21 problems

Let ${\mathcal A}_{2m}^\mu$ be the reconstruction operator in Algorithm N/2, acting from $C(B^2)$ to $C(B^2)$, where $\mu$ is the attenuation parameter and $m$ tends to infinity. W…

Let $\lambda$ be fixed, let $U_{\lambda}$ be the subgroup associated with $\lambda$, and let the Radon transform $\operatorname{\mathcal{R}}$ act on the complex-valued function spa…

Componentwise inversion conjecture. The inversion formula … .

Let $R_{+}^{\alpha}$ be the fractional Radon transform considered in the paper, with $(1-n)/2\leq \operatorname{Re}\alpha\leq 0$ and $\alpha\neq 0$. The relevant $L^p$-to-$L^q$ bou…

The paper studies higher-rank Radon transforms and related transform theories in Euclidean, elliptic, and hyperbolic spaces. Hyperbolic analogues conjecture. Similar theories can b…

Let $0\leq j<k\leq n-1$ satisfy $j+k=n-1$, and let $r>0$. Let $S(\Gamma_H(n,d))$ denote the space of rapidly decreasing smooth functions on the hyperbolic Grassmannian, with all de…

Let $0\leq j<k\leq n-1$ satisfy $j+k=n-1$. Let $R_A$ and $R_A^*$ be the affine Radon transform and its dual acting on the indicated smooth function spaces…

Finite-type extension conjecture. The conclusion of Theorem ftthm should remain true for every $p>4$, and the assumptions on $\pi_R$ should even be dispensable altogether. The curr…

Consider the operator $R_\beta$ and its adjoint $R_\beta^*$ in the model discussed in the source. For $\beta\ne0$, the right projection $\pi_R$ has only fold singularities, and The…

Let ${\mathcal M}\subset\Omega_L\times\Omega_R$ be a four-dimensional manifold whose projections are submersions, and let $({\mathcal N}^*{\mathcal M})'$ be the corresponding prime…

Let $n\ge 2$ and $1< p < \infty$. Let $M^{\beta}_*$ denote the generalized spherical maximal operator on $\mathbb{R}^n$, and let condition (cnd18) be … with the last inequality wea…

The sharp norm conjecture for the $(j,k)$-plane transform. One has

Maximal-operator conjecture. The operator $B_{P,Q}^*$ is bounded from $l^2\times l^2$ into $l^{1,\infty}$. This is a maximal analogue of the boundedness questions for discrete bili…

Let $S$ be a smooth compact surface in any dimension, let $\phi(x,y)=1$, and let $M$ denote the associated maximal averaging operator. For a hyperplane $H$ not passing through the…

Let $f$ satisfy … and suppose that its Radon transform $(Rf)(\xi)$ vanishes for almost all hyperplanes in $\mathbb{R}^n$. The additional condition … controls the growth of $f$ near…

Let $M$ be a Riemannian globally symmetric space of compact type. Its adjoint form is the unique symmetric space admitting $M$ as a Riemannian cover and not itself properly coverin…

Let $F_q$ be a finite field, let $dgeq 2$ and $1leq kleq d-1$ be integers, and let $T_{Pi_k}$ denote the $k$-plane transform on the collection $Pi_k$ of all $k$-planes in $F_q^d$,…

The Semyanistyi-type integrals are the analytic family … where … $\operatorname{Re}\alpha>0$, $1\leq k<n$, $x\in\mathbb{R}^n$, $\tau\in\Pi_{n,k}$, and $|x-\tau|$ is the Euclidean d…

The broken-ray transform, also called the V-line transform, maps information about a medium to measurements along rays that may change direction at specified points. Its inversion…

Let $R_\chi$ be the Radon transform associated with the Euler characteristic in the double-fibration setting above, and let the Khovanskii–Pukhlikov constructible functions be fini…

Let $X$ and $Y$ be smooth manifolds in a double fibration … where $q_1$ and $q_2$ are smooth submersions, $q_1\times q_2$ is a closed imbedding, and $q_2$ is proper. Let $R_\chi$ b…