On mixed and transverse ray transforms on orientable surfaces

If, the geodesic X-ray transform of a tensor field f of order m. See Section 2.4 and equations (2.1) and (2.3).

I S ⁢ M ⁢ h , the geodesic ray transform of a function h : S ⁢ M → ℝ . See Section 2.4 and equation (2.2).

I A , r ⁢ f , the (abstract) mixing ray transform with a mixing A of degree m, operating on a tensor field f of order m. See Section 3.2 and equation (3.6).

L k , l ⁢ f = I A k , l ⁢ f , the mixed ray transform of a tensor field f of order k + l on a two-dimensional orientable Riemannian manifold. See Section 2.5 and equations (2.7) and (2.8).

I ⊥ ⁢ f , the transverse ray transform of a tensor field f of order k, corresponding to the mixed ray transform with l = 0 . See Section 2.5 and equation (2.7).

I A , r q ⁢ [ f ] = I A , r ⁢ f , the quotient transform of an equivalence class of tensor field f of degree m. See Section 3.2.

ℒ β ⁢ f , the light ray transform of a (compactly supported) tensor field of order m. See Section 3.3.3 and equation (3.9).

ℒ β q ⁢ [ f ] = ℒ β ⁢ f , the quotient light ray transform of an equivalence class of a (compactly supported) tensor field f of degree m. See Section 3.3.3.

A, a mixing composed of automorphisms of the tangent bundle. See Section 3.2 and equation (3.5).

A i , automorphisms (fiberwise linear bijections) of the tangent bundle. See the beginning of Section 3.2.

λ and λ x , operators converting m-tensor field and m-tensor into a function on the tangent bundle and tangent space. See Section 3.1 and equation (3.3).

λ r = r ∘ λ and λ r , x = r x ∘ λ x , where r and r x are the restriction operators on the tangent bundle and tangent space. See Section 3.1.

A k , l , the mixing corresponding to the mixed ray transform L k , l . See Section 2.5 and equation 2.7.

σ, the usual symmetrization operator of tensor fields. See Section 3.1 and equation (3.1).

σ ^ A , r , the projection operator onto A - 1 ( Ker ( λ r ) ⊥ ) , related to the mixing ray transform I A , r . See Sections 3.1 and 3.2, and equations (3.4) and (3.7).

ℋ = Id - σ ^ A , r , an operator projecting m-tensor field onto Ker ⁡ ( λ r ∘ A ) . See Sections 3.2 and 3.3, and Theorem 3.3.

𝒟 = A - 1 ∘ A ~ , an auxiliary operator related to two admissible mixings A and A ~ of degree m. See Section 3.2 and Theorem 3.3.

∇ A = A - 1 ∘ ∇ , the weighted covariant derivative of an m-tensor field, where A is an admissible mixing of degree m. See Section 3.3.1.

N k , l , the normal operator of the mixed ray transform L k , l on compact simple surfaces. See Section 4.2 and Lemma 4.4.

ℱ ⁢ ( X ) , the set of all functions X → ℂ .

M or ( M , g ) , a connected (pseudo-)Riemannian manifold of dimension n ≥ 2 .

SM, the sphere bundle whose fibers are unit spheres of the tangent spaces. See Section 2.4.

𝔛 ⁢ ( T m ⁢ M ) , the space of all covariant m-tensor fields. See Section 2.1.

S m ⁢ M , the space of symmetric m-tensor fields. See Sections 2.1 and 3.1.

C q ⁢ ( T m ⁢ M ) and C q ⁢ ( S m ⁢ M ) , the set of C q -smooth (symmetric) m-tensor fields, where q ∈ ℕ . See Section 2.1.

H k ⁢ ( T m ⁢ M ) and H k ⁢ ( S m ⁢ M ) , the L 2 -Sobolev space of (symmetric) m-tensor fields, where k ∈ ℕ . See Section 2.2.

P η ⁢ ( T m ⁢ M ) and P η 1 ⁢ ( T m ⁢ M ) , the spaces of polynomially decaying m-tensor fields on Cartan–Hadamard manifolds. See Section 2.4 and equation (2.4).

E η ⁢ ( T m ⁢ M ) and E η 1 ⁢ ( T m ⁢ M ) , the spaces of exponentially decaying m-tensor fields on Cartan–Hadamard manifolds. See Section 2.4 and equation (2.4).

[ f ] and [ f ] A , the equivalence class of the tensor field f, under the relation f ∼ h if and only if

See Sections 3.1, 3.2, 3.3.1 and 3.3.3.