The Boundedness Problem for Generalized Hilbert Operators on Hardy Spaces

Abstract: Let $g$ be analytic in the unit disc and consider the generalized Hilbert operator $$ \mathcal{H}_g(f)(z)=\int_0¹ f(t)g&#39;(tz)\, dt. $$ The boundedness of $\mathcal H_g$ on $H<sup>p$ is characterized by the mean Lipschitz condition $g\inΛ\left(p,\frac{1}{p}\right)$ when $1&lt;p\leq2$, while the problem remains open for $2&lt;p&lt;\infty$. It has been recently proved that the condition $g\inΛ\left(p,\frac{1}{p}\right)$ does not imply the boundedness of $\mathcal H_g$ on $H<sup>p$, $2&lt;p&lt;\infty$ \cite{GuoTang2026}. We show that this condition is far from sufficient in the latter range: for every $2&lt;p&lt;\infty$, there exists a function $g\inΛ\left(p,\frac{1}{p}\right)$ such that $\mathcal H_g$ is not bounded even from $H<sup>p$ into $H<sup>1$. The main ingredient is an exact characterization of the boundedness of $\mathcal{H}_g:H^p\to H<sup>2$ for all $1\leq p\leq\infty$. In particular, when $2&lt;p&lt;\infty$, this mapping is bounded if and only if $g&#39;$ belongs to a certain mixed-norm space. For lacunary symbols, the same mixed-norm condition also characterizes the boundedness of $\mathcal H_g$ on $H<sup>p$, and hence gives a complete solution of the open problem within this class of symbols. We also show that, for $1\leq q\leq\infty$, boundedness of $\mathcal H_g:H^1\to H<sup>q$ is characterized by the condition $g&#39;\in H<sup>q$. We also characterize compactness of $\mathcal H_g$ in the aforementioned cases.

• The paper establishes exact boundedness and compactness criteria for operators H_g:H^p H^2 across all 1p, with mean Lipschitz conditions governing 1p2 and the mixed-norm space A^{2,\tilde p}_{\tilde p/2} governing 2p.
• For every 2<p<, boundedness from H^p to H^2 is equivalent to compactness and to g'A^{2,\tilde p}_{\tilde p/2}, while the H^1 endpoint is characterized by g'H^p, with explicit two-sided operator-norm estimates.
• The results show that g(p,1/p) is insufficient for boundedness when p>2 and completely resolve the H^p boundedness problem for lacunary symbols through Zygmund-type norm comparisons.

The paper "The Boundedness Problem for Generalized Hilbert Operators on Hardy Spaces" (2608.19086) by Norrbo, Peláez, and Wu advances the study of generalized Hilbert operators $\mathcal H_g(f)(z)=\int_0^1 f(t)g'(tz)\,dt$ on Hardy spaces $H^p$. The central contribution is an exact characterization of boundedness and compactness of $\mathcal H_g:H^p\to H^2$ for all $1\le p\le\infty$, which the authors then exploit to show that the classical mean Lipschitz condition $g\in\Lambda(p,1/p)$ is far from sufficient for boundedness of $\mathcal H_g$ on $H^p$ when $p>2$, and to resolve the long-standing open problem completely within the class of lacunary symbols.

Background and context

For a symbol $g\in H(\mathbb D)$, the operator $\mathcal H_g$ generalizes the classical Hilbert matrix operator, recovered when $H^p$0. The foundational work of Galanopoulos–Girela–Peláez–Siskakis established that $H^p$1 is necessary for boundedness of $H^p$2 on $H^p$3 for all $H^p$4, and sufficient when $H^p$5. For $H^p$6, sufficiency fails: Guo and Tang (Guo et al., 30 Jul 2026) constructed symbols in $H^p$7 for which $H^p$8 is unbounded on $H^p$9. A complete characterization in this range has remained open. The present paper sharpens that negative result considerably and identifies a mixed-norm condition that is both necessary and sufficient in important special cases.

Boundedness from $\mathcal H_g:H^p\to H^2$0 to $\mathcal H_g:H^p\to H^2$1

The first main theorem gives a clean characterization at the endpoint domain $\mathcal H_g:H^p\to H^2$2: for $\mathcal H_g:H^p\to H^2$3, boundedness of $\mathcal H_g:H^p\to H^2$4 is equivalent to compactness, and both hold if and only if $\mathcal H_g:H^p\to H^2$5. For the codomain $\mathcal H_g:H^p\to H^2$6, boundedness is equivalent to $\mathcal H_g:H^p\to H^2$7, while compactness requires the stronger condition $\mathcal H_g:H^p\to H^2$8, the disc algebra. Moreover, the operator norm satisfies two-sided bounds

$\mathcal H_g:H^p\to H^2$9

The proof of necessity uses reproducing kernels $1\le p\le\infty$0, normalized in $1\le p\le\infty$1, together with the uniform-on-compacta convergence $1\le p\le\infty$2 as $1\le p\le\infty$3. Compactness follows from polynomial approximation of $1\le p\le\infty$4; the failure of automatic compactness into $1\le p\le\infty$5 reflects the non-density of polynomials there. A remark notes that the coefficient condition $1\le p\le\infty$6, which characterizes the associated Hadamard multiplier from $1\le p\le\infty$7 to $1\le p\le\infty$8, is strictly weaker than $1\le p\le\infty$9 — so boundedness of $g\in\Lambda(p,1/p)$0 is sufficient but not necessary for the corresponding Hadamard product to be bounded.

Boundedness from $g\in\Lambda(p,1/p)$1 to $g\in\Lambda(p,1/p)$2 for $g\in\Lambda(p,1/p)$3

In the sub-$g\in\Lambda(p,1/p)$4 range, the paper proves that $g\in\Lambda(p,1/p)$5 is bounded exactly when $g\in\Lambda(p,1/p)$6, with norm equivalence

$g\in\Lambda(p,1/p)$7

and compactness holds precisely when $g\in\Lambda(p,1/p)$8 belongs to the little-oh space $g\in\Lambda(p,1/p)$9. The sufficiency argument combines moment estimates $\mathcal H_g$0 with dyadic block decompositions of $\mathcal H_g$1 and the known $\mathcal H_g$2-boundedness of the sublinear operator $\mathcal H_g$3. Necessity is obtained by testing against explicit kernel-like functions $\mathcal H_g$4 whose moments are uniformly large on dyadic blocks $\mathcal H_g$5, forcing the dyadic norms $\mathcal H_g$6 to satisfy the mean Lipschitz decay. This extends the previously known case $\mathcal H_g$7 and shows that in this range the target-$\mathcal H_g$8 problem is governed entirely by the same Lipschitz condition that governs boundedness on $\mathcal H_g$9.

Boundedness from $H^p$0 to $H^p$1 for $H^p$2

The situation changes qualitatively above $H^p$3. Setting $H^p$4, the authors prove that for $H^p$5 the following are equivalent: boundedness of $H^p$6, its compactness, and the membership

$H^p$7

an analytic weighted mixed-norm space with norm comparable to $H^p$8. Furthermore,

$H^p$9

Notably, boundedness and compactness coincide here — there is no gap between them, in contrast with the $p>2$0 case. The proof of necessity is the technical core: it employs a lemma producing test functions $p>2$1 built from dilated symbols $p>2$2, for which $p>2$3 is controlled by $p>2$4 while $p>2$5 dominates the full symbol norm. Letting $p>2$6 yields the lower bound. Sufficiency uses Hölder interpolation between the $p>2$7-weighted dyadic sums of $p>2$8 and the $p>2$9 structure of the moments of $g\in H(\mathbb D)$0.

Consequences: counterexamples and lacunary symbols

Two significant consequences follow. First, since one can exhibit lacunary series such as $g\in H(\mathbb D)$1 belonging to $g\in H(\mathbb D)$2 but with $g\in H(\mathbb D)$3, the authors obtain, for every $g\in H(\mathbb D)$4, a symbol $g\in H(\mathbb D)$5 for which $g\in H(\mathbb D)$6 is not even bounded from $g\in H(\mathbb D)$7 into $g\in H(\mathbb D)$8. This strengthens the earlier result of Guo and Tang, who had only shown failure of boundedness on $g\in H(\mathbb D)$9 itself; the implication is that the necessary condition $\mathcal H_g$0 lies strictly below any sufficient condition in the supercritical range, quantifying how far it is from characterizing boundedness.

Second, for lacunary symbols $\mathcal H_g$1 with $\mathcal H_g$2, the open problem is solved completely: $\mathcal H_g$3 is bounded on $\mathcal H_g$4 ($\mathcal H_g$5) if and only if $\mathcal H_g$6, with norm equivalence. The key observation is that $\mathcal H_g$7 remains lacunary with the same step whenever $\mathcal H_g$8 is analytic, so Zygmund's theorem comparing $\mathcal H_g$9 norms of lacunary series reduces the $H^p$00 question to the already-characterized $H^p$01 case. The authors also observe that for lacunary $H^p$02 the conditions $H^p$03 and $H^p$04 coincide, whereas for general symbols the latter condition (shown sufficient but not necessary by Guo–Tang) is strictly stronger.

Limitations and open questions

The principal limitation is inherent to the subject: for general (non-lacunary) symbols and $H^p$05, boundedness of $H^p$06 on $H^p$07 itself remains uncharacterized. The mixed-norm condition $H^p$08 fully resolves the problem only for the target space $H^p$09 and, via lacunarity, for symbols with spectral gaps. Whether some modification of this condition, or an interpolation-theoretic framework bridging the $H^p$10-target result and the Lipschitz obstruction, yields a full characterization on $H^p$11 is left open. Additionally, the equivalence of boundedness and compactness established for $H^p$12 does not extend to all codomains, as the $H^p$13 case demonstrates, and the paper does not address Schatten-class membership or weighted Bergman analogues beyond citing prior work.

Conclusion

This paper delivers exact norm equivalences for generalized Hilbert operators mapping into $H^p$14 across the full range of source Hardy spaces, establishes sharp endpoint results from $H^p$15, and produces the strongest available counterexamples showing insufficiency of the mean Lipschitz condition for $H^p$16. By resolving the boundedness problem for lacunary symbols and identifying the mixed-norm space $H^p$17 as the correct governing condition in the supercritical regime, the work delineates precisely where current techniques succeed and isolates the remaining difficulty in the general $H^p$18 case.