Weighted norm inequalities for k-plane transforms

We obtain sharp inequalities for the k-plane transform, the"j-plane to k-plane"transform, and the corresponding dual transforms, acting on $L^p$ spaces with a radial power weight. The operator norms are explicitly evaluated. Some generalizations and open problems are discussed.


Introduction
Mapping properties of Radon-like transforms were studied by many authors, e.g., [3]- [7], [9,10,13,14,17,22,24,27,32,35,36], to mention a few. Most of the publications deal with L p -L q estimates or mixed norm inequalities, when the problem is to minimize a gap between necessary and sufficient conditions and find the best possible bounds. We also mention a series of works devoted to weighted norm estimates for Radon-like transforms of radial functions; see, e.g., [8,19,20].
In the present article we show that for the k-plane transform in R n [16,28], the more general "j-plane to k-plane" transform [35, p. 701], [6,12,29], and the corresponding dual transforms, sharp estimates can be obtained if the action of these operators is considered in the L p -L q setting with p = q and radial power weights. In this case the proofs are elementary and self-contained. Our approach is inspired by a series of publications on operators with homogeneous kernel dating back, probably, to Schur [31]; see, e.g., [15,30,34]. It is not surprising that the same ideas are applicable to some operators of integral geometry, because the common point is invariance under rotations and dilations.
The weighted L 2 estimate for the hyperplane Radon transform in R n , n ≥ 3, was obtained by Quinto [26, p. 410] via spherical harmonics and Hardy's inequality. This estimate was generalized by Kumar and Ray [18] to all n ≥ 2 and all p by making use of the similar spherical harmonics techniques combined with interpolation. Our approach is completely different, covers these results, and provides the best possible constants, which coincide with explicitly evaluated operator norms.
Let us proceed to details. We denote by Π n,k the manifold of all non-oriented k-planes τ in R n ; G n,k is the Grassmann manifold of kdimensional linear subspaces ξ of R n ; 1 ≤ k ≤ n − 1. Each k-plane τ is parameterized by the pair (ξ, u), where ξ ∈ G n,k and u ∈ ξ ⊥ (the orthogonal complement of ξ in R n ). Thus, Π n,k is a bundle over G n,k with an (n − k)-dimensional fiber. The manifold Π n,k is endowed with the product measure dτ = dξdu, where dξ is the O(n)-invariant probability measure on G n,k and du denotes the volume element on ξ ⊥ . We define where || · || p is the norm in L p (Π n,k ). If w(τ ) = |τ | ν , where |τ | denotes the Euclidean distance from the plane τ ∈ Π n,k to the origin, we also write L p (Π n,k ; w) = L p ν (Π n,k ) and ||f || p,w = ||f || p,ν . The k-plane transform of a sufficiently good function f on R n is a function R k f on Π n,k defined by where dy is the volume element in ξ. The more general "j-plane to k-plane" transform takes a function f on Π n,j to a function R j,k f on Π n,k , 0 ≤ j < k < n, by the formula Here d ξ η denotes the probability measure on the manifold of all jdimensional linear subspaces η of ξ.
Apart of these theorems, we obtain similar statements for the corresponding dual transforms; see

Preliminaries
Notation. In the following σ n−1 = 2π n/2 /Γ(n/2) is the area of the unit sphere S n−1 in R n ; dσ(θ) stands for the surface element of S n−1 ; S n−1 + = {x = (x 1 , · · · , x n ) ∈ S n−1 : x n > 0} is the "upper" hemisphere of S n−1 ; e 1 , . . . , e n are coordinate unit vectors; G = O(n) is the group of orthogonal transformations of R n endowed with the invariant probability measure. This group acts on G n,k transitively. For g, γ ∈ G, we denote We will need the following simple statements.
Lemma 2.1. The norm of a function ϕ(τ ) ≡ ϕ(ξ, u) ∈ L p ν (Π n,k ) can be computed by the formula Proof. The proof is straightforward and based on the definition Proof.
3. Mapping properties of the k-plane transform 3.1. Preparations. The following explicit equalities, reflecting action of R k on weighted L 1 spaces, were obtained in [28,Theorem 2.3]. Then provided that either side of the corresponding equality exists in the Lebesgue sense. Then Proof. If µ > k − n/p, the first statement follows from (3.3) by Hólder's inequality.
3) holds with the new parameterμ and, therefore, (R k f )(τ ) is finite for almost all τ ∈ Π n,k . The second statement of the lemma follows from the Abel type representation [28, p. 98]: The scaling argument (cf. [33, p. 118]) yields the following.
Lemmas 3.2 and 3.3 contain necessary conditions for the operator R k to be bounded from L p µ (R n ) to L p ν (Π n,k ). It will be shown that these conditions, except µ = k − n when p = 1, are also sufficient.
The next statement is obvious.
Passing to the limit as ε → 0, we obtain , as desired; cf. (3.9). If p = 1, then ν = µ. We choose g 0 (r) = r µ and proceed as above. If p = ∞, we choose f 0 (r) = r −µ . Then ||f || ∞,µ = 1 and, by (3.7), Let g 0 (r) = 0 if r < 1 and g 0 (r) = r −δ , where δ is big enough. Then Letting δ → ∞, we obtain the result. Some comments are in order. In the case p = 1, the constant (1.3) coincides with (3.1). In the case p = 2 it differs from that in [26, p. 410]. The case p = 1, µ = ν = k −n, is skipped in Theorem 1.1, though it is included in Lemma 3.2. The reason is that the boundedness of R k from L 1 k−n (R n ) to L 1 k−n (Π n,k ) fails to be hold. Take, for instance, f (x) = f 0 (|x|) with f 0 (r) ≡ 0 if r < 10 and f 0 (r) = r k−δ , δ > 0, otherwise. Clearly, f ∈ L 1 k−n (R n ), however, by (3.6), 3.3. The dual k-plane transform. The dual k-plane transform of a function ϕ on Π n,k is defined by the formula and satisfies the duality relation [16,28] (3.12) The following lemma gives precise information about the L 1 case.
Then (3.14) provided that either side of the equality exists in the Lebesgue sense.
The following statement is dual to Theorem 1.1.
Theorem 3.8. Let 1 ≤ p ≤ ∞, 1/p + 1/p ′ = 1, ν = µ − k/p, µ < (n−k)/p ′ . Then the dual k-plane transform R * k is a linear bounded operator from L p µ (Π n,k ) to L p ν (R n ) with the norm Proof. By the duality (3.12), operators R k : L p µ (R n ) → L p ν (Π n,k ) and R * k : L p ′ −ν (Π n,k ) → L p ′ −µ (R n ) are bounded simultaneously and their norms coincide. Hence, replacing p by p ′ and making obvious changes in the statement of Theorem 1.1, we obtain the result.

The j-plane to k-plane transform
This transform is defined by (1.2). The reader is referred to [29] for additional information.

Preparations.
Lemma 4.1. (cf. Theorem 2.5 in [29]) Let 0 ≤ j < k < n, Then provided that either side of the corresponding equality exists in the Lebesgue sense.
Proof. The proof mimics that of Lemma 3.2, using Hölder's inequality in (4.3) and the known formula for radial functions [29, p. 5051]: The scaling argument (in the fibers) yields the following statement. To obtain an analogue of (3.8) for R j,k f , we denote Lemma 4.4. Let (R j,k f )(ξ, u) be the transformation (1.2) with u = 0. If g ∈ G satisfies g R k = ξ and g e k+1 = u/|u|, then Proof. Changing variables, we write (R j,k f )(ξ, u) as It remains to transform the inner integral using (2.3).

4.2.
Proof of Theorem 1.2. Denote by c j,k the constant on the right-hand side of (1.4). STEP 1. Let us show that ||R j,k || ≤ c j,k . By Lemmas 2.1 and 4.4, for 1 ≤ p < ∞ we have Hence, ||R j,k f || p,ν ≤ c j,k ||f || p,µ , This result also covers the case p = ∞, when the calculation is straightforward. The last integral gives the constant in (1.4). STEP 2. To prove that ||R j,k || ≥ c j,k , we proceed as in the proof of Theorem 1.1 and use the relevant analogue of (3.7). Let f and g be nonnegative radial functions, f (ζ) ≡ f 0 (|ζ|), g(τ ) ≡ g 0 (|τ |). Then r n−j−1 dr.
The dual of Theorem 1.2 is the following.
The proof of this statement is similar to the proof of Theorem 3.8.

Some generalizations and open problems
1. Owing to projective invariance of the Radon transforms [11, p. xi], all theorems of the present article can be transferred to totally geodesic Radon transforms on the hyperbolic and elliptic spaces. Almost all formulas, which are needed for this transition, are available in the literature. Specifically, for the hyperplane Radon transforms and the k-plane transform see [1,2,21]. The correspondence between the affine j-plane to k-plane transform and the similar transform for planes through the origin was established in [29]. We believe that a similar transition holds to the hyperbolic space.
2. It might be challenging to establish connection between weighted L p inequalities of the present article and known L p -L q or mixed norm estimates.