inf v∈V |v| > 0 is not required). v. Under the condition σ − σp ≥ ν > 0, the existence of the albedo operator A : ˜ → L1 (Γ+ , dξ) ˜ is proved in [4] (Proposition 2.4) when V is an open L1 (Γ− , dξ) subset of Rn (the condition inf v∈V |v| > 0 is not required). Finally under (4.7), we also obtain a decomposition of the albedo operator A similar to that of A given in Lemma 2.8. 4.2 Stability estimates We assume that X is convex and the function 0 ≤ σ is continous and bounded on X × V, the function 0 ≤ k is continous and bounded on X × V × V. (4.12) Let (σ̃, k̃) be a pair of absorption and scattering coefficients that also satisfy (4.12), (4.2) and (4.7). Let à be the albedo operator from L1 (Γ− , dξ) to L1 (Γ+ , dξ) associated to (σ̃, k̃). We can now obtain stability results similar to those in Lemma 3.1 and Theorem 3.2. Consider  n M := (σ(x), k(x, v ′ , v)) ∈ H 2 +r̃ (X) × C(X × V × V ) | (σ, k) satisfies (4.12), (4.7), kσkH n2 +r̃ (X) ≤ M, kσp k∞ ≤ M (4.13) for some r̃ > 0 and M > 0. We obtain the following theorem. Theorem 4.3. Assume n ≥ 3. Under conditions (3.20), for any (σ, k) ∈ M and (σ̃, k̃) ∈ M, the following stability estimates are valid: kσ − σ̃kH s (X) ≤ C1 kA − ÃkθL(L1 (Γ− ,dξ),L1 (Γ+ ,dξ)) , where − 12 ≤ s < n2 + r̃, θ = Z Z τ+ (x′0 ,v0′ ) n+2(r̃−s) , n+1+2r̃ (4.14) and C1 = C1 (X, v0 , V0 , M, s, r̃); k(x′0 + t′ vˆ0′ , v0′ , v) − k̃(x′0 + t′ vˆ0′ , v0′ , v) dt′ dv V 0   1−θ θ ≤ C2 kA − ÃkL(L1 (Γ− ,dξ),L1 (Γ+ ,dξ)) 1 + kA − ÃkL(L1 (Γ− ,dξ),L1 (Γ+ ,dξ)) , 16 (4.15) for (x′0 , v0′ ) ∈ Γ− and where θ = a consequence, we have 2(r̃−r) , n+1+2r̃ 0 < r < r̃, and C2 = C2 (X, v0 , V0 , M, r, r̃). As kk − k̃kL1 (X×V ×V )   1−θ , ≤ C3 kA − ÃkθL(L1 (Γ− ,dξ),L1 (Γ+ ,dξ)) 1 + kA − ÃkL(L 1 (Γ ,dξ),L1 (Γ ,dξ)) − + where θ = 2(r̃−r) , n+1+2r̃ (4.16) 0 < r < r̃, and C3 = C3 (X, v0 , V0 , M, r, r̃). The proof of Theorem 4.3 is similar to that of Theorem 3.4. Remark 4.4. Theorem 4.3 can also be extended to the case σ = σ(x, |v|) and V = {v ∈ Rn | 0 < λ1 ≤ |v| ≤ λ2 < ∞}. In this case the class M is replaced by the class  N := (σ(x, |v|), k(x, v ′ , v)) ∈ C(X × V ) × C(X × V × V ) | (σ, k) satisfies (4.12), (4.7), kσp k∞ ≤ M, and for any λ ∈ (λ1 , λ2 ), n σ|X (., λ) ∈ H 2 +r̃ (X), sup kσ(., λ)kH n2 +r̃ (X) ≤ M . (4.17) λ∈(λ1 ,λ2 ) Then the left-hand side of (4.14) is replaced by supλ∈(λ1 ,λ2 ) kσ(., λ)kH n2 +r̃ (X) whereas the right-hand side of (4.14) and estimates (4.15)–(4.16) remain unchanged. 5 Proof of the stability results We now prove Lemma 3.1 and Theorems 3.1 and 3.2. Proof of Lemma 3.1. Using the fact that X is a convex subset of Rn with C 1 boundary and using (3.1), we obtain that Z t n σ(x − sv̂, v)ds is continous the function R × R × V ∋ (t, x, v) → −R at any point (t̄, x̄, v̄) such that x̄ + ηv̄ ∈ X for some real η. The same statement holds by replacing σ by σ̃. From (3.5), it follows that Z Z Φ1 (x, v)fε (x − Rv̂, v)|v|dxdv, I1 (φ, ε) = V xv=0 |x|<R where Φ1 is the bounded function on F defined for (x, v) ∈ F by   R R −1 R −1 R Φ1 (x, v) = φ(x + Rv̂, v) e−|v| −R σ(x−sv̂,v)ds − e−|v| −R σ̃(x−sv̂,v)ds . 17 (5.1) (5.2) (5.3) From (5.1) and the continuity of φ, it follows that Φ1 is continuous at the point (x′0 , v0′ ) in F . Therefore using (5.2) and the definition of the functions fε , we obtain limε→0+ I1 (φ, ε) = Φ1 (x′0 , v0′ ), which implies (3.11). Performing the change of variables x − tv̂ = x′ + t′ v̂ ′ with x′ v ′ = 0 (“dtdx = dt′ dx′ ”) in formula (3.6) and using (3.1), we obtain ! Z RZ Z Z Φ2,t′ ,v (x′ , v ′ )fε (x′ − Rv̂ ′ , v ′ )|v ′ |dx′ dv ′ dvdt′ , I2 (φ, ε) = (5.4) −R V Πv′ (R) V where Φ2,t′ ,v (x′ , v ′ ) = 0 if x′ + t′ v̂ ′ 6∈ X, (5.5) and 1 Φ2,t′ ,v (x , v ) = ′ |v | ′ ′  k(x′ + t′ v̂ ′ , v ′ , v) [φ(x + Rv̂, v)E(x, t, v, v ′ )] t=t(x′ ,v′ ,t′ ,v) x=x(x′ ,v ′ ,t′ ,v) ! h i − k̃(x′ + t′ v̂ ′ , v ′ , v) φ(x + Rv̂, v)Ẽ(x, t, v, v ′ ) t=t(x′ ,v′ ,t′ ,v) (5.6) x=x(x′ ,v ′ ,t′ ,v) if x′ + t′ v̂ ′ ∈ X, for (t′ , v) ∈ (−R, R) × V , where   (t(x′ , v ′ , t′ , v), x(x′ , v ′ , t′ , v)) := −(x′ + t′ v̂ ′ )v̂, x′ + t′ v̂ ′ − (x′ + t′ v̂ ′ )v̂ , (5.7) for x′ ∈ Rn , v, v ′ ∈ V , t′ ∈ R. Let t′ ∈ (−R, R) such that x′0 + t′ v0′ ∈ X, and let v ∈ V . Then from (5.1), (3.1)–(3.2) and (5.6)–(5.7) it follows that Φ2,t′ ,v is continuous at the point (x′0 , v0′ ). Hence Z Z Φ2,t′ ,v (x′ , v ′ )fε (x′ − Rv̂ ′ , v ′ )|v ′ |dx′ dv ′ → Φ2,t′ ,v (x′0 , v0′ ), as ε → 0+ . (5.8) ′ ′ V x v =0 |x′ |<R Moreover using (5.5)–(5.6) and using the estimate σ ≥ 0 (and (3.2)) and the equality kfε kF− = 1 we obtain Z Z V ′ ′ ′ ′ ′ ′ ′ ′ Φ 2,t′ ,v (x , v )fε (x − Rv̂ , v )|v |dx dv ≤ ′ x′ v =0 |x′ |<R kk + k̃k∞ kφkL∞ (F+ ) χsuppV φ (v) , v0 (5.9) for (t′ , v) ∈ (−R, R) × V , where suppV φ = {v ∈ V | ∃x ∈ Rn , xv̂ = R, φ(x, v) 6= 0}. RR R From (5.8), (5.9), (5.4) it follows that limε→0+ I2 (φ, ε) = −R V Φ2,t′ ,v′ (x′0 , v0′ )dv ′ dt′ , which implies (3.12). It remains to prove (3.16). We first estimate supε>0 |I31 (φ, ε)|. Using (2.28), the estimate kJfε kL1 (O,|v|dxdv) ≤ 2Rkfε kF− and the equality kfε kF− = 1, we obtain k(I + K)−1 Jfε kL1 (O,|v|dxdv) ≤ 2Rk(I + K)−1 kL(L1 (O,|v|dxdv)) , ε > 0. 18 (5.10) 1 Let ε′ > 0 and 1 < p < 1 + n−1 and p−1 + p′ −1 = 1. Using (2.31), (3.8), (2.32) and the estimate kφkL∞ (F+ ) ≤ 1 we obtain Z Z Z −1 ′ ′ 1 φ(x, v)β(x, v, x′ , v ′ )|v|dxdvdx′ dv ′ ((I + K) Jfε )(x , v ) |I3 (φ, ε)| = V O −1 ≤ k(I + K) Jfε kL1 (O,|v|dxdv) V ≤ 1 C1 v0 C(ε′ , δ, p) V 1 β(x, v, x′ , v ′ )dxdv ′| |v Rv̂+Πv (R) !  1′ p + ε′ , χsuppφ (x, v)dxdv Z Z Z Z Rv̂+Πv (R) Rv̂+Πv (R) φ(x, v)|v| L∞ (O) (5.11) where C(ε′ , δ, p) is the constant from (2.32), and C1 = 2Rk(I + K)−1 kL(L1 (O,|v|dxdv)) . (5.12) ˜ σ̃ and β̃ in (5.11)–(5.12), we obtain an estimate for Replacing K, J, σ and β by K̃, J, 2 supε>0 I3 (φ, ε). Combining these estimates with (3.7), we obtain (3.16). Proof of Theorem 3.2. Let ε1 > 0 and let φε1 be any compactly supported continuous function on F+ which satisfies 0 ≤ φε1 ≤ 1 and ε1 , 2 suppφε1 ⊆ {(x, v) ∈ F+ , |x − Rv̂ − x′0 | + |v − v0′ | < ε1 }. φε1 (x + Rv̂, v) = 1 for (x, v) ∈ F, |x − x′0 | + |v − v0′ | < (5.13) (5.14) From (3.11) and (5.13) it follows that ′ −1 lim+ lim+ I1 (ε, φε1 ) = e−|v0 | ε1 →0 ε→0 RR −R σ(x′0 −svˆ0′ ,v0′ )ds ′ −1 − e−|v0 | RR −R σ̃(x′0 −svˆ0′ ,v0′ )ds . (5.15) From (3.16) and (5.14) it follows that lim lim sup |I3 (ε, φε1 )| = 0. ε1 →0+ (5.16) ε→0+ From (3.13), (3.14), it follows that |I21 (φε1 ) + I22 (φε1 )| 2R (kkk∞ + kk̃k∞ ) ≤ v0 Z V χsuppV φε1 (v)dv, (5.17) where suppV φε1 = {v ∈ V | ∃x ∈ Rn , xv̂ = R, φε1 (x, v) 6= 0}. Note that using (5.14), we obtain Z Z χsuppV φε1 (v)dxdv ≤ dv → 0, as ε1 → 0+ . (5.18) V v∈V ′ |<ε |v−v0 1 19 From (5.18) and (5.17) it follows that |I21 (φε1 ) + I22 (φε1 )| → 0, as ε1 → 0+ . (5.19) Note also that from (3.4) and (3.10), it follows that |I1 (φε1 , ε)| ≤ kA − ÃkL(L1 (F− ),L1 (F+ )) + |I21 (φε1 , ε) + I22 (φε1 , ε)| + |I3 (φε1 , ε)|, (5.20) for ε > 0 and ε1 > 0. Combining (5.20) (with “φ”= φε1 ), (5.15), (5.16) and (5.19), we obtain (3.18). This provides us with a stability result for the absorption coefficient. It remains to obtain a stability result for the scattering coefficient. We first construct an appropriate set of functions “φ” (see (5.28) below). The objective is to construct a sequence of such (smooth) functions whose support converges to the line in F+ where single scattering is restricted; see Fig.1. Moreover, we want these functions to be good approximations of the sign of k − k̃ on that support. This is the main new ingredient that allows us to obtain stability for spatially dependent scattering coefficients. More precisely, let U := {(t′ , v) ∈ R × V | x′0 + t′ vˆ0′ ∈ X and (k − k̃)(x′0 + t′ vˆ0′ , v0′ , v) > 0}. Using (3.1), it follows that S U is an open subset of R × V . Let (Km ) a sequence of compact sets such that m∈N Km = U and Km ⊆ Km+1 for m ∈ N. For m ∈ N let χm ∈ C ∞ (R × V, R) such that χKm ≤ χm ≤ χU , and let ρm = 2χm − 1. (5.21) Thus using (3.1) we obtain lim (k − k̃)(x′0 + t′ vˆ0′ , v0′ , v)ρm (t′ , v) = |k − k̃|(x′0 + t′ vˆ0′ , v0′ , v), m→+∞ (5.22) for v ∈ V and t′ ∈ R such that x′0 + t′ v0′ ∈ X ∪ (Rn \X̄). For (x, v) ∈ F+ such that v and v0′ are linearly independent, we define v̂ − (vˆ0′ v̂)vˆ0′ v̂ − (vˆ0′ v̂)vˆ0′ )q . d(x, v) := x − x′0 − ((x − x′0 )v̂)v̂ − ((x − x′0 ) q 1 − (vˆ0′ v̂)2 1 − (vˆ0′ v̂)2 (5.23) For (x, v) ∈ F+ such that v and v0′ are linearly independent, we verify that d(x, v) = inf t,t′ ∈R |x + tv̂ − (x′0 + t′ vˆ0′ )| and the infimum is reached at (t, t′ ) =  (x′ − x)(v̂ − (v̂ vˆ′ )vˆ′ ) (x − x′ )(vˆ′ − (v̂ vˆ′ )v̂)  0 0 0 0 0 0 . , ′ 2 ′ 2 ˆ ˆ 1 − (v̂ v ) 1 − (v̂ v ) 0 (5.24) 0 Consider Vδ,l := {(x, v) ∈ F+ | |x−Rv̂| < R−δ, |v̂ − 1 1 v̂v0′ ′ v | > δ, |v| < , d(x, v) < }, ′2 0 δ l v0 20 (5.25) 1 1 v̂v ′ Ṽδ,l := {(x, v) ∈ F+ | |x − Rv̂| ≤ R − δ − , |v̂ − ′ 20 v0′ | ≥ δ + , l l v0 1 |v| ≤ δ −1 − l−1 , d(x, v) ≤ }, 2l (5.26) for 0 < δ < min(R, v0−1 ) and l ∈ N, l > (R − δ)−1 + δ. For 0 < δ < min(R, v0−1 ) and l ∈ N, l > (R − δ)−1 + δ, let χδ,l ∈ C0∞ (F+ ) be such that χṼδ,l ≤ χδ,l ≤ χVδ,l . (5.27) Finally, for 0 < δ < min(R, v0−1 ) and m, l ∈ N, l > (R − δ)−1 + δ, let φδ,m,l ∈ C0∞ (F+ ) be defined by (5.28) φδ,m,l (x, v) := χδ,l (x, v)ρm (t′ , v) (x−x′0 )(vˆ0′ −(v̂vˆ0′ )v̂) . t′ = ′ )2 1−(v̂ vˆ0 (See (5.22), (5.24) and (5.37)–(5.38) given below.) Note that from (5.28) and (5.27) it follows that suppφδ,m,l ⊆ Vδ,l . (5.29) Using (3.11), (5.29) and (5.25), it follows that lim I1 (φδ,m,l , ε) = 0, ε→0+ (5.30) for 0 < δ < min(R, v0−1 ) and m, l ∈ N, l > (R − δ)−1 + δ (we used that (x′0 + Rv̂0′ , v0′ ) 6∈ Vδ,l ). Note that using (5.25) we obtain χVδ,l (x, v) ≤ χG (x, v) ≤ 1, l ∈ N, l > (R − δ)−1 + δ, lim χVδ,l (x, v) = χVδ (x, v), l→∞ (5.31) (5.32) for (x, v) ∈ F+ and 0 < δ < min(R, v0−1 ), where G is the compact subset of F+ given by v̂v ′ G = {(x, v) ∈ F+ | |x − Rv̂| ≤ R − δ, |v̂ − v′ 20 v0′ | ≥ δ, |v| ≤ δ −1 } and 0  v̂v ′ Vδ := (x, v) ∈ F+ | |x − Rv̂| < R − δ, |v̂ − ′ 20 v0′ | > δ, |v| < δ −1 , d(x, v) = 0 . (5.33) v0 Note also that, as n ≥ 3, we obtain Z Z χVδ (x + Rv̂, v)dxdv = 0, 0 < δ < min(R, v0−1 ). V (5.34) Πv (R) From (3.16), (5.29), (5.31)–(5.34), it follows that lim sup lim sup |I3 (φδ,m,l , ε)| ≤ ε′ for any ε′ > 0. l→∞ ε→0+ 21 (5.35) Hence lim lim sup |I3 (φδ,m,l , ε)| = 0. l→∞ (5.36) ε→0+ Let 0 < δ < min(R, v0−1 ) and m ∈ N. Using (5.28), (5.24) and (3.13), we obtain Z Z R 1 fδ,m,l (t′ , v)dt′ dv, (5.37) I2 (φδ,m,l ) = V −R for l > (R − δ)−1 + δ where fδ,m,l (t′ , v) := 1 (k − k̃)(x′0 + t′ vˆ0′ , v0′ , v)ρm (t′ , v) |v0′ | × [χδ,l (x + Rv̂, v)E(x, t, v, v0′ )] t=t(x′0 ,v0′ ,t′ ,v) , (5.38) ′ ,t′ ,v) x=x(x′0 ,v0 for t′ ∈ R and v ∈ V such that v and v0′ are linearly independent, where (t(x′0 , v0′ , t′ , v), x(x′0 , v0′ , t′ , v)) is defined by (3.15) for v ∈ V and t′ ∈ R. Using the estimates σ ≥ 0, |v|−1 ≤ v0−1 , 0 ≤ χδ,l ≤ 1, we obtain |fδ,m,l (t′ , v)| ≤ 1 (k + k̃)(x′0 + t′ vˆ0′ , v0′ , v)), v0 (5.39) for l > (R − δ)−1 + δ, t′ ∈ R and v ∈ V such that v and v0′ are linearly independent. Using (2.9) and (3.1), we obtain that the function arising on the right-hand side of (5.39) is integrable on V × (−R, R). In addition from (5.25)–(5.27), (5.32) and (5.33), it follows that fδ,m,l (t′ , v) → fδ,m (t′ , v) as l → +∞, (5.40) for t′ ∈ R and v ∈ V such that v and v0′ are linearly independent, where fδ,m (t′ , v) := 1 (k − k̃)(x′0 + t′ vˆ0′ , v0′ , v)ρm (t′ , v) |v0′ | × [χVδ (x + Rv̂, v)E(x, t, v, v0′ )] t=t(x′0 ,v0′ ,t′ ,v) , (5.41) ′ ,t′ ,v) x=x(x′0 ,v0 for t′ ∈ R and v ∈ V such that v and v0′ are linearly independent. Therefore we obtain by the Lebesgue dominated convergence theorem that Z Z R 1 fδ,m (t′ , v)dt′ dv. (5.42) lim I2 (φδ,m,l ) = l→+∞ V −R Let 0 < δ < min(R, v0−1 ). We also have fδ,m (t′ , v) ≤ v0−1 (k + k̃)(x′0 + t′ vˆ0′ , v0′ , v)), for m ∈ N. From (5.42), (5.41) and (5.22), it follows that Z Z R 1 1 |k − k̃|(x′0 + t′ vˆ0′ , v0′ , v) (5.43) lim lim I2 (φδ,m,l ) = ′ m→+∞ l→+∞ |v | V −R 0 [χVδ (x + Rv̂, v)E(x, t, v, v0′ )] t=−(x′0 +t′0 vˆ0′ )v̂ dt′ dv. ′ x−tv̂=x′0 +t′ vˆ0 22 From (5.43), we deduce lim+ lim δ→0 lim m→+∞ l→+∞ I21 (φδ,m,l ) = R 1 |k − k̃|(x′0 + t′ vˆ0′ , v0′ , v) ′ |v | −R 0 ×E(x, v, t, v0′ ) t=−(x′0 +t′ vˆ0′ )v̂ dt′ dv. Z Z V (5.44) ′ −((x′ +t′ vˆ′ )v̂)v̂ x=x′0 +t′ vˆ0 0 0 From (3.14), it follows that |I22 (φδ,m,l )| ≤ 2R kσ̃p (x′0 + t′ vˆ0′ , v0′ )kL∞ (Rt′ ) sup (E − Ẽ)(x, t, v, v0′ ) , v0 (x,v)∈F (5.45) t∈R for 0 < δ < min(R, v0−1 ), m ∈ N and l ∈ N, l > (R − δ)−1 + δ. Using (3.4) and (3.10), we obtain |I2 (φδ,m,l , ε)| ≤ kA − ÃkL(L1 (F− ),L1 (F+ )) + |I1 (φδ,m,l , ε)| + |I3 (φδ,m,l , ε)|, (5.46) for 0 < δ < min(R, v0−1 ), m ∈ N and l ∈ N, l > (R − δ)−1 + δ. From (5.46), (3.11) and (3.12), it follows that |I21 (φδ,m,l )| ≤ kA − ÃkL(L1 (F− ),L1 (F+ )) + |I22 (φδ,m,l )| + | lim+ I1 (φδ,m,l , ε)| ε→0 + lim sup |I3 (φδ,m,l , ε)|, (5.47) ε→0+ for 0 < δ < min(R, v0−1 ), m ∈ N and l ∈ N, l > (R − δ)−1 + δ. Estimate (3.19) follows from (5.47), (5.30), (5.36), (3.12), (5.44) and (5.45). Proof of Theorem 3.4. The method we use to prove (3.22) is the same as in [13]. Let (σ, k), (σ̃, k̃) ∈ M. LetRf = σ − σ̃ and consider P f the X-ray transform of f = σ − σ̃ +∞ defined by P f (x, θ) := −∞ f (tθ + x)dt for a.e. (x, θ) ∈ T Sn−1 . n From (3.1) and f|X ∈ H 2 +r̃ (X), it follows that kf kH − 21 (X) ≤ D1 (n, X)kP f k∗ , where kP f k∗ := Z Sn−1 Z |P f (x, θ)|2 dxdθ Πθ (5.48)  21 and D1 (n, X) is a real constant which does not depend on f and Πθ := {x ∈ Rn | xθ = 0} for θ ∈ Sn−1 . Using (3.1) (and (σ, k), (σ̃, k̃) ∈ M), it follows that P f (x, θ) = 0 for (x, θ) ∈ T Sn−1 and |x| ≥ R. Therefore using also (5.48) we obtain kf kH − 12 (X) ≤ D2 (n, X)kP f kL∞ (T Sn−1 ) , 23 (5.49) where D2 (n, X) is a real constant which does not depend on σ. We also use the following interpolation inequality: 2s+1 n+2r̃ n+1+2r̃ kf kH s (X) ≤ kf k n+1+2r̃ , n +r̃ kf k −1 2 H for − 21 ≤ s ≤ n 2 H 2 (5.50) + r̃. As σ ∈ M, it follows that kσk∞ ≤ D3 (n, r̃)kσkH n2 +r̃ ≤ D3 (n, r̃)M. (5.51) Therefore, Z R −R σ(x′0 − svˆ0′ )ds ≤ 2RD3 (n, r̃)M, (5.52) for a.e. (x′0 , vˆ0′ ) ∈ T Sn−1 . From (5.52) it follows that ′ −1 e−|v0 | RR −1 ≥ e−2v0 −R σ(x′0 −svˆ0′ ,v0′ )ds RD3 (n,r̃)M V0 ′ −1 − e−|v0 | RR −R σ̃(x′0 −svˆ0′ ,v0′ )ds |P (σ − σ̃)(x′0 , vˆ0′ )|, (5.53) for a.e. (x′0 , v0′ ) ∈ Rn × V , x′0 v0′ = 0 (we used the equality et1 − et2 = ec (t2 − t1 ) for t1 < t2 ∈ R and for some c ∈ [t1 , t2 ] which depends on t1 and t2 ). (In fact, the estimate (5.53) is valid for any (x′0 , v0′ ) ∈ Rn × V , x′0 v0′ = 0, such that {x′0 + tv0′ | t ∈ R} ∩ X 6= ∅ or {x′0 + tv0′ | t ∈ R} ∩ X̄ = ∅.) Combining (5.53), (5.49), and (3.18), we obtain −1 e−2v0 RD3 (n,r̃)M kσ − σ̃kH − 21 (X) ≤ kA − ÃkL(L1 (F− ),L1 (F+ )) . D2 (n, X)V0 (5.54) Combining (5.54) and (5.50), we obtain (3.22). We now prove (3.23). Using |v0′ |−1 ≥ V0−1 for v ∈ V , (3.2), and (5.51), we obtain that Z Z R 1 k(x′0 + t′ vˆ0′ , v0′ , v) − k̃(x′0 + t′ vˆ0′ , v0′ , v) E(x, v, t, v0′ ) t=t(x′0 ,v0′ ,t′ ,v) dt′ dv ′ |v0 | V −R ′ ,t′ ,v) x=x(x′0 ,v0 −1 ≥ e−4v0 RD3 (n,r̃)M V0 Z Z V R −R |k − k̃|(x′0 + t′ vˆ0′ , v0′ , v)|dt′ dv, (5.55) for any (x′0 , v0′ ) ∈ Rn × V , x′0 v0′ = 0, such that x′0 + sv0′ ∈ X for some s ∈ R, and where (t(x′0 , v0′ , t′ , v), x(x′0 , v0′ , t′ , v)) is defined by (3.15) for v ∈ V and t′ ∈ (−R, R). 24 As (σ̃, k̃) ∈ M we have kσ̃p k∞ ≤ M . Using the latter estimate, (3.2), and |v|−1 ≤ v0−1 for all v ∈ V , we obtain −1 kσ̃p (x′0 + t′ vˆ0′ , v0′ )kL∞ (Rt′ ) sup |E − Ẽ|(x, v, t, v0′ ) ≤ M e4v0 RD3 (n,r̃)M (x,v)∈F t∈R × sup (x,v)∈F t∈R " 1 |v| ≤ 4Rv0−1 M e t 1 |σ − σ̃|(x − sv̂, v)ds + ′ |v0 | −R Z 4v0−1 RD3 (n,r̃)M Z 0 R+(x−tv̂)vˆ0′ |σ − σ̃|(x − tv̂ − svˆ0′ , v0′ )ds kσ − σ̃k∞ , # (5.56) for any (x′0 , v0′ ) ∈ Rn ×V , x′0 v0′ = 0, such that x′0 +t′ v0′ ∈ X for some t′ ∈ R. (We also used R R+(x−tv̂)vˆ0′ Rt σ̃(x− |eu −eũ | ≤ emax(|u|,|ũ|) |u− ũ| where u = −|v|−1 −R σ(x−sv̂, v)ds−|v0′ |−1 0 ′ ′ ˆ tv̂ − sv0 , v0 )ds and ũ denotes the real number obtained by replacing σ by σ̃ on the righthand side of the latter equality which defines u; using (5.51) (for σ and for σ̃) we obtain max(|u|, |ũ|) ≤ 4Rv0−1 D3 (n, r̃)M .) Note that kσ − σ̃k∞ ≤ D3 (n, r)kσ − σ̃kH n2 +r for 0 < r < r̃ (see (5.51)). Therefore, combining (5.55), (5.56), (3.19) and (3.22), we obtain (3.23). 2(r̃−r) . From (3.23) it follows Let us finally prove (3.24). Let 0 < r < r̃ and let θ = n+1+2r̃ that Z RZ Z k(x′0 + t′ vˆ0′ , v0′ , v) − k̃(x′0 + t′ vˆ0′ , v0′ , v) dvdt′ dx′0 x′ v ′ =0 0 0 |x′0 |<R −R ≤ D4 kA − where D4 = C2 R ′ =0 x′0 v0 |x′0 |<R V ÃkθL(L1 (F− ),L1 (F+ ))  1 + kA − Ãk1−θ L(L1 (F− ),L1 (F+ ))  , (5.57) dx′0 and C2 is the constant that appears on the right-hand side of (3.23). From (3.1), (5.57) and the change of variables “x = x′0 + t′ v0′ ”, it follows that Z k(x, v0′ , v) − k̃(x, v0′ , v) dvdx n R ×V   1−θ θ (5.58) ≤ D4 kA − ÃkL(L1 (F− ),L1 (F+ )) 1 + kA − ÃkL(L1 (F− ),L1 (F+ )) . Integrating on v0′ ∈ V both sides of (5.58), we obtain (3.24). 6 Decomposition of the albedo operator We now prove Lemmas 2.7 and 2.8. 25 Proof of Lemma 2.7. Let ψ− ∈ L1 (O, |v|dxdv). Using the definition of K, we obtain
(6.1) (Rψ− )(x, v) = K 2 ψ− |F+ (x, v) Z Z 2R Z R+(x−tv̂)v̂1 1 1 = k(x − tv̂, v1 , v) V ×V |v| |v1 | 0 0 ×k(x − tv̂ − t1 vˆ1 , v ′ , v1 )E0 (x, v, x − tv̂, v1 , x − tv̂ − t1 v̂1 ) ×ψ− (x − tv̂ − t1 vˆ1 , v ′ )dt1 dtdv ′ dv1 , for a.e. (x, v) ∈ F+ , where E0 (x, v, x − tv̂, v1 , x − tv̂ − t1 v̂1 ) = e−|v| −1 Rt 0 σ(x−sv̂,v)ds−|v1 |−1 R t1 0 σ(x−tv̂−svˆ1 ,v1 )ds , (6.2) for x ∈ Rn , t, t1 ∈ R and v, v1 ∈ V . We recall that R is a bounded operator from L1 (O, |v|dxdv) to L1 (F+ ), i.e. kRψkF+ ≤ Ck|v|ψkO , for any ψ ∈ L1 (O, |v|dxdv). (6.3) Hence we obtain, in particular, that the integral in t, t1 , v ′ and v1 , on the right-hand side of (6.1) is absolutely convergent for a.e. (x, v) ∈ F+ . Let us assume first that V = Sn−1 . Performing the changes of variables “x′ = x − tv − t1 v1 ” (“dx′ = tn−1 dt1 dv1 ”), we obtain 1 Z (Rψ− )(x, v) = β(x, v, x′ , v ′ )ψ− (x′ , v ′ )dx′ dv ′ , (6.4) O where β(x, v, x′ , v ′ ) := (6.5)  Z 2R  k(x − tv, v1 , v)k(x′ , v ′ , v1 ) dt, E0 (x, v, x − tv, v1 , x − tv − t1 v1 ) t1 =|x−tv−x′ | |x − tv − x′ |n−1 0 x−tv−x′ v1 = ′ t1 ′ for a.e. (x, v) ∈ F+ , (x , v ) ∈ O, where E0 is defined by (6.2). Now assume that V is an open subset of Rn , which satisfies v0 = inf v∈V |v| > 0. From (6.1), it follows that
(Rψ− )(x, v) = K 2 ψ− |F+ (x, v) Z +∞ Z Z 2R Z R+(x−tv̂)ω n−2 −1 k(x − tv̂, rω, v) = r χV (rω) |v| V ×Sn−1 v0 0 0 ′ ×k(x − tv̂ − t1 ω, v , rω)E0 (x, v, x − tv̂, rω, x − tv̂ − t1 ω) ×ψ− (x − tv̂ − t1 ω, v ′ )dt1 dtdrdv ′ dω, 26 (6.6) for (x, v) ∈ F+ , where for r > 0 and ω ∈ Sn−1 . Performing the changes of variables “x′ = x − tv̂ − t1 ω” (“dx′ = tn−1 dt1 dω”), we obtain 1 Z β(x, v, x′ , v ′ )ψ− (x′ , v ′ )dx′ dv ′ , (6.7) (Rψ− )(x, v) = O where Z +∞ Z 2R  1 k(x − tv̂, rω, v)k(x′ , v ′ , rω) n−2 β(x, v, x , v ) = r χV (rω) |v| v0 |x − x′ − tv̂|n−1 0 ×E0 (x, v, x − tv̂, rω, x − tv̂ − t1 ω)] t1 =|x−x′ −tv̂| dtdr, ′ ′ t1 ω=x−x′ −tv̂ (6.8) for a.e. (x, v) ∈ F+ , (x′ , v ′ ) ∈ O, where E0 is defined by (6.2). From (6.3), (6.4)–(6.5), and (6.7)–(6.8), it follows that for a.e. (x′ , v ′ ) ∈ O, β(x, v, x′ , v ′ ) ∈ L1 (F+ ). Moreover from (6.3), it follows that the function O ∋ (x′ , v ′ ) → β(x, v, x′ , v ′ )ψ(x′ , v ′ ) ∈ L1 (F+ ) belongs to L1 (O, |v|dxdv) for any ψ ∈ L1 (O, |v ′ |dx′ dv ′ ). Therefore |v ′ |−1 β ∈ L∞ (O, L1 (F+ )). (6.9) 1 , Now we prove (2.32). Assume k ∈ L∞ (Rn × V × V ) and let 1 < p < 1 + n−1 ′ −1 −1 p + p = 1, be fixed for the rest of the proof of Lemma 2.7. We use (6.10). Using Hölder estimate, the change of variables “y = x − tv̂” (dy = dxdt) and the spherical coordinates, we obtain p Z 2R Z Z 1 dt dxdv |x − x′ − tv̂|n−1 Vδ Rv̂+Πv (R) 0 ≤ (2R) p p′ Z Z Vδ = (2R) p p′ Z Z Vδ Rv̂+Πv (R) y∈Rn (y,v)∈O Z 2R 0 dtdxdv |x − x′ − tv̂|p(n−1) n−(n−1)p p dydv n−1 (4R) ′ p ≤ (2R) Vol(Vδ )Vol(S ) , |y − x′ |p(n−1) n − (n − 1)p (6.10) for x′ ∈ Rn , |x′ | < 2R, where Vδ := {v ∈ V | |v| < δ −1 }. (6.11) Assume first that V = Sn−1 and let φ be a continuous function on F+ . Then using 27 (6.5), σ ≥ 0, Hölder estimate and (6.10) (with δ = 21 ), we obtain Z Z V ′ ′ φ(x, v)β(x, v, x , v )dxdv ≤ Rv̂+Πv (R) ≤ kkk2∞ Z Z V Z Z V kkk2∞ ′ |φ(x, v)|p dxdv Rv̂+Πv (R)  1′ |φ(x, v)| Rv̂+Πv (R) Z 0 2R dtdxdv |x − tv − x′ |n−1 p p  p1 1 dt dxdv × |x − tv − x′ |n−1 V Rv̂+Πv (R) 0  1′ Z Z p p′ |φ(x, v)| dxdv ≤C , Z Z Z 2R (6.12) Rv̂+Πv (R) V   p1 1 2 n−(n−1)p , which proves (2.32) for V = Sn−1 . where C = (2R) p′ kkk2∞ Vol(Sn−1 ) p (4R) n−(n−1)p Now assume that V is an open subset of Rn which satisfies inf v∈V |v| > 0. Let ε′ > 0 an δ > 0 be positive real numbers. Let φ be a compactly supported and continuous function on F+ such that suppφ ⊆ {(x, v) ∈ F+ | |v| < δ −1 }. We use the following lemma, whose proof is postponed to the end of this section. Lemma 6.1. The nonnegative measurable function β1 defined for a.e. (x, v, x′ , v ′ ) ∈ F+ × O by Z +∞ Z 2R  1 k(x − tv̂, rω, v)k(x′ , v ′ , rω) n−1 ′ ′ β1 (x, v, x , v ) = r χV (rω) |v| v0 |x − x′ − tv̂|n−1 0 (6.13) ×E0 (x, v, x − tv̂, rω, x − tv̂ − t1 ω)] t1 =|x−x′ −tv̂| dtdr, t1 ω=x−x′ −tv̂ belongs to L∞ (O, L1 (F+ )), where E0 is defined by (6.2). Let Mε′ > v0 be defined by Mε′ = v0 + ε ′ −1 Z Z V β1 (x, v, x′ , v ′ )|v|dxdv Rv̂+Πv (R) . From (6.8), it follows that Z Z φ(x, v)|v|β(x, v, x′ , v ′ )dxdv = I1 (x′ , v ′ ) + I2 (x′ , v ′ ), V Rv̂+Πv (R) 28 (6.14) L∞ (O) (6.15) for a.e. (x′ , v ′ ) ∈ O and where Z Z ′ ′ I1 (x , v ) = (6.16) φ(x, v) Vδ Rv̂+Πv (R) Z Mε′ Z 2R n−2  k(x − tv̂, rω, v)k(x′ , v ′ , rω) χV (rω) r |x − x′ − tv̂|n−1 0 v0 ×E0 (x, v, x − tv̂, rω, x − tv̂ − t1 ω)]t1 ω=x−x′ −tv̂ dtdrdxdv, Z Z φ(x, v) I2 (x′ , v ′ ) = (6.17) Vδ Rv̂+Πv (R) Z +∞ Z 2R n−2  k(x − tv̂, rω, v)k(x′ , v ′ , rω) r χV (rω) |x − x′ − tv̂|n−1 Mε′ 0 ×E0 (x, v, x − tv̂, rω, x − tv̂ − t1 ω)]t1 ω=x−x′ −tv̂ dtdrdxdv. n−1 for v ∈ V and r ≥ Mε′ , and Using (6.17) and the estimates rn−2 = rn−1 r−1 ≤ Mε−1 ′ r using (6.14), we obtain Z Z −1 ′ ′ β1 (x, v, x′′ , v ′′ )|v|dxdv |I2 (x , v )| ≤ kφkL∞ (F+ ) Mε′ v∈V Rv̂+Πv (R) ′ ′ L∞ (O) ′ ≤ ε kφkL∞ (F+ ) , for a.e. (x , v ) ∈ F+ . From (6.16) and Hölder estimate, it follows that Z Z Mε′ Z Z n−2 ′ ′ 2 r dr |φ(x, v)| |I1 (x , v )| ≤ kkk∞ Vδ ≤ Rv̂+Πv (R) M n−1 ′ kkk2∞ ε × n−1 Z Z Vδ v0 Z Z Vδ Rv̂+Πv (R) p′ Z 2R 0 |φ(x, v)| dxdv Rv̂+Πv (R) (6.18) 2R 1 dtdxdv. |x − − tv̂|n−1 0 p  p1 1 dt dxdv |x − x′ − tv̂|n−1  1′ x′ p (6.19) , for a.e. (x′ , v ′ ) ∈ O. Combining (6.15), (6.18)–(6.19) and (6.10), we obtain (2.32) with ′ C(ε , δ, p) = 1 M n−1 ′ kkk2∞ ε (2R) p′ n−1 n−1 Vol(Vδ )Vol(S
1 ) p  (4R)n−(n−1)p n − (n − 1)p  p1 . Proof of Lemma 2.8. Let φ− ∈ C01 (F− ) (which denotes the spaces of C 1 compactly supported functions on F− ). Let φ := (I + K)−1 Jφ− . Then note that φ := Jφ− − KJφ− + K 2 (I + K)−1 Jφ− . 29 (6.20) Thus Aφ− = φ|F+ := (Jφ− )|F+ − (KJφ− )|F+ + R(I + K)−1 Jφ− . From (2.18) and (2.35), it follows that Z Z α1 (x, v, x′ , v ′ )φ− (x′ − Rv̂ ′ , v ′ )dx′ dv ′ , (x, v) ∈ F+ . (Jφ− )|F+ (x, v) = ′ ′ x v =0 |x′ |<R V (6.21) (6.22) From the definitions of K and J, we obtain Z Z α2 (x, v, x′ , v ′ )φ− (x′ −Rv̂ ′ , v ′ )dx′ dv ′ , (x, v) ∈ F+ . (6.23) − (KJφ− )|F+ (x, v) = ′ ′ V x v =0 |x′ |<R Lemma 2.8 follows from (6.21)–(6.23) and Lemma 2.7. Proof of Lemma 6.1. Using (6.13) and the estimate σ ≥ 0 and using the change of variables “y = x − tv̂” (dy = dtdx) and (2.9), and spherical coordinates, we obtain Z Z |v|β1 (x, v, x′ , v ′ )dxdv v∈V Rv̂+Πv (R) ≤ Z  Z2R Z+∞ k(x − tv̂, rω, v)k(x′ , v ′ , rω) n−1 χV (rω) r dtdrdxdv, ′ |x − x′ − tv̂|n−1 ω= x−tv̂−x′ Z = Z V = Z y∈Rn |y|<R Z Z Sn−1 R 1 ′ |x − y|n−1 Z+∞ rn−1 [χV (rω)k(y, rω, v)k(x′ , v ′ , rω)]ω= y−x′ drdydv |y−x′ | v0 Z+∞ rn−1 χV (rω)σp (x′ + r′ ω, rω)k(x′ , v ′ , rω)drdr′ dω χ|y|<R (x + r ω) ′ 0 |x−tv̂−x | 0 V Rv̂+Πv (R) v0 ′ v0 ≤ kσp k∞ R Z Z+∞ rn−1 χV (rω)k(x′ , v ′ , rω)drdω = Rkσp k∞ σp (x′ , v ′ ) ≤ Rkσp k2∞ , Sn−1 v0 for a.e. (x′ , v ′ ) ∈ O. The lemma is proved. 30 7 Proof of existence of the albedo operator In this section, we prove Lemma 2.4 and Propositions 2.5 and 2.6. Proof of Lemma 2.4. Using the definition of T1 −1 , the estimate σ ≥ 0 and (2.11), we have Z Z Z R Z R+w −1 k|v|T1 f kO ≤ |f (y + (w − t)v̂, v)|dtdwdydv V Πv (R) Z Z = V Πv (R) −R R Z −R 0 Z w |f (y + tv̂, v)|dtdwdydv ≤ 2Rkf kO , −R for f ∈ L1 (O). Using the definition of A2 and (2.11), we have Z Z Z R −1 k(y + tv̂, v ′ , v)|v ′ |−1 |f (y + tv̂, v ′ )|dtdydvdv ′ kA2 |v| f kO ≤ = V ×V Πv (R) Z Z Z Πv′ (R) ′ −1 V −R R σp (y ′ + t′ v̂ ′ , v ′ )|v ′ |−1 |f (y ′ + t′ v̂ ′ , v ′ )|dt′ dy ′ dv ′ −R ′ ≤ k|v | σp (x , v ′ )kL∞ (O) kf kO , for f ∈ L1 (O). We also used (2.9) and the change of variables Z Z +∞ Z Z +∞ f (y ′ + t′ v̂ ′ )dt′ dy ′ , f (y + tv̂)dtdy = yv=0 y ′ v ′ =0 −∞ (7.1) −∞ for f ∈ L1 (Rn ) and v, v ′ ∈ V . Using the definition of T1 −1 and A2 and Lemma 2.1, (2.23) and (7.1), we obtain kA2 T1 −1 f kO ≤ Z 1 |v ′ | Z 1 |v ′ | V ×V = V ×V Z Z R Z Z R Πv (R) Πv′ (R) ′ k(y + wv̂, v , v) Z e 0 −R R+(y+wv̂)vˆ′ R −|v ′ |−1 0t σp (y+wv̂−svˆ′ ,v ′ )ds ×|f (y + wv̂ − tv̂ ′ , v ′ )|dtdwdydvdv ′ ′ k(y + w′ v̂ ′ , v ′ , v) Z 0 −R ′ R+w′ ′ −1 e−|v | Rt 0 σp (y ′ +(w′ −s)vˆ′ ,v ′ )ds ×|f (y + (w′ − t)v̂ ′ , v ′ )|dtdw′ dy ′ dvdv ′  Z R Z R Z Z d −|v′ |−1 R w′ σp (y′ +svˆ′ ,v′ )ds ′ t = (− ′ e dw |f (y ′ + tv̂ ′ , v ′ )|dtdy ′ dv ′ dw V Πv′ (R) −R t 31 = Z  O −|v ′ |−1 1−e R R−xvˆ′ 0 σp (x+sv ′ ,v ′ )ds  −1 |f (x′ , v ′ )|dx′ dv ′ ≤ (1 − e−2Rv0 kσp k∞ )kf kO , for f ∈ L1 (O). Item iii follows from items i and ii (under (2.24), we also use that kA2 T1 −1 k ≤ kA2 |v|−1 kk|v|T1 −1 k). Proof of Proposition 2.5. We first prove item i. Assume (2.27). For all f ∈ D(T), Tf = (T1 + A2 )f = (I + A2 T1 −1 )T1 f. (7.2) From (2.27) it follows that T admits a bounded inverse in L1 (O) given by T−1 := T1 −1 (I + A2 T1 −1 )−1 . Using the latter equality, we obtain (I + K)(I − T−1 A2 )=I + T1 −1 A2 − T1 −1 (I + A2 T1 −1 )−1 A2 −T1 −1 (I + A2 T1 −1 − I)(I + A2 T1 −1 )−1 A2 = I. (7.3) The proof that (I − T−1 A2 )(I + K) = I is similar. We now prove that (2.26) implies (2.27). For f ∈ D(T), Tf = (T1 + A2 )f = T1 (I + T1 −1 A2 )f = T1 (I + K)f. (7.4) Let us prove (I + K)(D(T)) = D(T). From the latter equality and (7.4) it follows that T admits a bounded inverse in L1 (O) given by T−1 = (I + K)−1 T1 −1 . (7.5) As K = T1 −1 A2 , we have (I + K)(D(T)) ⊆ D(T). Let g ∈ D(T), and let f = (I + K)−1 T1 −1 g ∈ L1 (O). Then f = −KT1 −1 g + g = −T1 −1 A2 T1 −1 g + g ∈ D(T) (we recall that g ∈ D(T)). Equality (7.2) still holds. Using (7.2), (7.5) and the fact that T1 : D(T) → L1 (O) is one-to-one and onto L1 (O), we obtain (2.27). Item i is thus proved. Item ii follows from item iii of Lemma 2.4 and item i. We shall prove item iii. Note that (see (7.3)) (I + K)(I − T−1 A2 ) = I = (I − T−1 A2 )(I + K) in L(L1 (O)). (7.6) Note also that L1 (O, |v|dxdv) ⊆ L1 (O) and recall that K is a bounded operator in L1 (O, |v|dxdv). Therefore, we only have to prove that T−1 A2 defines a bounded operator in L1 (O, |v|dxdv). Note that T−1 = T1 −1 (I + A2 T1 −1 )−1 . (7.7) From item i, (7.7) and item i of Lemma 2.4, it follows that T−1 A2 defines a bounded operator in L1 (O, |v|dxdv). Thus item iii is proved. 32 Proof of Proposition 2.6. Let f− ∈ L1 (F− ). From (2.19), it follows that Jf− ∈ W. Hence Jf− ∈ L1 (O, |v|dxdv) and from (2.28) it follows that (2.25) is uniquely solvable in L1 (O, |v|dxdv) and its solution is given by (I + K)−1 Jf− which satisfies k(I + K)−1 Jf− kL1 (O,|v|dxdv) ≤ C0 kf− kF− , (7.8) where C0 = 2R(1 + v0−1 kσk∞ )k(I + K)−1 kL(L1 (O,|v|dxdv)) . Let f := (I + K)−1 Jf− . Hence by definition f = Jf− − Kf in L1 (O, |v|dxdv). (7.9) Using (7.9), we check that the following equality is valid in the sense of distributions: T0 f = −A1 f − A2 f. (7.10) Using (7.10), we obtain T0 f ∈ L1 (O) and kT0 f k ≤ (kσ|v|−1 k∞ + k|v|−1 σp k∞ )kf kL1 (O,|v|dxdv) . (7.11) Therefore f ∈ W (item i is thus proved), and using (2.14) and (7.11) we obtain kf|F+ kF+ ≤ max((2R)−1 , 1)(k|v|−1 σk∞ + k|v|−1 σp k∞ + 1)kf kL1 (O,|v|−1 dxdv) . (7.12) Item ii follow from (7.8) and (7.12). Acknowledgments This work was funded in part by the National Science Foundation under Grants DMS0239097 and DMS-0554097. References [1] G. Bal, Inverse problems for homogeneous transport equations. Part II: Multidimensional case, Inverse Problems, 16 (2000), pp. 1013–1028. [2] M. Choulli and P. Stefanov, Inverse scattering and inverse boundary value problems for the linear Boltzmann equation, Comm. Partial Diff. Equ., 21 (1996), pp. 763–785. 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