Support theorems for Funk-type isodistant Radon transforms on constant curvature spaces

A connected maximal submanifold in a constant curvature space is called isodistant if its points are in equal distances from a totally geodesic of codimension 1. The isodistant Radon transform of a suitable real function f on a constant curvature space is the function on the set of the isodistants that gives the integrals of f over the isodistants using the canonical measure. Inverting the isodistant Radon transform is severely overdetermined because the totally geodesic Radon transform, which is a restriction of the isodistant Radon transform, is invertible on some large classes of functions. This raises the admissibility problem that is about finding reasonably small subsets of the set of the isodistants such that the associated restrictions of the isodistant Radon transform are injective on a reasonably large set of functions. One of the main results of this paper is that the Funk-type sets of isodistants are admissible, because the associated restrictions of the isodistant Radon transform, we call them Funk-type isodistant Radon transforms, satisfy appropriate support theorems on a large set of functions. This unifies and sharpens several earlier results for the sphere, and brings to light new results for every constant curvature space.


Introduction
Given a totally geodesic G of codimension 1 in a constant curvature space n of dimension n ∈ ℕ 2≤ and of curvature ∈ {1, 0, −1} , a connected maximal submanifold D whose points have a fix distance ≥ 0 from G , the axis, is called an isodistant of radius . In the Research was supported by NFSR of Hungary (NKFIH) under grants K 116451 and KH_18 129630, and by the Ministry for Innovation and Technology of Hungary (MITH) under grant NKFIH-1279-2/2020 and TUDFO/47138-1/2019-ITM. constant curvature planes the isodistants are well known, they are the straight lines in the plane, the circles in the sphere, and the hypercycles in the hyperbolic plane [40].
We denote the set of the isodistants by , and its subset, the set of the totally geodesics of codimension 1, by .
The isodistant Radon transform of a suitable function f on n is defined as the function f on that gives the integral of f over every isodistant using the natural measure. The totally geodesic Radon transform of a suitable function f on n is defined as the function f on that gives the integral of f over every totally geodesic using the natural measure.
The isodistant Radon transform is injective on a large class of functions, because the totally geodesic Radon transform , which is a restriction of the isodistant Radon transform, is injective by [24,Theorem 3.2]. This shows that the inversion problem of the isodistant Radon transform is severely overdetermined, and hence, the admissibility problem [15,17] arises: We call such a submanifold of admissible 1 [15,17]. For instance is an admissible submanifold of [24,Theorem 3.2].
Let the hypersurface K n ⊂ ℝ n+1 of points p = (p 1 , … , p n , p n+1 ) satisfying be equipped with the Riemannian metric at every point p ∈ K n . Then one gets the so-called projective model K n of the constant curvature space n [10], and also the canonical correspondence by identifying the points of K n ⊂ ℝ n+1 that are symmetric in the origin. It is very well known that every 1-codimensional totally geodesic of n is the intersection of K n with a 1-codimensional subspace of ℝ n+1 [24]. It is less known (see Lemma 3.1) that every isodistant of n corresponds to a slice, i.e., a hyperplane section of K n .
The slice transform of a suitable real function f on K n is defined as the function f on the set of slices that gives the integral of f over every slice using the canonical measure. After giving explicit formulas for in Sect. 4, we prove intertwining relations between the slice transforms and the classical Euclidean Radon transform in Sect. 5.
We call a set of slices rotational if it contains all of its rotations about the (n + 1)th axis. The set of the hyperplanes of the slices in a rotational set of slices is clearly rotation invariant, so they pass through a common point P = (0, … , 0, p) of the (n + 1)th axis, hence they are determined by the tangent q = tan ∈ [0, ∞] of the angle the hyperplanes closes with (1.1) What are the reasonably small submanifolds of for which the restricted isodistant Radon transform is injective on a reasonably large space of functions?
(1.2) g ; p ∶ T p K n × T p K n ∋ (x, y) ↦ x 1 y 1 + ⋯ + x n y n + x n+1 y n+1 (1.3) ∶ K n ∋ E → {E, −E} ∈K n ≅ n the (n + 1)th axis. The pairs (p, q) form a subset of the upper half plane extended with ideal points. So, the admissibility problem for the rotational slice transform can be formulated as to We call these curves admissible. Some curves are known to be admissible or inadmissible. For = −1 , the straight line q = 1 belongs to the horocyclic Radon transform [8,9,19,22,29], and so it is admissible. For = 1 , the hyperbola r 2 (1 + q 2 ) = p 2 q 2 ( r ∈ (0, 1) ) belongs to the Radon transform associated with the subspheres of radius √ 1 − r 2 , and so, by [36], it is admissible if and only if r is not a root of any Gegenbauer polynomial of the weight (1 − x 2 ) n−3 2 . For = 1 , the curve 1 = p 2 − q 2 cosh 2 ( ∈ [0, ∞] ) belongs to the Radon transform associated with the subspheres whose hyperplanes are tangent to the spheroid , so, by [35], it is admissible. If C is a ray with fixed p ∈ ℝ ∪ {±∞} , then we call the associated restrictions of the slice transform p-shifted Funk transform 2 and denote it by p . For the sphere ( = 1 ) this was recently quite intensively investigated [3,4,6,7,18,20,25,26,30,[32][33][34]38], but there are also sporadic earlier results [1,16] as well. Surprisingly enough there seems to be no general results for = 0, −1.
The We prove sharp support theorems and explicit kernel descriptions for every p for each ∈ {0, ±1} in Sect. 6, where the main tool is the intertwining relations, (5.4) and (5.5), of the slice transform and the Euclidean Radon transform. It is interesting, that, depending on p, different speeds of decay on the functions are necessary to employ for the support theorems.
We define the Funk-type isodistant Radon transform ̂ p of a suitable function h on n as the shifted Funk transform of ĥ ∶= ⋅ h• , where is the indicator function of the open upper half space of ℝ n+1 . It is considered in Sect. 7, where again sharp support theorems and complete kernel descriptions are proved. These results considerably generalize the author's earlier support theorems [24] for the totally geodesic Radon transform.
In n every 1-codimensional totally geodesic has exactly two isodistants for every > 0 . We call the union of such a pair of isodistants a duplex isodistant and define the duplex Funk-type isodistant Radon transform p of a suitable function h on n as the shifted Funk transform of h ∶= h• . It is considered in Sect. 8, where again sharp support theorems and complete kernel descriptions are proved. When = 1 , these results give geometric reasoning for [5,6]. For = 0 we do not get too much new, but we observe a new kind of problem that is discussed and solved in a special case in Sect. 9.
(1.4) determine the curves C in the (p, q) plane (equipped with ideal elements) such that the slice transform associated with the rotational set of slices given by C is injective on a reasonably large set of functions.
The presented support theorems and kernel descriptions are new for both curved constant curvature spaces. Further, these results bring to light new problems for all constant curvature spaces that we discuss in last Sect. 9 where some possible generalizations, consequences and worthy details are also outlined.

Notations and preliminaries
Points of ℝ n are denoted as A, B, … or a, b, … , and vectors are given as ���� ⃗ AB or a, b, … . The straight line through A and B is AB, and the closed segment with endpoints A and B is AB.
We denote the Euclidean scalar product by ⟨⋅, ⋅⟩ , B n is the n-dimensional closed unit ball centered at the origin, and its boundary is S n−1 = B n . If n = 2 , then we use the notation u = (cos , sin ) for the elements of S 1 .
We parameterize the manifold of the hyperplanes, the 1-codimensional totally geodesics in ℝ n+1 , on S n × ℝ , so that P(w, r) = {x ∶ r = ⟨w, x⟩} . This is a double covering, but it will not cause trouble. Then we have so the classical Euclidean Radon transform [18,28] on the set of suitable functions on ℝ n is defined [24, (2.4)], for r > 0, by where w ∈ S n−1 , and S n−1 w,s = {u ∈ S n−1 ∶ ⟨w, u⟩ > s} ( s ∈ ℝ ). Let C ∞ (ℝ n ) be the space of all continuous functions f on ℝ n such that f (x)|x| k is bounded for each k > 0 (this is a special case of (6.6)). Then we have the following support theorem of Helgason that is crucial for our results.
Restricting Π 0 to K n essentially gives the so-called gnomonic projection that results in the so-called projective models, i.e., the Cayley-Klein models of the constant curvature spaces.
The domain M n ;1 of such a Cayley-Klein model, is A n 1 with the ideal hyperplane if = 1, 0 , and it is the interior of the unit ball centered to O + in A n 1 if = −1 . The geodesics are the chords of M n ;1 , the totally geodesics are the n-dimensional slices (hyperplanes) of B n ⊂ A n 1 [10], hence every totally geodesic of K n is the intersection of K n with a 1-codimensional subspace of ℝ n+1 [24].
The manifold K n is a rotational one [21], so it is determined by the size function giving the radius (r) of the Euclidean sphere that is isometric with the geodesic sphere of radius r in K n . This defines the function (⋅) = √ 1 − 2 (⋅) , while the projector function [24] is defined by Π 0 (rw) = (r)w. We often use the polar coordinatization of K n and Ǩ n with respect to the appropriate point O ± : the pair (u, r) means the point Exp O ± (ru) , where u ∈ S n−1 ⊂ T O ± K n is a unit vector, r ∈ ℝ + , and Exp is the usual exponential mapping, hence d (O ± , Exp O ± (ru)) = r for the metric d on K n determined by (1.2). The injectivity radius > 0 is then the upper limit of the second parameter until which the polar coordinatization keeps injectivity. Finally, the supremum > 0 of the distances a point can be from a geodesic is called the geodesic injectivity radius (Table 1).

Isodistants and hyperplanes
We parameterize the manifold ̃ of the totally geodesics of K n on S n−1 × [0, ) so that the totally geodesic G (w, g) is perpendicular to the geodesic t ↦ Exp O + (tw) and contains the point Exp O + (gw) (this leaves out K n 1 ∩ A n 0 ), where w ∈ S n−1 ⊂ T O +K n and g ∈ [0, ) . This is a double covering at g = 0 , but it will not cause problem. The manifold ̃ of the isodistants in K n is parameterized on S n−1 × {(g, ) ∶ g ∈ [0, ) and + g ∈ (− , )} so that The following lemma shows that the isodistants are plane sections of K n . . Let Figure 1 shows what we have.
For any unit vector w every point of ℝ n+1 can be uniquely written in the form xw + yw ⟂ + zb n+1 , where w ⟂ is a unit vector in the orthogonal complement of the plane spanned by w and b n+1 .
In this form a point is in C ∶=K This means that Π −1 (C) is a sphere; hence, because Π −1 K n −1 is the Poincaré model, C belongs to an isodistant, so the lemma is proved. ◻

The slice transform
Lemma 3.1 gives rise to consider the slice transform . We call the intersections of K n with hyperplanes slices. The slice transform sends every suitable (not necessarily even) function h on K n to the function h on the set of slices so that h gives for every slice the integral of h over that slice.
To determine , firstly we define some special slice transforms ± , for which we need the "inverses" Π ;± p of the mappings . Then it is easy to see that where From now on Fix a unit vector u ∈ S n ∩ A n 0 . Then every point of M n ;p+1 can be uniquely ( pb n+1 is an exception) written in the form pb n+1 + eu , where e ∈ [0, ∞) . So there are functions .
we do not differentiate between the vectors corresponding through Γ.
Thus 2 (e)e 2 + (p + (e)) 2 = 1 , hence we obtain This and (4.1) allow us to define the mapping Observe that ;± ±1 vanishes, so Π ;± ±1 (xv + (±1 + 1)b n+1 ) = ±b n+1 . Further, the mapping , du is the standard surface measure of S n−1 , and ;± p;q is the density pulled back by Π p from the hypersurface K n ∩ P ;± . where ;± p is given by (4.3). Since u = q e q (cos ) , So letting e = e q (⟨w, u⟩) , we conclude that (4.7) E ;± p (e q (cos )u ) = pb n+1 + ;± p (e q (cos )) b n+1 + e q (cos )u , where the expression ( ‡) under the square root sign can be simplified as follows: To get ;± p;q for higher dimension n, we only have to multiply its 2-dimensional version with n−2 . Then every point of Π ∞ (K n ) in the plane spanned by u and b n+1 can be uniquely written in the form ∞b n+1 + eu , where e ∈ ℝ . So there are functions ∶ ℝ + → ℝ such that the point (e)b n+1 + (e)u is in K n . (See Fig. 3.) Thus 2 (e) + 2 (e) = 1 , hence we obtain ;± ±∞ (e) = ± √ 1 − e 2 . This and (4.1) allow us to define the mapping We define the special slice transforms ± for suitable functions h in C(K n ) by , du is the standard surface measure on S n ∩ A n 0 , and ;± ∞;q is the density pulled back by Π ∞ from the hypersurface K n ;∞;± ∩ P(w, q).
To make their later use easier, we extend definitions (4.5) and (4.11) of the special slice transforms ± by setting ± h(p;w, q) ∶= 0 for p ∈ ℝ and q > 0 if the hyperplane does not intersect K n , and by setting ± h(∞;w, q) ∶= 0 for q > 0 if the hyperplane P(w, q) does not intersect K n . With this understanding, the slice transform is where p ∈ ℝ ∪ {±∞} , w ∈ S n−1 , and q ≥ 0.

Shifted Funk transforms: support theorems and kernels
The p-shifted Funk transform ( p ∈ ℝ ∪ {±∞} ) of a suitable function h on K n is The proofs of the following support theorems in this section follow the method used in the proof of [24, Theorem 3.2]: we pull Support Theorem 2.1 back to K n through the adequate intertwining relation of (5.4) and (5.5). Following (4.2), let Then, since K n is a rotational manifold, we can define the nonnegative functions ± h ± (p;w, q) = ∫ S n−1 f (e(u)u)e n (u) u, f (e(u)u)e n (u) u, Since is the size function of K n , and so we have we deduce that for p ∈ ℝ , and Substituting ;± p from (4.3) into (6.4), it is easy to see that ;± p is strictly monotone increasing for ≥ 0 , and if = −1 , then it is strictly monotone increasing in [0, 1) and decreasing in (1, ∞) . It is clear from (6.5) that ;± ∞ is strictly monotone increasing. Let L ⊂ K n be a nonempty, open domain, and define the set C m (K n , L) ( m ∈ ℕ ) of all continuous functions h on K n that satisfy where P ∈ K n is any fixed point, and the usual big-O notation is in use. We use the abbreviations C m (K n ) ∶= C m (K n , �) , and C m (K n , p) ∶= C m (K n , K ;± p ) (see (5.2)).

Support theorems
We start with the elliptic case.
Proof We prove the statements one after the other.
4) ( ;± p (e)) = ;± p (e)e, ( ;± p (e)) = p + ;± p (e), (6.5) ( ;± ∞ (e)) = e, ( ;± ∞ (e)) = If h ∈ C ∞ (K n 0 , ±1) and 0 ±1 h(w, q) = 0 for every q > s > 0 and w ∈ S n−1 , then h(eu) vanishes for every e > 2s and u ∈ S n−1 . ⟨sp2⟩ If |p| ≠ 1 , h ∈ C ∞ (K n 0 , p) vanishes on K 0;∓ p , and 0 p h(w, q) = 0 for every q > s > 0 and w ∈ S n−1 , then h(eu) vanishes for every e > | − p ± 1|q and u ∈ S n−1 . ⟨sp3⟩ If h ∈ C ∞ (K n 0 ) vanishes on K 0;∓ ∞ , and 0 ∞ h(w, q) = 0 for every q > s > 0 and w ∈ S n−1 , then h(eu) vanishes for every e > s and u ∈ S n−1 . Theorem 6.2 is a direct application of Support Theorem 2.1, so we only notice that in statement ⟨sp1⟩ we can also deduce that 0 ± h(±1;w, 0) = 0. Finally, we deal with the hyperbolic case. Observe that for p = ±1 there cannot exist support theorem in the usual sense. The reason behind this is that if, say, p = −1 , then Ψ −1;− −1 maps the lower sheet Ǩ n −1 of the hyperboloid K n −1 onto the complement of the unit ball in A n 0 in such a way that the points of "infinity" in Ǩ n −1 are sent to the unit sphere in A n 0 . In the same way, if p = 1 , then Ψ 1;+ −1 maps the upper sheet K n −1 of the hyperboloid K n −1 onto the complement of the unit ball in A n 2 in such a way that the points of "infinity" in K n −1 are sent to the unit sphere in A n 2 .
Proof We prove the statements one after the other.

Kernel descriptions
Let h be a continuous function on K n . Using (5.2) we define the functions We start considering the kernels in the elliptic case. This makes a direct generalization of Funk's result [13] and leads to kernel descriptions different than the ones in [7,15,16,20,26,27,30,31,33,34]. Figure 4 shows what is at stake.

Theorem 6.4 Kernels of some shifted Funk transform on the sphere
Proof Statement ⟨ks1⟩ comes immediately from ⟨se1⟩ .
Although the result does not add very much new to the theory, we continue with the parabolic case for the sake of completeness. Let H u be the 1-codimensional subspace of A n −1 orthogonal to u ∈ S n ∩ A n −1 . Let g(u) be the integral of h over H u . Then h + (u, r) + g(u) ≡ 0 by (2.2). However, by (2.2), h + (u, r) → 0 if r → ∞ , so we deduce g(u) ≡ 0 . Then h + (u, r) ≡ 0 follows from which Support Theorem 2.1 implies the statement.
We do not give the proof of ⟨kp2⟩ and ⟨kp3⟩ here, because the procedures are very much analogous to the proof given for the elliptic case. ◻ We finish this section with the hyperbolic case. There seems to be no previous results in the literature about the shifted Funk transform for the hyperbolic case. However, for the hyperbolic Funk transform and for the hyperbolic slice transforms, the results seem greatly analogous to the spherical case. Figure 5 shows what the next theorem is about.

Proof
We prove the statements one after the other.
⟨kh3⟩ : Equation (4.2) gives M n −1;p = 1 √ 1−p 2 B n for |p| ∈ (0, 1) . Since this is a compact set, the reasoning given for ⟨kh2⟩ works very well for this statements too, but needs only a decay of order n − 1 , so we leave the details to the reader.
if |p| < 1 , |p| > 1 , and p = ∞ ⟨kh4⟩ and ⟨kh5⟩ : Equation (4.2) gives M n −1;p = ℝ n for |p| > 1 and M n −1;∞ = ℝ n . It is clear that the reasoning given for ⟨kh2⟩ works very well in these cases too, but with infinite decay condition, so we leave the details to the interested reader. ◻ According to ⟨kh1⟩ the hyperbolic slice transform is injective on C ∞ (K n −1 , ±1) , while the kernel of the hyperbolic Funk transform is the set of odd functions in C n (K n −1 ) by ⟨kh2⟩ . These results are totally analogous to the case of the sphere.

Funk-type isodistant Radon transforms
The double covering of n given by (1.3) can be reduced by considering the identifying mapping ̂∶K n ∋ E → (E, −E) ∈K n ≅ n . Then ̂ is bijective for ≤ 0 as well as for = 1 if the totally geodesic corresponding to K n 1 ∩ A n 0 is left out. If h ∈ C( n ) , then the corresponding function on K n is We define the Funk-type isodistant Radon transform ̂ p of a suitable function h ∈ C( n ) by where w ∈ S n−1 , g, ∈ [0, ) . So the Funk-type isodistant Radon transform ̂ p is essentially the restriction of the shifted Funk transform to the set of hyperplanes intersecting K n in isodistants.
For our considerations we will need decay conditions, so we introduce the function space C m ( n ,̂(L)) so that h ∈ C m ( n , . Additionally, analogously to the notations after (6.6), we use the notation C m ( n , p) ∶= C m ( n ,̂(K ;± p )) too (see (5.2)). We also need to define the functions for every h ∈ n , where p ∈ ℝ ∪ {±∞} , and Ψ ;± p is given by (5.1). Observe that if h ∈ C k ( n , p) , then both functions ĥ ± p are in C k (K n , p) for every k ∈ ℕ.
In the elliptic case, every slice of K n 1 is a part of an isodistant in K n 1 , so the properties of ̂ 1 p are essentially similar to that of 1 p . We give these properties without proof, because they follow directly from theorems 6.1 and 6.4 with the use of the functions ;± p defined in (6.2).
with n . These slices of K n −1 and the corresponding submanifolds in n −1 , that are not isodistant, are called virtual isodistants (Fig. 6). 4 Thus the properties of ̂ −1 p have significant differences from that of −1 p while they easily follow from the statements of theorems 6.3 and 6.6. Theorem 7.2 The Funk-type isodistant Radon transforms in the hyperbolic space have the following properties: ⟨eh1⟩ If p < 0, d ∈ [0, ∞), h ∈ C n−1 ( n −1 , p), and ̂ −1 p h(w, g) = 0 for every g > d and w ∈ S n−1 , then h(Exp O + (eu)) vanishes for every e > −1;+ p (tanh(d)) and u ∈ S n−1 .
Notice that no statement in this theorem is analogous to statements ⟨ee4⟩ , ⟨ee5⟩ , and ⟨ee7⟩ of Theorem 7.1. This is due to the fact that no virtual isodistants exist on the elliptic space.

Duplex Funk-type isodistant Radon transforms
Instead of reducing the double covering of n so as we did in the previous section, we can restrict the function space to the space of even functions on K n . Then the isodistants of n correspond to some of the slices of K n .
We define the duplex Funk-type isodistant Radon transforms p for suitable functions h ∈ C( n ) by where w ∈ S n−1 , g, ∈ [0, ) , and h ∶ K n ∋ E ↦ h( (E)) with given in (1.3). Recall our formula (3.3). It shows that p and g determine the isodistant to integrate on, more exactly q = (g) . The same formula shows that q = ( ) if p = ∞ (or, which is the same, g = 0).