Laplace–Carleson embeddings and weighted infinite-time admissibility

Abstract

In this paper, we will establish necessary and sufficient conditions for a Laplace–Carleson embedding to be bounded for certain spaces of functions on the positive half-line. We will use these results to characterise weighted (infinite-time) admissibility of control and observation operators. We present examples of weighted admissibility criterion for one-dimensional heat equation with Neumann boundary conditions, and a cetrain parabolic diagonal system which was previously known to be not admissible in the unweighted setting.

1 Introduction

Although in most common physical applications the norms used are usually the unweighted $L^1$ or $L^2$ norms, it may sometimes be useful to consider weighted $L^p$ norms. The main purpose of this paper is to generalise admissibility criteria, obtained in [10] for weighted $L^2$- and unweighted $L^p$-admissibility (given in terms of Carleson measures and Laplace–Carleson embeddings described in [9]) to weighted $L^p(0, \, \infty )$ case, applying and generalising recent results from [13], concerning the spaces defined and studied in [12]. A powerful boundedness criterion for the Laplace–Carleson embeddings for weighted $L^p$ spaces, containing the earlier version from [9] as a special case, is also proved here.

This article is structured as follows. In Sect.  2, the theory and definitions of admissibility for diagonal semigroups are outlined. Two important theorems, linking admissibility to Laplace–Carleson embeddings, are also cited there. In Sect. 3, some results concerning reproducing kernel Hilbert spaces are given. As a special case, the definitions of so-called Zen spaces and their generalisation are provided, presenting their connection to the admissibility concept. In Sect. 4, boundedness of Carleson embeddings for these generalised spaces is studied, following a similar analysis from [13]. And finally, in Sect. 5 boundedness of Laplace–Carleson embeddings for sectorial measures is characterised there, and we believe that Theorem 6 is the most important result of this paper. This theorem is followed by two examples illustrating the weighted admissibility for diagonal systems.

2 Admissibility for diagonal semigroups

Let H be a Hilbert space and let $({\mathbb {T}}_t)_{t \ge 0}$ be a strongly continuous (or $C_0$-) semigroup of bounded linear operators on H with the infinitesimal generator $A : {\mathcal {D}}(A) \longrightarrow H$, defined by

$$\begin{aligned} Ax := \lim _{t \rightarrow 0^+} \frac{{\mathbb {T}}_t x - x}{t}, \qquad \text { where } \qquad {\mathcal {D}}(A) := \left\{ x \in X \; : \; \lim _{t \rightarrow 0^+} \frac{{\mathbb {T}}_t x - x}{t} \text { exists} \right\} . \end{aligned}$$

Let us consider the linear system

$$\begin{aligned} \dot{x}(t) = Ax(t)+Bu(t), \qquad \qquad x(0) = x_0, \qquad \qquad t \ge 0, \end{aligned}$$

(1)

where $u(t) \in {\mathbb {C}}$ is the input at time t and $B : {\mathbb {C}} \longrightarrow {\mathcal {D}}(A^*)'$ is the control operator. Here ${\mathcal {D}}(A^*)'$ is the completion of H with respect to the norm

$$\begin{aligned} \Vert x\Vert _{{\mathcal {D}}(A^*)'} := \left\| (\beta -A)^{-1} x \right\| _H, \end{aligned}$$

for any fixed $\beta \in \rho (A)$ (the resolvent set of A). An operator B is said to be finite-time $L^2$ -admissible, if for every $\tau >0$ and all $u \in L^2[0, \, \infty )$ the Bochner integral $\int _0^\tau {\mathbb {T}}_{\tau -t} Bu(t) \, \mathrm{d}t$ lies in H (see Definition 4.2.1, p. 116 in [18]). Consequently, there exists $m_\tau >0$ such that

$$\begin{aligned} \left\| \int _0^\tau {\mathbb {T}}_{\tau -t} Bu(t) \, \mathrm{d}t \right\| _H \le m_\tau \Vert u\Vert _{L^2[0, \, \infty )} \qquad \quad (\forall u \in L^2[0, \, \infty )). \end{aligned}$$

(see Proposition 4.2.2, also in [18]). The admissibility criterion guarantees that Eq. (1) has continuous (mild) solution

$$\begin{aligned} x(\tau ) = {\mathbb {T}}_\tau x_0 + \int _0^\tau {\mathbb {T}}_{\tau -t} Bu(t) \, \mathrm{d}t\qquad \qquad (\forall \tau \ge 0), \end{aligned}$$

with values in H (see Proposition 4.2.5 in [18]). If the constant $m_\tau$ can be chosen independently of $\tau >0$, then we say that B is (infinite-time) $L^2$ -admissible. It follows that B is $L^2$-admissible if and only if there exists a constant $m_0>0$ such that

$$\begin{aligned} \left\| \int _0^\infty {\mathbb {T}}_t Bu(t) \, \mathrm{d}t \right\| _H \le m_0 \Vert u\Vert _{L^2[0, \, \infty )} \qquad \qquad (\forall u \in L^2[0, \, \infty )), \end{aligned}$$

(see Remark 4.6.2 in [18] and Remark 2.2 in [5]). This is a necessary condition for the state x(t) to lie in H. For more details, see, for example [8], and for the non-Hilbertian analogue: [6, 20, 21].

We may also consider the system

$$\begin{aligned} \dot{x}(t) = Ax(t), \qquad \qquad y(t) = Cx(t), \qquad \qquad x(0) = x_0, \end{aligned}$$

where $C : {\mathcal {D}}(A) \longrightarrow {\mathbb {C}}$ is an A-bounded observation operator, that is, there exist $m_1, \, m_2 >0$ such that

$$\begin{aligned} \Vert Cx\Vert \le m_1 \Vert x\Vert + m_2 \Vert Ax\Vert \qquad \qquad (\forall x \in {\mathcal {D}}(A)). \end{aligned}$$

The operator C is said to be (infinite-time) admissible if there exists $m_0>0$ such that

$$\begin{aligned} \Vert y\Vert _{L^2[0, \, \infty )} \le m_0 \Vert x_0\Vert _H. \end{aligned}$$

There is duality between these two conditions, namely: B is an admissible control operator for $({\mathbb {T}}_t)_{t\ge 0}$ if and only if $B^*$ is an admissible observation operator for the dual semigroup $({\mathbb {T}}^*_t)_{t\ge 0}$.

More detailed treatment of admissibility of control and observation semigroup operators and the theory of well-posed linear evolution equations is given the survey[8] and the book [18].

For diagonal semigroups (see Example 2.6.6 in [18]), the admissibility condition is linked to the theory of Carleson measures in the following way (see [7, 19]). Suppose that A has a Riesz basis of eigenvectors $(\phi _k)_{k\in {\mathbb {N}}}$ (i.e. there exists invertible $Q \in {\mathscr {L}}(H, \ell ^2)$ such that $Q\phi _k = e_k$, for all $k \in {\mathbb {N}}$, where $(e_k)_{k\in {\mathbb {N}}}$ is the standard basis for $\ell ^2$, i.e. each $e_k$ has 1 as its kth entry and zeros elsewhere), with eigenvalues $(\lambda _k)_{k\in {\mathbb {N}}}$, each of them lying in the open left complex half-plane ${\mathbb {C}}_- := \left\{ z \in {\mathbb {C}} \; : \; {{\mathrm{Re}}}(z)<0\right\}$. Then a scalar control operator B, corresponding to a sequence $(b_k)_{k\in {\mathbb {N}}}$, is admissible if and only if the measure

$$\begin{aligned} \mu := \sum _{k=1}^\infty |b_k|^2 \delta _{-\lambda _k} \end{aligned}$$

is a Carleson measure for the Hardy space $H^2({\mathbb {C}}_+)$ on the right complex half-plane, which means the canonical embedding $H^2({\mathbb {C}}_+) \longrightarrow L^2({\mathbb {C}}_+, \, \mu )$ is bounded. An extension to normal semigroups has also been made in [22].

In some applications, requiring the input u to lie $L^2[0, \, \infty )$ might be unsuitable, and hence a more general setting ought to be considered. In [3] and [23], a concept of $\alpha$-admissibility, in which u must lie in weighted $L^2_{t^\alpha }(0, \,\infty )$, for $\alpha >-1$ in the first and $-1< \alpha < 0$ in the latter article, was studied. The second paper linked admissibility to the Carleson measures, using the fact that the Laplace transform maps $L^2_{t^\alpha }(0, \, \infty )$ onto a weighted Bergman space—in this article, we adopt a similar approach. Papers [4] and [6] discuss the same problem in non-Hilbertian setting. Further generalisations, to $L^2_w(0, \, \infty )$, for any positive measurable weight w and unweighted $L^p(0, \, \infty ), \, 1 \le p < \infty$, have been obtained in [10]. In this paper, we shall present it for the weighted $L^p_w(0, \, \infty )$ case, for certain weights (measurable self-maps on $(0, \, \infty )$) w, and by the weighted $L^p_w(0, \, \infty )$, we mean the Banach space of all functions $f : (0, \, \infty ) \rightarrow {\mathbb {C}}$ satisfying

$$\begin{aligned} \Vert f\Vert _{L^p_w(0, \, \infty )} := \left( \int _0^\infty |f(t)|^p w(t) \, \mathrm{d}t \right) ^{1/p}< \infty \qquad \qquad (1 \le p < \infty ). \end{aligned}$$

Given $1 \le q < \infty$, assume that the semigroup $({\mathbb {T}}_t)_{t \ge 0}$ acts on a Banach space X with a $q-$Riesz basis (having the same definition as above, but with ${\mathscr {L}}(H, \ell ^2)$ replaced by ${\mathscr {L}}(X, \ell ^q)$), of eigenvectors $(\phi _k)_{k \in {\mathbb {N}}}$ with corresponding eigenvalues $(\lambda _k)_{k \in {\mathbb {N}}}\subset {\mathbb {C}}_-$, that is,

$$\begin{aligned} {\mathbb {T}}_t \phi _k = e^{\lambda _k t} \phi _k \qquad \qquad (\forall k \in {\mathbb {N}}), \end{aligned}$$

Suppose that $(\phi _k)_{k \in {\mathbb {N}}}$ is also a Schauder basis of X such that there exist constants $c, C >0$ such that

$$\begin{aligned} c \sum _{k=1}^\infty |a_k|^q \le \left\| \sum _{k=1}^\infty a_k \phi _k \right\| ^q \le C \sum _{k=1}^\infty |a_k|^q \qquad \qquad (\forall (a_k)_{k=1}^\infty \in \ell ^q). \end{aligned}$$

This means that we can effectively identify X with $\ell ^q$ and this shall be our standing assumption for the whole paper.

The following two theorems, proved in [10], link admissibility of control and observation operators with Laplace–Carleson embeddings (i.e. Carleson embeddings induced by the Laplace transform). These results were presented there for weighted $L^2$ spaces and unweighted $L^p$ spaces on $(0, \, \infty )$, but it is easy to check that their proofs remain valid even for weighted $L^p$ spaces.

Theorem 1

(Theorem 2.1 in [10]) Let $1 \le p, \, q < \infty$, and suppose X is defined as above. Let w be a measurable self-map on $(0, \, \infty )$, and let B be a bounded linear map from ${\mathbb {C}}$ to ${\mathcal {D}}(A^*)'$ corresponding to the sequence $(b_k)_{k \in {\mathbb {N}}}$. The control operator B is $L^p_w$-admissible for $({\mathbb {T}}_t)_{t \ge 0}$, that is, there exists a constant $m_0>0$ such that

$$\begin{aligned} \left\| \int _0^\infty {\mathbb {T}}_t Bu(t) \, \mathrm{d}t \right\| _X \le m_0 \Vert u\Vert _{L^p_w(0, \, \infty )} {{\mathrm{\mathop {=}\limits ^{def^{\underline{n}}}}}}m_0 \left( \int _0^\infty |u(t)|^p w(t) \, \mathrm{d}t \right) ^{1/p}, \end{aligned}$$

for all $u \in L^p_w(0, \, \infty )$, if and only if the Laplace transform induces a continuous mapping from $L^p_w(0, \, \infty )$ into $L^q({\mathbb {C}}_+, \, \mu )$, where $\mu$ is the measure $\sum _{k=1}^\infty |b_k|^q \delta _{-\lambda _k}$.

Note that for $1<p<\infty$, we can associate the dual space of $L^p_w(0, \, \infty )$ with $L^{p'}_{w^{-p'/p}}(0, \, \infty )$, where $p':=p/(p+1)$ is the conjugate index of p via the pairing

$$\begin{aligned} \left\langle f, \, g \right\rangle = \int _0^\infty f(t)g(t) \, \mathrm{d}t \qquad \qquad (f \in L^p_w(0, \, \infty ), \, g \in L^{p'}_{w^{-p'/p}}(0, \, \infty )) \end{aligned}$$

(see Remark 1.4 in [4] with $t^\alpha$ replaced by $w^{1/p}$), and hence the following result follows.

Theorem 2

(Theorem 2.2 in [10]) Let C be a bounded linear map from ${\mathcal {D}}(A)$ to ${\mathbb {C}}$. The observation operator C is $L^p_w$-admissible for $({\mathbb {T}}_t)_{t \ge 0}$, that is, there exists a constant $m_0>0$ such that

$$\begin{aligned} \left\| C{\mathbb {T}}_. x \right\| _{L^p_w(0, \, \infty )} {{\mathrm{\mathop {=}\limits ^{def^{\underline{n}}}}}}\left( \int _0^\infty |C{\mathbb {T}}_tx(t)|^p w(t)\, \mathrm{d}t \right) ^{1/p} \le m_0 \Vert x\Vert _X \qquad (\forall x \in {\mathcal {D}}(A)), \end{aligned}$$

if and only if the Laplace transform induces a continuous mapping from

$L^{p'}_{w^{-p'/p}}(0, \, \infty )$ into $L^{q'}({\mathbb {C}}_+, \, \mu )$, where $\mu$ is the measure $\sum _{k=1}^\infty |c_k|^{q'} \delta _{-\lambda _k}$,

$c_k := C\phi _k$, for all $k \in {\mathbb {N}}$, and $q':=q/(q-1)$ is the conjugate index of q.

So in order to test admissibility of a control operator, we need to determine when the embedding

$$\begin{aligned} {\mathfrak {L}} : L^p_w(0, \, \infty ) \longrightarrow L^q({\mathbb {C}}_+, \, \mu ) \qquad f \mapsto {\mathfrak {L}}f {{\mathrm{\mathop {=}\limits ^{def^{\underline{n}}}}}}\int _0^\infty f(t) e^{-t \cdot } \, \mathrm{d}t \end{aligned}$$

is bounded. Or, in other words, whether there exists a constant $C>0$ such that

$$\begin{aligned} \left( \int _{{\mathbb {C}}_+} \left| {\mathfrak {L}}f\right| ^q \, \mathrm{d}\mu \right) ^{p/q} \le C \int _0^\infty |f(t)|^p w(t) \, \mathrm{d}t \qquad \qquad (\forall f \in L^p_w(0, \, \infty )). \end{aligned}$$

If $p=q$ and this embedding is indeed bounded, then we shall refer to $\mu$ as a Carleson measure for ${\mathfrak {L}}(L^p_w(0, \, \infty ))$ (a space which we equip with the $L^p_w$ norm). Because the observation operator version of this problem is analogous, from now on we shall only state our results for control operators, leaving the observation operator case to be derived from duality by an interested reader.

3 Carleson measures for Hilbert spaces of analytic functions on ${\mathbb {C}}_+$

Throughout this section, the operator B and the measure $\mu$ will be as defined in Sect. 2. We begin by considering the most elementary case, that is, when $p~=~q~=~2$. Then ${\mathfrak {L}}(L^2_w(0, \, \infty ))$, equipped with the $L^2_w$ inner product, is a Hilbert space of analytic functions. If for all $z \in {\mathbb {C}}_+$, $e^{-t\overline{z}}/w(t)$ belongs to $L^p_w(0, \, \infty )$, we can easily verify that it is also a reproducing kernel Hilbert space:

$$\begin{aligned} {\mathfrak {L}}f(z) {{\mathrm{\mathop {=}\limits ^{def^{\underline{n}}}}}}\int _0^\infty f(t) e^{-tz} \, \mathrm{d}t = \int _0^\infty f(t) \overline{\frac{e^{-t\overline{z}}}{w(t)}} w(t) \, \mathrm{d}t {{\mathrm{\mathop {=}\limits ^{def^{\underline{n}}}}}}\left\langle {\mathfrak {L}}f, \, {\mathfrak {L}}\frac{e^{-t\overline{z}}}{w(t)}\right\rangle _{{\mathfrak {L}}(L^2_w(0, \, \infty ))}, \end{aligned}$$

for each $f \in L^2_w(0, \, \infty )$. Suppose that for each $(z, \, \zeta ) \in {\mathbb {C}}_+^2$, $k_z(\zeta )$ is the reproducing kernel of ${\mathfrak {L}}(L^2_w(0, \, \infty ))$. In this case, Lemma 24 from [2] can be rephrased as the following proposition.

Proposition 1

The control operator B is $L^2_w$-admissible if and only if the linear map

$$\begin{aligned} f \mapsto \int _{{\mathbb {C}}_+} {{\mathrm{Re}}}(k_z(\cdot )) f(z) \, \mathrm{d}\mu (z) = \sum _{k=1}^\infty |b_k|^2 {{\mathrm{Re}}}(k_{-\lambda _k}(\cdot ))f(-\lambda _k) \end{aligned}$$

is bounded on $L^2({\mathbb {C}}_+, \, \mu )$.

Corollary 1

• 1. If

  $$\begin{aligned} \sum _{k=1}^\infty \sum _{l=1}^\infty \left| b_k b_l{{\mathrm{Re}}}\left( k_{-\lambda _k} (-\lambda _l) \right) \right| ^2 < \infty , \end{aligned}$$

  (2)

  then B is $L^2_w$-admissible.

• 2. If B is $L^2_w$-admissible, then there exists $C>0$ such that

  $$\begin{aligned} \sum _{k \in \Gamma } \sum _{l \in \Gamma } \left| b_k b_l{{\mathrm{Re}}}\left( k_{-\lambda _k}(-\lambda _l) \right) \right| ^2 \le C \sum _{n \in \Gamma } |b_n|^2 \qquad \qquad (\forall \Gamma \subset {\mathbb {N}}). \end{aligned}$$

Proof

To prove 1, we notice that by Hölder’s inequality

$$\begin{aligned} \int _{{\mathbb {C}}_+}&\left| \int _{{\mathbb {C}}_+} {{\mathrm{Re}}}\left( k_z(\zeta ) \right) G(z) \, \mathrm{d}\mu (z) \right| ^2 \, \mathrm{d}\mu (\zeta ) \\&\le \left( \int _{{\mathbb {C}}_+} |G|^2 \, \mathrm{d}\mu \right) \left( \int _{{\mathbb {C}}_+} \int _{{\mathbb {C}}_+} \left| {{\mathrm{Re}}}\left( k_z(\zeta ) \right) \right| ^2 \, \mathrm{d}\mu (z) \mathrm{d}\mu (\zeta )\right) , \end{aligned}$$

for all $G \in L^2({\mathbb {C}}_+, \, \mu )$, and the result follows from the previous proposition. Also, by the proof of Lemma 26 from [2], we have

$$\begin{aligned} \int _{{\mathbb {C}}_+} \int _{{\mathbb {C}}_+} {{\mathrm{Re}}}\left( k_z\right) G(z) \overline{G(\zeta )} \, \mathrm{d}\mu (z) \mathrm{d}\mu (\zeta ) \le C(\mu ) \int _{{\mathbb {C}}_+} |G|^2 \, \mathrm{d}\mu , \qquad (\forall G \in L^2({\mathbb {C}}_+, \, \mu )) \end{aligned}$$

And then we apply it to $G=\chi _\Omega$, the characteristic function of $\Omega =\{-\lambda _k\}_{k \in \Gamma }$. $\square$

If ${\mathfrak {L}}(L^2_w(0, \, \infty ))$ is a Banach algebra with respect to the pointwise multiplication (e.g. ${\mathfrak {L}}(L^2_{1+t^2}(0, \, \infty ))$, then, by Theorem 3 from [12], we know that ${\mathfrak {L}}(L^2_w(0, \, \infty ))$ must also be a reproducing kernel Hilbert space (with kernel $k_z$, say) and

$$\begin{aligned} \sup _{z \in {\mathbb {C}}_+} \Vert k_z\Vert _{{\mathfrak {L}}(L^2_w(0, \, \infty ))} \le 1. \end{aligned}$$

(3)

Proposition 2

Suppose that ${\mathfrak {L}}(L^2_w(0, \, \infty ))$ is a Banach algebra with respect to the pointwise multiplication. If $(b_k)_{k=1}^\infty \in \ell ^2$, then B is $L^2_w$-admissible

Proof

By the Cauchy–Schwarz inequality, we get

$$\begin{aligned} \sum _{k=1}^\infty \sum _{l=1}^\infty \left| b_k b_l {{\mathrm{Re}}}\left( k_{-\lambda _k} (-\lambda _l) \right) \right| ^2&{{\mathrm{\mathop {=}\limits ^{def^{\underline{n}}}}}}\int _{{\mathbb {C}}_+} \int _{{\mathbb {C}}_+} \left| {{\mathrm{Re}}}\left( k_z(\zeta ) \right) \right| ^2 \, \mathrm{d}\mu (z) \mathrm{d}\mu (\zeta ) \\&\le \int _{{\mathbb {C}}_+} \int _{{\mathbb {C}}_+} \left| k_z(\zeta ) \right| ^2 \, \mathrm{d}\mu (z) \mathrm{d}\mu (\zeta ) \\&\le \int _{{\mathbb {C}}_+} \int _{{\mathbb {C}}_+} \left\| k_z \right\| ^2_{{\mathfrak {L}} (L^2_w(0, \, \infty ))}\\&\quad \left\| k_\zeta \right\| ^2_{{\mathfrak {L}}(L^2_w(0, \, \infty ))} \mathrm{d}\mu (z) \mathrm{d}\mu (\zeta ) \\&\le \left( \int _{{\mathbb {C}}_+} \left\| k_z \right\| ^2_{{\mathfrak {L}} (L^2_w(0, \, \infty ))} \, \mathrm{d}\mu (z)\right) ^2 \\&\le \left( \sup _{z \in {\mathbb {C}}_+} \left\| k_z \right\| ^2_{{\mathfrak {L}}(L^2_w (0, \, \infty ))} \int _{{\mathbb {C}}_+} \, \mathrm{d}\mu \right) ^2 \\&\le \left( \sum _{k=1}^\infty |b_k|^2 \right) ^2 < \infty , \end{aligned}$$

and the result follows from the previous corollary.$\square$

More examples of ${\mathfrak {L}}(L^2_w(0, \, \infty ))$ can be easily produced from criteria given, for example, in [12, 14] or [15].

Let us now consider another type of spaces of analytic functions on ${\mathbb {C}}_+$. Let $\tilde{\nu }$ be a positive regular Borel measure on $[0, \, \infty )$ satisfying so-called $\Delta _2$ -condition:

and let $\lambda$ denote the Lebesgue measure on $i{\mathbb {R}}$. We define $\nu$ to be the positive regular Borel measure $\tilde{\nu } \otimes \lambda$ on $\overline{{\mathbb {C}}_+} := [0, \, \infty ) \times i{\mathbb {R}}$. For this measure and $1 \le p < \infty$, a Zen space (see [9]) is defined to be:

$$\begin{aligned} A^p_\nu := \left\{ F : {\mathbb {C}}_+ \longrightarrow {\mathbb {C}} \, \text {analytic} \; : \; \left\| F\right\| ^p_{A^p_\nu } := \sup _{\varepsilon >0} \int _{{\mathbb {C}}_+} |F(z+\varepsilon )|^p \, d\nu < \infty \right\} . \end{aligned}$$

The Zen spaces generalise Hardy spaces on ${\mathbb {C}}_+$ (these correspond to $\tilde{\nu } = \frac{1}{2\pi }\delta _0$) and weighted Bergman spaces (corresponding to $d\tilde{\nu }=r^\alpha dr, \, \alpha >-1$). If $p=2$, then Zen spaces are Hilbert spaces (see [16]) and in fact a reproducing kernel Hilbert spaces (see [12]). In [9], it was proved that the Laplace transform defines an isometric map

$$\begin{aligned} {\mathfrak {L}} : L^2_w(0, \, \infty ) \longrightarrow A^2_\nu , \end{aligned}$$

where w is given by

$$\begin{aligned} w(t) := 2\pi \int _0^\infty e^{-2rt} \, d\tilde{\nu }(r) \qquad \qquad (t > 0). \end{aligned}$$

The article [9] also contains a full characterisation of Carleson measures for Zen spaces, which was also presented in terms of admissibility in [10]. In [12], a generalisation of Zen spaces was defined, namely

$$\begin{aligned} A^p\left( {\mathbb {C}}_+, \, (\nu _n)_{n=0}^m \right) := \left\{ F : {\mathbb {C}}_+ \longrightarrow {\mathbb {C}} \, \text {analytic} \, : \, \left\| F\right\| ^p_{A^p\left( {\mathbb {C}}_+, \, (\nu _n)_{n=0}^m \right) } := \sum _{n=0}^m \left\| F^{(n)}\right\| ^p_{A^p_{\nu _n}} < \infty \right\} , \end{aligned}$$

where each $\nu _n=\tilde{\nu }_n \otimes \lambda$, and $\tilde{\nu }_n$ is defined as $\nu$ above, and $0\le m \le \infty$. It is also proved there that if $p=2$, then the Laplace transform again defines an isometric map

$$\begin{aligned} {\mathfrak {L}} : L^2_{w_{(m)}}(0, \, \infty ) \longrightarrow A^2\left( {\mathbb {C}}_+, \, (\nu _n)_{n=0}^m \right) , \end{aligned}$$

where w is given by

$$\begin{aligned} w_{(m)}(t) := 2\pi \sum _{n=0}^m t^{2n} \int _0^\infty e^{-2rt} \, d\tilde{\nu }_n(r) \qquad \qquad (t >0). \end{aligned}$$

The image of $L^2_{w_{(m)}}(0, \, \infty )$ is denoted by $A^2_{(m)}$. This is a large class of spaces, containing, for example, Hardy and weighted Bergman spaces mentioned earlier, but also the Dirichlet space on ${\mathbb {C}}_+$ (when we have $\tilde{\nu _0}=\frac{1}{2\pi }\delta _0$ and $\tilde{\nu }_1$ being the Lebesgue measure with the weight $1/\pi$) which has not been studied often in the complex half-plane context before. And our problem of determining admissibility of control or observation operators is reduced to the characterisation of Carleson measures for $A^2_{(m)}$, allowing us to consider $L^2_w$-admissibility for non-decreasing weights, which were not included in the Zen space context. This has been partially done in [13], and we aim to extend the results obtained there to the non-Hilbertian case of $A^p\left( {\mathbb {C}}_+, \, (\nu _n)_{n=0}^m \right)$ in the next section.

4 $A^p({\mathbb {C}}_+, \, (\nu _n)_{n=0}^m) \hookrightarrow L^q({\mathbb {C}}_+, \, \mu )$ embeddings

The boundedness of canonical embeddings into $L^q({\mathbb {C}}_+, \, \mu )$ (in this context also called Carleson embeddings), for some Borel measure $\mu$, and characterisations of Carleson measures is very often given in terms of Carleson squares (sometimes called Carleson boxes). On the half-plane, these are defined as follows. In this section, we prove general version of Theorems 2 and 4 from [13]. Note that for $p=2$ these can be used to describe corresponding $L^2_{w_{(m)}}$-admissibility. This is left for the interested reader.

Definition 1

Let $a \in {\mathbb {C}}_+$. A Carleson square centred at a is defined to be the set

$$\begin{aligned} Q(a) := \left\{ z = x+iy \; : \; 0 \le x < 2{{\mathrm{Re}}}(a), \, |y-{{\mathrm{Im}}}(a)| \le {{\mathrm{Re}}}(a)\right\} . \end{aligned}$$

(4)

Theorem 3

Suppose that $m < \infty$. If the embedding

$$\begin{aligned} A^p({\mathbb {C}}_+, \, (\nu _n)_{n=0}^m) \hookrightarrow L^q({\mathbb {C}}_+, \, \mu ) \end{aligned}$$

is bounded, then there exists a constant $C(\mu )>0$, such that

$$\begin{aligned} \mu (Q(a)) \le C(\mu ) \left[ \sum _{n=0}^m \frac{\nu _n\left( \overline{Q(a)}\right) }{({{\mathrm{Re}}}(a))^{np}}\right] ^{\frac{q}{p}}, \end{aligned}$$

(5)

for each Carleson square Q(a).

Proof

Let $a \in {\mathbb {C}}_+$, and choose $\gamma > \sup _{0\le n \le m}(\log _2 R_n -np +1)/p$, where $R_n$ denotes the supremum obtained from the ($\Delta _2$)-condition for each $\tilde{\nu }_n, \, 0~\le ~n~\le ~m$. Then for all z in Q(a) we have $|z+\overline{a}| \le \sqrt{10}{{\mathrm{Re}}}(a)$, and hence

$$\begin{aligned} \frac{\mu \left( Q(a)\right) }{(\sqrt{10}{{\mathrm{Re}}}(a))^{\gamma q}} \le \int _{{\mathbb {C}}_+} \frac{\mathrm{d}\mu (z)}{|z+\overline{a}|^{\gamma q}}. \end{aligned}$$

(6)

Similarly

$$\begin{aligned} |z+\overline{a}| \ge \sqrt{{{\mathrm{Re}}}(a)^2} = {{\mathrm{Re}}}(a) > \frac{{{\mathrm{Re}}}(a)}{2} \qquad \qquad (\forall z \in Q(a)). \end{aligned}$$

Also, given $k \in {\mathbb {N}}_0$, for all $z \in Q(2^{k+1}{{\mathrm{Re}}}(a)+i{{\mathrm{Im}}}(a)) \setminus Q(2^k({{\mathrm{Re}}})+i{{\mathrm{Im}}}(a))$, with $0 < {{\mathrm{Re}}}(z) \le 2^{k+1}{{\mathrm{Re}}}(a)$ we have

$$\begin{aligned} |z+\overline{a}| \ge \sqrt{{{\mathrm{Re}}}(a)^2+ (2^k{{\mathrm{Re}}}(a))^2} \ge 2^k {{\mathrm{Re}}}(a), \end{aligned}$$

and even if $2^{k+1}{{\mathrm{Re}}}(a) < {{\mathrm{Re}}}(z) \le 2^{k+2}{{\mathrm{Re}}}(a)$, we also have

$$\begin{aligned} |z+\overline{a}| \ge \sqrt{(2^{k+1}{{\mathrm{Re}}}(a)+{{\mathrm{Re}}}(a))^2} \ge 2^k {{\mathrm{Re}}}(a). \end{aligned}$$

And

$$\begin{aligned} {\begin{matrix} \nu _n&{}\left( Q(2^{k+1}{{\mathrm{Re}}}(a)+i{{\mathrm{Im}}}(a)) \setminus Q(2^k({{\mathrm{Re}}})+i{{\mathrm{Im}}}(a))\right) \\ &{}\le \nu _n\left( Q(2^{k+1}{{\mathrm{Re}}}(a)+i{{\mathrm{Im}}}(a))\right) \\ &{}\le \tilde{\nu }_n \left[ 0, \, 2^{k+2}{{\mathrm{Re}}}(a)\right) \cdot 2^{k+1}{{\mathrm{Re}}}(a) \mathop {\le }\limits ^{(\Delta _2)} \left( 2R_n\right) ^{k+1} \tilde{\nu }_n [0, \, 2{{\mathrm{Re}}}(a)) \cdot 2{{\mathrm{Re}}}(a) \\ &{}\le \left( 2R_n\right) ^{k+1} \nu _n\left( \overline{Q(a)}\right) , \end{matrix}} \end{aligned}$$

(7)

so

$$\begin{aligned} \int _{{\mathbb {C}}_+} \frac{d\nu _n(z)}{|z + \overline{a}|^{(\gamma +n)p}}&\le \left( \frac{2}{{{\mathrm{Re}}}(a)}\right) ^{(\gamma +n)p} \nu _n\left( Q(a)\right) \\&\qquad + \sum _{k=0}^\infty \frac{\nu _n\left( Q(2^{k+1}{{\mathrm{Re}}}(a)+i{{\mathrm{Im}}}(a)) \setminus Q(2^k({{\mathrm{Re}}}(a))+i{{\mathrm{Im}}}(a))\right) }{(2^k{{\mathrm{Re}}}(a))^{(\gamma +n)p}} \\&\mathop {\le }\limits ^{(7)} \left( \frac{2}{{{\mathrm{Re}}}(a)}\right) ^{(\gamma +n)p} \nu _n\left( \overline{Q(a)}\right) \left( 1+\sum _{k=0}^\infty \frac{(2R_n)^{k+1}}{2^{(k+1)(\gamma +n)p}}\right) \\&\le \left( \frac{2}{{{\mathrm{Re}}}(a)}\right) ^{(\gamma +n)p} \nu _n\left( \overline{Q(a)}\right) \sum _{k=0}^\infty \left( \frac{R_n}{2^{(\gamma +n)p-1}}\right) ^k \end{aligned}$$

and the sum converges for all $0 \le n \le m$. Hence $(z+\overline{a})^{-\gamma } \in A^p({\mathbb {C}}_+, \, (\nu _n)_{n=0}^m)$. Now, if the embedding is bounded, with constant $C'(\mu )>0$ say, then

$$\begin{aligned} \mu (Q(a))&\mathop {\le }\limits ^{(6)} (\sqrt{10}{{\mathrm{Re}}}(a))^{\gamma q} \int _{{\mathbb {C}}_+} \frac{\mathrm{d}\mu (z)}{\left| z+\overline{a}\right| ^{\gamma q}} \\&\le C'(\mu )(\sqrt{10}{{\mathrm{Re}}}(a))^{\gamma q} \left[ \sum _{n=0}^m \int _{{\mathbb {C}}_+} \frac{d\nu _n(z)}{\left| \left[ \left( z+\overline{a}\right) ^\gamma \right] ^{(n)}\right| ^p} \right] ^{\frac{q}{p}} \\&\le C'(\mu )(\sqrt{10}{{\mathrm{Re}}}(a))^{\gamma q} \left[ \sum _{n=0}^m \left( \prod _{l=1}^n (\gamma +l-1) \right) \int _{{\mathbb {C}}_+} \frac{d\nu _n(z)}{|z+\overline{a}|^{(\gamma +n)p}}\right] ^{\frac{q}{p}} \\&\le C(\mu ) \left[ \sum _{n=0}^m \frac{\nu _n\left( \overline{Q(a)}\right) }{({{\mathrm{Re}}}(a))^{np}} \right] ^{\frac{q}{p}}, \end{aligned}$$

where

$$\begin{aligned} C(\mu ) := 2^{q(n+3\gamma /2)}5^{\gamma q/2} \left[ \left( \prod _{l=0}^m (\gamma +l-1)\right) \max _{0\le n \le m} \sum _{k=0}^\infty \left( \frac{R_n}{2^{(\gamma +n)p-1}}\right) ^k\right] ^{\frac{q}{p}} \, C'(\mu ), \end{aligned}$$

(and we adopted the convention that the product $\prod (\gamma +l-1)$ is defined to be 1, if the lower limit is a bigger number than the upper limit). $\square$

Remark

For $p=q$ and $m=0$ (i.e. Carleson measures for Zen spaces), this result was stated in [10] and proved to be necessary as well as sufficient. An extension to $A^2({\mathbb {C}}_+, \, (\nu _n)_{n=0}^m)$ was made in [13], but only as a necessary condition. In the last section of this paper, we will prove that for some sequences of measures $(\nu _n)_{n=0}^m$ and sectorial measures $\mu$ it is also sufficient. However, it still remains unclear whether this could be true for a general case.

A version of the next theorem (for Carleson measures for $A^2_{(m)})$ has been proved in [13], following closely earlier version for analytic Besov spaces in [1] on the open unit disc ${\mathbb {D}}$ of the complex plane and Drury–Arveson Hardy spaces and other Besov–Sobolev spaces on complex balls from [2].

Theorem 4

Let $1<p \le q < \infty$ and let $\mu$ be a positive Borel measure on ${\mathbb {C}}_+$. If $\rho$ is a regular weight such that

$$\begin{aligned} \int _{{\mathbb {C}}_+} |F'(z)|^p ({{\mathrm{Re}}}(z))^{p-2} \rho (z)\, \mathrm{d}z \le \Vert F\Vert ^p_{A^p({\mathbb {C}}_+, \, (\nu _n)_{n=0}^m)}, \end{aligned}$$

(8)

for all $F \in A^p({\mathbb {C}}_+, \, (\nu _n)_{n=0}^m)$ and there exists a constant $C(\mu , \rho )>0$ such that

$$\begin{aligned} \left( \int _{Q(a)} \frac{(\mu (Q(a) \cap Q(z))^{p'}}{({{\mathrm{Re}}}(z))^2} \rho (z)^{1-p'} \, \mathrm{d}z \right) ^{q'/p'} \le C(\mu , \rho ) \mu (Q(a)) \qquad (\forall a \in {\mathbb {C}}_+), \end{aligned}$$

(9)

then the embedding

$$\begin{aligned} A^p({\mathbb {C}}_+, \, (\nu _n)_{n=0}^m) \hookrightarrow L^q({\mathbb {C}}_+, \, \mu ) \end{aligned}$$

is bounded.

Proof

Let $\zeta \in {\mathbb {C}}_+$. We use a representation of ${\mathbb {C}}_+$ as an ordered tree $T(\zeta )$, namely, we decompose the complex half-plane into a set of rectangles

$$\begin{aligned} R_{(k,l)}(\zeta ) := \left\{ z \in {\mathbb {C}}_+ \; : \; 2^{k-1}< \frac{{{\mathrm{Re}}}(z)}{{{\mathrm{Re}}}(\zeta )} \le 2^{k}, \, 2^k l \le \frac{{{\mathrm{Im}}}(z)-{{\mathrm{Im}}}(\zeta )}{{{\mathrm{Re}}}{\zeta }} < 2^k (l+1)\right\} , \end{aligned}$$

for all $(k, \, l) \in {\mathbb {Z}}^2$, and we identify each of these rectangles with a vertex of an abstract graph $T(\zeta )$. We put an order relation “$\le$” on the set of vertices of $T(\zeta )$ by saying that $x \le y$ whenever the area of the rectangle corresponding to x is greater or equal to the area of the rectangle corresponding to y and there is a sequence of horizontally adjacent rectangles $(R_{k, \, l}(\zeta ))$ forming a path connecting the rectangles corresponding to x and y. This decomposition is detailed in [13].

Given $F \in A^p({\mathbb {C}}_+, \, (\nu _n)_{n=0}^m)$, for each $\alpha \in T(\zeta )$ let $w_\alpha , \, z_\alpha \in \overline{\alpha } \subset {\mathbb {C}}_+$ be such that

$$\begin{aligned} z_\alpha := \sup _{z \in \alpha } \{|F(z)|\} \qquad \text { and } \qquad w_\alpha := \sup _{w \in \alpha } \{|F'(w)|\}. \end{aligned}$$

Define a weight $\tilde{\rho }$ on $T(\zeta )$ by $\tilde{\rho }(\alpha ) := \rho (z_\alpha )$. And also: $r_\alpha = {{\mathrm{Re}}}(w_\alpha )/4, \, \Phi (\alpha ):= F(z_\alpha ), \, \varphi (\alpha ) = \Phi (\alpha )-\Phi (\alpha ^-)$, for all $\alpha \in T(\zeta )$. Note that ${\mathcal {I}}\varphi = \Phi$. This is because if F is in $A^p({\mathbb {C}}_+, \, (\nu _n)_{n=0}^m$, then it is in the Zen space $A^p_{\nu _0}$, and hence in the Hardy space $H^p({\mathbb {C}}_+)$ (or its shifted version, see [16]), and hence $\lim _{\alpha \longrightarrow - \infty } |F(z_\alpha )| = \lim _{{{\mathrm{Re}}}(z) \longrightarrow \infty }|F(z)|=0$. Since (9) holds, we can apply Lemmata 3 and 4 from [13] to $\varphi , \, \tilde{\rho }, \, \tilde{\mu }$ (where $\tilde{\mu }(\alpha ):= \mu (\alpha )$, for all $\alpha \in T(\zeta )$) in the following way

$$\begin{aligned} \int _{{\mathbb {C}}_+} |F|^q \, \mathrm{d}\mu&= \sum _{\alpha \in T(\zeta )} \int _\alpha |F|^q \, \mathrm{d}\mu \le \sum _{\alpha \in T(\zeta )} |\Phi (\alpha )|^q \tilde{\mu }(\alpha ) \\&\lessapprox \left( \sum _{\alpha \in T(\zeta )} |\varphi (\alpha )|^p \tilde{\rho } (\alpha )\right) ^{q/p} {{\mathrm{\mathop {=}\limits ^{def^{\underline{n}}}}}}\left( \sum _{\alpha \in T(\zeta )} |\Phi (\alpha )-\Phi (\alpha ^-)|^p \tilde{\rho }(\alpha )\right) ^{q/p} \\&\mathop {\le }\limits ^{\mathop {\text {of Calculus}}\limits ^{\text {Fundamental Thm}}} \left( \sum _{\alpha \in T(\zeta )} \left| \int _{z_{\alpha ^-}}^{z_\alpha } F'(w) \, \mathrm{d}w \right| ^p \tilde{\rho }(\alpha )\right) ^{q/p} \\&\lessapprox \left( \sum _{\alpha \in T(\zeta )} \text {diam}(\alpha )^p |F'(w_\alpha )+F'(w_{\alpha ^-})|^p \tilde{\rho }(\alpha )\right) ^{q/p} \\&\lessapprox \left( \sum _{\alpha \in T(\zeta )} \text {diam}(\alpha )^p |F'(w_\alpha ) |^p \tilde{\rho }(\alpha )\right) ^{q/p} \\&\mathop {\le }\limits ^{\text {Mean-Value Property}} \left( \sum _{\alpha \in T(\zeta )} \text {diam}(\alpha )^p \left| \frac{1}{\pi r^2_\alpha } \int _{B(w_\alpha , \, r_\alpha )} F'(z) \, \mathrm{d}z \right| ^p \tilde{\rho }(\alpha )\right) ^{q/p} \\&\mathop {\le }\limits ^{\text {H}\ddot{\text {o}}\text {lder's}} \left( \sum _{\alpha \in T(\zeta )} \frac{\text {diam}(\alpha )^p}{(\pi r_\alpha ^2)^{p(1-1/p')}} \int _{B(w_\alpha , \, r_\alpha )} |F'(z)|^p \, \mathrm{d}z \tilde{\rho }(\alpha )\right) ^{q/p} \\&\lessapprox \left( \sum _{\alpha \in T(\zeta )} \text {diam}(\alpha )^{p-2} \int _{\bigcup _{\beta \in T(\zeta ) \; : \; \beta \cap B(w_\alpha , \, r_\alpha ) \ne \varnothing }} |F'(z)|^p \, \mathrm{d}z \, \tilde{\rho }(\alpha ) \right) ^{q/p} \\&\lessapprox \left( \sum _{\alpha \in T(\zeta )} \int _{\bigcup _{\beta \in T(\zeta ) \; : \; \beta \cap B(w_\alpha , \, r_\alpha ) \ne \varnothing }} |F'(z)|^p \frac{\rho (z)}{({{\mathrm{Re}}}(z))^{2-p}}\, \mathrm{d}z \right) ^{q/p} \\&\lessapprox \left( \sum _{\alpha \in T(\zeta )} \int _{\alpha } |F'(z)|^p \frac{\rho (z)}{({{\mathrm{Re}}}(z))^{2-p}} \, \mathrm{d}z\right) ^{q/p}, \end{aligned}$$

which is less than $\Vert F\Vert ^q_{A^p({\mathbb {C}}_+, \, (\nu _n)_{n=0}^m)}$ by the assumption of the theorem. $\square$

Remark

Although condition (8) looks unnecessarily artificial and very restrictive, it simply means that $A^p({\mathbb {C}}_+, \, (\nu )_{n=0}^m)$ is (or is contained within) some analytic Besov space on ${\mathbb {C}}_+$. For example, if $p=2$ and $\rho \equiv 1$, then $A^p({\mathbb {C}}_+, \, (\nu )_{n=0}^m) \subseteq {\mathcal {D}}({\mathbb {C}}_+)$, the Dirichlet space on ${\mathbb {C}}_+$. Condition (9), expressed in terms of Carleson boxes on ${\mathbb {D}}$ and distance from $\partial {\mathbb {D}}$, is known to be necessary and sufficient for the disc equivalent of the above theorem. It is not clear whether the same could be true for ${\mathbb {C}}_+$.

5 Laplace–Carleson embeddings for sectorial measures

Testing the boundedness of Laplace–Carleson embedding for arbitrary

$1 \le p, \, q < \infty$ is generally very difficult. We can however obtain some partial results if we consider measures with some restrictions imposed on their support

Proposition 3

Let $1<p< \infty , \, 1 \le q < \infty$, let w be a measurable self-map on $(0, \, \infty )$ and suppose that $\mu$ be a positive Borel measure supported on $(0, \infty )$. If the Laplace–Carleson embedding ${\mathfrak {L}} : L^p_w (0, \, \infty ) \longrightarrow L^q ({\mathbb {C}}_+, \, \mu )$ is well defined and bounded, then

$$\begin{aligned} \mu (I) \le C \left( \int _0^\infty \frac{e^{-|I|p't}}{w^{\frac{1}{p-1}}(t)} \, \mathrm{d}t \right) ^{-\frac{q}{p'}}, \end{aligned}$$

for all intervals $I = (0, \, |I|]$, provided that the integral on the right exists.

Proof

Let $0 < x \le |I|$ and $a>0$, then

$$\begin{aligned} \left| {\mathfrak {L}}\left[ \frac{e^{-\cdot a}}{w^{\frac{1}{p-1}}(\cdot )}\right] (x)\right| = \int _0^\infty \frac{e^{-t(a+x)}}{w^{\frac{1}{p-1}}(t)} \, \mathrm{d}t \ge \int _0^\infty \frac{e^{-t(a+|I|)}}{w^{\frac{1}{p-1}}(t)} \, \mathrm{d}t. \end{aligned}$$

And hence

$$\begin{aligned} \mu (I)&\le \left( \int _0^\infty \frac{e^{-t(a+|I|)}}{w^{\frac{1}{p-1}}(t)} \, \mathrm{d}t\right) ^{-q} \int _I \left| {\mathfrak {L}}\left[ \frac{e^{-\cdot a}}{w^{\frac{1}{p-1}} (\cdot )}\right] (x)\right| ^q \, \mathrm{d}\mu (x) \\&\le \left( \int _0^\infty \frac{e^{-t(a+|I|)}}{w^{\frac{1}{p-1}}(t)} \, \mathrm{d}t\right) ^{-q} \int _{{\mathbb {C}}_+} \left| {\mathfrak {L}}\left[ \frac{e^{-\cdot a}}{w^{\frac{1}{p-1}} (\cdot )}\right] (x)\right| ^q \, \mathrm{d}\mu (x) \\&\le C(\mu ) \left( \int _0^\infty \frac{e^{-t(a+|I|)}}{w^{\frac{1}{p-1}}(t)}\, \mathrm{d}t\right) ^{-q} \left\| \frac{e^{-\cdot a}}{w^{\frac{1}{p-1}}(\cdot )}\right\| ^q_{L^p_w (0, \, \infty )} \\&= C(\mu ) \left( \int _0^\infty \frac{e^{-t(a+|I|)}}{w^{\frac{1}{p-1}}(t)}\, \mathrm{d}t\right) ^{-q} \left( \int _0^\infty \frac{e^{-apt}}{w^{\frac{p}{p-1}}(t)} w(t) \, \mathrm{d}t \right) ^{\frac{q}{p}} \\&= C(\mu ) \left( \int _0^\infty \frac{e^{-t(a+|I|)}}{w^{\frac{1}{p-1}}(t)}\, \mathrm{d}t\right) ^{-q} \left( \int _0^\infty \frac{e^{-apt}}{w^{\frac{1}{p-1}}(t)} \, \mathrm{d}t \right) ^{\frac{q}{p}}, \end{aligned}$$

where $C(\mu )>0$ is the constant from the Laplace–Carleson embedding. Choosing $a = |I|/(p-1)$ gives us the desired result. $\square$

Theorem 5

Given $0<a\le b< \infty$, let

$$\begin{aligned} S_{(a,\, b]} := \{z \in {\mathbb {C}}_+ \; : \; a < {{\mathrm{Re}}}(z) \le b\}. \end{aligned}$$

If there exists a partition

$$\begin{aligned} P : 0 < \cdots \le x_{-n} \le \cdots \le x_{-1} \le x_0 \le x_1 \le \cdots \le x_n \le \cdots \qquad \qquad n \in {\mathbb {N}} \end{aligned}$$

of $(0, \, \infty )$ and sequence $(c_n) \in \ell ^1_{\mathbb {Z}}$ such that

$$\begin{aligned} \mu (S_{(x_n, \, x_{n+1}]}) \le |c_n| \left( \int _0^\infty \frac{e^{-p'tx_n}}{w^{\frac{1}{p-1}}(t)} \, \mathrm{d}t \right) ^{-\frac{q}{p'}} \qquad \qquad (\forall n \in {\mathbb {Z}}), \end{aligned}$$

then the Laplace–Carleson embedding ${\mathfrak {L}} : L^p_w (0, \, \infty ) \longrightarrow L^q ({\mathbb {C}}_+, \, \mu )$ is well defined and bounded.

Proof

For any$z \in S_{(x_k, \, x_{k+1}]}$ and $f \in L^p_w(0, \, \infty )$, we have

$$\begin{aligned} |{\mathfrak {L}}f(z)| \le \int _0^\infty e^{-tx_n} |f(t)| \, \mathrm{d}t \mathop {\le }\limits ^{\text {H}\ddot{\text {o}}\text {lder's}} \left( \int _0^\infty \frac{e^{-p'tx_n}}{w^{\frac{1}{p-1}}(t)} \, \mathrm{d}t \right) ^{\frac{1}{p'}} \Vert f\Vert _{L^p_w (0, \, \infty )}, \end{aligned}$$

so

$$\begin{aligned} \int _{{\mathbb {C}}_+} |{\mathfrak {L}}f|^q \, \mathrm{d}\mu&\le \Vert f\Vert ^q_{L^p_w (0, \, \infty )} \sum _{n=-\infty }^\infty \left( \int _0^\infty \frac{e^{-p'tx_n}}{w^{\frac{1}{p-1}}(t)} \, \mathrm{d}t \right) ^{\frac{q}{p'}} \mu (S_{(x_n, \, x_{n+1}]}) \\&\le \Vert (c_n)\Vert _{\ell ^1_{\mathbb {Z}}} \Vert f\Vert ^q_{L^p_w (0, \, \infty )}. \end{aligned}$$

$\square$

Definition 2

Let $1 \le p \le \infty$ and let $f \in L^p({\mathbb {R}})$. We define the maximal function of f to be

$$\begin{aligned} Mf(x) = \sup _{r>0} \frac{1}{2r} \int _{|y| \le r} |f(x-y)| \, \mathrm{d}y. \end{aligned}$$

The maximal function of f is finite almost everywhere. The book [17] by E. M. Stein offers extensive description of the maximal function and its properties, such as the link between Mf and the $L^p$ norm of f used in the arguments below.

Lemma 1

Let f be in $L^p_w(0, \, \infty ), \, 1 \le p < \infty$. Then for all $x>0$ and any partition

$$\begin{aligned} P \, : \, 0 \le \cdots \le t_{-k} \le \cdots \le t_0 = 1 \le t_1 \le \cdots \le t_k \le \cdots \qquad k \in {\mathbb {N}}_0 \end{aligned}$$

of $[0, \, \infty )$, such that $\inf _{k \in {\mathbb {N}}} t_{-k} =0$. We then have

$$\begin{aligned} \int _0^\infty e^{-\frac{t}{x}} |f(t)| \, \mathrm{d}t \le \Theta (P, w, x) xMg(x), \end{aligned}$$

(10)

where

$$\begin{aligned} g(t) = {\left\{ \begin{array}{ll} w^{1/p}(t)f(t), &{}\quad t >0, \\ 0 &{}\quad t\le 0, \end{array}\right. } \end{aligned}$$

$g \in L^p({\mathbb {R}})$, and

$$\begin{aligned} \Theta (P, w, x) = 2\left[ \sum _{k=-\infty }^{-1} \frac{e^{-t^*_k}}{w^{\frac{1}{p}}(t^*_k x)} \left( 1-t_k\right) + \sum _{k=0}^{\infty } \frac{e^{-t^*_k}}{w^{\frac{1}{p}}(t^*_k x)} \left( t_{k+1}-1\right) \right] , \end{aligned}$$

where each $t^*_k$ is such that

$$\begin{aligned} \frac{e^{-t^*_k}}{w^{\frac{1}{p}}(t^*_k x)} \ge \frac{e^{-t}}{w^{\frac{1}{p}}(t x)} \qquad \qquad (\forall t \in \left( t_k, \, t_{k+1} \right) ). \end{aligned}$$

Proof

Let $r_k := \max \left\{ |1-t_k|, \, |1-t_{k+1}|\right\}$, for each k. Given $x>0$, we have

$$\begin{aligned} \int _0^\infty e^{-\frac{t}{x}} |f(t)| \, \mathrm{d}t&= x \int _0^\infty e^{-t} |f(tx)| \, \mathrm{d}t \le x\sum _{k=-\infty }^\infty \frac{e^{-t^*_k}}{w^{\frac{1}{p}}(t^*_k x)} \int _{t_k}^{t_{k+1}} |g(tx)| \, \mathrm{d}t \\&=\sum _{k=-\infty }^\infty \frac{e^{-t^*_k}}{w^{\frac{1}{p}}(t^*_k x)} \int _{(1-t_{k+1})x}^{(1-t_k)x} |g(x-y)| \, \mathrm{d}y \\&\le \sum _{k=-\infty }^\infty \frac{e^{-t^*_k}}{w^{\frac{1}{p}}(t^*_k x)} \frac{r_k x}{r_k x} \int _{|y| \le r_k x} |g(x-y)| \, \mathrm{d}y \\&\le 2\left[ \sum _{k=-\infty }^\infty \frac{e^{-t^*_k}}{w^{\frac{1}{p}}(t^*_k x)} r_k \right] xMg(x). \end{aligned}$$

To get the required result, note that if $k \le -1$, then $t_{k+1} \le t_0 = 1$ and hence

$$\begin{aligned} 1-t_k \ge 1-t_{k+1} \ge 0 \quad \Longrightarrow \quad r_k=|1-t_k| = 1-t_k, \end{aligned}$$

otherwise $t_{k+1} > 1$, so $t_k \ge 1$, and so

$$\begin{aligned} 0 \ge 1-t_k \ge 1-t_{k+1} \quad \Longrightarrow \quad r_k=|1-t_{k+1}| = t_{k+1}-1. \end{aligned}$$

$\square$

The following theorem has been proved in [9] (Theorem 3.3, p. 801) for unweighted $L^p(0, \, \infty )$ case. We use the above lemma to obtain a weighted version.

Theorem 6

Let $1< p \le q < \infty$, let $\mu$ be a positive Borel measure on ${\mathbb {C}}_+$ supported only in the sector

$$\begin{aligned} S(\theta ) := \{z \in {\mathbb {C}}_+ \, : \, |\arg (z)| < \theta \}, \end{aligned}$$

for some $0 \le \theta < \pi /2$, and let $\alpha <p-1$. For an interval $I=(0, \, |I|) \subset {\mathbb {R}}$, we define

$$\begin{aligned} \Delta _I: = \left\{ z \in S(\theta ) \; : \; {{\mathrm{Re}}}(z) \le |I| \right\} . \end{aligned}$$

The Laplace–Carleson embedding ${\mathfrak {L}} : L^p_{t^\alpha }(0, \, \infty ) \longrightarrow L^q({\mathbb {C}}_+, \, \mu )$ is well defined and bounded if and only if there exists a constant $C(\mu ) >0$ such that

$$\begin{aligned} \mu (\Delta _I) \le C(\mu ) |I|^{\frac{q}{p'}\left( 1-\frac{\alpha }{p-1}\right) } \end{aligned}$$

(11)

for all intervals $I=(0, \, |I|) \subset {\mathbb {R}}$.

Proof

Suppose first that (11) holds. Let

$$\begin{aligned} T_n := \left\{ z \in S(\theta ) \; : \; 2^{n-1} < {{\mathrm{Re}}}(z) \le 2^n \right\} \subset \Delta _{(0, \, 2^n)} \qquad \qquad (n \in {\mathbb {Z}}), \end{aligned}$$

and let also $x_n = 2^{-n+1}$. Clearly

$$\begin{aligned} S(\theta ) = \bigcup _{n \in {\mathbb {Z}}} T_n \qquad \text { and } \qquad \mu (T_n) \le \mu (\Delta _{(0, \, 2^n)}) \mathop {\le }\limits ^{(11)} C(\mu ) x_n^{-\frac{q}{p'}\left( 1-\frac{\alpha }{p-1}\right) }. \end{aligned}$$

By the previous lemma, we have that

$$\begin{aligned} |{\mathfrak {L}}f(z)| \le \int _0^\infty e^{-\frac{t}{x_n}} |f(t)| \, \mathrm{d}t \mathop {\le }\limits ^{(10)} \Theta (P, t^\alpha , x_n) x_n Mg(x_n), \end{aligned}$$

for all $z \in T_n$ ($\Theta$ and g are as in Lemma 1). Note that the choice of $t_k^*$ does not depend on $x_n$, since

$$\begin{aligned} \frac{e^{-t^*_k}}{(t^*_k x_n)^{\frac{\alpha }{p}}} \ge \frac{e^{-t}}{(t x_n)^{\frac{\alpha }{p}}} \; \; \forall t \in (t_k, \, t_{k+1}) \quad \Longleftrightarrow \quad \frac{e^{-t^*_k}}{(t^*_k)^{\frac{\alpha }{p}}} \ge \frac{e^{-t}}{t^{\frac{\alpha }{p}}} \; \; \forall t \in (t_k, \, t_{k+1}), \end{aligned}$$

and there exists a partition P of $(0, \, \infty )$, for which $\Theta (P, t^\alpha , x_n)$ converges (since $\alpha < p$), so fixing P we can define $D_\Theta = x_n^{\frac{\alpha }{p}} \Theta (P, t^\alpha , x_n)$, which, by the definition of $\Theta$, is a constant depending on P and $\alpha$ only. Thus we have

$$\begin{aligned} \int _{S(\theta )} |{\mathfrak {L}}f|^q \, \mathrm{d}\mu&\le D_\Theta \sum _{n=-\infty }^\infty \left( x_n^{1-\frac{\alpha }{p}} Mg(x_n)\right) ^q \mu (T_n) \\&\le C(\mu )D_\Theta \sum _{n=-\infty }^\infty x_n^{q\left( 1-\frac{\alpha }{p}\right) -\frac{q}{p'}\left( 1-\frac{\alpha }{p-1}\right) } Mg(x_n)^q \\&= C(\mu )D_\Theta \sum _{n=-\infty }^\infty x_n^{q\left( 1-\frac{\alpha }{p}-\frac{1}{p'}+\frac{\alpha }{p}\right) } Mg(x_n)^q \\&= C(\mu )D_\Theta \sum _{n=-\infty }^\infty \left( x_n Mg(x_n)^p\right) ^{\frac{q}{p}} \\&\le C(\mu )D_\Theta \left( \sum _{n=-\infty }^\infty x_n Mg(x_n)^p\right) ^{\frac{q}{p}} \\&\lessapprox \left\| g\right\| ^q_{L^p(0, \, \infty )} = \left\| f\right\| ^q_{L^p_{t^\alpha }(0, \, \infty )}. \end{aligned}$$

Now suppose that the converse is true. For each $z \in \Delta _I$, we have $|z| \le |I|\sec (\theta )$, so

$$\begin{aligned} \left| {\mathfrak {L}} \left[ \frac{e^{-|I|\sec (\theta )t}}{t^{\frac{\alpha }{p-1}}}\right] (z)\right| = \frac{\Gamma \left( 1-\frac{\alpha }{p-1}\right) }{\left| z+|I|\sec (\theta )\right| ^{1-\frac{\alpha }{p-1}}} \ge \frac{\Gamma \left( 1-\frac{\alpha }{p-1}\right) }{\left( 2|I|\sec (\theta )\right) ^{1-\frac{\alpha }{p-1}}}. \end{aligned}$$

And therefore, we have

$$\begin{aligned} \mu (\Delta _I)&\lessapprox |I|^{q\left( 1-\frac{\alpha }{p-1}\right) } \int _{S(\theta )} \left| {\mathfrak {L}} \left[ \frac{e^{-|I|\sec (\theta )t}}{t^{\frac{\alpha }{p-1}}}\right] (z)\right| ^q \, d\mu (z) \\&\lessapprox |I|^{q\left( 1-\frac{\alpha }{p-1}\right) } \left\| \frac{e^{-|I| \sec (\theta )t}}{t^{\frac{\alpha }{p-1}}}\right\| ^q_{L^p_{t^\alpha }(0, \, \infty )} \\&= |I|^{q\left( 1-\frac{\alpha }{p-1}\right) } \left( \int _0^\infty \frac{e^{-|I|p \sec (\theta )t}}{t^{\frac{\alpha }{p-1}}} \, \mathrm{d}t \right) ^{\frac{q}{p}} \\&\lessapprox |I|^{q\left( 1-\frac{\alpha }{p-1}\right) } |I|^{-\frac{q}{p}\left( 1-\frac{\alpha }{p-1}\right) } \\&= |I|^{\frac{q}{p'}\left( 1-\frac{\alpha }{p-1}\right) }, \end{aligned}$$

as required. $\square$

Corollary 2

Let $1< p \le q < \infty$, let w be a measurable self-map on $(0, \, \infty )$, and let $\mu$ be a positive Borel measure on ${\mathbb {C}}_+$ supported only in the sector $S(\theta )$, $0~\le ~\theta ~<~\pi /2$. Suppose that

$$\begin{aligned} \sup _{t>0} \frac{t^\alpha }{w(t)} < \infty \end{aligned}$$

for some $\alpha < p-1$. If for some family of intervals $(I_n)_{n \in {\mathbb {Z}}} = ((0, \, 2^n|I|))_{n \in {\mathbb {Z}}}$ there exists $C(\mu )>0$ such that

$$\begin{aligned} \mu (\Delta _{I_n}) \le C(\mu ) (|I_n|)^{\frac{q}{p'}\left( 1-\frac{\alpha }{p-1}\right) } \qquad \qquad (\forall n \in {\mathbb {Z}}), \end{aligned}$$

then the Laplace–Carleson embedding ${\mathfrak {L}} : L^p_w (0, \, \infty ) \longrightarrow L^q ({\mathbb {C}}_+, \, \mu )$ is well defined and bounded.

Proof

By the previous theorem, we get that

$$\begin{aligned} \int _{S(\theta )} |{\mathfrak {L}}f|^q \, d\mu \lessapprox \left\| f\right\| ^q_{L^p_{t^\alpha }(0, \, \infty )} \le \left( \sup _{t>0}\frac{t^\alpha }{w(t)}\right) ^{\frac{q}{p}} \left\| f\right\| ^q_{L^p_w(0, \, \infty )}. \end{aligned}$$

$\square$

Corollary 3

Let B and $\mu$ be defined as in Theorem 1, let $1<p \le q < \infty$ and $\alpha < p-1$, and suppose that there exists $0< \theta < \pi /2$ such that

$$\begin{aligned} {{\mathrm{Im}}}(-\lambda _k) < {{\mathrm{Re}}}(-\lambda _k)\tan \theta \qquad \qquad (\forall k \in {\mathbb {N}}). \end{aligned}$$

Then the control operator B is $L^p_{t^\alpha }$-admissible if and only if there exists a constant $C(\mu )>0$ such that

$$\begin{aligned} \sum _{k \in \Gamma } |b_k|^q \le C(\mu ) \max _{k \in \Gamma }\left[ {{\mathrm{Re}}}(-\lambda _k)\right] ^{\frac{q}{p'}\left( 1-\frac{\alpha }{p-1}\right) } \qquad \qquad (\forall \Gamma \subset {\mathbb {N}}). \end{aligned}$$

Example 1

Consider the following one-dimensional heat PDE on the interval $[0, \, 1]$:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial z}{\partial t} (\zeta , \, t) = \frac{\partial ^2 z}{\partial \zeta ^2} (\zeta , \, t) \\ \frac{\partial z}{\partial \zeta } (0, \, t) = 0 \\ \frac{\partial z}{\partial \zeta } (1, \, t) = u(t) \\ z(\zeta ,0) = z_0(\zeta ) \end{array}\right. } \qquad \qquad \zeta \in (0, 1), \, t \ge 0. \end{aligned}$$

This system can be expressed in the form (1) with $H=\ell ^2, \, Ae_n = -n^2\pi ^2e_n$, and $b_n = 1$, for each $n \in {\mathbb {N}}$ (see Example 3.6 in [10]). For $1<p \le 2$ and $\alpha < p-1$, by the previous corollary we have that B is $L^p_{t^\alpha }$-admissible if and only if $p \ge \frac{4}{3}(\alpha +1)$.

Example 2

Let $1<p\le 2$. Consider the following parabolic diagonal system: let $X~=~\ell ^2, \, \lambda _n~=~-2^n, \, b_n~=~2^n/n$, and let A be defined by $Ae_n~=~\lambda _n e_n, \, n~\in ~{\mathbb {N}}$. By the previous corollary, if $\alpha \le -1$, then the control operator B is $L^2_{t^\alpha }$-admissible, and if $-1< \alpha < p-1$, then B is not $L^2_{t^\alpha }$-admissible. This contrasts with the unweighted setting, in which for all $1<p< \infty$, the control operator B is not $L^p$-admissible. This was proved only very recently in [11] (Example 5.2).

Theorem 7

Let $\mu$ be a positive Borel measure supported only in the sector $S(\theta ), \, 0< \theta < \pi /2$. If there exists an interval $I \subset i{\mathbb {R}}$, centred at 0, and a constant $C(\mu )>0$ such that

$$\begin{aligned} \mu \left( Q(2^k|I|)\right) \le C(\mu ) \left[ \left( \nu _0\left( Q(2^k|I|)\right) \right) ^{-\frac{1}{2}} + \left( \sum _{n=0}^m \frac{\nu _n\left( Q(2^k|I|)\right) }{(2^k |I|)^{2n}} \right) ^{-\frac{1}{2}}\right] ^{-2}, \end{aligned}$$

(12)

for all $k \in {\mathbb {Z}}$, then $\mu$ is a Carleson measure for $A^2_{(m)}$.

Proof

For all $t, \, x >0$, we have

$$\begin{aligned} w_{(m)}(tx)&{{\mathrm{\mathop {=}\limits ^{def^{\underline{n}}}}}}2\pi \sum _{n=0}^m (tx)^{2n} \int _0^\infty e^{-2rtx} \, d\tilde{\nu }_n(r) \\&\ge 2\pi \sum _{n=0}^m t^{2n} 2^{2n} \left( \frac{x}{2}\right) ^{2n} e^{-t} \tilde{\nu }_n\left[ 0, \, \frac{1}{2x}\right) \\&\ge 2\pi \sum _{n=0}^m t^{2n} \left( \frac{x}{2}\right) ^{2n} e^{-t} \frac{\tilde{\nu }_n\left[ 0, \, \frac{2}{x}\right) }{R^2_n}, \end{aligned}$$

where each $R_n$ is the supremum obtained from the ($\Delta _2$)-condition, corresponding to $\tilde{\nu }_n$. Clearly we have that

$$\begin{aligned} w_{(m)}(tx) \ge 2\pi e^{-t} \frac{\tilde{\nu }_0\left[ 0, \, \frac{2}{x}\right) }{R^2_0}, \qquad \qquad (\forall t, x > 0), \end{aligned}$$

and

$$\begin{aligned} w_{(m)}(tx) \ge 2\pi \sum _{n=0}^m \left( \frac{x}{2}\right) ^{2n} e^{-t} \frac{\tilde{\nu }_n\left[ 0, \, \frac{2}{x}\right) }{R^2_n}, \qquad \qquad (\forall x > 0, \, t \ge 1). \end{aligned}$$

Let

$$\begin{aligned} P \, : \, 0 = \cdots = t_{-k}= \cdots = t_{-1} < t_0 = 1 \le t_1 \le \cdots \le t_k \le \cdots , \qquad \qquad (k \in {\mathbb {N}}), \end{aligned}$$

be a partition of $[0, \, \infty )$, and let $x_k = 2^{-k+1}|I|^{-1}, \, k \in {\mathbb {Z}}$. Then

$$\begin{aligned} \Theta (P, \, w_{(m)}, x_k)&{{\mathrm{\mathop {=}\limits ^{def^{\underline{n}}}}}}2\left[ \frac{e^{-t^*_{-1}}}{\sqrt{w_{(m)}(t^*_{-1} x_k)}} + \sum _{l=0}^{\infty } \frac{e^{-t^*_l}}{\sqrt{w_{(m)}(t^*_l x_k)}} (t_{l+1}-1)\right] \\&\le \sqrt{\frac{2}{\pi }} \left[ \frac{R_0}{\sqrt{\tilde{\nu }_0\left[ 0, \, \frac{2}{x_k}\right) }} + \frac{\sum _{l=0}^\infty e^{-\frac{t_l}{2}}t_{l+1}}{\sqrt{\sum _{n=0}^m \left( \frac{x_k}{2}\right) ^{2n} \frac{\tilde{\nu }_n\left[ 0, \, \frac{2}{x_k}\right) }{R^2_n}}} \right] . \end{aligned}$$

And by Lemma 1, we get that for any $z \in T_k$

$$\begin{aligned} |{\mathfrak {L}}f(z)|&\le \sqrt{\frac{2}{\pi }} \left[ \frac{R_0}{\sqrt{\tilde{\nu }_0 \left[ 0, \, \frac{2}{x_k}\right) }} + \frac{\sum _{l=0}^\infty e^{-\frac{t_l}{2}} t_{l+1}}{\sqrt{\sum _{n=0}^m \left( \frac{x_k}{2}\right) ^{2n} \frac{\tilde{\nu }_n \left[ 0, \, \frac{2}{x_k}\right) }{R^2_n}}} \right] x_k Mg(x_k)\\&= \sqrt{\frac{2}{\pi }} \left[ \frac{R_0}{\sqrt{\frac{1}{x_k}\tilde{\nu }_0\left[ 0, \, \frac{2}{x_k}\right) }} + \frac{\sum _{l=0}^\infty e^{-\frac{t_l}{2}}t_{l+1}}{\sqrt{\sum _{n=0}^m \left( \frac{x_k}{2}\right) ^{2n-1} \frac{\tilde{\nu }_n\left[ 0, \, \frac{2}{x_k}\right) }{R^2_n}}} \right] \sqrt{x_k} Mg(x_k)\\&\lessapprox \left[ \left( \nu _0\left( Q(2^k|I|)\right) \right) ^{-\frac{1}{2}} + \left( \sum _{n=0}^m \frac{\nu _n\left( Q(2^k|I|)\right) }{(2^k |I|)^{2n}} \right) ^{-\frac{1}{2}}\right] \sqrt{x_k}Mg(x), \end{aligned}$$

so for any ${\mathfrak {L}}f=F \in A^2_{(m)}$, we have

$$\begin{aligned} \int _{{\mathbb {C}}_+}|F|^2 \, \mathrm{d}\mu = \int _{S(\theta )}|{\mathfrak {L}}f|^2 \, \mathrm{d}\mu \lessapprox \sum _{k=-\infty }^\infty x_k (Mg(x_k))^2 \lessapprox \Vert f\Vert ^2_{L^2_{w_{(m)}}(0, \infty )} = \Vert F\Vert ^2_{A^2_{(m)}}, \end{aligned}$$

as required. $\square$

Remark

Note that condition (12), although it looks somehow superficial, actually almost matches condition (5) from Theorem 3 (with $m=1$ and $p=q=2$). It suggests that if there exists a criterion characterising Carleson measures for $A^2_{(m)}$, which is both necessary and sufficient, then it must be expressible in a very similar form. This, however, still remains to be done.