Characterization of bounded generalized Hilbert operators on Hardy spaces for p greater than 2

Establish a complete characterization of the symbols g for which the generalized Hilbert operator H_g, defined by H_g(f)(z)=\int_0^1 f(t)g'(tz)\,dt, is bounded on the Hardy space H^p when 2<p<\infty.

Background

The paper studies generalized Hilbert operators H_g on Hardy spaces, where the operator associated with an analytic symbol g is given by H_g(f)(z)=\int_01 f(t)g'(tz)\,dt. For 1<p\leq 2, boundedness on Hp is characterized by the mean Lipschitz condition g\in\Lambda(p,1/p).

For 2<p<\infty, the same condition is necessary but is not sufficient, and the paper cites a counterexample demonstrating this failure. The paper obtains a complete characterization for the mapping H_g:Hp\to H2 and, for lacunary symbols, a complete characterization of boundedness on Hp itself; however, the general boundedness problem on Hp remains unresolved.

References

The boundedness of $H$ on $Hp$ is well understood when $1<p\leq 2$, whereas the case $2<p<\infty$ remains open. More precisely, the condition $g\in\Lambda\left(p,\frac{1}{p}\right)$ is necessary for the boundedness of $H$ on $Hp$ whenever $1<p<\infty$, and it is also sufficient when $1<p\leq 2$ cite[Theorem 1]{GaGiPeSis}. For $2<p<\infty$, however, this condition is no longer sufficient cite[Theorem~3.5 and Corollary~3.6]{GuoTang2026}, and no complete characterization is currently known.

— The Boundedness Problem for Generalized Hilbert Operators on Hardy Spaces  (2608.19086 - Norrbo et al., 19 Aug 2026) in Section 1, Introduction and main results

Looking at Theorem \ref{summing up}, Theorem \ref{summing up2}, Theorem \ref{Rhaly Hardy} and Theorem \ref{Hg Hardy} it is natural to wander what happens with case $\mathcal R_{\eta}: Hp \to Hq,\,\, 2<p\leq q <\infty.$

— Rhaly and generalized Hilbert operators between spaces of analytic functions  (2609.03837 - Galanopoulos et al., 3 Sep 2026) in Section 4, Final remarks

Additionally, the authors proposed an unresolved conjecture stating that the equality

|\mathcal{H}b|{H{p}\to H{p} = B\Big(\frac{1}{p},~b+1-\frac{1}{p}\Big)

holds for $b \geq 0$ and $1 < p < \infty$.

— Norm of the generalized Hilbert operator on weighted Bergman spaces  (2609.04833 - Li et al., 4 Sep 2026) in Section 1, subsection “Generalized Hilbert operator on Hardy spaces”