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The Boundedness Problem for Generalized Hilbert Operators on Hardy Spaces · arXivDesk

2608.19086Aug 19, 2026

The Boundedness Problem for Generalized Hilbert Operators on Hardy Spaces

David Norrbo, José Ángel Peláez, Fanglei Wu

Abstract

Let $g$ be analytic in the unit disc and consider the generalized Hilbert operator $\mathcal{H}_g(f)(z)=\int_0^1 f(t)g'(tz)\, dt.$

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The boundedness of

\mathcal H_g

H^p

is characterized by the mean Lipschitz condition

g\inΛ\left(p,\frac{1}{p}\right)

when

1<p\leq2

, while the problem remains open for

2<p<\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

H^p

2<p<\infty

GuoTang2026. We show that this condition is far from sufficient in the latter range: for every

2<p<\infty

, there exists a function

g\inΛ\left(p,\frac{1}{p}\right)

such that

\mathcal H_g

is not bounded even from

H^p

into

H^1

. The main ingredient is an exact characterization of the boundedness of

\mathcal{H}_g:H^p\to H^2

for all

1\leq p\leq\infty

. In particular, when

2<p<\infty

, this mapping is bounded if and only if

g'

belongs to a certain mixed-norm space. For lacunary symbols, the same mixed-norm condition also characterizes the boundedness of

\mathcal H_g

H^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^q

is characterized by the condition

g'\in H^q

. We also characterize compactness of

\mathcal H_g

in the aforementioned cases.

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