HIGHER-RANK RADON TRANSFORMS ON CONSTANT CURVATURE SPACES

Abstract

We study higher-rank Radon transforms of the form $f(\tau ) \rightarrow \int _{\tau \subset \zeta } f(\tau )$, where $\tau$ is a j-dimensional totally geodesic submanifold in the n-dimensional real constant curvature space and $\zeta$ is a similar submanifold of dimension $k >j$. The corresponding dual transforms are also considered. The transforms are explored in the Euclidean case (affine Grassmannian bundles), the elliptic case (compact Grassmannians), and the hyperbolic case (the hyperboloid model, the Beltrami-Klein model, and the projective model). The main objectives are sharp conditions for the existence and injectivity of the Radon transforms in Lebesgue spaces, transition from one model to another, support theorems, and inversion formulas. Conjectures and open problems are discussed.

8. Appendix. The Erdélyi–Kober-type fractional integrals

We recall some elementary facts from Fractional Calculus [41, Section “2.6.2”], [47]. The following Erdélyi–Kober-type fractional integrals on $\mathbb {R}_+ =(0, \infty )$ of order $\alpha >0$ arise in numerous integral-geometric considerations:

$$\begin{aligned} (I^{\alpha }_{+, 2} f)(t)= & {} \frac{2}{\Gamma (\alpha )}\int \limits _{0}^{t} (t^{2} -r^{2})^{\alpha -1}f (r) \, r\, dr,\\ (I^{\alpha }_{-, 2} f)(t)= & {} \frac{2}{\Gamma (\alpha )}\int \limits _{t}^{\infty }(r^{2} - t^{2})^{\alpha -1}f (r) \, r\, dr. \end{aligned}$$

Lemma 8.1

[41, p. 65] Let $\alpha >0$.

(i) The integral $(I^{\alpha }_{+, 2} f)(t)$ is absolutely convergent for almost all $t>0$ whenever $r\mapsto rf(r)$ is a locally integrable function on $\mathbb {R}_{+}$.

(ii) If

$$\begin{aligned} \int \limits _{a}^{\infty }|f(r)|\, r^{2\alpha -1}\, dr <\infty ,\qquad a>0, \end{aligned}$$

(8.1)

then $(I^{\alpha }_{-, 2} f)(t)$ is finite for almost all $t>a$. If f is non-negative, locally integrable on $[a,\infty )$, and (8.1) fails, then $(I^{\alpha }_{-, 2} f)(t)=\infty$ for every $t\ge a$.

The corresponding Erdélyi–Kober fractional derivatives are defined as the left inverses ${\mathcal D^{\alpha }_{\pm , 2} = (I^{\alpha }_{\pm , 2})^{-1}}$. For example, if $\alpha = m + \alpha _{0}$, $0 \le \alpha _{0} < 1$, $m = \lfloor \alpha \rfloor$, the integer part of $\alpha$, then, formally,

$$\begin{aligned} \mathcal D^{\alpha }_{\pm , 2} \varphi =(\pm D)^{m +1}\, I^{1 - \alpha _{0}}_{ \pm , 2}\varphi , \qquad D=\frac{1}{2t}\,\frac{d}{dt}. \end{aligned}$$

(8.2)

More precisely, the following statements hold.

Theorem 8.2

(cf. [41, formula (2.6.22)]) Let $\varphi = I^{\alpha }_{+, 2} f$, where rf(r) is locally integrable on $\mathbb {R}_{+}$. Then, $f(t)= (\mathcal D^{\alpha }_{+, 2} \varphi )(t)$ for almost all $t\in \mathbb {R}_{+}$, as in (8.2).

Theorem 8.3

[41, Theorem 2.44] If f satisfies (8.1) for every $a>0$ and $\varphi \!= \!I^{\alpha }_{-, 2} f$, then $f(t)= (\mathcal D^{\alpha }_{-, 2} \varphi )(t)$ for almost all $t\in \mathbb {R}_{+}$, where $\mathcal D^{\alpha }_{-, 2} \varphi$ can be represented as follows.

(i) If $\alpha =m$ is an integer, then

$$\begin{aligned} \mathcal D^{\alpha }_{-, 2} \varphi =(- D)^{m} \varphi , \qquad D=\frac{1}{2t}\,\frac{d}{dt}. \end{aligned}$$

(8.3)

(ii) If $\alpha = m +\alpha _{0}, \; m = \lfloor \alpha \rfloor , \; 0< \alpha _{0} <1$, then

$$\begin{aligned} \mathcal D^{\alpha }_{-, 2} \varphi = t^{2(1-\alpha +m)} (- D)^{m +1} t^{2\alpha }\psi , \quad \psi =I^{1-\alpha +m}_{-,2} \,t^{-2m-2}\, \varphi . \end{aligned}$$

(8.4)

In particular, for $\alpha =k/2$, k odd,

$$\begin{aligned} \mathcal D^{k/2}_{-, 2} \varphi = t\,(- D)^{(k+1)/2} t^{k}I^{1/2}_{-,2} \,t^{-k-1}\,\varphi . \end{aligned}$$

(8.5)

In the above theorem, powers of t are interpreted as the corresponding multiplication operators.

Theorems 8.2 and 8.3 can be used for explicit inversion of diverse Radon-like transforms of radial (or zonal) functions.