Holomorphic stability for Carleman pairs of function spaces

Let $\Omega$ be a planar domain and let $\mathcal{O}(\Omega)$ be the space of holomorphic functions on $\Omega$. In what follows, we shall always consider a pair $(X,Y)$ of vector spaces both consisting of functions on $\Omega$.

The Venn diagram in Figure 1 illustrates a holomorphically stable pair $(X,Y)$.

For any $f\in X$ and $g\in\mathcal{O}(\Omega)$, we interpret $f+g$ as a holomorphic perturbation of $f$. Then the holomorphic stability of a pair $(X,Y)$ means that a holomorphic perturbation of any element in $X$ remains in $X$ provided that it is contained in $Y$ after the perturbation.

Note that a holomorphically stable pair $(X,Y)$ always satisfies the condition

Such pairs will be referred as Carleman pairs. This terminology comes from the classical Carleman embeddings of some classical holomorphic-function spaces on the open unit disk $\mathbb{D}$ in the complex plane (see [Car21, Sai79, GK89, Vuk03]). For its generalization to complex domains of arbitrary dimension, see Hörmander [Hor67]. Note that if $Y\subset X$, then the pair $(X,Y)$ is trivially holomorphically stable. Therefore, the notion of the holomorphic stability is of interests only for the pairs $(X,Y)$ satisfying

Our research is inspired by Da Lio, Rivière and Wettstein’s very recent work [DLRW21] on Bourgain-Brezis type inequalities, where they essentially proved that the pair

of the harmonic Bergman space $B^{2}(\mathbb{D})$ and the classical harmonic Hardy space $h^{1}(\mathbb{D})$ is holomorphically stable (the precise definitions of $B^{2}(\mathbb{D})$ and $h^{1}(\mathbb{D})$ are given in §1.2). The notion of holomorphic stability leads to many natural questions. Generalization of our work in more general planar domains (including the non-simply connected ones) is more involved and will be given in the sequel to this paper.

The harmonic Hardy space $h^{1}(\mathbb{D})$ is defined by

where $\mathcal{O}_{h}(\mathbb{D})$ denotes the space of all harmonic functions on $\mathbb{D}$. The Hardy space $H^{1}(\mathbb{D})$ is then defined as $H^{1}(\mathbb{D}):=h^{1}(\mathbb{D})\cap\mathcal{O}(\mathbb{D})$.

Throughout the paper, all measures are assumed to be positive measures. Given a measure $\mu$ on $\mathbb{D}$, the associated weighted harmonic Bergman space $B^{2}(\mathbb{D},\mu)$ and weighted Bergman space $A^{2}(\mathbb{D},\mu)$ are defined as

both of which inherit the norm of $L^{2}(\mathbb{D},\mu)$. If $\mu$ is the Lebesgue measure on $\mathbb{D}$, then we use the simplified notation $B^{2}(\mathbb{D})$ and $A^{2}(\mathbb{D})$.

A measure $\mu$ on $\mathbb{D}$ is called boundary-accessable if its support is not relatively compact in $\mathbb{D}$. Note that if $\mu$ is boundary-inaccessable, then it is trivial to verify that the pair $(B^{2}(\mathbb{D},\mu),h^{1}(\mathbb{D}))$ is holomorphically stable. Therefore, in what follows, we always assume that $\mu$ is boundary-accessable.

A measure $\mu$ on $\mathbb{D}$ is called a $(1,2)$-Carleson measure if there exists a constant $C>0$ such that

A natural conjecture is

For a finite measure $\mu$ on $\mathbb{D}$ which is not necessarily radial, a simple situation (which in general is rather different from the situation in Theorem 1.1, see Theorem 1.2 below for more details) for the holomorphic stability of $(B^{2}(\mathbb{D},\mu),h^{1}(\mathbb{D}))$ is provided as follows. Consider the linear map $\mathcal{Q}_{+}$ defined on the space $\mathcal{O}_{h}(\mathbb{D})$ by

Then the pair $(B^{2}(\mathbb{D},\mu),h^{1}(\mathbb{D}))$ is holomorphically stable if both

and

are bounded linear operators.

It is not hard to see that the operator (1.3) is bounded if and only if

Hence the boundedness of the operator (1.3) in general fails even for a radial $(1,2)$-Carleson measure. On the other hand, the boundedness of (1.4) holds for all radial finite measure $\mu$ on $\mathbb{D}$. For general weights, a clear sufficient condition for the boundedness of the operator (1.4) is that the Bergman projection being bounded on $L^{2}(\mathbb{D},\mu)$ (which then is equivalent to the condition that $\mu$ is a $B_{2}$-weight à la Békollé-Bonami, see [BB78] for more details on Bergman projections). Consequently, for any $B_{2}$-weight $\mu$ on $\mathbb{D}$ satisfying (1.5), the pair $(B^{2}(\mathbb{D},\mu),h^{1}(\mathbb{D}))$ is holomorphically stable.

For a radial boundary-accessable finite measure $\mu$ on $\mathbb{D}$, the space $B^{2}(\mathbb{D},\mu)$ is complete and $B^{2}(\mathbb{D},\mu)+h^{1}(\mathbb{D})$ is a Banach space equipped with the norm:

Recall that a closed subspace $B_{1}$ of a Banach space $B$ is called complemented in $B$ if there exists a bounded linear projection from $B$ onto $B_{1}$.

In general, the notion of holomorphic stability for the pair $(B^{2}(\Omega,\mu),h^{1}(\Omega))$ is not conformally invariant, where $\Omega$ is assumed to have a nice boundary and $h^{1}(\Omega)$ is then defined as the set of all the Poisson convolutions of functions in $L^{1}(\partial\Omega,ds)$ (where $ds$ is the arc-length measure on $\partial\Omega$). In particular, our following result for the upper half plane does not seem to be a direct consequence of the result on the unit disk and its proof requires more efforts.

Let $\mathbb{H}=\{z\in\mathbb{C}|\Im(z)>0\}$ denote the upper half plane. In this case, the suitable spaces for studying the holomorphic stability are the harmonic Zen-type spaces (which reduces to the ordinary harmonic Bergman space when the weight is the Lebesgue measure on $\mathbb{H}$), see [Har09, JPP13].

A measure $\mu$ on $\mathbb{H}$ is called boundary-accessable if its support is not contained in

for any $\varepsilon>0$. Given a horizontal translation-invariant boundary-accessable measure $\mu$ on $\mathbb{H}$, define the harmonic Zen-type space by

where $\mathcal{O}_{h}(\mathbb{H})$ denotes the set of all harmonic functions on $\mathbb{H}$. It is easy to see that the above space $\mathscr{B}^{2}(\mathbb{H},\mu)$ is complete and thus is a Hilbert space.

The relation between $\mathscr{B}^{2}(\mathbb{H},\mu)$ and the ordinary weighted harmonic Bergman space

is given as follows. For any $g\in\mathcal{O}_{h}(\mathbb{H})$ and any $y>0$, define $g_{y}:\mathbb{R}\rightarrow\mathbb{C}$ by

Recall the Poisson kernel for $\mathbb{H}$ at the point $z=x+iy\in\mathbb{H}$:

Set

One can easily check that, for any horizontal translation-invariant boundary-accessable measure $\mu$ on $\mathbb{H}$,

The harmonic Hardy space $h^{1}(\mathbb{H})$ is defined by

and the Hardy space $H^{1}(\mathbb{H})$ is defined as $H^{1}(\mathbb{H}):=h^{1}(\mathbb{H})\cap\mathcal{O}(\mathbb{H})$. A measure $\mu$ on $\mathbb{H}$ is called a $(1,2)$-Carleson measure if there exists a constant $C>0$ such that

Here we give a sketch of the proof of Theorem 1.1. The proof is based on a generalization of the following one-dimensional Bourgain-Brezis-type inequality due to Da Lio-Rivière-Wettstein: there exists a universal constant $C>0$ such that for any smooth function $u\in C^{\infty}(\mathbb{T})$ with $\int u(e^{i\theta})d\theta=0$,

where the Hilbert transform of $u$ is given by

and the $1/4$-fractional Laplace transform $(-\Delta)^{1/4}u$ is given by

The inequality (1.13) implies

where, slightly by abusing the notation, $\mathcal{H}f$ is defined by

In our situation, we are able to prove that, if $\mu$ is a radial boundary-accessable measure, then $\mu$ is a $(1,2)$-Carleson measure on $\mathbb{D}$ if and only if there exists a constant $C_{\mu}>0$ such that for any bounded harmonic function $f\in\mathcal{O}_{h}(\mathbb{D})\cap L^{\infty}(\mathbb{D})$,

The inequality (1.16) applied to holomorphic functions immediately gives the result stated in Theorem 1.1.

The proof of the inequality (1.16) relies on a weighted version of Bourgain-Brezis-type inequality obtained in Theorem 1.4 below. More precisely, define a Fourier multiplier operator by

where

Define also a Sobolev-type space corresponding to the radial weight $\mu$ by

If $(B^{2}(\mathbb{D},\mu),h^{1}(\mathbb{D}))$ is a holomorphically stable pair, then we have set-theoretical inclusion

It follows that $\mu$ is a finite measure, which, when combined with the assumption of the theorem, implies that $\mu$ is a radial boundary-accessable finite measure on $\mathbb{D}$. Therefore, $A^{2}(\mathbb{D},\mu)$ is complete. Hence the embedding $H^{1}(\mathbb{D})\subset A^{2}(\mathbb{D},\mu)$ is continuous by the Closed Graph Theorem. In other words, $\mu$ is a $(1,2)$-Carleson measure on $\mathbb{D}$.

Now assume that $\mu$ is a radial boundary-accessable $(1,2)$-Carleson measure $\mu$ on $\mathbb{D}$. To prove the holomorphic stability of the pair $(B^{2}(\mathbb{D},\mu),h^{1}(\mathbb{D}))$, it suffices to show that there exists a constant $C>0$ such that

Indeed, assuming (3.7) and let $u\in B^{2}(\mathbb{D},\mu),f\in\mathcal{O}(\mathbb{D})$ with $u+f\in h^{1}(\mathbb{D})$, we obtain

and hence $f\in A^{2}(\mathbb{D},\mu)\subset B^{2}(\mathbb{D},\mu)$. It follows that $u+f\in B^{2}(\mathbb{D},\mu)$ and this gives the holomorphic stability of the pair $(B^{2}(\mathbb{D},\mu),h^{1}(\mathbb{D}))$.

It remains to prove (3.7). For any $f\in\mathcal{O}(\mathbb{D})$ and any $0<r<1$, write $f_{r}(z):=f(rz)$. Then, since $\mu$ is radial,

and

Therefore, it suffices to show that (3.7) holds for all $f$ belonging to the following class:

We now proceed to the derivation of the inequality (3.7) for any $f\in\mathcal{O}(\overline{\mathbb{D}})$ from the inequality (1.19) obtained in Theorem 1.4. Observe that, any $v\in B^{2}(\mathbb{D},\mu)$ has the form

where $e_{n}$ is defined as in (3.6). Then, by the radial assumption on $\mu$,

Comparing (3.8) and (1.18), we obtain a natural isometric isomorphism $H_{\mu}(\mathbb{T})\rightarrow B^{2}(\mathbb{D},\mu)$ that associates any $v\in H_{\mu}(\mathbb{T})$ to its harmonic extension in $\mathbb{D}$ defined by (3.6). Similarly, there is a natural isometric isomorphism $L^{1}(\mathbb{T})\rightarrow h^{1}(\mathbb{D})$. Therefore, we get a natural identification of the Banach spaces

Hence, for any $f\in\mathcal{O}(\overline{\mathbb{D}})$, by taking $u=f|_{\mathbb{T}}\in C^{\infty}(\mathbb{T})$ in (1.19), we obtain

with $\mathcal{A}_{\mu}$ defined in (1.17), $\mathcal{H}f$ defined in (1.15). Since $\mathcal{H}f=f-f(0)$ for all $f\in\mathcal{O}(\overline{\mathbb{D}})$ and $|f(0)|\leq\|f\|_{h^{1}(\mathbb{D})}\leq\|f\|_{B^{2}(\mathbb{D})+h^{1}(\mathbb{D})}$, we get

Finally, notice that, for any $f=\sum_{n\geq 0}c_{n}z^{n}\in\mathcal{O}(\overline{\mathbb{D}})$,

Thus we obtain the desired inequality (3.7) for all $f\in\mathcal{O}(\overline{\mathbb{D}})$ and complete the derivation of Theorem 1.1 from Theorem 1.4.

A pair $(a,b)$ of sequences $a=(a(n))_{n\in\mathbb{Z}}$ and $b=(b(n))_{n\in\mathbb{Z}}$ is called $\mu$-adapted if the following conditions are satisfied:

• (i)

  $a(0)\neq 0$;

• (ii)

  for any $n\in\mathbb{Z}^{*}=\mathbb{Z}\setminus\{0\}$,

  (3.9)

  $\displaystyle|a(n)|^{2}b(n)\mathrm{sgn}(n)=\sigma_{n}^{-1},\quad\text{where\,\,}\sigma_{n}:=\int_{0}^{1}r^{2|n|}\sigma(dr)>0;$

• (iii)

  there exists a constant $C_{b}$ such that

  (3.10)

  $\displaystyle 0<\frac{1}{C_{b}}\leq|b(n)|\leq C_{b},\quad n\in\mathbb{Z}^{*}.$

Note that the condition (3.9) implies in particular that $b(n)\in\mathbb{R}$ for all $n\in\mathbb{Z}^{*}$.

The next criterion of radial $(1,2)$-Carleson measures will be useful for us.

We postpone the proof of Lemma 3.2 for a while and proceed to the proof of Theorem 1.4.