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ON RADON TRANSFORMS BETWEEN LINES AND HYPERPLANES 6 1 BORIS RUBIN AND YINGZHAN WANG* 0 2 p Abstract. WeobtainnewinversionformulasfortheRadontrans- e formanditsdualbetweenlinesandhyperplanesinRn. TheRadon S transforminthissettingisnon-injectiveandtheconsiderationisre- 2 strictedtotheso-calledquasi-radialfunctionsthatareconstanton 2 symmetric clusters oflines. For the correspondingdualtransform, which is injective, explicit inversionformulas are obtained both in ] A the symmetric case and in full generality. The main tools are the Funk transform on the sphere, the Radon-John d-plane transform F . in Rn, the Grassmannian modification of the Kelvin transform, h and the Erdélyi-Koberfractional integrals. t a m [ 3 1. Introduction v 6 Let L and H be the manifolds of all non-oriented lines ℓ and all 2 non-oriented hyperplanes h in Rn, respectively. In the present article 8 3 we consider the Radon-like transform that takes functions on L to 0 functions on H . We also consider the corresponding dual transform . 1 acting in the opposite direction. Both transforms are defined by the 0 integrals 6 1 : (Rf)(h) = f(ℓ)d ℓ, (R∗ϕ)(ℓ) = ϕ(h)d h, (1.1) v h ℓ Z Z i X ℓ⊂h h⊃ℓ r where the integration is performed with respect to the corresponding a canonical measures. Our aim is to obtain explicit inversion formulas for R and R∗ in the cases when these operators are injective. The manifolds L and H are important representatives of the more general vector bundles G(n,k) over the Grassmann manifolds G of n,k k-dimensional linear subspaces of Rn. Elements of G(n,k) are non- oriented k-dimensional affine planes in Rn. The corresponding Radon 2010 Mathematics Subject Classification. Primary 44A12; Secondary 47G10. Key words and phrases. Radon transforms, Grassmann manifolds, Funk transform, Erdélyi-Koberoperators. * Supported by the National Natural Science Foundation of China under Grant #11271091and the China Scholarship Council under Grant #201308440083. 1 2 BORIS RUBINANDYINGZHANWANG* transforms Rk,k′, Rk∗,k′ that take functions on G(n,k) to functions on G(n,k′), 1 k < k′ n 1, and backwards, were considered by ≤ ≤ − Gonzalez and Kakehi [5, 6] who studied these operators on smooth functions in the group-theoretic terms. The paper [5] contains an explicit inversion formula for Rk,k′ in the case of k′ k even. This formula − was obtained by applying the Fourier transform over fibers and using the corresponding inversion formula for compact Grassmannians due to Kakehi [10]. We also mention the paper [4] by Gonzalez that contains the range description of the plane-to-line transform for smooth functions on R3. InversionformulasofdifferentkindforbothRk,k′ andRk∗,k′ inLebesgue spaceswere obtainedbythefirst-namedco-author[11]who reduced the problem to the compact case with the aid of a certain analogue of the stereographic projection. The dimensions k and k′ in [11] can be of arbitrary parity and the corresponding inversion formula for the compact Grassmannians was borrowed from Grinberg and Rubin [7]. One should also mention the work by Strichartz [18], who developed harmonic analysis on Grassmannian bundles, and a series of publica- tions related to integral geometric problems on compact Grassmanni- ans; see, e.g., [1, 2, 7, 10, 13, 17, 19, 20] and references therein. The boundedness of the operators Rk,k′ and Rk∗,k′ in Lp spaces with power weights was studied in [15]. It is a challenging open problem to find an alternative approach to inversion formulas for Rk,k′ and Rk∗,k′ that would be straightforward (without stereographic projection), available for all admissible k and k′, and applicable to large classes of functions (not only to C∞ and rapidly decreasing ones). Here one should take into account that for 1 ≤ k < k′ ≤ n−1, theoperatorsRk,k′ andRk∗,k′ areinjectiveifandonly ifk+k′ n 1 andk+k′ n 1, respectively; see[11, Propositions1.3 ≤ − ≥ − and 1.4] for precise statements. Because dimG(n,k) = (n k)(k+1), − the condition k+k′ n 1 is equivalent to dimG(n,k) dimG(n,k′). ≤ − ≤ In the present article we give a solution to the aforementioned inversion problem in the case 1 = k < k′ = n 1. This simple case is − less technical and reflects basic features of Radon transforms on Grass- mannian bundles. The paper is organized as follows. Section 2 contains necessary preliminaries. It is worth noting that planes in Rn can be parametrized in different ways andour mainconcern in this section is to choose suitable parametrizations. Section 3 is devoted to the operator R R . Because 1,n−1 ≡ dimG(n,1) = 2n 2 > n = dimG(n,n 1), − − ON RADON TRANSFORMS 3 this operator is non-injective. The situation changes if we restrict R to functions f satisfying additional symmetry. Specifically, we assume f to be constant on symmetric clusters of parallel lines in every direction. We call such functions quasi-radial. It will be shown that if f is quasi- radial, then Rf is a tensor product of the classical Funk transform on the sphere [3, 9, 16] and a one-dimensional fractional integration operator of the Erdélyi-Kober type [16]. Both can be explicitly inverted. The main result of this section is presented by Theorem 3.2. In Section 4 we obtain explicit inversion formulas for the dual transform R∗ R∗ . Here we consider three different approaches. The first one d≡eals1,wn−it1h even locally integrable functions ϕ on H such that ϕ(h) = ϕ( h) for all h H , and relies on averaging over the line − ∈ clusters. The second approach covers the general case and invokes a Grassmannian modification of the Kelvin transform introduced in [11]. Hereϕ is assumed to becontinuous with prescribed behavior at theori- gin and at infinity or belonging to some weighted Lp-space. The third approach is based on a certain alternative parametrization of line and hyperplanes. Main results of this section are presented by Theorems 4.3, 4.11, 4.13, and 4.17. The methods of the present article can be generalized to the Radon transforms Rk,k′ and Rk∗,k′ for arbitrary 0 < k < k′ < n. We plan to consider this case in the forthcoming publication. Acknowledgements. Themainpartoftheworkwasdonewhenthe second-named author was visiting Louisiana State University in 2014- 2015. Both authors are grateful to the China Scholarship Council for the support and the administration of the Department of Mathematics at Louisiana State University for the hospitality. 2. Preliminaries 2.1. Notation. We recall that G(n,k) is the affine Grassmann manifold of non-oriented k-dimensional planes in Rn, 0 < k < n; G is n,k the compact Grassmann manifold of k-dimensional linear subspaces of Rn. If ξ G , then ξ⊥ denotes the orthogonal complement of ξ n,k ∈ in Rn. Each element τ of G(n,k) is parameterized by the pair (ξ,u), where ξ G and u ξ⊥. In the following τ denotes the Euclidean n,k ∈ ∈ | | distance from the origin to τ τ(ξ,u) so that τ = u is the Eu- ≡ | | | | clidean norm of u. We write C (G(n,k)) for the space of all continuous 0 functions f on G(n,k) satisfying lim f(τ) = 0. We also set |τ|→∞ C (G(n,k)) = f C(G(n,k)) : f(τ) = O( τ −µ) , µ { ∈ | | } C (Rn) = f C(Rn) : f(x) = O( x −µ), µ { ∈ | | 4 BORIS RUBINANDYINGZHANWANG* where C(G(n,k)) and C(Rn) are the space of continuous functions on G(n,k) and Rn, respectively. The manifold G(n,k) will be endowed with the product measure dτ = dξdu, where dξ is the O(n)-invariant probability measure on G n,k and du is the usual volume element on ξ⊥. For k′ > k and η Gn,k′, we denote by Gk(η) the Grassmann ∈ manifold of all k-dimensional linear subspaces of η. In the following, Sn−1 = x Rn : x = 1 is the unit sphere in Rn. For θ Sn−1, θ { ∈ | | } ∈ { } denotes the one-dimensional linear subspace spanned by θ, dθ stands forthesurfaceelement onSn−1, andσ = 2πn/2 Γ(n/2)is thesurface n−1 area of Sn−1. We set d θ = dθ/σ for the norma(cid:14)lized surface element ∗ n−1 on Sn−1. We write e ,...,e for the coordinate unit vectors in Rn. Given 1 n 1 k < k′ < n, the following notations are used for the coordinate ≤ planes: Rk=Re1 ... Rek, Rk′=Re1 ... Rek′, Rn−k=Rek+1 ... Ren. ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 2.2. Radon Transforms on Affine Grassmannians. Let G(n,k) and G(n,k′) be a pair of affine Grassmann manifolds of non-oriented k-planes τ and k′-planes ζ in Rn, respectively; 1 k < k′ n 1. We ≤ ≤ − write τ τ(ξ,u) G(n,k), ζ ζ(η,v) G(n,k′), (2.1) ≡ ∈ ≡ ∈ where ξ Gn,k, u ξ⊥, η Gn,k′, v η⊥. The Radon transform of ∈ ∈ ∈ ∈ a function f(τ) on G(n,k) is a function (Rk,k′f)(ζ) on G(n,k′) defined by (Rk,k′f)(ζ) = f(τ)dζτ = dηξ f(ξ,v+x)dx. (2.2) Z Z Z τ⊂ζ ξ⊂η ξ⊥∩η Here d ξ denotes the probability measure on the Grassmannian G (η). η k This transform integrates f(τ) over all k-planes τ in the k′-plane ζ. ThedualRadontransformofafunctionϕ(ζ)onG(n,k′)isafunction (R∗ ϕ)(τ) (R∗ ϕ)(ξ,u) on G(n,k) defined by k,k′ ≡ k,k′ (R∗ ϕ)(τ) = ϕ(ζ)d ζ = ϕ(η+u)d η k,k′ τ ξ Z Z ζ⊃τ η⊃ξ = ϕ(η,Prη⊥u)dξη. (2.3) Z η⊃ξ Here Prη⊥u is the orthogonal projection of u ( ξ⊥) onto η⊥( ξ⊥), ∈ ⊂ d η is the relevant probability measure. This transform integrates ϕ(ζ) ξ over all k′-planes ζ containing the k-plane τ. In order to give (2.3) ON RADON TRANSFORMS 5 precise meaning, we choose an orthogonal transformation g O(n) so ξ ∈ that g Rk = ξ, and let O(n k) be the subgroup of O(n) that consists ξ − of orthogonal transformations preserving the coordinate plane Rn−k. Then (2.3) means (R∗ ϕ)(τ) (R∗ ϕ)(ξ,u) = ϕ(g ρRk′ +u)dρ. (2.4) k,k′ ≡ k,k′ Z ξ O(n−k) Proposition 2.1. [11, Lemma 2.1] The equality Z (Rk,k′f)(ζ)ϕ(ζ)dζ = Z f(τ)(Rk∗,k′ϕ)(τ)dτ (2.5) G(n,k′) G(n,k) holds provided that at least one of these integrals exists in the Lebesgue sense (i.e., it is finite if f and ϕ are replaced by f and ϕ , respec- | | | | tively). Proposition 2.2. (i) If f Lp(G(n,k)), 1 p < (n k)/(k′ k), then (Rk,k′f)(ζ) ∈ ≤ − − is finite for almost all ζ G(n,k′). If f C (G(n,k)), µ > k′ k, µ ∈ ∈ − then (Rk,k′f)(ζ) is finite for all ζ G(n,k′). The conditions p < ∈ (n k)/(k′ k) and µ > k′ k are sharp. − − − (ii) The dual transform (R∗ ϕ)(τ) is finite a.e. on G(n,k) for every k,k′ locally integrable function ϕ. The statement (i) is proved in [11, Corollary 2.6]. The statement (ii) follows from the equality (R∗ ϕ)(τ)dτ = const ϕ(ζ)(a2 ζ 2)(k′−k)/2dζ (2.6) Z k,k′ Z −| | |τ|<a |ζ|<a which is a particular case of the formula (2.19) from [11]. Proposition 2.3. [11, Lemma 2.3] For τ G(n,k) and ζ G(n,k′), ∈ ∈ let r = τ , s = ζ . If f(τ) = f (r) and ϕ(ζ) = ϕ (s), then 0 0 | | | | ∞ (Rk,k′f)(ζ) = σk′−k−1 f0(r)(r2 s2)(k′−k)/2−1rdr, (2.7) Z − s r (R∗ ϕ)(τ) = σk′−k−1σn−k′−1 ϕ (s)(r2 s2)(k′−k)/2−1sn−k′−1ds, k,k′ σ rn−k−2 Z 0 − n−k−1 0 (2.8) provided that the corresponding integrals exist in the Lebesgue sense. 6 BORIS RUBINANDYINGZHANWANG* 2.3. Alternative Parametrization of Lines and Hyperplanes. In this subsection we consider the case when the Radontransform R takes a function f on L = G(n,1) to a function Rf on H = G(n,n 1). For the future purposes we parametrize the manifolds L and H−in a slightly different way in comparison with subsection 2.2. Specifically, let L˜ = (ω,u) : ω Sn−1,u Rn,u ω . (2.9) { ∈ ∈ ⊥ } Every line ℓ has the form ℓ = ω + u, where (ω,u) L˜, ω = span(ω). Setting ℓ = ℓ(ω,u), eve{ry}function f on L can∈be reg{ar}ded as a function f˜(ω,u) = f( ω +u)on L˜satisfying f˜(ω,u) = f˜( ω,u). { } − WeequipL˜withtheproductmeasured ωdu, whered ω isthenormal- ∗ ∗ ized surface measure on Sn−1 and du is the Euclidean volume element on ω⊥. Then, for f L1(L), ∈ ˜ ˜ f(ℓ)dℓ = f(ω,u)d ωdu = d ω f(ω,u)du, (2.10) ∗ ∗ Z Z Z Z L L˜ Sn−1 ω⊥ where the integral on the left-hand side has the same meaning as in subsection 2.2. For the hyperplane case h H = G(n,n 1), in parallel with the ∈ − parametrization h = h(η,v), where η G and v η⊥ (cf. (2.1)), n,n−1 ∈ ∈ we set h = h(θ,t) = x Rn : x θ = t , where θ Sn−1,t R. Let { ∈ · } ∈ ∈ H˜ = (θ,t) : θ Sn−1,t R . { ∈ ∈ } Every function ϕ on H can be thought of as a function ϕ˜(θ,t) = ϕ(θ⊥,tθ) on the cylinder H˜, so that ϕ˜(θ,t) = ϕ˜( θ, t). Endowing − − H˜ with the measure d θdt, we get ∗ ϕ(h)dh dη ϕ(η,v)dv = dγ ϕ(γe⊥,tγe )dt Z ≡ Z Z Z Z n n H Gn,n−1 η⊥ O(n) R = d θ ϕ(θ⊥,tθ)dt = ϕ˜(θ,t)d θdt. (2.11) ∗ ∗ Z Z Z Sn−1 R H˜ Abusing notation, we identify L L˜, H H˜, f f˜, ϕ ϕ˜. (2.12) ≡ ≡ ≡ ≡ According to (2.2) and the identification (2.12), the Radon transform (2.2) can be written in the new parametrization as (Rf)(θ,t) = d ω f(ω,tθ+x)dx, (θ,t) H˜, (2.13) θ Z Z ∈ Sn−1∩θ⊥ ω⊥∩θ⊥ ON RADON TRANSFORMS 7 (set η = θ⊥, v = tθ, ξ = ω ). Here d ω is the probability measure on θ { } Sn−1 θ⊥ that is invariant under orthogonal transformations leaving θ ∩ fixed. Similarly, (2.3) yields (R∗ϕ)(ω,u) = ϕ(θ,θ u)d θ, (ω,u) L˜, (2.14) ω Z · ∈ Sn−1∩ω⊥ (usetheequalityPr u = θθTu, where θ isinterpreted asamatrixwith {θ} one column and θT is its transpose). By Proposition 2.1 and equalities (2.10) and (2.11), (Rf)(θ,t)ϕ(θ,t)d θdt = f(ω,u)(R∗ϕ)(ω,u)d ωdu (2.15) ∗ ∗ Z Z H˜ L˜ provided that at least one of these integrals exists in the Lebesgue sense. 3. Inversion of the Radon Transform of Quasi-Radial Functions As we mentioned in Introduction, the Radon transform that takes functions on L to functions on H is noninjective because dimL > dimH . To remedy the situation, we need to reduce the dimension of the source space or increase the dimension of the target space. Of course, the geometrical meaning of the problem should remain un- changed. Below we pursue the first approach. Suppose that the value of f at ℓ = ℓ(ω,u) depends only on ω and u , that is, f(ω, ) is constant on | | · the set of all lines equidistant from the central line ω . We call such { } a set a line cluster and denote cl(ω,r) = ℓ(ω,u) L : u = r , ω Sn−1, r > 0. (3.1) { ∈ | | } ∈ A line function f is called quasi-radial if it is constant on all clusters (3.1), that is, f(ω,u) = f (ω, u )for some function f andall or almost 0 0 | | all (ω,u) L˜. A more restrictive, radial case, when f(ω,u) = f ( u ), 0 ∈ | | that is, f(ℓ) depends only on the distance from the origin to ℓ, was studied in [11]; cf. (2.7). The set of all clusters (3.1) has the same dimension n, as the set H of all hyperplanes. Hence it is natural to expectthattherestrictionoftheRadontransform(2.13)toquasi-radial functions is injective. 8 BORIS RUBINANDYINGZHANWANG* Lemma 3.1. If f is quasi-radial, f(ω,u) = f (ω, u ), then 0 | | ∞ (Rf)(θ,t) = σ (r2 t2)(n−4)/2rdr f (ω,r)d ω (3.2) n−3 0 θ Z − Z |t| Sn−1∩θ⊥ provided that the integral on the right-hand side exists in the Lebesgue sense. Proof. Passing to polar coordinates in (2.13), we obtain ∞ (Rf)(θ,t) = σ d ω f (ω,√s2 +t2)sn−3ds. n−3 θ 0 Z Z Sn−1∩θ⊥ 0 (cid:3) This coincides with (3.2). The formula (3.2)isa generalizationof(2.7)for k = 1andk′ = n 1. − Itshowsthatonquasi-radialfunctions, (Rf)(θ,t)isaconstant multiple of the tensor product of the classical Funk transform (Fψ)(θ) = ψ(ω)d ω, θ Sn−1, (3.3) θ Z ∈ Sn−1∩θ⊥ and the fractional integration operator of the Erdélyi-Kober type ∞ 2 (Iα χ)(t) = (r2 t2)α−1χ(r)rdr, t > 0, (3.4) −,2 Γ(α) Z − t with α = n/2 1. Both operators were studied systematically in [16, − Sections 5.1 and 2.6.2]. Thus, for t > 0, we can write (Rf)(θ,t) = πn/2−1(R˜f )(θ,t), R˜ = In/2−1 F, (3.5) 0 −,2 ⊗ n/2−1 where F acts in the ω-variable and I in the r-variable, as in (3.2). −,2 By Lemma 2.42 from [16, p. 65] and the existence results for the Funk transform (see, e.g., [16, p. 281], the integral (3.2) is absolutely convergent for almost all (θ,t) H˜ provided that ∈ ∞ dω f (ω,r) rn−3dr < a > 0, (3.6) 0 Z Z | | ∞ ∀ Sn−1 a where the exponent n 3 in (3.6) is exact. − A variety of inversion formulas for F and Iα can be found in [16, −,2 subsections 5.1.6-5.1.8, 2.6.2]. For example, the following statement is an immediate consequence of the formulas (2.6.23), (2.6.25), and (5.1.96) from [16]. ON RADON TRANSFORMS 9 Theorem 3.2. Let ϕ = Rf, f(ω,u) = f (ω, u ), where f satisfies 0 0 | | (3.6). Then f (ω,r) = π1−n/2(R˜−1ϕ)(ω,r), R˜−1 = n/2−1 F−1. (3.7) 0 D−,2 ⊗ n/2−1 The Erdélyi-Kober derivative is defined by the following formu- D−,2 las: 1 d n/2−1χ = ( D)n/2−1χ, D = , (3.8) D−,2 − 2r dr if n is even, and n/2−1χ = r( D)(n−1)/2rn−2I1/2r1−nχ, (3.9) D−,2 − −,2 if n is odd, where the powers of r stand for the corresponding multipli- cation operators. Furthermore, t n−2 1 ∂ 2 (F−1ψ)(ω)=lim  (t2 s2)n/2−2Φ (s)sn−2ds, ω t→1(cid:18)2t ∂t(cid:19) (n 3)! Z − −  0  (3.10) Φ (s) = ψ(sξ +√1 s2ω)d ξ, 1 s 1. ω ω Z − − ≤ ≤ Sn−1∩ω⊥ The limit in (3.10) is understood in the L1(Sn−1)-norm. Remark 3.3. Ifthefunctionf inTheorem3.2issmooth, then(3.10)can be replaced by the corresponding expression in terms of the Beltrami- Laplace operator on Sn−1; see [11, Theorem 5.37]. If f is a radial function, i.e., f(ω,u) = f ( u ), the spherical component F in (3.5) 0 | | and (3.7) disappears and we simply have R˜−1 = n/2−1. The last D−,2 formula agrees with [11, Lemma 2.3]. 4. Inversion of the Dual Transform 4.1. The Dual Transform of Even Functions. In this subsection we confine to hyperplane functions ϕ with the property ϕ(h) = ϕ( h) h H . (4.1) − ∀ ∈ Apair(h, h)ofhyperplanes isanaturalcounterpart ofthelinecluster − (3.1). It is convenient to take the dual transform R∗ in the form (2.14), namely, (R∗ϕ)(ω,u) = ϕ(θ,θ u)d θ, (4.2) ω Z · Sn−1∩ω⊥ 10 BORIS RUBINANDYINGZHANWANG* where ω Sn−1 and u ω⊥. In this notation, the function ϕ on H is identified∈withafunctio∈nonthecylinder H˜ = (θ,t) : θ Sn−1,t R { ∈ ∈ } satisfying ϕ(θ,t) = ϕ( θ, t) (θ,t) H˜. (4.3) − − ∀ ∈ Moreover, if ϕ is even in the sense of (4.1), we additionally have ϕ(θ,t) = ϕ(θ, t) = ϕ( θ,t) (θ,t) H˜. (4.4) − − ∀ ∈ Remark 4.1. We pay attention to the following interesting fact. Unlike the Radon transform R that takes quasi-radial functions on L to even functions on H (cf. Lemma 3.1 and (4.4)), the dual transform R∗ of an even function on H is not necessarily a quasi-radial function on L. Let, for example, n = 3,k = 1, and choose ϕ(θ,t) = tθ , where 2 | | θ = (θ ,θ ,θ ) S2 and t R. Obviously, ϕ satisfies (4.4). By (4.2), 1 2 3 ∈ ∈ 1 (R∗ϕ)(e ,e ) = ϕ(θ, θ e )dθ. 1 2 2 2π Z · S2∩e⊥1 To evaluate this integral, we set 1 0 0 θ = γ(α)e , γ(α) = 0 cosα sinα, 0 α 2π. 2 ≤ ≤ 0 sinα cosα  −  Then 2π 0 1 (R∗ϕ)(e ,e ) = ϕ cosα ,cosα dα 1 2 2π Z sinα 0 −   2π 1 1 = cos2αdα = . 2π Z 2 0 Similarly, 2π 1 1 (R∗ϕ)(e ,e ) = sinαcosα dα = . 1 3 2π Z | | π 0 Thus (R∗ϕ)(e ,e ) = (R∗ϕ)(e ,e ) which means that R∗ϕ is not quasi- 1 2 1 3 6 radial. Now we proceed to reconstruction of ϕ(h) ϕ(θ,t) from(R∗ϕ)(ω,u) ≡ assuming that ϕ enjoys the symmetry (4.1) (or (4.4)). Suppose u = 0 6