Momentum ray transforms

The momentum ray transform $I^k$ integrates a rank $m$ symmetric tensor field $f$ over lines with the weight $t^k$: $ (I^k\!f)(x,\xi)=\int_{-\infty}^\infty t^k\langle f(x+t\xi),\xi^m\rangle\,dt. $ In particular, the ray transform $I=I^0$ was studied by several authors since it had many tomographic applications. We present an algorithm for recovering $f$ from the data $(I^0\!f,I^1\!f,\dots, I^m\!f)$. In the cases of $m=1$ and $m=2$, we derive the Reshetnyak formula that expresses $\|f\|_{H^s_t({\mathbb{R}}^n)}$ through some norm of $(I^0\!f,I^1\!f,\dots, I^m\!f)$. The $H^{s}_{t}$-norm is a modification of the Sobolev norm weighted differently at high and low frequencies. Using the Reshetnyak formula, we obtain a stability estimate.


Introduction
The ray transform integrates symmetric tensor fields over straight lines. Let ·, · be the standard dot-product on R n and | · |, the corresponding norm. The family of oriented straight lines in R n is parameterized by points of the manifold T S n−1 = {(x, ξ) ∈ R n × R n | |ξ| = 1, x, ξ = 0} ⊂ R n × R n that is the tangent bundle of the unit sphere S n−1 . Namely, a point (x, ξ) ∈ T S n−1 determines the line {x + tξ | t ∈ R}. Along with the space C ∞ (T S n−1 ) of smooth functions, we use the Schwartz space S(T S n−1 ). Observe that the space S(E) is well defined for a smooth vector bundle E → M over a compact manifold M.
Let S m R n be the complex vector space of rank m symmetric tensors on R n . The dimension of S m R n is n+m−1 m . In particular, S 0 R n = C and S 1 R n = C n . Let S(R n ; S m R n ) be the Schwartz space of S m R n -valued functions that are called rank m smooth fast decaying symmetric tensor fields on R n . The ray transform is the linear bounded operator I : S(R n ; S m R n ) → S(T S n−1 ) (1.1) vectors and tensors. For instance, ξ i = ξ i in (1.2). There is no difference between covariant and contravariant tensors since we use Cartesian coordinates only. Being initially defined by (1.2) on smooth fast decaying tensor fields, the operator (1.1) then extends to some wider spaces of tensor fields. The study of the ray transform is motivated by several applications. In the case of m = 0 (when f is a function), the ray transform is the main mathematical tool of Computer Tomography. In the case of m = 1 (when f is a vector field), the operator I is called the Doppler transform and serves as the main mathematical tool of Doppler Tomography. In the cases of m = 2 and of m = 4, the operator I and some its relatives are applied to various problems of tomography of anisotropic media, see [3,Chapters 6,7] and [2,5].
The operator I has a big null-space in the case of m > 0. A symmetric tensor field can be uniquely decomposed into its solenoidal and potential parts [3, Theorem 2.6.2], and the potential part lies in the null-space. Given If , one can recover only the solenoidal part of f , and there is a reconstruction formula [3,Theorem 2.12.2]. This naturally leads to the question of what additional information should be added to the data If for the unique recovery of the entire tensor field f . One possibility is to consider the momentum ray transforms I k : S(R n ; S m R n ) → S(T S n−1 ) that are defined for k = 0, 1, . . . as follows: ( In particular, I 0 = I. A rank m symmetric tensor field f is uniquely determined by the functions (I 0 f, I 1 f, . . . , I m f ), see [3,Theorem 2.17.2] and [1]. The momentum ray transforms are primary objects of study of this paper, and we have three goals: (1) To obtain an algorithm for recovering a rank m symmetric tensor field f from the data (I 0 f, I 1 f, . . . , I m f ).
(2) To derive a version of the Reshetnyak formula [4] that expresses the norm f H s t through some norms of the functions (I 0 f, I 1 f, . . . I m f ). The H s t -norm is a modification of the Sobolev norm weighted differently at high and low frequencies, see Section 2 for the precise definition.
(3) To obtain stability estimates in terms of H s t -norms. The first goal is achieved for arbitrary m in any dimension n ≥ 2. The Reshetnyak formula and stability estimate are obtained in the cases of m = 1 and m = 2 only. We believe our approach works for any m, but the bulkiness of the Reshetnyak formula grows very fast with m.
The paper is organized as follows. In Section 2, we discuss basic properties of momentum ray transforms and state a few preliminaries. Section 3 presents the inversion algorithm. Section 4 is devoted to the Reshetnyak formula and stability estimates. Finally, in Section 5, we restrict ourselves to the 2-dimensional case and propose an alternate approach based on the fact that there are natural coordinates on T S 1 = S 1 × R.

Preliminaries
2.1. Basic properties of momentum ray transforms. First of all we observe that the right-hand side of (1.3) makes sense for all (x, ξ) ∈ R n × (R n \ {0}). We define the continuous linear operators The data (I 0 f, I 1 f, . . . , I m f ) and (J 0 f, J 1 f, . . . , J m f ) are equivalent as we will demonstrate right now. Therefore the operators (2.1) are also called momentum ray transforms. The function J k f is sometimes more convenient than I k f because the partial derivatives ∂(J k f ) On the other hand, the function (I k f )(x, ξ) obeys good decay conditions in the second argument.
For a tensor field f ∈ S(R n ; S m R n ), the function (J k f )(x, ξ) possesses the following homogeneity in the second argument and has the following property in the first argument On the other hand, Formulas (2.4) and (2.6) mean in particular that the operator I k must always be considered together with lower order momenta (I 0 , . . . , I k−1 ), i.e., the data (I 0 f, . . . , I k f ) must always be used instead of I k f .
There are two important first order differential operators on symmetric tensor fields: the inner derivative operator d : C ∞ (R n ; S m R n ) → C ∞ (R n ; S m+1 R n ) and the divergence operator δ : C ∞ (R n ; S m R n ) → C ∞ (R n ; S m−1 R n ), see the definitions in [3,4]. Operators I k are related to the inner derivative by the formula which is valid at least for f ∈ S(R n ; S m R n ). This easily follows from (2.2) with the help of integration by parts. In particular, I k (d ℓ f ) = 0 for k < ℓ.
Let us also mention the transformation law for operators I k under a change of the origin in R n . Given a ∈ R n , set f a (x) = f (x + a). As easily follows from (2.5) and (2.6), We are going to derive a formula that expresses f H s t through (I 0 f, I 1 f, . . . I m f ). Since f H s t = f a H s t , the expression must be invariant under the transformation (2.7).

2.2.
Momentum ray transforms and the Fourier transform. We use the Fourier transform F : S(R n ) → S(R n ), f → f in the following form (hereafter i is the imaginary unit and y is the Fourier dual variable of x): The Fourier transform F : S(R n ; S m R n ) → S(R n ; S m R n ), f → f on symmetric tensor fields is defined componentwise, i.e., f i 1 ...im = f i 1 ...im (we use Cartesian coordinates only). Introduce also the Fourier transform F : S(T S n−1 ) → S(T S n−1 ), ϕ → ϕ on T S n−1 by where dx is the (n − 1)-dimensional Lebesgue measure on the hyperplane ξ ⊥ = {x ∈ R n | ξ, x = 0}. It is the standard Fourier transform in the (n − 1)-dimensional variable x while ξ ∈ S n−1 is considered as a parameter. The Fourier transform of the momentum ray transform is given by the formula for (y, ξ) ∈ T S n−1 . Indeed, On assuming (y, ξ) ∈ T S n−1 , change the integration variables as z = x + tξ On using the equality we obtain This coincides with (2.9).
2.3. The Reshetnyak formula for scalar functions. Recall [4] that the Hilbert space H s t (R n ) is defined for s ∈ R and t > −n/2 as the completion of S(R n ) with respect to the norm (2.10) Similarly, the Hilbert space H s t (T S n−1 ) is defined for s ∈ R and t > −(n − 1)/2 as the completion of S(T S n−1 ) with respect to the norm where dξ is the volume form on the sphere S n−1 induced by the Euclidean metric of R n .
The following statement is a particular case of m = 0 in [4,Theorem 4.2]. Given a function f ∈ S(R n ), the Reshetnyak formula holds for every s ∈ R and every t > −n/2, where a n = Γ (n − 1)/2 2π (n−1)/2 . (2.13) (Unfortunately, there is a misprint in the formula for the coefficient a k = a k (m, n) in Theorem 4.2 of [4]. The right formula is presented in [3, Formula (2.15.3)], the coefficients are independent of (s, t).) Formula (2.12) will be used throughout the paper.

Inversion algorithm for momentum ray transforms
Let us recall the definition of the partial symmetrization where the summation is performed over the set Π r of all permutations of the set {1, . . . r}.
The following statement is the main ingredient of our algorithm for recovering a rank m symmetric tensor field f from the data (I 0 f, . . . , I m f ).
hold for all indices (i 1 , . . . , i m ), where the left-hand side is the result of applying the ray transform J = J 0 to the coordinate f i 1 ...im considered as a scalar function on R n (we use Cartesian coordinates only).
Proof. Equality (3.1) trivially holds for m = 0. We proceed by induction on m. Assume (3.1) to be valid for tensor fields f ∈ S(R n ; S m R n ) with some m. Now assume f ∈ S(R n ; S m+1 R n ). Definition (2.2) can be written as Differentiate this equality with respect to ξ i m+1 Let us fix a value of the index i m+1 and define the tensor fieldf ∈ S(R n ; S m R n ) bỹ Then (3.2) can be written as By the induction hypothesis,

Substitute value (3.3) into the last formula
We have replaced the symmetrization σ(i 1 . . . i m ) by the stronger operator σ(i 1 . . . i m+1 ) because the left-hand side is symmetric in the indices (i 1 , . . . , i m+1 ). In the second sum on the right-hand side, we change the summation index as k = k ′ − 1. After the change, we again use the notation k instead of k ′ . In such the way, we transform the formula to the form We assume binomial coefficients m k = m! k!(m−k)! to be defined for all integers m and k under the agreement: m k = 0 if either m < 0 or k < 0 or k > m. Therefore both summations can be extended to the limits 0 ≤ k ≤ m+1. Besides this, we can write indices (i 1 , . . . , i m+1 ) in an arbitrary order on the right-hand side because of the presence of the symmetrization σ(i 1 . . . i m+1 ). With the help of the Pascal relation m k + m k−1 = m+1 k , the formula takes the form This finishes the induction step.
Let us recall the inversion formula for recovering a scalar function f ∈ S(R n ) from If : The stability of the recovery procedure for a scalar function is completely described by the Reshetnyak formula (2.12). For higher rank tensor fields, the stability question is more delicate because of the presence of m th order derivatives in formula (3.1). We will investigate the stability question in the next section.

Reshetnyak formula for momentum ray transforms
In this section, we derive the Reshetnyak formula and stability estimate for m = 1 and for m = 2.
4.1. Operators X i and Ξ i . In the cases of m = 1 and of m = 2, formula (3.1) takes the forms and respectively. We are going to rewrite these formulas in intrinsic terms of the manifold The notationsX i andΞ i are chosen because the derivatives ∂ ∂x i and ∂ ∂ξ i are in some sense leading terms on the right-hand sides of (4.3).
are tangent to T S n−1 . Let X i and Ξ i be the restrictions of vector fieldsX i andΞ i to the manifold T S n−1 respectively. Thus, X i and Ξ i are smooth vector fields on T S n−1 and can be considered as first order differential operators

The operators satisfy
Proof. Definition (4.3) implies Right-hand sides of these equalities vanish on T S n−1 . This proves the first statement. From definition (4.3), Right-hand sides of these equalities vanish on T S n−1 . This proves (4.7).
Recall the well known formula: for vector fields X, Y and for functions f, g. On using this formula, one easily derives from definition (4.3) On the other hand, Comparing (4.8) and (4.9), we see that [X i ,Ξ j ] = ξ iXj . This proves (4.6). Formulas (4.4)-(4.5) are proved in a similar way. Finally we prove that, at a point (x, ξ) ∈ T S n−1 , the vectors X i (x, ξ), Ξ i (x, ξ) (1 ≤ i ≤ n) generate the tangent space T (x,ξ) (T S n−1 ). To this end we have to demonstrate that any linear dependence between these vectors is actually a corollary of (4.7). Assume that with some coefficients α i , β i . Substitute values (4.3) into this equality This can be written in the form where δ i j is the Kronecker tensor. Since vectors ∂ ∂x p , ∂ ∂ξ p (p = 1, . . . , n) are linearly independent, the last equation implies At a point (x, ξ) ∈ T S n−1 , the rank of the matrix (δ p i − ξ i ξ p ) of system (4.12) is equal to n − 1 and any solution to the system is of the form β i = β 0 ξ i . System (4.11) takes now the form (δ p i − ξ i ξ p )α i = 0 (p = 1, . . . , n). As before, this implies α i = α 0 ξ i . Thus, equation (4.10) is actually of the form This means that any linear dependence between vectors is actually a corollary of (4.7).
We can now present the intrinsic forms of equations (4.1) and (4.2).

Theorem 4.2.
For a vector field f = (f i ) ∈ S(R n ; C n ), the equality holds on T S n−1 for every i, where the left-hand side is the result of applying the ray transform I = I 0 to the coordinates f i considered as scalar functions on R n .
For a tensor field f = (f ij ) ∈ S(R n ; S 2 R n ), the equality holds on T S n−1 for all indices (i, j), where δ ij is the Kronecker tensor and the left-hand side is the result of applying the ray transform I = I 0 to the coordinates f ij considered as scalar functions on R n .
We present the proof of Theorem 4.2. Theorem 4.3 is proved in the same way although all involved calculations are more cumbersome.
Proof of Theorem 4.2. By the very definition of X i and Ξ i , the equalities hold on T S n−1 for an arbitrary function ϕ ∈ C ∞ (T S n−1 ), where ψ is an arbitrary smooth extension of ϕ to some neighborhood of T S n−1 in R n ×(R n \{0}). For every k, the function J k f is an extension of I k f . Therefore

Substitute values (4.3) to obtain
Let us specify (4.15) for k = 0. As is seen from (2.3), the function (J 0 f )(x, ξ) is positively homogeneous of zero degree in the second argument. By the Euler equation for homogeneous functions, Besides this, J 0 f satisfies (see (2.4)) Differentiating this identity with respect to t and then setting t = 0, we obtain In virtue of (4.16)-(4.17), equalities (4.15) for k = 0 are simplified to the following ones: Let us specify (4.15) for k = 1. As is seen from (2.3), the function (J 1 f )(x, ξ) is positively homogeneous of degree −1 in the second argument. By the Euler equation for homogeneous functions, Besides this, J 1 f satisfies (see (2.4)) Differentiating this identity with respect to t and then setting t = 0, we obtain In virtue of (4.19)-(4.20), equalities (4.15) for k = 1 are simplified to the following ones: Inserting value (4.18) of ∂(J 0 f ) ∂ξ i and value (4.21) of ∂(J 1 f ) ∂x i into (4.1), we arrive to (4.13).
Recall that y i is the Fourier dual variable of x i . The formulas for commuting the Fourier transform with operators X i and Ξ i are described by the following hold for every function ϕ ∈ S(T S n−1 ) and for every i, 1 ≤ i ≤ n.
By (4.3), With the help of (4.23), this is simplified to Applying the Fourier transform to this equality, we obtain From this with the help of integration by parts This proves the first of formulas (4.22). Differentiate equality (4.24) with respect to y i From this In virtue of (4.26), this is simplified to The differentiation of equality (4.24) with respect to ξ i is not very easy because the integration hyperplane ξ ⊥ depends on ξ. We first transform integral (4.24) into an integral over a hyperplane independent of ξ. Fix a vector ξ 0 ∈ S n−1 . For an arbitrary vector ξ ∈ S n−1 sufficiently close to ξ 0 , the orthogonal projection is one-to-one. We change the integration variable in (4.24) according to (4.28). The Jacobian of the change is ξ 0 , ξ −1 . After the change, formula (4.24) takes the form We can now differentiate this equality with respect to ξ i Substituting the values On assuming (y, ξ 0 ) ∈ T S n−1 , we set ξ = ξ 0 in the latter formula. The formula simplifies to the following one: By (4.23), the integrand of the last integral is identically equal to zero. Thus, replacing the notations ξ 0 and x ′ with ξ and x respectively, we obtain With the help of (2.8), we obtain from (4.29) for (x, ξ) ∈ T S n−1 . Therefore formula (4.31) can be written as This proves the second of formulas (4.22).
By the definition of X i , X iφ = ∂φ ∂y i − ξ i ξ p ∂φ ∂y p . In virtue of (4.26), this is simplified to With the help of (4.25), this gives This proves the last of formulas (4.22).
Theorem 4.5. Given a vector field f ∈ S(R n ; C n ), the equality holds for every s ∈ R and every t > −n/2, where the constant a n is defined by (2.13) and Z is the first order pseudodifferential operator Proof. Apply the Fourier transform to formula (4.13)

With the help of Lemma 4.4, this gives
For a vector field f = (f i ), Applying the Reshetnyak formula (2.12) for scalar functions, we obtain On using the definition (2.11) of the H s+1/2 t+1/2 (T S n−1 )-norm, this is written as Now, we calculate the integrand on (4.35). By (4.34), After opening parentheses and using the equalities |ξ| 2 = 1, ξ, y = 0, this gives By (4.7), ξ i Ξ i = 0, and the formula takes the form On using the operator (4.33), this can be written as Substituting the value (4.36) for On using the statement Ξ i I 0 f = Ξ i (I 0 f ) of Lemma 4.4, we rewrite the latter formula as In view of (2.11), this is equivalent to (4.32).
Formula (4.32) suggests the idea of introducing the norm on the space S(T S n−1 ) ⊕ S(T S n−1 ). Let us demonstrate that the right-hand side of (4.37) is positive for (ϕ, ψ) = (0, 0). Indeed, the inequality easily follows from the definition of H s t -norms with the help of the Schwartz inequality. In particular, easily follows from definition (4.33) of the operator Z. Together with the previous inequality, it gives (4.39) On using the last inequality, we derive from (4.37) . The right-hand side of this inequality is non-negative and equals zero only if ϕ = 0. In the latter case the right-hand side of (4.37) is non-negative and equals zero only if ψ = 0.
We define the Hilbert space H 1,s 4.3. Reshetnyak formula for second rank symmetric tensor fields. Along with the operator Z defined by (4.33), we need two similar operators Q, W : S(T S n−1 ) → C ∞ (T S n−1 ) that are defined by the formulas (y i y j Ξ i Ξ jφ )(y, ξ) = |y| 2 Qϕ(y, ξ) and (y i X iφ )(y, ξ) = |y| W ϕ(y, ξ) respectively. We will sometimes write Ξ i instead of Ξ i in order to adopt our formulas to the Einstein summation rule.
Theorem 4.8. Given a tensor field f ∈ S(R n ; S 2 R n ), the equality (4.43) holds for every s ∈ R and every t > −n/2, where the constant a n is defined by (2.13).
As before, the abbreviated notation · H s ′ t ′ for the norm · H s ′ t ′ (T S n−1 ) is used on the right-hand side of (4.43) Sketch of the proof. The proof follows the same line as the proof of Theorem 4.5 although all calculations are more cumbersome. We will present key formulas only.
The Reshetnyak formula (2.12) for scalar functions implies (4.44) Applying the Fourier transform to equation (4.14) and using Lemma 4.4, we obtain the following analog of formula (4.34):
We define the Hilbert space H 2,s t (T S n−1 ) as the completion of S(T S n−1 ) ⊕ S(T S n−1 ) ⊕ S(T S n−1 ) with respect to the norm (4.47).
takes the following form in the two-dimensional case: