Mixed-norm estimates for a restricted X-ray transform in \(\mathbb{R}^4\) and \(\mathbb{R}^5\) (Q2729312)

Let \(Xf\) be the \(X\)-ray transform of \(f \in L^p(R^d), \;d>2\). This is the special case of the \(k\)-plane transform corresponding to \(k=1\). By taking into account that the operator \(X\) is overdetermined, the argument of \(Xf\) is assumed to belong to a certain \(d\)-dimensional well-curved complex of lines. The paper is devoted to \(L^q (L^r)\) mixed-norm estimates for \(X\)-ray transforms in this set-up. By making use of counterexamples, the author establishes necessary conditions for \(p, q, r, d\) under which the aforementioned estimates hold. It is conjectured that these conditions are also sufficient. The main result of the paper is that the conjecture is true for \(d=3,4,5\) exept the endpoint issues. The techniques of the paper are based on the so-called ``bush'' construction, which was introduced by J. Bourgain and used by several authors.