Radon Transforms for Mutually Orthogonal Affine Planes
aa r X i v : . [ m a t h . F A ] J a n RADON TRANSFORMS FOR MUTUALLYORTHOGONAL AFFINE PLANES
BORIS RUBIN AND YINGZHAN WANG
Abstract.
We study a Radon-like transform that takes functionson the Grassmannian of j -dimensional affine planes in R n to func-tions on a similar manifold of k -dimensional planes by integrationover the set of all j -planes that meet a given k -plane at a rightangle. The case j = 0 gives the classical Radon-John k -planetransform. For any j and k , our transform has a mixed structurecombining the k -plane transform and the dual j -plane transform.The main results include action of such transforms on rotationinvariant functions, sharp existence conditions, intertwining prop-erties, connection with Riesz potentials and inversion formulas ina large class of functions. The consideration is inspired by the pre-vious works of F. Gonzalez and S. Helgason who studied the case j + k = n − n odd, on smooth compactly supported functions. Introduction
Let G ( n, j ) and G ( n, k ) be a pair of Grassmannian bundles of affine j -dimensional and k -dimensional unoriented planes in R n , respectively.In the present paper we study a Radon-like transform that takes afunction f on G ( n, j ) to a function R j,k f on G ( n, k ) when the value( R j,k f )( ζ ) for ζ ∈ G ( n, k ) is defined as an integral of f over the setof all planes τ ∈ G ( n, j ), which meet ζ at a right angle. Our aimis to study properties of this transform and obtain explicit inversionformulas.In the limiting case j = 0, when G ( n, j ) is identified with R n , ourtransform is the well-known Radon-John k -plane transform, which wasstudied in many books and papers; see, e.g., [4, 14, 16, 20] and ref-erences therein. Another limiting case k = 0 corresponds to the dualRadon-John transform, which averages a given function on G ( n, j ) overall j -dimensional planes passing through a fixed point x ∈ R n . Thesetransforms are usually studied in parallel with the j -plane transforms,but have special features; see [8, 14, 20, 31]. Thus R j,k f is a kind of Mathematics Subject Classification.
Primary 44A12; Secondary 47G10.
Key words and phrases.
Radon transforms, Grassmann manifolds, inversionformulas. mixture of these limiting cases. An inversion formula for the Radontransform R j,k f , when j + k = n − n is odd and f ∈ C ∞ c ( G ( n, j )),was obtained by Gonzalez [7, 8] in the form f = c R k,j ( − ∆ n − k ) ( n − / R j,k f, (1.1)where R k,j is the dual of R j,k , ∆ n − k is the Laplace operator on thefiber of the Grassmannian bundle G ( n, k ), and c is a constant, whichis explicitly evaluated; see also Helgason [14, p. 90], where the resultsfrom [7, 8] are announced.The present paper conains new results for R j,k f for all j + k < n anda large class of functions f . In particular, we establish sharp conditionsof convergence of the integrals R j,k f , their relation to Riesz potentialsand Radon-John transforms, and obtain new inversion formulas.A great deal has been written about Radon transforms on Grassmannmanifolds; see, e.g., [2, 3, 5, 6, 10–13, 15, 17, 21, 25, 27, 28, 32, 33]. Theclasses of Radon transforms and the methods of these works differ fromthose in the present paper. More information about Radon transformsand their applications can be found, e.g., in the books [4, 14, 26] andreferences therein. Plan of the Paper and Main Results.
Section 2 contains neces-sary preliminaries. Besides the notation, it includes basic facts aboutErd´elyi–Kober fractional integrals and derivatives, Radon-John k -planetransforms and Riesz potentials. In Section 3 we give precise definitionof the mixed j -plane to k -plane Radon transforms and prove the corre-sponding duality relation. In Section 4 we show that these transformson radial (i.e., rotation invariant) functions are represented as compo-sitions of Erd´elyi–Kober fractional integrals and give some examples.The results of Section 4 are used in Section 5 to establish sharp condi-tions, under which the integral R j,k f exists in the Lebesgue sense. InSection 6 we derive new formulas connecting Radon transforms R j,k f with Riesz potentials. Section 7 is devoted to inversion formulas for R j,k f . Here, by the dimensionality argument, the natural setting ofthe problem corresponds to j + k < n . The most complete informationis obtained in the following cases:(a) any j + k < n , when f is radial;(b) j + k = n − j + k < n , when f belongs to the range of the j -plane Radon-John transform.Regarding other cases, we have the following Conjecture. If j + k < n − , then R j,k is non-injective on the set ofall infinitely smooth rapidly decreasing functions. ADON TRANSFORMS 3
Acknowledgements.
The study of the operators R j,k f for all j + k Notation. Let G n,j and G ( n, j ) be the sets of all j -dimensionallinear subspaces and j -dimensional unoriented affine planes in R n , re-spectively. Each “point” in G n,j represents a j -plane passing throughthe origin. Every j -plane τ ∈ G ( n, j ) is naturally parameterized bythe pair ( ξ, u ), where ξ ∈ G n,j and u ∈ ξ ⊥ , the orthogonal complementof ξ in R n . Under this parametrization, the manifold G ( n, j ) is a fiberbundle with the base G n,j and the canonical projection π : τ ( ξ, u ) → ξ .The fiber π − ξ over the point ξ ∈ G n,j is the set of all j -dimensionalplanes parallel to ξ . This set is ( n − j )-dimensional and indexed by u ∈ ξ ⊥ . We equip G ( n, j ) with the product measure dτ = dξdu , where dξ is the standard probability measure on G n,j and du is the Euclideanvolume element on ξ ⊥ . Abusing notation, we write | τ | for the Eu-clidean distance between τ ∈ G ( n, j ) and the origin. If τ ≡ τ ( ξ, u ),then, clearly, | τ | = | u | = ( u + · · · + u n ) / . Given a subspace X of R n ,we write G j ( X ) and G ( j, X ) for the Grassmannians of all j -dimensionallinear subspaces and j -dimensional unoriented affine planes in X , re-spectively.In the following, S n − denotes the unit sphere in R n . For θ ∈ S n − , dθ stands for the Riemannian measure on S n − so that the area of S n − is σ n − ≡ R S n − dθ = 2 π n/ (cid:14) Γ( n/ e , . . . , e n be the coordinate unit vectors in R n . We set E j = R e ⊕ · · · ⊕ R e j , E k = R e n − k +1 ⊕ · · · ⊕ R e n ; (2.1) E ℓ = R e j +1 ⊕ · · · ⊕ R e j + ℓ , ℓ = n − j − k ; (2.2) R n − j = R e j +1 ⊕ · · · ⊕ R e n , R n − k = R e ⊕ · · · ⊕ R e n − k . (2.3)The notation O ( n ) for the orthogonal group of R n is standard. For ρ ∈ O ( n ), dρ stands for the O ( n )-invariant probability measure on O ( n ); M ( n ) = R n ⋊ O ( n ) is the group of rigid motions in R n .All integrals are understood in the Lebesgue sense. The letter c (sometimes with subscripts) stands for an unessential positive constantthat may be different at each occurrence. BORIS RUBIN AND YINGZHAN WANG Erd´elyi–Kober Fractional Integrals. The following Erd´elyi–Kober type fractional integrals on R + = (0 , ∞ ) arise in numerousintegral-geometric problems:( I α + , f )( t ) = 2Γ( α ) t Z ( t − r ) α − f ( r ) r dr, (2.4)( I α − , f )( t ) = 2Γ( α ) ∞ Z t ( r − t ) α − f ( r ) r dr. (2.5)We review basic facts about these integrals. More information can befound in [26, Subsection 2.6.2]. Lemma 2.1. (i) The integral ( I α + , f )( t ) is absolutely convergent for almost all t > whenever r → rf ( r ) is a locally integrable function on R + . (ii) If ∞ Z a | f ( r ) | r α − dr < ∞ , a > , (2.6) then ( I α − , f )( t ) is finite for almost all t > a . If f is non-negative,locally integrable on [ a, ∞ ) , and (2.6) fails, then ( I α − , f )( t ) = ∞ forevery t ≥ a . Fractional derivatives of the Erd´elyi–Kober type are defined as theleft inverses D α ± , = ( I α ± , ) − and have different analytic expressions.For example, if α = m + α , m = [ α ] , ≤ α < 1, then, formally, D α ± , ϕ = ( ± D ) m +1 I − α ± , ϕ, D = 12 t ddt . (2.7)More precisely, the following statements hold. Theorem 2.2. Let ϕ = I α + , f , where rf ( r ) is locally integrable on R + .Then f ( t ) = ( D α + , ϕ )( t ) for almost all t ∈ R + , as in (2.7). Theorem 2.3. If f satisfies (2.6) for every a > and ϕ = I α − , f ,then f ( t ) = ( D α − , ϕ )( t ) for almost all t ∈ R + , where D α − , ϕ can berepresented as follows. (i) If α = m is an integer, then D α − , ϕ = ( − D ) m ϕ, D = 12 t ddt . (2.8) ADON TRANSFORMS 5 (ii) If α = m + α , m = [ α ] , < α < , then D α − , ϕ = t − α ) ( − D ) m +1 t α ψ, ψ = I − α + m − , t − m − ϕ. (2.9) In particular, for α = k/ , k odd, D k/ − , ϕ = t ( − D ) ( k +1) / t k I / − , t − k − ϕ. (2.10)Fractional integrals and derivatives of the Erd´elyi–Kober type pos-sess the semi-group property D α ± , D β ± , = D α + β ± , , I α ± , I β ± , = I α + β ± , (2.11)in suitable classes of functions, which guarantee the existence of thecorresponding expressions.2.3. Radon-John k -Plane Transforms and Riesz Potentials. Werecall some facts from [20, 24]; see also [4, 14]. The k -plane transform of a function f on R n is defined by the formula( R k f )( ζ ) = Z ζ f ( x ) d ζ x, ζ ∈ G ( n, k ) , ≤ k ≤ n − , (2.12)where d ζ x stands for the Euclidean measure on ζ . Using parametriza-tion ζ ≡ ζ ( η, v ), η ∈ G n,k , v ∈ η ⊥ , we have( R k f )( ζ ) ≡ ( R k f )( η, v ) = Z η f ( v + y ) d η y, (2.13)where d η y is the Euclidean volume element on η . The dual k -planetransform R ∗ k averages a function ϕ on G ( n, k ) over all k -planes passingthrough the fixed point x ∈ R n . Specifically,( R ∗ k ϕ )( x ) = Z O ( n ) ϕ ( γη + x ) dγ, (2.14)where η is an arbitrary fixed k -plane through the origin. The dualityrelation Z G ( n,k ) ( R k f )( ζ ) ϕ ( ζ ) dζ = Z R n f ( x ) ( R ∗ k ϕ )( x ) dx (2.15)holds provided that either side of this equality exists in the Lebesguesense.The following inequality, which is a particular case of Lemma 2.6from [20], shows for which f the integral ( R k f )( ζ ) is well defined for BORIS RUBIN AND YINGZHAN WANG almost all ζ ∈ G ( n, k ). We have Z G ( n,k ) | ( R k f )( ζ ) | | ζ | | ζ | k − n +1 dζ ≤ c Z R n | f ( x ) | | x | log(2 + | x | ) dx. (2.16)If f and ϕ are radial functions, then R k f and R ∗ k ϕ can be expressedthrough the Erd´elyi–Kober fractional integrals as follows. Lemma 2.4. (cf. [24, Lemma 3.1, Theorem 3.2], [20, Lemma 2.1])(i) If f ( x ) = f ( | x | ) , then ( R k f )( ζ ) = F ( | ζ | ) , where F ( s ) = π k/ ( I k/ − , f )( s ) . (2.17)(ii) If ϕ ( ζ ) = ϕ ( | ζ | ) , then ( R ∗ k ϕ )( x ) = Φ ( | x | ) , where Φ ( r ) = Γ( n/ n − k ) / r − n ( I k/ , s n − k − ϕ )( r ) . (2.18) The formulas (2.17) and (2.18) hold provided that either side of thecorresponding equality exists in the Lebesgue sense. The duality (2.15) yields the following existence result. Lemma 2.5. (cf. [24, Theorem 3.2])(i) If f is locally integrable in R n \ { } and satisfies Z | x | >a | f ( x ) || x | n − k dx < ∞ for some a > , (2.19) then ( R k f )( ζ ) is finite for almost all ζ ∈ G ( n, k ) . If f is nonnegative,radial, and (2.19) fails, then ( R k f )( ζ ) ≡ ∞ . (ii) If ϕ ∈ L loc ( G ( n, k ) , then R ∗ k ϕ ∈ L loc ( R n ) . There is a remarkable connection between the operators R k , R ∗ k andthe Riesz potential( I αn f )( x ) = 1 γ n ( α ) Z R n f ( y ) dy | x − y | n − α , γ n ( α ) = 2 α π n/ Γ( α/ n − α ) / , (2.20) α > , α = n, n + 2 , n + 4 , . . . . If f ∈ L p ( R n ), 1 ≤ p < n/α , then ( I αn f )( x ) < ∞ for almost all x , andthe bounds for p are sharp.We recall that formally I αn f = ( − ∆ n ) − α/ f , where ∆ n is the Laplaceoperator in R n . The corresponding Riesz’s fractional derivative is de-fined as the left inverse D αn = ( I αn ) − ∼ ( − ∆ n ) α/ (2.21) ADON TRANSFORMS 7 and has many different analytic expressions, depending on the class offunctions; see [26, Section 3.5] for details. For example, if ϕ = I αn f , f ∈ L p ( R n ), 1 ≤ p < n/α , then, by Theorem 3.44 from [26], D αn ϕ canbe represented as a hypersingular integral( D αn ϕ )( x ) ≡ d n,ℓ ( α ) Z R n (∆ ℓy ϕ )( x ) | y | n + α dy (2.22)= lim ε → d n,ℓ ( α ) Z | y | >ε (∆ ℓy ϕ )( x ) | y | n + α dy, (2.23)where (∆ ℓy ϕ )( x ) = ℓ X k =0 ( − k (cid:18) ℓk (cid:19) ϕ ( x − ky )is the finite difference of ϕ of order ℓ with step y at the point x , d n,ℓ ( α ) = π n/ α Γ(( n + α ) / × Γ( − α/ B l ( α ) if α = 2 , , , . . . ,2( − α/ − ( α/ ddα B l ( α ) , if α = 2 , , , . . . ,B l ( α ) = l X k =0 ( − k (cid:18) lk (cid:19) k α . The integer ℓ is arbitrary with the choice ℓ = α if α = 1 , , , . . . and any ℓ > α/ 2] (the integer part of α/ L p -norm and in the almost everywhere sense. If,moreover, f is continuous, then the convergence in (2.23) is uniformon R n .The inversion formula (2.22) is non-local. If α is an even integer,then a local inversion formula ( − ∆ n ) α/ I αn f = f is available underadditional smoothness assumptions for f ; see [26, Theorem 3.32] fordetails.The following theorem establishes remarkable connection betweenthe Riesz potentials and the k -plane transform. Theorem 2.6. For any ≤ k ≤ n − , R ∗ k R k f = c k,n I kn f, c k,n = 2 k π k/ Γ( n/ n − k ) / , (2.24) provided that either side of this equality exists in the Lebesgue sense. BORIS RUBIN AND YINGZHAN WANG The formula (2.24) is due to Fuglede [1]. Its derivation is straight-forward and relies on the Fubini theorem. The following formulas canbe proved in a similar way (cf. [26, Proposition 4.38, Lemma 4.100],[20, Section 3]): R k I αn f = I αn − k R k f, I αn R ∗ k ϕ = R ∗ k I αn − k ϕ. (2.25)Here I αn − k stands for the Riesz potential on the ( n − k )-dimensionalfiber of the Grassmannian bundle G ( n, k ). As above, it is assumedthat either side of the corresponding equality exists in the Lebesguesense. If f and ϕ are good enough, these formulas extend by analyticityto all complex α ; cf. [19, Theorem 2.6]. It suffices to assume that f belongs to the Semyanistyi-Lizorkin space Φ( R n ) of Schwartz functionsorthogonal to all polynomials and ϕ ∈ R k (Φ( R n )). Corollary 2.7. The inversion formula for the dual k -plane transform ϕ = c − k,n R k D kn R ∗ k ϕ, c k,n = 2 k π k/ Γ( n/ n − k ) / , (2.26) holds provided that ϕ = R k f for some f ∈ L p ( R n ) , ≤ p < n/k .Proof. By (2.24), R k D kn R ∗ k ϕ = R k D kn R ∗ k R k f = c k,n R k D kn I kn f = c k,n R k f = c k,n ϕ. (cid:3) For further purposes, we also mention the following factorizationformula, which connects the Riesz potential (2.20) with the Erd´elyi-Kober fractional integrals. Specifically, if f is a radial function on R n , f ( x ) = f ( | x | ), then ( I αn f )( x ) = F ( | x | ), where F ( r ) = 2 − α r − n ( I α/ , s n − α − I α/ − , f )( r ); (2.27)see [18], [26, formula (3.4.11)]. It is assumed that either side of (2.27)exists in the Lebesgue sense.2.4. The Hyperplane Radon Transform, Its Dual, and the FunkTransform. We will need explicit relations connecting the hyperplaneRadon transform, its dual, the Funk transform, and their inverses.Some of these facts are new. Others are scattered in the literature indifferent forms.Every hyperplane τ ∈ G ( n, n − 1) can be parametrized by the pair( η, t ) ∈ S n − × R , so that τ ≡ τ ( η, t ) = { x ∈ R n : x · η = t } , τ ( − η, − t ) = τ ( η, t ) . (2.28) ADON TRANSFORMS 9 The hyperplane Radon transform g → Rg takes a function g on R n toa function Rg on G ( n, n − 1) (or on S n − × R ) by the formula( Rg )( τ ) ≡ ( Rg )( η, t ) = Z η ⊥ g ( tη + y ) d η y, (2.29)where d η y denotes the Euclidean measure on η ⊥ . The dual transform h → R ∗ h averages a function h on G ( n, n − 1) over the set of allhyperplanes passing through a fixed point x ∈ R n . Specifically,( R ∗ h )( x ) = 1 σ n − Z S n − h ( η, x · η ) dη. (2.30)The Funk transform of an even function f on the unit sphere S n in R n +1 , n ≥ 2, is defined by the formula( F f )( θ ) = Z S n ∩ θ ⊥ f ( σ ) d θ σ, θ ∈ S n , (2.31)where d θ σ stands for the standard probability measure on the ( n − S n ∩ θ ⊥ . The integral operator F is bounded from L ( S n ) to L ( S n ) and every integrable even function f can be explic-itly reconstructed from F f . A variety of different inversion formulasdepending on the class of functions can be found in [26, Section 5.1];see also [4, 14]. For example, the following statement holds. Theorem 2.8. (cf. [26, Theorem 5.40]) Let ϕ = F f , f ∈ L even ( S n ) , Φ θ ( s ) = 1 σ n − Z S n ∩ θ ⊥ ϕ ( sσ + √ − s θ ) dσ, − ≤ s ≤ . Then f ( θ ) = lim t → (cid:18) t ∂∂t (cid:19) n − n − t Z ( t − s ) ( n − / Φ θ ( s ) s n − ds . (2.32) In particular, for n odd, f ( θ ) = lim t → π / Γ( n/ (cid:18) t ∂∂t (cid:19) ( n − / [ t n − Φ θ ( t )] . (2.33) Altenatively, for all n ≥ we have f ( θ ) = lim t → (cid:18) ∂∂t (cid:19) n − n − t Z ( t − s ) ( n − / Φ θ ( s ) s ds . (2.34) The limit in these formulas is understood in the L -norm. If f ∈ C even ( S n ) , it can be interpreted in the sup -norm. Numerous properties of the hyperplane Radon transform, its dual,and the Funk transform are described in the literature; see, e.g., [4,14, 26] and references therein. Our main concern is explicit inversionformulas for these transforms under possibly minimal assumptions forfunctions.We set R n = R e ⊕ · · · ⊕ R e n , S n + = { θ = ( θ , . . . , θ n +1 ) ∈ S n : 0 < θ n +1 ≤ } . (2.35)Consider the projection map R n ∋ x µ −→ θ ∈ S n + , θ = µ ( x ) = x + e n +1 | x + e n +1 | , (2.36)for which x = µ − ( θ ) = θ ′ θ n +1 , θ ′ = ( θ , . . . , θ n ) . (2.37)The map µ extends to the bijection ˜ µ from the affine Grassmannian G ( n, n − 1) onto the set˜ S n + = { ω = ( ω , . . . , ω n +1 ) ∈ S n : 0 ≤ ω n +1 < } . (2.38)cf. (2.35). Specifically, if τ = τ ( η, t ) ∈ G ( n, n − η ∈ S n − ⊂ R n , t ≥ 0, and ˜ τ is the n -dimensional subspace containing the lifted plane τ + e n +1 , then ω is a normal vector to ˜ τ , so that ω ≡ ˜ µ ( τ ) = − η cos α + e n +1 sin α, tan α = t. (2.39) Theorem 2.9. Let ( Af )( τ ) = σ n − p | τ | f (˜ µ ( τ )) , τ ∈ G ( n, n − , (2.40)( Bg )( θ ) = 1 | θ n +1 | n g (cid:18) θ ′ θ n +1 (cid:19) , θ ∈ S n , θ n +1 = 0 . (2.41) If Z R n | g ( x ) | dx p | x | < ∞ , (2.42) then Bg ∈ L even ( S n ) and ( Rg )( τ ) = ( AF Bg )( τ ) , (2.43) where F is the Funk transform (2.31). ADON TRANSFORMS 11 Proof. We make use of Theorem 5.48 from [26], according to which( F f )( ω ) = ( A RB f )( ω ) , ω ∈ ˜ S n + , (2.44)where the operators A and B have the form( A h )( ω ) = 2 σ n − | ω ′ | h (˜ µ − ( ω )) , ω = ( ω ′ , ω n +1 ) , ω ′ = ( ω , . . . , ω n ) , ( B f )( x ) = (1 + | x | ) − n/ f ( µ ( x )) , x ∈ R n . Extending (2.44) by the evenness to all ω ∈ S n and passing to inverses,we obtain ( Rg )( τ ) = ( A − F B − g )( τ ) , τ ∈ G ( n, n − , where A − and B − are defined by the formulas( A − f )( τ ) = σ n − p | τ | f (˜ µ ( τ )) ( ≡ ( Af )( τ )) , (2.45)(here we use (2.39), so that | ω ′ | = cos α = (1+ t ) − = (1+ | τ | ) − ),( B − g )( θ ) = 1 | θ n +1 | n g (cid:18) θ ′ θ n +1 (cid:19) ( ≡ ( Bg )( θ )) , (2.46)(here we use (2.37) and extend the right-hand side as an even functionof θ ). This gives (2.43).The relation Bg ∈ L even ( S n ) that guarantees the existence of theFunk transform in (2.43) and justifies the above reasoning, is a con-sequence of (2.42). Indeed, using, e.g., [26, formula (1.12.17)], andassuming g ≥ 0, we have Z S n ( Bg )( θ ) dθ = 2 Z S n g (cid:18) θ ′ θ n +1 (cid:19) dθθ nn +1 = 2 Z R n g ( x ) dx p | x | < ∞ . (cid:3) Corollary 2.10. If g satisfies (2.42), then the Radon transform h ( τ ) =( Rg )( τ ) exists in the Lebesgue sense for almost all τ ∈ G ( n, n − andcan be inverted by the formula g ( x ) = ( B − F − A − h )( x ) , (2.47) where the inverse Funk transform F − is evaluated by Theorem 2.8 andthe operators A − and B − are defined by the formulas ( A − h )( ω ) = 2 σ n − | ω ′ | h (˜ µ − ( ω )) , | ω ′ | 6 = 0 , (2.48)( B − f )( x ) = (1 + | x | ) − n/ f ( µ ( x )) . (2.49) Our next aim is to obtain similar statements for the dual Radontransform (2.30). Theorem 2.11. Let ( A ∗ f )( x ) = 1 p | x | f − x + e n +1 p | x | ! , x ∈ R n , (2.50)( B ∗ h )( θ ) = 1 | θ ′ | n h (cid:18) θ ′ | θ ′ | , θ n +1 | θ ′ | (cid:19) , θ ∈ S n , | θ ′ | 6 = 0 . (2.51) If Z G ( n,n − | h ( τ ) | dτ p | τ | < ∞ , (2.52) then B ∗ h ∈ L even ( S n ) and ( R ∗ h )( x ) = ( A ∗ F B ∗ h )( x ) , (2.53) where F is the Funk transform (2.31).Proof. We make use Lemma 4.16 from [26], according to which( R ∗ h )( x ) = ( ˜ AR ˜ Bh )( x ) , (2.54)( ˜ A ˜ h )( x ) = 2 | x | σ n − ˜ h (cid:18) x | x | , | x | (cid:19) , (2.55)( ˜ Bh )( x ) = 1 | x | n h (cid:18) x | x | , | x | (cid:19) , x ∈ R n \ { } . (2.56)Here ˜ h is a function on G ( n, n − 1) parametrized by the pair ( η, t ) ∈ S n − × R + and extended to G ( n, n − ∼ S n − × R , so that ˜ h ( − η, − t ) =˜ h ( η, t ). In our case, η = x/ | x | , t = 1 / | x | . Combining (2.54) with (2.43),we obtain R ∗ = ˜ AAF B ˜ B . This formally gives (2.53) with A ∗ = ˜ AA , B ∗ = B ˜ B .Let us write ˜ AA and B ˜ B in the desired form. By (2.55) and (2.40),( A ∗ f )( x ) = ( ˜ AAf )( x ) = 2 | x | σ n − ( Af ) (cid:18) x | x | , | x | (cid:19) = 1 p | x | f (˜ µ ( τ )) , where τ = τ ( η, t ) with η = x/ | x | , t = 1 / | x | . By (2.39) with tan α = t = 1 / | x | , we have˜ µ ( τ ) = − η cos α + e n +1 sin α = − x + e n +1 p | x | . ADON TRANSFORMS 13 This gives (2.50). Further, by (2.41) and (2.56),( B ∗ h )( θ ) = ( B ˜ Bh )( θ ) = 1 | θ n +1 | n ( ˜ Bh ) (cid:18) θ ′ θ n +1 (cid:19) = 1 | θ ′ | n h (cid:18) θ ′ | θ ′ | , θ n +1 | θ ′ | (cid:19) provided that θ n +1 > | θ ′ | 6 = 0. Here h ( · , · ) ≡ h ( η, t ) with η = θ ′ / | θ ′ | , t = θ n +1 / | θ ′ | . Because h ( η, t ) = h ( − η, − t ), the last expressiongives (2.51).To complete the proof, it remains to show that g = ˜ Bh satisfies(2.42). For h ≥ 0, passing to polar coordinated and changing variables,we have Z R n ( ˜ Bh )( x ) dx p | x | = Z R n | x | n h (cid:18) x | x | , | x | (cid:19) dx p | x | = ∞ Z drr √ r Z S n − h ( η, /r ) dη = ∞ Z dt √ t Z S n − h ( η, t ) dη = Z G ( n,n − h ( τ ) dτ p | τ | < ∞ . (cid:3) Corollary 2.12. If h satisfies (2.52), then the dual Radon transform g ( x ) = ( R ∗ h )( x ) can be inverted by the formula h ( τ ) = ( B − ∗ F − A − ∗ g )( τ ) , τ ∈ G ( n, n − , (2.57) where the inverse Funk transform F − is evaluated by Theorem 2.8 andthe operators A − ∗ and B − ∗ are defined by the formulas ( A − ∗ g )( ω ) = 1 | θ n +1 | g (cid:18) − θ ′ θ n +1 (cid:19) , θ ∈ S n , θ n +1 = 0 , (2.58)( B − ∗ f )( τ ) = (1+ | τ | ) − n/ f (˜ µ ( − τ )) , − τ = { x ∈ R n : − x ∈ τ } . (2.59)The analytic expressions for A − ∗ and B − ∗ are consequences of theformulas (2.50) and (2.51) for A ∗ and B ∗ and the definitions of themaps µ and ˜ µ . Remark . If h is infinitely differentiable and rapidly decreasing,then the inversion formula for R ∗ can be written in the form (2.26).Specifically, h = c n R D n − n R ∗ h, c n = 2 − n π − n/ / Γ( n/ . (2.60)Here D n − n = ( − ∆ n ) ( n − / if n is odd. If n is even, then D n − n = Λ n − ,where Λ = P nj =1 H j ∂ j is the Calderon-Zygmund operator, H j , be-ing some singular integral operators, called the Riesz transforms. In this case, D n − n R ∗ h ∈ C ∞ ( R n ) and has the order O ( | x | − n ); see Sol-mon [31, p. 340] for details. The modification of (2.60) in the form h = c n D n − RR ∗ h was proved by Helgason [13, Theorem 4.5] in theframework of the Semyanistyi-Lizorkin spaces of Schwartz functionsorthogonal to polynomials; see also [26, formula (4.6.38)]. When n isodd, the formula (2.60) was obtained by Gonzalez in [7] (see also [8,Theorem 2.1]). More information on this subject, including furtherreferences, can be found in [26, p. 275, Notes 4.6].Our formula (2.57) does not contain singular integral operators andis applicable to a much larger class of functions.3. Mixed j -Plane to k -Plane Radon Transforms Setting of the Problem. Let τ ∈ G ( n, j ), ζ ∈ G ( n, k ), so that τ ≡ τ ( ξ, u ) = ξ + u, ξ ∈ G n,j , u ∈ ξ ⊥ ,ζ ≡ ζ ( η, v ) = η + v, η ∈ G n,k , v ∈ η ⊥ . The planes τ ( ξ, u ) and ζ ( η, v ) are called perpendicular (we write τ ⊥ ζ )if a · b = 0 for all a ∈ ξ and b ∈ η . We define the set of incidence I = { ( τ, ζ ) ∈ G ( n, j ) × G ( n, k ) : τ ⊥ ζ , τ ∩ ζ = ∅ } and denoteˆ ζ = { τ ∈ G ( n, j ) : ( τ, ζ ) ∈ I } , ˇ τ = { ζ ∈ G ( n, k ) : ( τ, ζ ) ∈ I } . The corresponding mixed j -plane to k -plane Radon transform has theform ( R j,k f )( ζ ) = Z ˆ ζ f ( τ ) d ζ τ, ( R k,j ϕ )( τ ) = Z ˇ τ ϕ ( ζ ) d τ ζ . (3.1) Remark . It is clear that if ( τ, ζ ) ∈ I , then, necessarily, j + k ≤ n ,because otherwise, there exist a ∈ ξ and b ∈ η such that a · b = 0. If j + k = n , then ξ = η ⊥ and( R j,k f )( ζ ) ≡ ( R j,k f )( η, v ) = Z η f ( η ⊥ + u ) d η u ∀ v ∈ η ⊥ . This integral operator is in general non-injective. Indeed, if f is a radialfunction, that is, f ( ξ, u ) ≡ ˜ f ( | u | ) for some single-variable function ˜ f ,then R j,k f ≡ const on the set of all functions f of the form f ( ξ, u ) = ADON TRANSFORMS 15 ˜ f λ ( | u | ) = λ − k ˜ f ( | u | /λ ), λ > 0. Specifically, by rotation and dilationinvariance,( R j,k f )( ζ ) = Z η ˜ f λ ( | u | ) du = Z R k ˜ f λ ( | u | ) du = Z R k ˜ f ( | u | ) du ≡ const , R k = R e ⊕ · · · ⊕ R e k , provided that this integral converges. Because of the lack of injectivity,the case j + k = n will be excluded from our consideration and wealways assume j + k < n .The operators (3.1) can be explicitly written as( R j,k f )( ζ ) ≡ ( R j,k f )( η, v ) = Z G j ( η ⊥ ) d η ⊥ ξ Z η f ( ξ + u + v ) d η u, (3.2)( R k,j ϕ )( τ ) = ( R k,j ϕ )( ξ, u ) = Z G k ( ξ ⊥ ) d ξ ⊥ η Z ξ f ( η + u + v ) d ξ v. (3.3)Here d η ⊥ ξ and d ξ ⊥ η denote the canonical probability measures on thecorresponding Grassmannians G j ( η ⊥ ) and G k ( ξ ⊥ ), d η u and d ξ v standfor the Euclidean measures on η and ξ , respectively.If j = 0 and k ≥ 1, then R j,k is the Radon-John transform (2.13).If j ≥ k = 0, then R j,k is the dual j -plane transform; cf. (2.14)with k replaced by j .Our aim is to investigate the operators (3.2) and (3.3) under theassumption j > , k > , j + k < n. Duality.Lemma 3.2. The duality relation Z G ( n,k ) ( R j,k f )( ζ ) ϕ ( ζ ) dζ = Z G ( n,j ) f ( τ )( R k,j ϕ )( τ ) dτ (3.4) holds provided that either integral exists in the Lebesgue sense.Proof. Let I r and I l denote the right-hand side and the left-hand sideof (3.4), respectively; E j = R e ⊕ · · · ⊕ R e j , E k = R e n − k +1 ⊕ · · · ⊕ R e n . (3.5) We set I = Z M ( n ) f ( g E j ) ϕ ( g E k ) dg = Z O ( n ) dρ Z R n f ( a + ρ E j ) ϕ ( a + ρ E k ) da . (3.6)It suffices to show that I = I r = I l . Let O j and O k be the stationarysubgroups of O ( n ) at E j and E k , respectively. We replace ρ by ρρ j ( ρ j ∈ O j ), then integrate in ρ j ∈ O j , and, after that, replace a by ρa . Changing the order of integration and taking into account that ρ j E j = E j , we obtain I = Z O ( n ) dρ Z O j dρ j Z R n f ( ρ ( a + E j )) ϕ ( ρ ( a + ρ j E k )) da. (3.7)Then we set a = t + b , where t ∈ E j , b ∈ E ⊥ j . Because t + E j = E j ,(3.7) gives I = Z O ( n ) dρ Z E ⊥ j f ( ρ ( b + E j )) db Z E j dt Z O j ϕ ( ρb + ρ ( t + ρ j E k )) dρ j = Z O ( n ) dρ Z E ⊥ j f ( ρ ( b + E j )) ( R k,j ϕ )( ρ ( b + E j )) db = I r . The proof of I = I l is similar. (cid:3) Mixed Radon Transforms of Radial Functions We recall that a function f on G ( n, j ) is radial, if there is a function f on R + , such that f ( τ ) = f ( | τ | ). If f is radial, then, by (3.2),( R j,k f )( η, v ) = Z G k ( η ⊥ ) d η ⊥ ξ Z η f ( | u + Pr ξ ⊥ v | ) d η u, (4.1)where Pr ξ ⊥ v denotes the orthogonal projection of v onto ξ ⊥ . The right-hand side of this equality is the k -plane transform of a radial functionrestricted to the ( n − j )-dimensional subspace ξ ⊥ combined with thedual j -plane transform restricted to the ( n − k )-dimensional space η ⊥ . Lemma 4.1. If f ( τ ) ≡ f ( | τ | ) satisfies the conditions a Z | f ( t ) | t n − j − dt < ∞ and ∞ Z a | f ( t ) | t k − dt < ∞ (4.2) ADON TRANSFORMS 17 for some a > , then ( R j,k f )( ζ ) = ( I j,k f )( | ζ | ) , (4.3) where ( I j,k f )( s ) = c s n − k − s Z ( s − r ) j/ − r ℓ − dr ∞ Z r f ( t )( t − r ) k/ − tdt = ˜ c s n − k − ( I j/ , r ℓ − I k/ − , f )( s ) , l = n − j − k ≥ , (4.4) c = σ j − σ k − σ ℓ − σ n − k − , ˜ c = π k/ Γ(( n − k ) / ℓ/ . (4.5) Moreover, β Z α | ( I j,k f )( s ) | ds < ∞ for all < α < β < ∞ . (4.6) Proof. It is straightforward to show that the assumptions in (4.2) imply(4.6); see Appendix. Then ( I j,k f )( s ) < ∞ for almost all s > 0, andtherefore the proof of (4.3) presented below is well-justified.We transform the integral( R j,k f )( ζ ) ≡ ( R j,k f )( η, v ) = Z G j ( η ⊥ ) d η ⊥ ξ Z η f ( ξ + u + v ) d η u by changing variable ξ = γξ ′ , where γ ∈ O ( n ) is an orthogonal trans-formation satisfying γ E k = η . Then, by (2.3), ξ ′ ⊂ E ⊥ k = R n − k = R e ⊕ · · · ⊕ R e n − k , so that ξ ′ ∈ G n − k,j = G j ( R n − k ). We also set u = γu ′ , v = γv ′ , where u ′ ∈ E k , v ′ ∈ E ⊥ k = R n − k ). This gives( R j,k f )( η, v ) = Z G j ( R n − k ) dξ ′ Z E k ( f ◦ γ )( ξ ′ + u ′ + v ′ ) du ′ (4.7)= Z G j ( R n − k ) dξ ′ Z E k f ( | ξ ′ + u ′ + v ′ | ) du ′ = Z G j ( R n − k ) dξ ′ Z E k f (cid:16)q | u ′ | + | Pr ξ ′⊥ v ′ | (cid:17) du ′ , = σ k − ∞ Z r k − dr Z G j ( R n − k ) f (cid:16)q r + | Pr ξ ′⊥ v ′ | (cid:17) dξ ′ , where Pr ξ ′⊥ v ′ denotes the orthogonal projection of v ′ onto ξ ′⊥ . We set ξ ′ = α E j , α ∈ O ( n − k ). Then (cf. (2.3)) ξ ′⊥ = α E ⊥ j , E ⊥ j = R n − j = R e j +1 ⊕ · · · ⊕ R e n , and therefore( R j,k f )( η, v ) = σ k − ∞ Z r k − dr Z O ( n − k ) f (cid:16)p r + | Pr α R n − j v ′ | (cid:17) dα. Because v ′ ∈ R n − k = R e ⊕ · · · ⊕ R e n − k , we havePr α R n − j v ′ = Pr α R n − j ∩ R n − k v ′ = Pr α ( R n − j ∩ R n − k ) v ′ = Pr α E ℓ v ′ , where E ℓ = R e j +1 ⊕ · · · ⊕ R e n − k , ℓ = n − k − j. Thus( R j,k f )( η, v ) = σ k − ∞ Z r k − dr Z O ( n − k ) f (cid:18)q r + | Pr α E ℓ v ′ | (cid:19) dα. Keeping in mind that | Pr α E ℓ v ′ | = | Pr E ℓ α − v ′ | and setting s = | v | , wecan write the last expression as( R j,k f )( η, v ) = σ k − σ n − k − ∞ Z r k − dr Z S n − k − f ( q r + s | Pr E ℓ θ | ) dθ, ADON TRANSFORMS 19 where S n − k − is the unit sphere in R n − k . The inner integral can betransformed by making use of the bi-spherical coordinates θ = a cos ψ + b sin ψ, a ∈ S n − ∩ E ℓ , b ∈ S n − ∩ E j , < ψ < π/ ,dθ = sin j − ψ cos ℓ − ψ da db dψ, see, e.g., [26, p. 31]. Setting c = σ k − σ ℓ − σ j − /σ n − k − , we obtain( R j,k f )( η, v ) = c ∞ Z r k − dr π/ Z f (cid:16)p r + s cos ψ (cid:17) sin j − ψ cos ℓ − ψ dψ = c ∞ Z r k − dr Z f (cid:16) √ r + s λ (cid:17) (1 − λ ) j/ − λ ℓ − dλ = c s n − k − ∞ Z r k − dr s Z f (cid:16) √ r + t (cid:17) ( s − t ) j/ − t ℓ − dt = c s n − k − s Z ( s − r ) j/ − r ℓ − dr ∞ Z r f ( t ) ( t − r ) k/ − t dt . (cid:3) The following analogue of Lemma 4.1 for the dual transform R k,j ϕ follows from Lemma 4.1 by the symmetry. Lemma 4.2. If ϕ ( ζ ) ≡ ϕ ( | ζ | ) satisfies the conditions a Z | ϕ ( s ) | s n − k − ds < ∞ and ∞ Z a | ϕ ( s ) | s j − ds < ∞ (4.8) for some a > , then ( R k,j ϕ )( τ ) = ( I ∗ j,k ϕ )( | τ | ) , (4.9) where ( I ∗ j,k ϕ )( t ) = c t n − j − t Z ( t − r ) k/ − r ℓ − dr ∞ Z r ϕ ( s )( s − r ) j/ − s ds = ˜ c t n − j − ( I k/ , r ℓ − I j/ − , ϕ )( t ) , l = n − j − k ≥ , (4.10) c = σ j − σ k − σ ℓ − σ n − j − , ˜ c = π j/ Γ(( n − j ) / ℓ/ . (4.11) Moreover, β Z α | ( I ∗ j,k ϕ )( t ) | dt < ∞ for all < α < β < ∞ . (4.12) Remark . By Lemma 2.1 (ii), the finiteness of the second integralsin (4.2) and (4.8) is necessary for the existence of the correspondingintegrals ( I j,k f )( s ) and ( I ∗ j,k ϕ )( t ). Example 4.4. The following formulas can be easily obtained from(4.4) and (4.10) using tables of integrals (see, e.g., [9]):(i) If f ( τ ) = | τ | − λ , k < λ < n − j , then ( R j,k f )( ζ ) = c j,k | ζ | k − λ , where c j,k = π k/ Γ (cid:0) n − k (cid:1) Γ (cid:0) λ − k (cid:1) Γ (cid:0) n − j − λ (cid:1) Γ (cid:0) n − j − k (cid:1) Γ (cid:0) λ (cid:1) Γ (cid:0) n − λ (cid:1) . (4.13)(ii) If ϕ ( ζ ) = | ζ | − λ , j < λ < n − k , then ( R k,j ϕ )( τ ) = c k,j | τ | j − λ , where c k,j = π j/ Γ (cid:0) n − j (cid:1) Γ (cid:0) λ − j (cid:1) Γ (cid:0) n − k − λ (cid:1) Γ (cid:0) n − j − k (cid:1) Γ (cid:0) λ (cid:1) Γ (cid:0) n − λ (cid:1) . (4.14)(iii) If f ( τ ) = (1 + | τ | ) − n/ , then ( R j,k f )( ζ ) = c k (1 + | ζ | ) ( j + k − n ) / ,where c k = π k/ Γ (cid:0) n − k (cid:1) Γ (cid:0) n (cid:1) . (4.15)(iv) If ϕ ( ζ ) = (1 + | ζ | ) − n/ , then ( R k,j ϕ )( τ ) = c j (1 + | τ | ) ( j + k − n ) / ,where c j = π j/ Γ (cid:0) n − j (cid:1) Γ (cid:0) n (cid:1) . (4.16)5. Existence of the Mixed Radon Transforms Example 4.4 in conjunction with duality (3.4) gives informationabout the existence in the Lebesgue sense of the corresponding Radontransforms R j,k f and R k,j ϕ . Theorem 5.1. The following formulas hold provided that the integralin either side of the corresponding equality exists in the Lebesgue sense. Z G ( n,k ) ( R j,k f )( ζ ) | ζ | λ dζ = c k,j Z G ( n,j ) f ( τ ) | τ | λ − j dτ, j < λ < n − k ; (5.1) Z G ( n,k ) ( R j,k f )( ζ )(1 + | ζ | ) n/ dζ = c j Z G ( n,j ) f ( τ )(1 + | τ | ) ( n − j − k ) / dτ ; (5.2) ADON TRANSFORMS 21 Z G ( n,j ) ( R k,j ϕ )( τ ) | τ | λ dτ = c j,k Z G ( n,k ) ϕ ( ζ ) | ζ | λ − k dζ , k < λ < n − j ; (5.3) Z G ( n,j ) ( R k,j ϕ )( τ )(1 + | τ | ) n/ dτ = c k Z G ( n,k ) ϕ ( ζ )(1 + | ζ | ) ( n − j − k ) / dζ . (5.4) Here c k,j , c j , c j,k , and c k have the same meaning as in Example 4.4. The next theorems characterize the existence of the Radon trans-forms R j,k f an R k,j ϕ in different terms. Theorem 5.2. Let j + k < n . If f is a locally integrable function on G ( n, j ) satisfying Z | τ | >a | f ( τ ) | | τ | k + j − n dτ < ∞ (5.5) for some a > , then ( R j,k f )( ζ ) is finite for almost all ζ ∈ G ( n, k ) .If for a nonnegative, radial, locally integrable function f , the condition(5.5) fails, then ( R j,k f )( ζ ) = ∞ for all ζ ∈ G ( n, k ) .Proof. It suffices to show that I α,β ≡ Z α< | ζ | <β | ( R j,k f )( ζ ) | dζ < ∞ for all 0 < α < β < ∞ . (5.6)Because R j,k commutes with orthogonal transformations, I α,β ≡ Z O ( n ) dγ Z α< | ζ | <β | ( R j,k f )( γζ ) | dζ ≤ Z α< | ζ | <β ( R j,k ˜ f )( ζ ) dζ , where ˜ f ( τ ) = R O ( n ) | f ( γτ ) | dγ is a radial function. We set ˜ f ( τ ) = f ( | τ | ). The assumptions for f in the lemma imply (4.2) for f . Indeed, a Z f ( t ) t n − j − dt = a Z t n − j − dt Z O ( n ) | f ( γ ( E j + te j +1 )) | dγ = a Z t n − j − dt Z O ( n − j ) dω Z O ( n ) | f ( γ ( E j + tωe j +1 )) | dγ = 1 σ n − j − a Z t n − j − dt Z O ( n ) dγ Z S n − j − | f ( γ ( E j + tσ )) | dσ = 1 σ n − j − Z O ( n ) dγ Z y ∈ E ⊥ j , | y | a | f ( τ ) | | τ | k + j − n dτ < ∞ . Hence, by (4.3) and (4.6), I α,β ≤ Z α< | ζ | <β ( R j,k ˜ f )( ζ ) dζ = Z α< | ζ | <β ( I j,k f )( | ζ | ) dζ = σ n − j − β Z α ( I j,k f )( s ) s n − j − ds ≤ c β Z α ( I j,k f )( s ) ds < ∞ , as desired.To complete the proof, suppose that (5.5) fails for some nonnegative,radial, locally integrable function f . If f ( τ ) ≡ f ( | τ | ) then the inner ADON TRANSFORMS 23 integral in (3.2) becomes I ( ξ, η, v ) ≡ Z η f ( ξ + u + v ) d η u = Z η f (cid:16)q | u | + | Pr ξ ⊥ v | (cid:17) d η u = σ k − ∞ Z s f ( t )( t − s ) k/ − tdt, s = | Pr ξ ⊥ v | . The condition (5.5) is equivalent to ∞ Z a f ( t ) t k − dt = ∞ for all a > . By Lemma 2.1(ii), it follows that I ( ξ, η, v ) = ∞ for all triples ( ξ, η, v ),and therefore ( R j,k f )( ζ ) = ∞ for all ζ ∈ G ( n, k ). (cid:3) The following statement is an analogue of Theorem 5.2 for the dualtransform R k,j ϕ and holds by the symmetry. Theorem 5.3. Let j + k < n . If ϕ is a locally integrable function on G ( n, k ) satisfying Z | ζ | >a | ϕ ( ζ ) | | ζ | k + j − n dζ < ∞ (5.7) for some a > , then ( R k,j ϕ )( τ ) is finite for almost all τ ∈ G ( n, j ) .If for a nonnegative, radial, locally integrable function ϕ , the condition(5.7) fails, then ( R k,j ϕ )( τ ) = ∞ for all τ ∈ G ( n, j ) . Corollary 5.4. (i) If f ∈ L p ( G ( n, j )) , ≤ p < ( n − j ) /k , then ( R j,k f )( ζ ) is finite foralmost all ζ ∈ G ( n, k ) . (ii) If ϕ ∈ L q ( G ( n, k )) , ≤ q < ( n − k ) /j , then ( R k,j ϕ )( τ ) is finite foralmost all τ ∈ G ( n, j ) .The bounds p < ( n − j ) /k and q < ( n − k ) /j in these statements aresharp.Proof. (i) By Theorem 5.2, it suffices to check (5.5). For any a > Z | τ | >a | f ( τ ) | | τ | k + j − n dτ ≤ || f || p Z | τ | >a | τ | ( k + j − n ) p ′ dτ ! /p ′ , p + 1 p ′ = 1 . Because the integral in bracket is finite whenever 1 ≤ p < ( n − j ) /k ,the result follows. The proof of (ii) is similar. If p ≥ ( n − j ) /k , thefunction f ( τ ) = (2 + | τ | ) ( j − n ) /p (log(2 + | τ | )) − (5.8)provides a counter-example. Indeed, this function belongs to L p ( G ( n, j ))and does not obey (5.5). The case q ≥ ( n − k ) /j is similar. (cid:3) Remark . It is interesting to note that the same bounds for p and q can be obtained from (5.2) and (5.4) if we apply H¨older’s inequalityto the right-hand sides.6. Mixed Radon Transforms and Riesz Potentials It is known that the classical hyperplane Radon transform, its k -plane generalization, and their duals intertwine Laplace operators onthe source space and the target space. More general intertwining for-mulas can be obtained if we replace the Laplace operators by the cor-responding Riesz potentials; cf. (2.25). Our aim in this section is toextend these formulas to the mixed Radon transforms (3.1). We alsoobtain certain Grassmannian analogues of Fuglede’s formula (2.24) andits generalizations (2.25). Throughout this section, we keep the nota-tion from Subsection 2.3.6.1. Intertwining Formulas.Theorem 6.1. If < α < n − k − j , then ( I αn − k R j,k f )( ζ ) = ( R j,k I αn − j f )( ζ ) , ζ ∈ G ( n, k ) , (6.1) provided that either side of this equality exists in the Lebesgue sense.Remark . Before we prove this theorem, some comments are in or-der. In the limiting cases j = 0 (the k -plane transform) and k = 0(the dual j -plane transform), the formula (6.1) agrees with (2.25). Thecorresponding formulas for the hyperplane Radon transform and itsdual can be found in [26, Proposition 4.38]. It is natural to conjecture that for sufficiently good f , (6.1) extendsby analyticity to all complex α . If α = − m , m ∈ { , , . . . } , then (6.1)reads ( − ∆ n − k ) m R j,k f = R j,k ( − ∆ n − j ) m f, (6.2)where ∆ n − k and ∆ n − j stand for the Laplace operators on the corre-sponding fibers. ADON TRANSFORMS 25 In the case j + k = n − m = 1, the formula (6.2) was proved byGonzalez under the assumption f ∈ C ∞ c ( G ( n, k )); cf. [8, Lemma 3.3].Similar formulas for the Radon-John transform and its dual are due toHelgason [13, Lemma 8.1]. For the hyperplane Radon transform on R n and its dual, an analogueof (6.1) for all α ∈ C in the corresponding Semyanistyi-Lizorkin spacescan be found in [26, Proposition 4.58]. Proof of Theorem 6.1. We assume ζ ≡ ζ ( E k , 0) and set I = ( I αn − k R j,k f )( E k , , J = ( R j,k I αn − j f )( E k , . Because the operators in (6.1) commute with rigid motions, it sufficesto show that I = J . We recall the notation for the subspaces: E j = R e ⊕ · · · ⊕ R e j , E k = R e n − k +1 ⊕ · · · ⊕ R e n ; R n − j = R e j +1 ⊕ · · · ⊕ R e n , R n − k = R e ⊕ · · · ⊕ R e n − k . Then I = 1 γ n − k ( α ) Z R n − k | v | α − n + k dv Z G j ( R n − k ) dξ Z E k ∩ ξ ⊥ f ( ξ + u + v ) du = 1 γ n − k ( α ) Z R n − k | v | α − n + k dv Z O ( n − k ) dγ Z E k f ( γ ( E j + u + v )) du = 1 γ n − k ( α ) Z E k du ∞ Z r α − dr Z S n − k − dθ Z O ( n − k ) f ( γ ( E j + u + rθ )) dγ. This expression can be transformed by making use of the bi-sphericalcoordinates [26, p. 31] θ = ϕ cos ω + ψ sin ω, ϕ ∈ S j − , ψ ∈ S n − k − j − , < ω < π/ ,dθ = cos j − ω sin n − k − j − ω dϕ dψ dω. We obtain I = 1 γ n − k ( α ) Z E k du ∞ Z r α − dr π/ Z cos j − ω sin n − k − j − ω dω × Z S j − dϕ Z S n − k − j − dψ Z O ( n − k ) f ( γ ( E j + u + rψ sin ω )) dγ. The integral over O ( n − k ) depends only on r sin ω and u . We denoteit by ˜ f ( r sin ω, u ) and continue: I = σ j − σ n − k − j − γ n − k ( α ) Z E k du π/ Z cos j − ω sin n − k − j − ω dω ∞ Z r α − ˜ f ( r sin ω, u ) dr = c Z E k du ∞ Z s α − ˜ f ( s, u ) ds, (6.3)where c = σ j − σ n − k − j − γ n − k ( α ) π/ Z cos j − ω sin n − k − j − α − ω dω = 2 − a Γ(( n − k − j − α ) / α/ 2) Γ(( n − k − j ) / . Let us show that J has the form (6.3) too. We have J = Z G j ( R n − k ) dξ Z E k ∩ ξ ⊥ ( I αn − j f )( ξ, u ) du (note that E k ∩ ξ ⊥ = E k )= 1 γ n − j ( α ) Z O ( n − k ) dγ Z E k du Z R n − j | y | α − n + j f ( γ ( E j + u + y )) dy = 1 γ n − j ( α ) Z E k du ∞ Z r α − dr Z S n − j − dθ Z O ( n − k ) f ( γ ( E j + u + rθ )) dγ. Setting θ = ϕ cos ω + ψ sin ω, ϕ ∈ S n − k − j − , ψ ∈ S k − , < ω < π/ , (cf. the first part of the proof), we continue J = 1 γ n − j ( α ) Z E k du ∞ Z r α − dr π/ Z cos n − j − k − ω sin k − ω dω × Z S n − k − j − dϕ Z S k − dψ Z O ( n − k ) f ( γ ( E j + u + rϕ cos ω + rψ sin ω )) dγ. ADON TRANSFORMS 27 The integral over O ( n − k ) depends only on r cos ω and u . We denoteit by ˜ f ( r cos ω, u ). Then J = σ k − σ n − k − j − γ n − j ( α ) Z E k du π/ Z cos n − j − k − ω sin k − ω dω ∞ Z r α − ˜ f ( r cos ω, u ) dr = c Z E k du ∞ Z s α − ˜ f ( s, u ) ds, (6.4)where c = σ k − σ n − k − j − γ n − j ( α ) π/ Z cos n − j − k − α − ω sin k − ω dω = 2 − a Γ(( n − k − j − α ) / α/ 2) Γ(( n − k − j ) / 2) = c . Comparing (6.3) and (6.4), we complete the proof.6.2. Fuglede Type Formulas. We introduce the following integraloperators acting on functions h : R n → C :Λ j,k h = R ∗ k R j,k R j h, Λ k,j h = R ∗ j R k,j R k h. (6.5)If h is a radial function, h ( x ) = h ( | x | ), then (4.3) together with (2.17)and (2.18) givesΛ j,k h = ˜ c π j/ Γ( n/ n − k ) / r − n I k/ , I j/ , s n − j − k − I k/ − , I j/ − , h = ˜ c π j/ Γ( n/ n − k ) / r − n I ( j + k ) / , s n − j − k − I ( j + k ) / − , h . By (2.27), the last expression is a constant multiple of the Riesz po-tential I j + kn h .This observation paves the way to the following general result. Theorem 6.3. If j + k < n , then R ∗ k R j,k R j h = R ∗ j R k,j R k h = c I j + kn h, c = 2 j + k π ( j + k ) / Γ( n/ n − j − k ) / , (6.6) provided that the Riesz potential I j + kn h exists in the Lebesgue sense. Proof. Because all operators in (6.6) commute with rigid motions and j and k are interchangeable, it suffices to show that ( R ∗ k R j,k R j h )(0) = c ( I j + kn h )(0). By (2.14) and (3.2),( R ∗ k R j,k R j h )(0) = Z O ( n ) ( R j,k R j h ) ( γ E k ) dγ = Z O ( n ) dγ Z G j ( R n − k ) dξ Z E k ( R j h ) ( γξ + γy ) dy = Z O ( n ) dγ Z O ( n − k ) dα Z E k ( R j h ) ( γα E j + γy ) dy = Z O ( n ) dγ Z E k ( R j h ) ( γ ( E j + y )) dy. Using (2.13) (with k replaced by j ), we write the last expression asfollows. Z O ( n ) dγ Z E k dy Z E j h ( γ ( y + z )) dz = Z O ( n ) dγ Z E j ⊕ E k h ( γ ˜ y ) d ˜ y = σ j + k − σ n − ∞ Z r j + k − dr Z S n − h ( rθ ) dθ = σ j + k − σ n − Z R n h ( x ) | x | j + k − n dx = c ( I j + kn h )(0) , as desired. (cid:3) The formula (6.6) is a generalization of Fuglede’s formula (2.24). Thelatter can be obtained from (6.6) if we formally set j = 0 or k = 0.7. Inversion formulas for R j,k f The following preliminary discussion explains the essence of the mat-ter and the plan of the section. It might be natural to expect that R j,k is injective on some standard function space, like C ∞ c ( G ( n, j )),if dim G ( n, j ) ≤ dim G ( n, k ), which is equivalent to ( j + 1)( n − j ) ≤ ( k + 1)( n − k ). The latter splits in two cases:(a) j + k = n − j and k ;(b) j + k < n − j ≤ k . ADON TRANSFORMS 29 In the present paper we do not investigate both (a) and (b) in fullgenerality and proceed as follows. We first consider R j,k f for radial f , when an explicit inversion formula is available for all j + k < n .Then we address to the case (a) and obtain an inversion formula in asufficiently large class of functions f , including functions in Lebesguespaces and continuous functions. The case of all j + k < n for suchfunctions remains open. Some progress can be achieved if we restrictthe class of functions f to the range of the j -plane transform. Underthis assumption, R j,k f can be explicitly inverted, no matter whether j ≤ k or vice versa.7.1. The Radial Case. If f is radial, then R j,k f is radial too and wehave the following result. Theorem 7.1. Let f ( τ ) ≡ f ( | τ | ) be a locally integrable radial functionon G ( n, j ) satisfying (5.5). The function f can be recovered from theRadon transform ( R j,k f )( ζ ) ≡ ( I j,k f )( | ζ | ) for all j + k < n by theformula f ( t ) = ˜ c − ( D k/ − , r − n + j + k D j/ , s n − k − I j,k f )( t ) , (7.1) where ˜ c = π k/ Γ(( n − k ) / / Γ(( n − j − k ) / and the Erd´elyi–Koberfractional derivatives D k/ − , and D j/ , are defined by (2.7)-(2.10).Proof. By (4.4),( I j,k f )( s ) = ˜ c s k +2 − n ( I j/ , r n − j − k − I k/ − , f )( s ) . (7.2)The assumption (5.5) is equivalent to R ∞ a | f ( t ) | t k − dt < ∞ for some a > 0. The local integrability of f implies r n − j − k − I k/ − , f ∈ L loc ( R + )(a simple calculation is left to the reader). Hence the conditions ofTheorems 2.2 and 2.3 are satisfied and both fractional integrals in (7.2)can be inverted to give (7.1). (cid:3) Interchanging j and k , the reader can easily obtain a similar state-ment for the dual transform R k,j ϕ .7.2. The Case j + k = n − . The Structure of R j,k f . We consider the flag F = { ( ξ, η, v ) : ξ ∈ G n,j , η ∈ G k ( ξ ⊥ ) , v ∈ ξ ⊥ ∩ η ⊥ } . Given a function f on G ( n, j ), we define a function ˜ f on F by theformula ˜ f ( ξ, η, v ) = Z η f ( ξ + u + v ) d η u. This function is the inner integral in the definition (3.2) of R j,k f andhas two interpretations. On the one hand, for every fixed ξ ∈ G n,j , ˜ f isthe k -plane transform of the function u → f ( ξ, u ) in the ( n − j )-space ξ ⊥ :˜ f ξ ( ζ ) ≡ ˜ f ( ξ, η, v ) = ( R k,ξ ⊥ [ f ( ξ, · )])( ζ ) , ζ = ζ ( η, v ) ∈ G ( k, ξ ⊥ ) . (7.3)On the other hand, for every fixed η ∈ G n,k , ˜ f is a function on theaffine Grassmannian G ( j, η ⊥ ):˜ f η ( τ ′ ) ≡ ˜ f ( ξ, η, v ) = Z η f ( τ ′ + u ) du, τ ′ = τ ′ ( ξ, v ) ∈ G ( j, η ⊥ ) . (7.4)The dual j -plane transform of ˜ f η ( · ) in the ( n − k )-space η ⊥ at the point v ∈ η ⊥ has the form( R ∗ j,η ⊥ ˜ f η )( v ) = Z O ( η ⊥ ) ˜ f η ( γξ + v ) dγ (7.5)= Z G j ( η ⊥ ) d η ⊥ ξ Z η f ( ξ + u + v ) d η u = ( R j,k f )( η, v ) , where O ( η ⊥ ) is the subgroup of O ( n ), which consists of orthogonaltransformations in η ⊥ .The above reasoning shows that R j,k is a certain mixture (but not acomposition) of the k -plane transform and the dual j -plane transform.7.2.2. Inversion Procedure. According to the structure of the operator R j,k , to reconstruct f from R j,k f , we first invert the dual j -plane trans-form (7.5) in the ( n − k )-space η ⊥ and then the k -plane transform (7.3)in the ( n − j )-space ξ ⊥ . Because, in general, the dual j -plane transformis injective only in the co-dimension one case (cf. [21, Theorem 4.4]),we restrict to the case j = n − k − 1, when the injectivity can be provedon a pretty large class of functions.STEP 1. We make use of Corollary 2.12 with R n replaced by η ⊥ , whenthe condition (2.52) becomes Z G ( j,η ⊥ ) | ˜ f η ( τ ′ ) | | τ ′ | dτ ′ < ∞ for almost all η ∈ G n,k . (7.6)Under this condition,˜ f η ( τ ′ ) = ( R ∗ η ⊥ ) − [( R j,k f )( η, · )]( τ ′ ) , τ ′ = ξ + v ∈ G ( j, η ⊥ ) , (7.7) ADON TRANSFORMS 31 where ( R ∗ η ⊥ ) − stands for the inverse dual j -plane transform in the( j + 1)-subspace η ⊥ . An explicit formula for ( R ∗ η ⊥ ) − can be obtainedfrom the equality (2.57) adapted for our case.Our next aim is to find sufficient conditions for (7.6) in terms of f .We observe that (7.6) will be proved if we show that I = Z G n,k dη Z G j ( η ⊥ ) d η ξ Z ξ ⊥ ∩ η ⊥ | ˜ f η ( ξ, v ) | | v | dv < ∞ . Changing the order of integration and using (7.3), we can write I as I = Z G n,j dξ Z G k ( ξ ⊥ ) d ξ η Z ξ ⊥ ∩ η ⊥ | ( R k,ξ ⊥ [ f ( ξ, · )])( η, v ) | | v | dv, where R k,ξ ⊥ stands for the k -plane transform in the ( k + 1)-space ξ ⊥ .Using (2.16) with n = k + 1, we have Z G k ( ξ ⊥ ) d ξ η Z ξ ⊥ ∩ η ⊥ | ( R k,ξ ⊥ [ f ( ξ, · )])( η, v ) | | v | dv ≤ c Z ξ ⊥ | f ( ξ, u ) | | u | log(2 + | u | ) du, whence I ≤ c Z G n,j dξ Z ξ ⊥ | f ( ξ, u ) | | u | log(2 + | u | ) du = c Z G ( n,j ) | f ( τ ) | | τ | log(2 + | τ | ) dτ. Thus the inversion formula (7.6) is valid if Z G ( n,j ) | f ( τ ) | | τ | log(2 + | τ | ) dτ < ∞ . (7.8) Definition 7.2. The class of all functions f satisfying (7.8) will bedenoted by L log ( G ( n, j )).STEP 2. By Corollary 2.10, the function f ( ξ, · ) on ξ ⊥ can be recon-structed from its k -plane transform ˜ f ξ ( ζ ) = ( R k,ξ ⊥ [ f ( ξ, · )])( ζ ) if Z ξ ⊥ | f ( ξ, u ) | | u | d u < ∞ . The latter is guaranteed for almost all ξ if Z G ( n,j ) | f ( τ ) | | τ | dτ < ∞ . (7.9) Thus Step 2 gives one more assumption for f which is, however, weakerthan (7.8).Combining Step 1 and Step 2, we arrive at the following statement. Theorem 7.3. If j + k = n − , then every function f ∈ L log ( G ( n, j )) can be reconstructed from ϕ = R j,k f by the formula f ( ξ, u ) = ( R − k,ξ ⊥ [ ˜ f ξ ])( u ) , (7.10) where ˜ f ξ ( η, v ) ≡ ˜ f η ( ξ, v ) = ( R ∗ η ⊥ ) − [ ϕ ( η, · )]( ξ, v ) , (7.11) with the inverse transforms R − k,ξ ⊥ and ( R ∗ η ⊥ ) − being defined accordingto Corollaries 2.10 and 2.12, respectively.Remark . The condition f ∈ L log ( G ( n, j )) in Theorem 7.3 falls intothe scope of the Existence Theorem 5.2 and differs from the latteronly by the logarithmic factor. Thus Theorem 7.3 provides inversion of R j,k f under almost minimal assumptions. Note also that by H¨older’sinequality, any function in L p ( G ( n, j )), 1 ≤ p < ( n − j ) /k , and anycontinuous function of order O ( | τ | − µ ), µ > k , belong to L log ( G ( n, j )),where the bounds for p and µ are sharp; cf. (5.8).7.3. Inversion of R j,k f on the Range of the j -Plane Transform. Theorem 6.3 implies the following inversion result, which resemblesCorollary 2.7 for the dual k -plane transform. Theorem 7.5. Let f = R j h , h ∈ L p ( R n ) . If ≤ p < n/ ( j + k ) , then f = c − R j D j + kn R ∗ k R j,k f, c = 2 j + k π ( j + k ) / Γ( n/ n − j − k ) / , (7.12) where D j + kn is the Riesz fractional derivative (2.21). More generally, if < α < n − j − k and ≤ p < n/ ( j + k + α ) , then f = c − R j D j + k + αn R ∗ k I αn − k R j,k f = c − R j D j + k + αn R ∗ k R j,k I αn − j f (7.13) with the same constant c .Proof. By (6.6), c − R j D j + kn R ∗ k R j,k f = c − R j D j + kn R ∗ k R j,k R j h = R j D j + kn I j + kn h = R j h = f. Further, combining (6.6) with the semigroup property of Riesz poten-tials, we obtain I αn R ∗ k R j,k f = I αn R ∗ k R j,k R j h = I αn I j + kn h = I j + k + αn h. However, by (2.25) and (6.1), I αn R ∗ k R j,k f = R ∗ k I αn − k R j,k f = R ∗ k R j,k I αn − j f. This gives (7.13). (cid:3) ADON TRANSFORMS 33 The formula (7.13) can be used if we want to replace the nonlocalRiesz fractional derivative by the local one. For example, if j + k isodd and j + k < n − 1, we can apply (7.13) with α = 1. Remark . If f is good enough, then (7.12) formally agrees with theinversion formula of Gonzalez [8, Theorem 3.4]. In our notation hisformula reads f = ˜ c − R k,j D n − n − k R j,k f, ˜ c = 2 n − π ( n − / Γ (cid:18) j +12 (cid:19) Γ (cid:18) k +12 (cid:19) . (7.14)The following non-rigorous reasoning shows the consistency of (7.12)and (7.14). It suffices to show that c − R j D n − n R ∗ k ϕ = ˜ c − R k,j D n − n − k ϕ, (7.15)where c is the constant from (7.12) with j + k = n − 1. We set ϕ = R k h and apply R ∗ j to both sides of (7.15). For the left-hand side, (2.24)yields c − R ∗ j R j D n − n R ∗ k R k h = c − c j,n c k,n I jn D n − n I kn h = π / Γ( n/ n − j ) / 2) Γ(( n − k ) / h. (7.16)For the right-hand side, the formula D n − n − k R k h = R k D n − n h (cf. (2.25))in conjunction with (6.6) gives˜ c − R ∗ j R k,j D n − n − k R k h = ˜ c − R ∗ j R k,j R k D n − n h = c I n − n D n − n h = c h, where c is exactly the same as in (7.16). Thus, if R ∗ j is injective andthe class of functions f is good enough, we are done.Note that the proof of convergence of the expression on right-handside of (7.14) is rather nontrivial, even for f ∈ C ∞ c ( G ( n, j )).8. Appendix Proof of (4.6). Let us show that (4.6) follows from (4.2). It sufficesto assume f ≥ 0. We recall that ℓ = n − j − k ≥ c stands for a constant that can be different at each occurrence. Let F ( r ) = ∞ Z r f ( t ) ( t − r ) k/ − t dt . For any 0 < α < β < ∞ we have β Z α ( I j,k f )( s ) ds ≤ c β Z α ds s Z ( s − r ) j/ − r ℓ − F ( r ) dr ≤ c β Z α ds s Z ( s − r ) j/ − r ℓ − F ( r ) dr = c α Z r ℓ − F ( r ) dr β Z α ( s − r ) j/ − ds + c β Z α r ℓ − F ( r ) dr β Z r ( s − r ) j/ − ds ≤ c β Z r ℓ − ( β − r ) j/ F ( r ) dr − c α Z r ℓ − ( α − r ) j/ F ( r ) dr. (8.1)Because the integrals in (8.1) have the same form, it suffices to showthat I ( α ) ≡ α Z r ℓ − ( α − r ) j/ F ( r ) dr < ∞ ∀ α > . We have I ( α ) = α Z r ℓ − ( α − r ) j/ dr ∞ Z r f ( t )( t − r ) k/ − t k/ dt ≤ c α Z f ( t ) t k/ ℓ − dt t Z ( t − r ) k/ − dr + c ∞ Z α f ( t ) t k/ dt α Z ( t − r ) k/ − dr = c α Z f ( t ) t k + ℓ − dt + c ∞ Z α f ( t ) t k/ ( t k/ − ( t − α ) k/ ) dt ≤ c α Z f ( t ) t n − j − dt + c ∞ Z α f ( t ) t k − dt < ∞ . The last expression is finite by (4.2). References [1] B. Fuglede, An integral formula, Math. Scand., 6 , 207–212 (1958). ADON TRANSFORMS 35 [2] F. B. Gonzalez and T. Kakehi, Pfaffian systems and Radon trans-forms on affine Grassmann manifolds, Math. Ann., (2) 326 , 237–273(2003).[3] F. B. Gonzalez, Dual Radon transforms on affine Grassmann man-ifolds, Trans. Amer. Math. Soc., (10) 356, 4161-4180 (2004).[4] I. M. Gelfand, S. G. Gindikin and M. I. Graev, Selected topics inintegral geometry, Translations of Mathematical Monographs, AMS,Providence, Rhode Island (2003).[5] I. M. Gel’fand, M. I. Graev and R. Ro¸su, The problem of integralgeometry and intertwining operators for a pair of real Grassmannianmanifolds, J. Operator Theory, 12 , 339–383 (1984).[6] I. M. Gel’fand, M. I. Graev and Z. Ja. Shapiro, A problem of integralgeometry connected with a pair of Grassmann manifolds, Dokl. Akad.Nauk SSSR, 193, No. 2, 892-896 (1970).[7] F. B. Gonzalez, Radon transform on Grassmann manifolds, Thesis,MIT, Cambridge, MA (1984).[8] F. B. Gonzalez, Radon transform on Grassmann manifolds, Journalof Func. Anal., 71, 339-362 (1987).[9] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series andproducts, Academic Press (1980).[10] M. I. Graev, A problem of integral geometry related to a tripleof Grassmann manifolds, Functional Analysis and its Applications,(4)34, 299-301 (2000).[11] E. L. Grinberg, Radon transforms on higher rank Grassmannians,J. Differential Geometry, 24, 53-68 (1986).[12] E. Grinberg and B. Rubin, Radon inversion on Grassmannians viaG˚arding-Gindikin fractional integrals, Annals of Math., 159, 809-843(2004).[13] S. Helgason, The Radon transform on Euclidean spaces, com-pact two-point homogeneous spaces and Grassmann manifolds, ActaMath., 113, 153-180 (1965).[14] S. Helgason, Integral geometry and Radon transform, Springer,New York-Dordrecht-Heidelberg-London (2011).[15] T. Kakehi, Integral geometry on Grassmann manifolds and calcu-lus of invariant differential operators, J. Funct. Anal., 168, 1-45(1999).[16] A. Markoe, Analytic tomography, Cambridge University Press(2006).[17] E. E. Petrov, The Radon transform in spaces of matrices and inGrassmann manifolds, Dokl. Akad. Nauk SSSR, 177, No. 4, 1504-1507(1967).[18] B. Rubin, One-dimensional representation, inversion and certainproperties of Riesz potentials of radial functions[in Russian], Mat. Zametki, (4) 34, 521-533 (1983).[19] B. Rubin, Inversion of k -plane transforms via continuous wavelettransforms, J. Math. Anal. Appl., 220, 187-203 (1998).[20] B. Rubin, Reconstruction of functions from their integrals over k -planes, Israel J. of Math., 141, 93-117 (2004).[21] B. Rubin, Radon transforms on affine Grassmannians, Trans.AMS, 356, 5045-5070(2004).[22] B. Rubin, Combined Radon transforms, Preprint (2006).[23] B. Rubin, Weighted norm inequalities for k-plane transforms,Proc. Amer. Math. Soc., 142, 3455-3467 (2014).[24] B. Rubin, On the Funk-Radon-Helgason inversion method in inte-gral geometry, Contemp. Math., 599, 175-198 (2013).[25] B. Rubin, Funk, cosine, and sine transforms on Stiefel and Grass-mann manifolds, J. of Geometric Analysis (3) , 1441-1497 (2013).[26] B. Rubin, Introduction to Radon transforms: With elements offractional calculus and harmonic analysis, Cambridge University Press(2015).[27] B. Rubin and Y. Wang, On Radon transforms between lines andhyperplanes, Internat. J. Math., 28, 93-117 (2017).[28] B. Rubin and Y. Wang, New inversion formulas for Radon trans-forms on affine Grassmannians, J. Funct. Anal., 274 , no. 10, 2792-2817(2018).[29] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional inte-grals and derivatives, theory and applications, Gordon and BreachSc. Publ., New York (1993).[30] V. I. Semyanistyi, On some integral transformations in Euclideanspace [in Russian], Dokl. Akad. Nauk SSSR, 134, 536-539 (1960).[31] D. C. Solmon, Asymptotic formulas for the dual Radon transformand applications, Math. Z., 195, 321-343 (1987).[32] R. S. Strichartz, Harmonic analysis on Grassmannian bundles,Trans. of the Amer. Math. Soc., 296, 387-409 (1986).[33] G. Zhang, Radon transform on real, complex, and quaternionicGrassmannians, Duke Math. J., 138, 137-160 (2007). Department of Mathematics, Louisiana State University, Baton Rouge,LA, 70803, USA E-mail address : [email protected] School of Mathematics and Information Science, Guangzhou Uni-versity, Guangzhou 510006, China; E-mail address ::