Local embedding inequality for non-even exponents greater than two

Determine whether the local embedding inequality \(\int_0^1 |f(1/2+it)|^q\,dt\lesssim \|f\|_{\mathscr H^q}^q\) holds for every \(q>2\) with \(q\notin 2\mathbb N\), for Hardy-space functions of Dirichlet series.

Background

The paper discusses the local embedding inequality for Hardy spaces of Dirichlet series, which is known in the $q=2$ case. Earlier investigations showed that the inequality fails for $1\leq q<2$. The unresolved range identified by the author is the set of exponents $q>2$ that are not even integers; even exponents are excluded because the corresponding estimates are accessible through established methods.