The Berezin Transform and BMO

Abstract

In this chapter, we study the Berezin transform on F _α ² and certain spaces of functions of bounded mean oscillation (BMO) on the complex plane. We first consider the Berezin symbol of a bounded linear operator on F _α ² and show that this is a Lipschitz function in the Euclidean metric. We then consider the Berezin transform of a function and show that there is a semigroup property with respect to the parameter α. We also consider the action of the Berezin transform on L ^p spaces and the behavior of the Berezin transform when it is iterated.

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In this chapter, we study the Berezin transform on F _α ² and certain spaces of functions of bounded mean oscillation (BMO) on the complex plane. We first consider the Berezin symbol of a bounded linear operator on F _α ² and show that this is a Lipschitz function in the Euclidean metric. We then consider the Berezin transform of a function and show that there is a semigroup property with respect to the parameter α. We also consider the action of the Berezin transform on L ^p spaces and the behavior of the Berezin transform when it is iterated.

For every exponent p ∈ [1, ∞), we define a space BMO^p of functions of bounded mean oscillation, based on Euclidean disks of a fixed radius, and study the structure of these spaces. When 1 < p < ∞, we will show that the Berezin transform of every function in BMO^p is Lipschitz in the Euclidean metric.

As is well known, the Berezin transform is closely related to the notion of Carleson measures. So we include the discussion of Fock–Carleson measures in this chapter as well.

3.1 The Berezin Transform of Operators

Recall that for each $z \in \mathbb{C}$, we use k _z to denote the normalized reproducing kernel at z, namely,

$${k}_{z}(w) = K(w,z)/\sqrt{K(z, z)} ={ \mathrm{e}}^{\alpha w\overline{z}-\frac{\alpha } {2} \vert z{\vert }^{2} }.$$

These are unit vectors in F _α ².

If T is any linear operator on F _α ² whose domain contains all the normalized reproducing kernels, then we can define a function $\widetilde{T}$ on $\mathbb{C}$ as follows:

$$\widetilde{T}(z) =\langle T{k}_{z},{k}_{z}\rangle, \qquad z \in \mathbb{C},$$

(3.1)

where ⟨ ,  ⟩ is the inner product in F _α ². We are going to call $\widetilde{T}$ the Berezin transform (or sometimes the Berezin symbol) of T. In particular, if T is a bounded linear operator on F _α ², then the Berezin transform $\widetilde{T}$ is well defined and is actually real analytic in $\mathbb{C}$.

Proposition 3.1.

Let L(F _α ² ) be the Banach space of all bounded linear operators on F _α ² . Then $T\mapsto \widetilde{T}$ is a bounded linear mapping from L(F _α ² ) into ${L}^{\infty }(\mathbb{C})$ . Furthermore, the mapping is one-to-one and order preserving.

Proof.

Everything is obvious except the one-to-one part. To see this, assume that T is a bounded linear operator on F _α ² and that ⟨Tk _z , k _z ⟩ = 0 for all $z \in \mathbb{C}$. Then ⟨TK _z , K _z ⟩ = 0 for all $z \in \mathbb{C}$, where K _z (w) = K(w, z). The function F(z, w) = ⟨TK _z , K _w ⟩ is real analytic on $\mathbb{C} \times \mathbb{C}$, holomorphic in w, and conjugate holomorphic in z. Also, F vanishes on the diagonal of $\mathbb{C} \times \mathbb{C}$. It follows from a well-known theorem in several complex variables (see [142] for example) that F is identically zero on $\mathbb{C} \times \mathbb{C}$. Consequently, TK _z (w) = 0 for all z and w, or TK _z  = 0 for all $z \in \mathbb{C}$. Since the set of finite linear combinations of kernel functions is dense in F _α ², we conclude that T = 0.

Note that the proof above concerning the one-to-one property of the Berezin transform works for certain unbounded operators as well. More specifically, if T is an unbounded linear operator on F _α ² such that its domain contains all finite linear combinations of kernel functions and ⟨TK _z , K _w ⟩ is real analytic, then $\widetilde{T} = 0$ implies that T = 0.

Proposition 3.2.

If T is compact on F _α ² , then $\widetilde{T}(z) \rightarrow 0$ as z →∞.

Proof.

It is easy to see that k _z  → 0 weakly in F _α ² as z → ∞. This gives the desired result.

It is a classical result in functional analysis that if T is positive and compact on a Hilbert space H, then there exists an orthonormal set {e _n } in H and a nonincreasing sequence {s _n } of positive numbers such that

$$T(x) =\sum \limits_{n}{s}_{n}\langle x,{e}_{n}\rangle {e}_{n},\qquad x \in H.$$

The numbers s _n are uniquely determined by T and are called the singular values of T.

Let T be a positive and compact operator with singular values {s _n }, and let 0 < p < ∞. We say that the operator T belongs to the Schatten class S _p if the sequence {s _n } belongs to l ^p. For a more general operator T, we say that it belongs to the Schatten class S _p if | T |  = (T ^∗ T)^1 ∕ 2 belongs to S _p . If {s _n } is the sequence of singular values for | T | , we write

$$\|{T\|}_{{S}_{p}} ={ \left [\sum \limits_{n}{s}_{n}^{p}\right ]}^{1/p}.$$

Two special cases are worth mentioning: S ₁ is called the trace class, and S ₂ is called the Hilbert–Schmidt class. We refer the reader to [250] for more information about the Schatten classes.

Proposition 3.3.

If S is a trace-class operator or a positive operator, then

$$\mathrm{tr}\,(S) = \frac{\alpha } {\pi }\int \nolimits \nolimits \mathbb{C}\widetilde{S}(z)\,\mathrm{d}A(z).$$

(3.2)

Furthermore, a positive operator S belongs to the trace class if and only if the integral in (3.2) converges.

Proof.

First, assume that S is positive, say S = T ² for some T ≥ 0. Then for any orthonormal basis {e _n }, it follows from Fubini’s theorem that

$$\begin{array}{rcl} \mathrm{tr}\,(S)& =& \sum \limits_{n=1}^{\infty }\langle S{e}_{ n},{e{}_{n}\rangle }_{\alpha } =\sum \limits_{n=1}^{\infty }\|T{e{}_{ n}\|}_{2,\alpha }^{2} =\sum \limits_{n=1}^{\infty }\int \nolimits \nolimits \mathbb{C}\vert T{e}_{n}(z){\vert }^{2}\,\mathrm{d}{\lambda }_{ \alpha }(z) \\ & =& \int \nolimits \nolimits \mathbb{C}\left [\sum \limits_{n=1}^{\infty }\vert T{e}_{ n}(z){\vert }^{2}\right ]\,\mathrm{d}{\lambda }_{ \alpha }(z) = \int \nolimits \nolimits \mathbb{C}\left [\sum \limits_{n=1}^{\infty }\langle T{e}_{ n},{K{}_{z}\rangle }_{\alpha }^{2}\right ]\,\mathrm{d}{\lambda }_{ \alpha }(z) \\ & =& \int \nolimits \nolimits \mathbb{C}\left [\sum \limits_{n=1}^{\infty }\langle {e}_{ n},T{K{}_{z}\rangle }_{\alpha }^{2}\right ]\,\mathrm{d}{\lambda }_{ \alpha }(z) = \int \nolimits \nolimits \mathbb{C}\|T{K{}_{z}\|}_{2,\alpha }^{2}\,\mathrm{d}{\lambda }_{ \alpha }(z) \\ & =& \int \nolimits \nolimits \mathbb{C}\langle S{K}_{z},{K{}_{z}\rangle }_{\alpha }\,\mathrm{d}{\lambda }_{\alpha }(z) = \int \nolimits \nolimits \mathbb{C}\widetilde{S}(z)K(z,z)\,\mathrm{d}{\lambda }_{\alpha }(z) \\ & =& \frac{\alpha } {\pi }\int \nolimits \nolimits \mathbb{C}\widetilde{S}(z)\,\mathrm{d}A(z).\end{array}$$

Next, assume that S is self-adjoint and belongs to the trace class. Then we can write

$$\begin{array}{rcl} S = \frac{\vert S\vert + S} {2} -\frac{\vert S\vert - S} {2}, & & \\ \end{array}$$

where each of the two quotients above is a positive operator in the trace class. The desired trace formula then follows from the corresponding ones for positive trace-class operators.

Finally, an arbitrary trace-class operator S can be written as

$$S = \frac{S + {S}^{{_\ast}}} {2} + \mathrm{i}\,\frac{S - {S}^{{_\ast}}} {2\mathrm{i}}, $$

where each of the two quotients above is a self-adjoint operator in the trace class. The desired trace formula for S follows from the corresponding ones for self-adjoint trace-class operators.

Lemma 3.4.

Suppose T is a positive operator on a Hilbert space H and x is a unit vector in H. Then ⟨T ^p x,x⟩≥⟨Tx,x⟩ ^p for p ≥ 1 and ⟨T ^p x,x⟩≤⟨Tx,x⟩ ^p for all 0 < p ≤ 1.

Proof.

See Proposition 1.31 of [250].

Proposition 3.5.

If p ≥ 1 and T is in the Schatten class S _p , then $\widetilde{T}$ belongs to ${L}^{p}(\mathbb{C},\mathrm{d}A)$.

Proof.

If T is in the trace class, then we can write

$$T = {T}_{1} - {T}_{2} + \mathrm{i}({T}_{3} - {T}_{4}),$$

where each T _k is a positive trace-class operator. By Proposition 3.3 above, the function

$$\widetilde{T} =\widetilde{ {T}}_{1} -\widetilde{ {T}}_{2} + \mathrm{i}\widetilde{{T}}_{3} -\mathrm{i}\widetilde{{T}}_{4}$$

is in ${L}^{1}(\mathbb{C},\mathrm{d}A)$.

If T is a bounded linear operator on F _α ², the function $\widetilde{T}$ is in ${L}^{\infty }(\mathbb{C},\mathrm{d}A)$. It follows from complex interpolation that if T is any operator in the Schatten class S _p , 1 < p < ∞, then the function $\widetilde{T}$ is in ${L}^{p}(\mathbb{C},\mathrm{d}A)$.

Alternatively, if 1 ≤ p < ∞ and T is in the Schatten class S _p , then by the decomposition T = T ₁ − T ₂ + i(T ₃ − T ₄), we may assume that T is positive. But when T is positive, it is in the Schatten class S _p if and only if T ^p is in the trace class, so the desired result follows from Proposition 3.3 and Lemma 3.4.

Note that we did not need the positivity of T above, while this is necessary in the next proposition.

Proposition 3.6.

Suppose 0 < p ≤ 1 and T is a positive operator on F _α ² . If $\widetilde{T} \in {L}^{p}(\mathbb{C},\mathrm{d}A)$ , then T belongs to the Schatten class S _p.

Proof.

Since T is positive, it belongs to the Schatten class S _p if and only if S ^p is in the trace class. The desired result then follows from Proposition 3.3 and Lemma 3.4.

Theorem 3.7.

Let T be any bounded linear operator on F _α ² . We have

$$\vert \widetilde{T}(z) -\widetilde{ T}(w)\vert \leq 2\|T\|\,{\left [1 -\vert \langle {k}_{z},{k}_{w}\rangle {\vert }^{2}\right ]}^{1/2}$$

for all z and w in $\mathbb{C}$.

Proof.

For any $z \in \mathbb{C}$, let P _z denote the rank-one projection from F _α ² onto the one-dimensional subspace spanned by k _z . More specifically,

$${P}_{z}(f) =\langle f,{k}_{z}\rangle {k}_{z},\qquad f \in {F}_{\alpha }^{2}.$$

It is clear that P _z is a positive operator with tr (P _z ) = 1.

Let {e _k } be an orthonormal basis of F _α ² with e ₁ = k _z . Then

$$\mathrm{tr}\,(T{P}_{z}) =\sum \limits_{n=1}^{\infty }\langle T{P}_{ z}{e}_{n},{e}_{n}\rangle =\langle T{P}_{z}{k}_{z},{k}_{z}\rangle =\langle T{k}_{z},{k}_{z}\rangle =\widetilde{ T}(z).$$

It follows that

$$\vert \widetilde{T}(z) -\widetilde{ T}(w)\vert = \left \vert \mathrm{tr}\,\left (T({P}_{z} - {P}_{w})\right )\right \vert \leq \| T\|\|{P}_{z} - {P{}_{w}\|}_{{S}_{1}},$$

where S ₁ denotes the trace class as a Banach space. Note that we have just used the well-known inequality

$$\left \vert \mathrm{tr}\,(TS)\right \vert \leq \| T\|\|{S\|}_{{S}_{1}}$$

from operator theory.

For any two different complex numbers z and w, the operator P _z  − P _w is a rank-two self-adjoint operator with trace 0. So there is an orthonormal basis in which P _z  − P _w is diagonal with two nonzero eigenvalues λ and − λ, where $\lambda =\| {P}_{z} - {P}_{w}\| > 0$. Consequently, the positive rank-two operator (P _z  − P _w )² has a single nonzero eigenvalue λ² of multiplicity 2, and its trace equals 2λ². It follows that the positive operator | P _z  − P _w  | has a single positive eigenvalue λ with multiplicity 2, and its trace is 2λ, which is also the value of $\|{P}_{z} - {P{}_{w}\|}_{{S}_{1}}$.

Since

$$\mathrm{tr}\,{({P}_{z} - {P}_{w})}^{2} = \mathrm{tr}\,({P}_{ z} - {P}_{z}{P}_{w} - {P}_{w}{P}_{z} + {P}_{w}) = 2 - 2\mathrm{tr}\,({P}_{z}{P}_{w}),$$

we can expand the unit vector k _w to an orthonormal basis of F _α ² and calculate the trace of P _z P _w with respect to this basis to obtain

$$\mathrm{tr}\,({P}_{z}{P}_{w}) =\langle {P}_{z}{P}_{w}{k}_{w},{k}_{w}\rangle =\langle {P}_{z}{k}_{w},{k}_{w}\rangle.$$

But P _z k _w  = ⟨k _w , k _z ⟩k _z , we have

$$\mathrm{tr}\,{({P}_{z} - {P}_{w})}^{2} = 2\left [1 -\vert \langle {k}_{ z},{k}_{w}\rangle {\vert }^{2}\right ].$$

It follows that λ² = 1 − | ⟨k _z , k _w ⟩ | ², which gives the desired result.

Corollary 3.8

Let T be any bounded linear operator on F _α ² . Then

$$\vert \widetilde{T}(z) -\widetilde{ T}(w)\vert \leq 2\sqrt{\alpha }\|T\|\vert z - w\vert $$

for all z and w in $\mathbb{C}$.

Proof.

It is easy to see that

$$1 -\vert \langle {k}_{z},{k}_{w}\rangle {\vert }^{2} = 1 -{\mathrm{e}}^{-\alpha \vert z-w{\vert }^{2} } \leq \alpha \vert z - w{\vert }^{2}$$

for all z and w. The desired Lispchitz estimate is then obvious.

Every bounded linear operator on F _α ² also induces a function on $\mathbb{C} \times \mathbb{C}$. More specifically, if S is a bounded linear operator on F _α ² and $z \in \mathbb{C}$, then

$$Sf(z) =\langle Sf,{K{}_{z}\rangle }_{\alpha } =\langle f,{S}^{{_\ast}}{K{}_{ z}\rangle }_{\alpha }$$

for all f ∈ F _α ². We then define

$${K}_{S}(w,z) = {S}^{{_\ast}}{K}_{ z}(w) =\langle {S}^{{_\ast}}{K}_{ z},{K{}_{w}\rangle }_{\alpha } =\langle {K}_{z},S{K{}_{w}\rangle }_{\alpha }$$

(3.3)

for all z and w in $\mathbb{C}$. It is easy to see that the function K _S (w, z) is uniquely determined by the following two properties:

• (a) $Sf(z) = \int \nolimits \nolimits \mathbb{C}f(w)\overline{{K}_{S}(w,z)}\,\mathrm{d}{\lambda }_{\alpha }(w)$ for all f ∈ F _α ² and $z \in \mathbb{C}$.

• (b) K _S (  ⋅ , z) ∈ F _α ² for all $z \in \mathbb{C}$.

We collect in the following proposition some of the elementary properties of the kernel function K _S (w, z) induced by S.

Proposition 3.9.

The mapping S↦K _S has the following properties:

• (1) K _S+T = K _S + K _T , K _cS = cK _S.

• (2) K _S (  ⋅  ,z) ∈ F _α ².

• (3) ${K}_{{S}^{{_\ast}}}(w,z) = \overline{{K}_{S}(z,w)}$.

• (4) K _I (w,z) = K(w,z).

• (5) ${K}_{{S}_{n}} \rightarrow {K}_{S}$ pointwise whenever S _n → S weakly.

• (6) $\vert {K}_{S}(w,z)\vert \leq \| S\|\sqrt{K(w, w)K(z, z)}$.

• (7) ${K}_{{S}_{n}} \rightarrow {K}_{S}$ uniformly on compacta whenever S _n → S in norm.

• (8) ${K}_{S}(z,z) = K(z,z)\widetilde{{S}^{{_\ast}}}(z)$.

• (9) K _S (w,w) ≡ 0 if and only if S = 0.

Proof.

Properties (1)–(5) and (8) are direct consequences of the definition of K _S in (3.3) and the definition of the Berezin transform. Property (6) follows from (3.3) and the Cauchy–Schwarz inequality, and it implies property (7). Since the Berezin transform $S\mapsto \widetilde{S}$ is one-to-one, we see that (9) follows from (8).

Proposition 3.10.

Let S and T be bounded operators on F _α ² . Then

$${K}_{ST}(w,z) = \int \nolimits \nolimits \mathbb{C}{K}_{S}(u,z){K}_{T}(w,u)\,\mathrm{d}{\lambda }_{\alpha }(u)$$

for all w and z in $\mathbb{C}$.

Proof.

It follows from (3.3) that

$$\begin{array}{rcl}{ K}_{ST}(w,z)& =& \langle {T}^{{_\ast}}{S}^{{_\ast}}{K}_{ z},{K{}_{w}\rangle }_{\alpha } =\langle {S}^{{_\ast}}{K}_{ z},T{K{}_{w}\rangle }_{\alpha } \\ & =& \int \nolimits \nolimits \mathbb{C}{S}^{{_\ast}}{K}_{ z}(u)\overline{T{K}_{w}(u)}\,\mathrm{d}{\lambda }_{\alpha }(u) \\ & =& \int \nolimits \nolimits \mathbb{C}\langle {S}^{{_\ast}}{K}_{ z},{K{}_{u}\rangle }_{\alpha }\langle {T}^{{_\ast}}{K}_{ u},{K{}_{w}\rangle }_{\alpha }\,\mathrm{d}{\lambda }_{\alpha }(u) \\ & =& \int \nolimits \nolimits \mathbb{C}{K}_{S}(u,z){K}_{T}(w,u)\,\mathrm{d}{\lambda }_{\alpha }(u) \\ \end{array}$$

for all z and w in $\mathbb{C}$.

Proposition 3.11.

If S is a positive or trace-class operator, then

$$\mathrm{tr}\,(S) = \int \nolimits \nolimits \mathbb{C}\overline{{K}_{S}(z,z)}\,\mathrm{d}{\lambda }_{\alpha }(z).$$

Proof.

This follows from Proposition 3.3 and property (8) in Proposition 3.9.

Corollary 3.12.

Let S and T be bounded linear operators on F _α ² such that ST is trace class. Then

$$\mathrm{tr}\,(ST) = \int \nolimits \nolimits \mathbb{C}\mathrm{d}{\lambda }_{\alpha }(w)\int \nolimits \nolimits \mathbb{C}\overline{{K}_{S}(z,w)}\,\overline{{K}_{T}(w,z)}\,\mathrm{d}{\lambda }_{\alpha }(z).$$

Proof.

This is a direct consequence of Propositions 3.10 and 3.11.

3.2 The Berezin Transform of Functions

We say that a Lebesgue measurable function φ satisfies condition (I _p), where 0 < p < ∞, if $\varphi \circ {t}_{a} \in {L}^{p}(\mathbb{C},\mathrm{d}{\lambda }_{\alpha })$ for every $a \in \mathbb{C}$. In particular, any function satisfying condition (I _p ) must be in ${L}^{p}(\mathbb{C},\mathrm{d}{\lambda }_{\alpha })$.

By a change of variables, we see that a Lebesgue measurable function φ on $\mathbb{C}$ satisfies condition (I _p ) if and only if

$$\int \nolimits \nolimits \mathbb{C}\vert K(z,a){\vert }^{2}\vert \varphi (z){\vert }^{p}\,\mathrm{d}{\lambda }_{ \alpha }(z) < \infty $$

(3.4)

for all $a \in \mathbb{C}$. By the exponential form of the kernel function K(w, z), the above condition is equivalent to

$$\int \nolimits \nolimits \mathbb{C}\vert K(z,a)\vert \vert \varphi (z){\vert }^{p}\,\mathrm{d}{\lambda }_{ \alpha }(z) < \infty, \qquad a \in \mathbb{C}.$$

(3.5)

We are mostly interested in two particular cases: p = 1 and p = 2. The case p = 1 is needed in this section, while the case p = 2 will be used in Chap. 6 when we study Toeplitz operators with unbounded symbols. It is clear that every function in ${L}^{\infty }(\mathbb{C})$ satisfies condition (I _p ).

Suppose f satisfies condition (I ₁). We can then define a function $\widetilde{f}$ on $\mathbb{C}$ as follows:

$$\widetilde{f}(z) =\langle f{k}_{z},{k}_{z}\rangle = \int \nolimits \nolimits \mathbb{C}\vert {k}_{z}(w){\vert }^{2}f(w)\,\mathrm{d}{\lambda }_{ \alpha }(w).$$

(3.6)

We will also call $\widetilde{f}$ the Berezin transform of f. It is clear that we can write

$$\widetilde{f}(z) = \frac{\alpha } {\pi }\int \nolimits \nolimits \mathbb{C}f(w){\mathrm{e}}^{-\alpha \vert z-w{\vert }^{2} }\,\mathrm{d}A(w) = \int \nolimits \nolimits \mathbb{C}f(z \pm w)\,\mathrm{d}{\lambda }_{\alpha }(w).$$

(3.7)

Sometimes, we will need to emphasize the dependence on α. In such situations, we will use the notation

$${B}_{\alpha }f(z) = \frac{\alpha } {\pi }\int \nolimits \nolimits \mathbb{C}f(w){\mathrm{e}}^{-\alpha \vert z-w{\vert }^{2} }\,\mathrm{d}A(w),\qquad z \in \mathbb{C}.$$

(3.8)

Thus, $\widetilde{f} = {B}_{\alpha }f$ if no parameter is specified.

Theorem 3.13.

Let H _t = B _1∕t for any positive parameter t. Then we have the following semigroup property: H _s H _t = H _s+t for all positive parameters s and t.

Proof.

We check the semigroup property on ${L}^{\infty }(\mathbb{C})$. For $f \in {L}^{\infty }(\mathbb{C})$, we have

$${H}_{t}f(z) = \frac{1} {\pi t}\int \nolimits \nolimits \mathbb{C}f(w){\mathrm{e}}^{-\frac{1} {t}\vert z-w{\vert }^{2} }\,\mathrm{d}A(w)$$

(3.9)

for $z \in \mathbb{C}$ and

$${H}_{s}{H}_{t}f(z) = \frac{1} {{\pi }^{2}st}\int \nolimits \nolimits \mathbb{C}{\mathrm{e}}^{-\frac{1} {s}\vert z-w{\vert }^{2} }\,\mathrm{d}A(w)\int \nolimits \nolimits \mathbb{C}f(u){\mathrm{e}}^{-\frac{1} {t}\vert w-u{\vert }^{2} }\,\mathrm{d}A(u)$$

for $z \in \mathbb{C}$. By Fubini’s theorem,

$${H}_{s}{H}_{t}f(z) = \int \nolimits \nolimits \mathbb{C}f(u)I(z,u)\,\mathrm{d}A(u),\qquad z \in \mathbb{C},$$

where

$$I(z,u) = \frac{1} {{\pi }^{2}st}\int \nolimits \nolimits \mathbb{C}{\mathrm{e}}^{-\frac{1} {s}\vert z-w{\vert }^{2}-\frac{1} {t}\vert w-u{\vert }^{2} }\,\mathrm{d}A(w).$$

Since

$$\begin{array}{rcl} -\frac{1} {s}\vert z - w{\vert }^{2} -\frac{1} {t}\vert w - u{\vert }^{2}& =& -\left (\frac{1} {s} + \frac{1} {t}\right )\vert w{\vert }^{2} -\frac{1} {s}\vert z{\vert }^{2} -\frac{1} {t}\vert u{\vert }^{2} \\ & & +\left (\frac{z} {s} + \frac{u} {t} \right )\overline{w} + \left (\frac{\overline{z}} {s} + \frac{\overline{u}} {t} \right )w, \\ \end{array}$$

we have

$$\begin{array}{rcl} I(z,u)& =& \frac{1} {{\pi }^{2}st}{\mathrm{e}}^{-\frac{1} {s}\vert z{\vert }^{2}-\frac{1} {t}\vert u{\vert }^{2} } \int \nolimits \nolimits \mathbb{C}{\left \vert {\mathrm{e}}^{\left (\frac{z} {s}+\frac{u} {t} \right )\overline{w}}\right \vert }^{2}{\mathrm{e}}^{-\left (\frac{1} {s}+\frac{1} {t}\right )\vert w{\vert }^{2} }\,\mathrm{d}A(w) \\ & =& \frac{{\mathrm{e}}^{-\frac{1} {s}\vert z{\vert }^{2}-\frac{1} {t}\vert u{\vert }^{2} }} {\pi (s + t)} \cdot \frac{\frac{1} {s} + \frac{1} {t} } {\pi } \int \nolimits \nolimits \mathbb{C}{\left \vert {\mathrm{e}}^{\left (\frac{1} {s}+\frac{1} {t}\right )\frac{tz+su} {s+t} \overline{w}}\right \vert }^{2}{\mathrm{e}}^{-\left (\frac{1} {s}+\frac{1} {t}\right )\vert w{\vert }^{2} }\,\mathrm{d}A(w).\end{array}$$

Applying the reproducing formula in ${F}_{\frac{1} {s}+\frac{1} {t} }^{2}$, we obtain

$$I(z,u) = \frac{1} {\pi (s + t)}{\mathrm{e}}^{-\frac{1} {s}\vert z{\vert }^{2}-\frac{1} {t}\vert u{\vert }^{2}+\left (\frac{1} {s}+\frac{1} {t}\right ){\left \vert \frac{tz+su} {s+t} \right \vert }^{2} }.$$

Elementary calculations then show that

$$I(z,u) = \frac{1} {\pi (s + t)}{\mathrm{e}}^{- \frac{1} {s+t}\vert z-u{\vert }^{2} }.$$

Therefore,

$${H}_{s}{H}_{t}f(z) = \frac{1} {\pi (s + t)}\int \nolimits \nolimits \mathbb{C}f(u){\mathrm{e}}^{- \frac{1} {s+t}\vert z-u{\vert }^{2} }\,\mathrm{d}A(u) = {H}_{s+t}f(z).$$

This proves the desired result.

Because of the following result, the operator H _t is sometimes called the heat transform.

Theorem 3.14.

The function u(x,y,t) = H _t f(z), where z = x + i y, satisfies the heat equation

$$\frac{{\partial }^{2}u} {\partial {x}^{2}} + \frac{{\partial }^{2}u} {\partial {y}^{2}} = 4\,\frac{\partial u} {\partial t}.$$

(3.10)

Moreover, if f is bounded and continuous on $\mathbb{C}$ , then u also satisfies the initial condition

$${ \lim }_{t\rightarrow {0}^{+}}{H}_{t}f(z) = f(z),\qquad z \in \mathbb{C}.$$

(3.11)

Proof.

With z = x + iy and w = u + iv, we have

$$u(x,y,t) = \frac{1} {\pi t}{\int \nolimits \nolimits }_{{\mathbb{R}}^{2}}f(u,v){\mathrm{e}}^{-\frac{1} {t} [{(x-u)}^{2}+{(y-v)}^{2}] }\,\mathrm{d}u\,\mathrm{d}v.$$

Differentiating under the integral sign, we obtain

$$\begin{array}{rcl} \frac{\partial u} {\partial t} & =& - \frac{1} {\pi {t}^{2}}{ \int \nolimits \nolimits }_{{\mathbb{R}}^{2}}f(u,v){\mathrm{e}}^{-\frac{1} {t} [{(x-u)}^{2}+{(y-v)}^{2}] }\,\mathrm{d}u\,\mathrm{d}v \\ & & + \frac{1} {\pi {t}^{3}}{ \int \nolimits \nolimits }_{{\mathbb{R}}^{2}}[{(x - u)}^{2} + {(y - v)}^{2}]f(u,v){\mathrm{e}}^{-\frac{1} {t} [{(x-u)}^{2}+{(y-v)}^{2}] }\,\mathrm{d}u\,\mathrm{d}v.\end{array}$$

Similarly,

$$\frac{\partial u} {\partial x} = - \frac{2} {\pi {t}^{2}}{ \int \nolimits \nolimits }_{{\mathbb{R}}^{2}}(x - u)f(u,v){\mathrm{e}}^{-\frac{1} {t} [{(x-u)}^{2}+{(y-v)}^{2}] }\,\mathrm{d}u\,\mathrm{d}v,$$

and

$$\begin{array}{rcl} \frac{{\partial }^{2}u} {\partial {x}^{2}}& =& - \frac{2} {\pi {t}^{2}}{ \int \nolimits \nolimits }_{{\mathbb{R}}^{2}}f(u,v){\mathrm{e}}^{-\frac{1} {t} [{(x-u)}^{2}+{(y-v)}^{2}] }\,\mathrm{d}u\,\mathrm{d}v \\ & & + \frac{4} {\pi {t}^{3}}{ \int \nolimits \nolimits }_{{\mathbb{R}}^{2}}{(x - u)}^{2}f(u,v){\mathrm{e}}^{-\frac{1} {t} [{(x-u)}^{2}+{(y-v)}^{2}] }\,\mathrm{d}u\,\mathrm{d}v.\end{array}$$

Combining this with a similar calculation for ∂ ² u ∕ ∂y ² gives

$$\begin{array}{rcl} \Delta u& =& - \frac{4} {\pi {t}^{2}}{ \int \nolimits \nolimits }_{{\mathbb{R}}^{2}}f(u,v){\mathrm{e}}^{-\frac{1} {t} [{(x-u)}^{2}+{(y-v)}^{2}] }\,\mathrm{d}u\,\mathrm{d}v \\ & & + \frac{4} {\pi {t}^{3}}{ \int \nolimits \nolimits }_{{\mathbb{R}}^{2}}[{(x - u)}^{2} + {(y - v)}^{2}]f(u,v){\mathrm{e}}^{-\frac{1} {t} [{(x-u)}^{2}+{(y-v)}^{2}] }\,\mathrm{d}u\,\mathrm{d}v \\ & =& 4\,\frac{\partial u} {\partial t}, \\ \end{array}$$

where

$$\Delta u = \frac{{\partial }^{2}u} {\partial {x}^{2}} + \frac{{\partial }^{2}u} {\partial {y}^{2}}$$

is the Laplacian of u. Thus, u satisfies the heat equation (3.10).

To show that u also satisfies the initial condition (3.11), assume that f is bounded and continuous on $\mathbb{C}$. Fix a point $z \in \mathbb{C}$ and write

$$\begin{array}{rcl}{ H}_{t}f(z) - f(z)& =& \frac{1} {\pi t}\int \nolimits \nolimits \mathbb{C}(f(w) - f(z)){\mathrm{e}}^{-\frac{1} {t}\vert z-w{\vert }^{2} }\,\mathrm{d}A(w) \\ & =& \frac{1} {\pi t}{\int \nolimits \nolimits }_{\vert w-z\vert <\delta } + \frac{1} {\pi t}{\int \nolimits \nolimits }_{\vert w-z\vert >\delta } \\ & =& \!\!: {I}_{1} + {I}_{2}.\end{array}$$

Given any positive ε, we can choose a positive δ such that

$$\vert f(w) - f(z)\vert < \epsilon, \qquad w \in B(z,\delta ).$$

It follows that

$$\vert {I}_{1}\vert \leq \frac{\epsilon } {\pi t}{\int \nolimits \nolimits }_{\vert w-z\vert <\delta }{\mathrm{e}}^{-\frac{1} {t}\vert z-w{\vert }^{2} }\,\mathrm{d}A(z) < \frac{\epsilon } {\pi t}\int \nolimits \nolimits \mathbb{C}{\mathrm{e}}^{-\frac{1} {t}\vert z-w{\vert }^{2} }\,\mathrm{d}A(w) = \epsilon.$$

On the other hand,

$$\begin{array}{rcl} \vert {I}_{2}\vert & \leq & 2\|{f\|}_{\infty } \frac{1} {\pi t}{\int \nolimits \nolimits }_{\vert z-w\vert >\delta }{\mathrm{e}}^{-\frac{1} {t}\vert z-w{\vert }^{2} }\,\mathrm{d}A(w) \\ & =& 2\|{f\|}_{\infty } \frac{1} {\pi t}{\int \nolimits \nolimits }_{\vert w\vert >\delta }{\mathrm{e}}^{-\frac{1} {t}\vert w{\vert }^{2} }\,\mathrm{d}A(w) \\ & =& 2\|{f\|}_{\infty }{\mathrm{e}}^{-{\delta }^{2}/t } \rightarrow \end{array}$$

(0)

as t → 0⁺. It follows that

$${limsup}_{t\rightarrow {0}^{+}}\vert {H}_{t}f(z) - f(z)\vert \leq \epsilon.$$

Since ε is arbitrary, we must have

$${\lim }_{t\rightarrow {0}^{+}}{H}_{t}f(z) = f(z),$$

which completes the proof of the theorem.

Note that in the heat equation (3.10), the value u(x, y, t) represents the temperature at the point $(x,y) \in \mathbb{C}$ at time t. Thus, the function f(z) represents the initial temperature distribution in the complex plane at time t = 0. With this interpretation, the assumption that f be bounded and continuous is reasonable. However, the initial condition in (3.11) can be shown to hold for certain functions that are more general than bounded and continuous ones.

The following result is a direct consequence of Theorems 3.13 and 3.14.

Corollary 3.15.

For any positive α and β, we have the identities

$${B}_{\alpha }{B}_{\beta } = {B}_{ \frac{\alpha \beta } {\alpha +\beta } } = {B}_{\beta }{B}_{\alpha }.$$

If f is bounded and continuous, then

$${\lim }_{\alpha \rightarrow +\infty }{B}_{\alpha }f(z) = f(z)$$

for every $z \in \mathbb{C}$.

We need the following result from Fourier analysis to generalize Proposition 3.1 to the Berezin transform of functions.

Lemma 3.16

Suppose that n is a positive integer and f is a function on ${\mathbb{R}}^{n}$ such that the function

$$x\mapsto f(x){\mathrm{e}}^{\vert tx\vert }{\mathrm{e}}^{-{x}^{2} }$$

is integrable on ${\mathbb{R}}^{n}$ with respect to Lebesgue measure d x for any $t \in {\mathbb{R}}^{n}$ . Here,

$$x = ({x}_{1},\ldots, {x}_{n}),\quad t = ({t}_{1},\ldots, {t}_{n}),\quad tx = {t}_{1}{x}_{1} + \cdots + {t}_{n}{x}_{n},$$

and

$${x}^{2} = {x}_{ 1}^{2} + \cdots + {x}_{ n}^{2},\quad \mathrm{d}x = \mathrm{d}{x}_{ 1}\cdots \mathrm{d}{x}_{n}.$$

If

$${\int \nolimits \nolimits }_{{\mathbb{R}}^{n}}f(x)P(x){\mathrm{e}}^{-{x}^{2} }\,\mathrm{d}x = 0$$

for every polynomial P, then f = 0 almost everywhere on ${\mathbb{R}}^{n}$.

Proof.

Since

$${\mathrm{e}}^{\mathrm{i}tx} =\sum \limits_{k=0}^{\infty }\frac{{(\mathrm{i}tx)}^{k}} {k!}$$

and

$$\left \vert \sum \limits_{k=0}^{N}\frac{{(\mathrm{i}tx)}^{k}} {k!} \right \vert \leq \sum \limits_{k=0}^{\infty }\frac{\vert tx{\vert }^{k}} {k!} ={ \mathrm{e}}^{\vert tx\vert }$$

for all N ≥ 0, we apply the dominated convergence theorem to partial sums to obtain

$${\int \nolimits \nolimits }_{{\mathbb{R}}^{n}}{\mathrm{e}}^{\mathrm{i}tx}f(x){\mathrm{e}}^{-{x}^{2} }\,\mathrm{d}x =\sum \limits_{k=0}^{\infty }\frac{{\mathrm{i}}^{k}} {k!}{\int \nolimits \nolimits }_{{\mathbb{R}}^{n}}{(tx)}^{k}f(x){\mathrm{e}}^{-{x}^{2} }\,\mathrm{d}x = 0$$

for all $t \in {\mathbb{R}}^{n}$. By the Fourier inversion theorem, we have $f(x){\mathrm{e}}^{-{x}^{2} } = 0$, and hence f(x) = 0 for almost every $x \in {\mathbb{R}}^{n}$.

Note that the integral condition (3.12) in the next proposition is slightly stronger than condition (I ₁) which was necessary for the definition of B _α f.

Proposition 3.17.

The Berezin transform B _α is linear and order preserving. Furthermore, if B _α f = 0 and f satisfies the condition that

$$\int \nolimits \nolimits \mathbb{C}\vert f(z)\vert {\mathrm{e}}^{\vert tz\vert }{\mathrm{e}}^{-\alpha \vert z{\vert }^{2} }\,\mathrm{d}A(z) < \infty $$

(3.12)

for all real t, then f(z) = 0 for almost every $z \in \mathbb{C}$.

Proof.

It is clear that each B _α is linear and order preserving.

If B _α f = 0 and f satisfies the integral condition (3.12), then differentiating under the integral sign gives

$$\frac{{\partial }^{n+m}} {\partial {z}^{n}\partial {\overline{z}}^{m}}{B}_{\alpha }f(0) = {c}_{m,n} \int \nolimits \nolimits \mathbb{C}f(w){w}^{m}{\overline{w}}^{n}{\mathrm{e}}^{-\alpha \vert w{\vert }^{2} }\,\mathrm{d}A(w),$$

where c _m, n is a nonzero constant. It follows that

$$\int \nolimits \nolimits \mathbb{C}f(w){w}^{m}{\overline{w}}^{n}{\mathrm{e}}^{-\alpha \vert w{\vert }^{2} }\,\mathrm{d}A(w) = 0$$

for all nonnegative integers m and n. The result then follows from Lemma 3.16.

In the next few results, we describe some of the mapping properties of the Berezin transform. In particular, we will compare B _α f and B _β f in various situations.

Theorem 3.18.

Let 1 ≤ p ≤∞. Suppose α, β, and γ are positive weight parameters. Then B _α L _β ^p ⊂ L _γ ^p if and only if γ(2α − β) ≥ 2αβ.

Proof.

First, assume that γ(2α − β) ≥ 2αβ. Then, in particular, $\alpha > \frac{\beta } {2}$. If f ∈ L _β ^∞, we write

$${B}_{\alpha }f(z) = \frac{\alpha } {\pi }{\mathrm{e}}^{-\alpha \vert z{\vert }^{2} } \int \nolimits \nolimits \mathbb{C}f(w){\mathrm{e}}^{-\frac{\beta } {2} \vert w{\vert }^{2} }\vert {\mathrm{e}}^{\alpha z\overline{w}}{\vert }^{2}{\mathrm{e}}^{-\left (\alpha -\frac{\beta } {2} \right )\vert w{\vert }^{2} }\,\mathrm{d}A(w).$$

It follows that

$$\begin{array}{rcl} \vert {B}_{\alpha }f(z)\vert & \leq & \frac{\alpha \|{f\|}_{\infty, \beta }} {\pi }{ \mathrm{e}}^{-\alpha \vert z{\vert }^{2} } \int \nolimits \nolimits \mathbb{C}{\left \vert {\mathrm{e}}^{(\alpha -\frac{\beta } {2} ) \frac{\alpha z} {\alpha -\frac{\beta }{2} } \overline{w} }\right \vert }^{2}{\mathrm{e}}^{-\left (\alpha -\frac{\beta } {2} \right )\vert w{\vert }^{2} }\,\mathrm{d}A(w) \\ & =& \frac{\alpha } {\alpha -\frac{\beta } {2} }\|{f\|}_{\infty, \beta }{\mathrm{e}}^{-\alpha \vert z{\vert }^{2} }{\mathrm{e}}^{(\alpha -\frac{\beta } {2} ){\left \vert \frac{\alpha z} {\alpha -\frac{\beta }{2} } \right \vert }^{2} }.\end{array}$$

Therefore,

$$\vert {B}_{\alpha }f(z)\vert {\mathrm{e}}^{-\frac{\gamma } {2} \vert z{\vert }^{2} } \leq \frac{2\alpha \|{f\|}_{\infty, \beta }} {2\alpha - \beta }{ \mathrm{e}}^{-(\alpha +\frac{\gamma } {2} - \frac{2{\alpha }^{2}} {2\alpha -\beta })\vert z{\vert }^{2} }.$$

It is elementary to check that the condition γ(2α − β) ≥ 2αβ is equivalent to

$$\alpha + \frac{\gamma } {2} - \frac{2{\alpha }^{2}} {2\alpha - \beta } \geq 0.$$

Thus, B _α maps L _β ^∞ into L _γ ^∞.

If f ∈ L _β ¹, the integral

$$I = \int \nolimits \nolimits \mathbb{C}\left \vert {B}_{\alpha }f(z){\mathrm{e}}^{-\frac{\gamma } {2} \vert z{\vert }^{2} }\right \vert \,\mathrm{d}A(z)$$

equals

$$\frac{\alpha } {\pi }\int \nolimits \nolimits \mathbb{C}\left \vert {\mathrm{e}}^{-(\alpha +\frac{\gamma } {2} )\vert z{\vert }^{2} } \int \nolimits \nolimits \mathbb{C}f(w)\vert {\mathrm{e}}^{\alpha z\overline{w}}{\vert }^{2}{\mathrm{e}}^{-\alpha \vert w{\vert }^{2} }\,\mathrm{d}A(w)\right \vert \,\mathrm{d}A(z),$$

which by Fubini’s theorem is less than or equal to

$$\frac{\alpha } {\pi }\int \nolimits \nolimits \mathbb{C}\vert f(w)\vert {\mathrm{e}}^{-\alpha \vert w{\vert }^{2} }\,\mathrm{d}A(w)\int \nolimits \nolimits \mathbb{C}\vert {\mathrm{e}}^{\alpha z\overline{w}}{\vert }^{2}{\mathrm{e}}^{-(\alpha +\frac{\gamma } {2} )\vert z{\vert }^{2} }\,\mathrm{d}A(z).$$

With the help of Corollary 2.5, we obtain

$$I \leq \frac{2\alpha } {2\alpha + \gamma }\int \nolimits \nolimits \mathbb{C}\vert f(w)\vert {\mathrm{e}}^{-\left (\alpha - \frac{2{\alpha }^{2}} {2\alpha +\gamma }\right )\vert w{\vert }^{2} }\,\mathrm{d}A(w).$$

Again, it is elementary to check that the condition γ(2α − β) ≥ 2αβ is equivalent to

$$\alpha - \frac{2{\alpha }^{2}} {2\alpha + \gamma } \geq \frac{\beta } {2}.$$

Thus, B _α maps L _β ¹ into L _γ ¹.

By complex interpolation, the Berezin transform B _α maps L _β ^p into L _γ ^p for all 1 ≤ p ≤ ∞ whenever γ(2α − β) ≥ 2αβ.

To prove the other direction, observe that

$${B}_{\alpha }f(z) ={ \mathrm{e}}^{-\alpha \vert z{\vert }^{2} }{Q}_{\alpha }f(2z).$$

It follows from this and a change of variables that B _α f ∈ L _γ ^p if and only if ${Q}_{\alpha }f \in {L}_{\frac{\gamma } {4} +\frac{\alpha } {2} }^{p}$. Therefore, B _α L _β ^p ⊂ L _γ ^p is equivalent to ${Q}_{\alpha }{L}_{\beta }^{p} \subset {L}_{\frac{\gamma } {4} +\frac{\alpha } {2} }^{p}$, which implies that ${P}_{\alpha }{L}_{\beta }^{p} \subset {L}_{\frac{\gamma } {4} +\frac{\alpha } {2} }^{p}$. Combining this with Theorem 2.31, we conclude that B _α L _β ^p ⊂ L _γ ^p implies that

$${\alpha }^{2} \leq (2\alpha - \beta )\left (\frac{\gamma } {4} + \frac{\alpha } {2} \right ),$$

which is equivalent to γ(2α − β) ≥ 2αβ. This completes the proof of the theorem.

Corollary 3.19.

Let α > 0 and β > 0. For 1 ≤ p ≤∞, we have

• (a) B _α : L _α ^p → L _β ^p if and only if β ≥ 2α.

• (b) B _α : L _β ^p → L _α ^p if and only if 2α ≥ 3β.

Proposition 3.20.

Let α > 0 and 1 ≤ p < ∞. Then

• (a) ${B}_{\alpha } : {L}^{\infty }(\mathbb{C}) \rightarrow {L}^{\infty }(\mathbb{C})$ is a contraction.

• (b) ${B}_{\alpha } : {C}_{0}(\mathbb{C}) \rightarrow {C}_{0}(\mathbb{C})$ is a contraction.

• (c) ${B}_{\alpha } : {L}^{p}(\mathbb{C},\mathrm{d}A) \rightarrow {L}^{p}(\mathbb{C},\mathrm{d}A)$ is a contraction.

Proof.

Part (a) is obvious. If $f \in {C}_{c}(\mathbb{C})$, namely, if f is a continuous function on $\mathbb{C}$ with compact support, then it is easy to see that ${B}_{\alpha }f \in {C}_{0}(\mathbb{C})$. Thus, part (b) follows from (a) and the fact that ${C}_{c}(\mathbb{C})$ is dense in ${C}_{0}(\mathbb{C})$ in the supremum norm.

To prove (c), we first consider the case p = 1. In this case, it follows from Fubini’s theorem that

$$\begin{array}{rcl} \int \nolimits \nolimits \mathbb{C}\vert {B}_{\alpha }f(z)\vert \,\mathrm{d}A(z)& \leq & \frac{\alpha } {\pi }\int \nolimits \nolimits \mathbb{C}\vert f(w)\vert \,\mathrm{d}A(w)\int \nolimits \nolimits \mathbb{C}{\mathrm{e}}^{-\alpha \vert z-w{\vert }^{2} }\,\mathrm{d}A(z) \\ & =& \int \nolimits \nolimits \mathbb{C}\vert f(w)\vert \,\mathrm{d}A(w).\end{array}$$

The case 1 < p < ∞ then follows from complex interpolation.

Proposition 3.21.

Let 0 < β < α and 1 ≤ p < ∞. Then

• (a) ${B}_{\alpha }f \in {L}^{\infty }(\mathbb{C})$ implies ${B}_{\beta }f \in {L}^{\infty }(\mathbb{C})$ with

  $$\|{B}_{\beta }{f\|}_{\infty }\leq \| {B}_{\alpha }{f\|}_{\infty }$$

  for all f.

• (b) ${B}_{\alpha }f \in {C}_{0}(\mathbb{C})$ implies that ${B}_{\beta }f \in {C}_{0}(\mathbb{C})$.

• (c) ${B}_{\alpha }f \in {L}^{p}(\mathbb{C},\mathrm{d}A)$ implies that ${B}_{\beta }f \in {L}^{p}(\mathbb{C},\mathrm{d}A)$ with

  $$\int \nolimits \nolimits \mathbb{C}\vert {B}_{\beta }f(z){\vert }^{p}\,\mathrm{d}A(z) \leq \int \nolimits \nolimits \mathbb{C}\vert {B}_{\alpha }f(z){\vert }^{p}\,\mathrm{d}A(z)$$

  for all f.

Proof.

Choose a positive γ such that 1 ∕ γ + 1 ∕ α = 1 ∕ β. By Corollary 3.15, we have B _β = B _γ B _α. The desired result then follows from Proposition 3.20.

Proposition 3.22.

If 0 < β < α, 0 < p < ∞, and f ≥ 0. Then

$${B}_{\alpha }f(z) \leq \frac{\alpha } {\beta }{B}_{\beta }f(z),\qquad z \in \mathbb{C}.$$

Consequently:

• (a) ${B}_{\beta }f \in {L}^{\infty }(\mathbb{C})$ implies that ${B}_{\alpha }f \in {L}^{\infty }(\mathbb{C})$.

• (b) ${B}_{\beta }f \in {C}_{0}(\mathbb{C})$ implies that ${B}_{\alpha }f \in {C}_{0}(\mathbb{C})$.

• (c) ${B}_{\beta }f \in {L}^{p}(\mathbb{C},\mathrm{d}A)$ implies that ${B}_{\alpha }f \in {L}^{p}(\mathbb{C},\mathrm{d}A)$.

Proof.

Since f ≥ 0 and 0 < β < α, we have

$$\begin{array}{rcl}{ B}_{\alpha }f(z)& =& \frac{\alpha } {\pi }\int \nolimits \nolimits \mathbb{C}f(w){\mathrm{e}}^{-\alpha \vert z-w{\vert }^{2} }\,\mathrm{d}A(w) \\ & \leq & \frac{\alpha } {\pi }\int \nolimits \nolimits \mathbb{C}f(w){\mathrm{e}}^{-\beta \vert z-w{\vert }^{2} }\,\mathrm{d}A(w) \\ & =& \frac{\alpha } {\beta } \cdot \frac{\beta } {\pi }\int \nolimits \nolimits \mathbb{C}f(w){\mathrm{e}}^{-\beta \vert z-w{\vert }^{2} }\,\mathrm{d}A(w) \\ & =& \frac{\alpha } {\beta }{B}_{\beta }f(z).\end{array}$$

This proves the desired results.

Theorem 3.23.

Suppose α and β are positive weight parameters and f ≥ 0 on $\mathbb{C}$ . For 0 < p ≤∞, we have

• (a) ${B}_{\alpha }f \in {L}^{p}(\mathbb{C},\mathrm{d}A)$ if and only if ${B}_{\beta }f \in {L}^{p}(\mathbb{C},\mathrm{d}A)$.

• (b) ${B}_{\alpha }f \in {C}_{0}(\mathbb{C})$ if and only if ${B}_{\beta }f \in {C}_{0}(\mathbb{C})$.

Proof.

Part (a) in the case 1 ≤ p ≤ ∞ and part (b) follow from Propositions 3.21 and 3.22. Part (a) in the case 0 < p < 1 will be proved in Chap. 6.

Recall that for any $a \in \mathbb{C}$, we have

$${t}_{a}(z) = z + a,\quad {\tau }_{a}(z) = z - a,\quad {\varphi }_{a}(z) = a - z.$$

The following result shows that the Berezin transform commutes with each of these maps.

Proposition 3.24.

If f is a function such that the Berezin transform B _α f is well defined, then for any $a \in \mathbb{C}$ , we have

• (i) B _α (f ∘ t _a ) = (B _α f) ∘ t _a.

• (ii) B _α (f ∘ τ _a ) = (B _α f) ∘ τ _a.

• (iii) B _α (f ∘ φ _a ) = (B _α f) ∘ φ _a.

Proof.

By (3.7), we have

$$\begin{array}{rcl} \widetilde{f \circ {t}_{a}}(z)& =& \int \nolimits \nolimits \mathbb{C}f \circ {t}_{a}(z + w)\,\mathrm{d}{\lambda }_{\alpha }(w) \\ & =& \int \nolimits \nolimits \mathbb{C}f(a + z + w)\,\mathrm{d}{\lambda }_{\alpha }(w) \\ & =& \widetilde{f}(a + z) =\widetilde{ f} \circ {t}_{a}(z) \\ \end{array}$$

for any $z \in \mathbb{C}$. This proves (i). Replacing a by − a in (i) leads to (ii).

Similarly, it follows from (3.7) that

$$\begin{array}{rcl} \widetilde{f \circ {\varphi }_{a}}(z)& =& \int \nolimits \nolimits \mathbb{C}f \circ {\varphi }_{a}(z + w)\,\mathrm{d}{\lambda }_{\alpha }(w) \\ & =& \int \nolimits \nolimits \mathbb{C}f(a - z - w)\,\mathrm{d}{\lambda }_{\alpha }(w) \\ & =& \widetilde{f}(a - z) =\widetilde{ f} \circ {\varphi }_{a}(z).\end{array}$$

This proves (iii).

For any positive integer n, we use B _α ⁿ f to denote the n-th iterate of the Berezin transform of f, that is, we take the Berezin transform of f repeatedly n times to obtain B _α ⁿ f.

Theorem 3.25.

Suppose $f \in {L}^{\infty }(\mathbb{C})$ and n is a positive integer. Then

$$\vert {B}_{\alpha }^{n}f(z) - {B}_{ \alpha }^{n}f(w)\vert \leq \frac{C\|{f\|}_{\infty }} {\sqrt{n}} \,\vert z - w\vert $$

(3.13)

for all z and w in $\mathbb{C}$ , where $C = 2\sqrt{\alpha /\pi }$.

Proof.

Recall that the Berezin transform of f is

$${B}_{\alpha }f(z) = \frac{\alpha } {\pi }\int \nolimits \nolimits \mathbb{C}f(u){\mathrm{e}}^{-\alpha \vert z-u{\vert }^{2} }\,\mathrm{d}A(u).$$

It follows that the difference

$$D = {B}_{\alpha }f(z) - {B}_{\alpha }f(w)$$

can be written as

$$\frac{\alpha } {\pi }\int \nolimits \nolimits \mathbb{C}f\left (u + \frac{z + w} {2} \right )\left [{\mathrm{e}}^{-\alpha \vert u-(z-w)/2{\vert }^{2} } -{\mathrm{e}}^{-\alpha \vert u+(z-w)/2{\vert }^{2} }\right ]\,\mathrm{d}A(u).$$

Let (z − w) ∕ 2 = re^iθ with r ≥ 0. By the rotation invariance of the area measure,

$$\begin{array}{rcl} \vert D\vert & \leq & \frac{\alpha \|{f\|}_{\infty }} {\pi } \int \nolimits \nolimits \mathbb{C}\left \vert {\mathrm{e}}^{-\alpha \vert u-r{\vert }^{2} } -{\mathrm{e}}^{-\alpha \vert u+r{\vert }^{2} }\right \vert \,\mathrm{d}A(u) \\ & =& \frac{\alpha \|{f\|}_{\infty }} {\pi } \int \nolimits \nolimits \mathbb{C}{\mathrm{e}}^{-\alpha (\vert u{\vert }^{2}+{r}^{2}) }\vert {\mathrm{e}}^{\alpha (u+\bar{u})r} -{\mathrm{e}}^{-\alpha (u+\bar{u})r}\vert \,\mathrm{d}A(u).\end{array}$$

Write u = x + iy and dA(u) = dxdy. We obtain

$$\begin{array}{rcl} \vert D\vert & \leq & \frac{\alpha \|{f\|}_{\infty }} {\pi } {\int \nolimits \nolimits }_{-\infty }^{\infty }{\mathrm{e}}^{-\alpha {y}^{2} }\,\mathrm{d}y{\int \nolimits \nolimits }_{-\infty }^{\infty }{\mathrm{e}}^{-\alpha ({x}^{2}+{r}^{2}) }\vert {\mathrm{e}}^{-2r\alpha x} -{\mathrm{e}}^{2r\alpha x}\vert \,\mathrm{d}x \\ & =& \frac{2\sqrt{\alpha }\|{f\|}_{\infty }} {\pi } {\int \nolimits \nolimits }_{-\infty }^{\infty }{\mathrm{e}}^{-{y}^{2} }\,\mathrm{d}y{\int \nolimits \nolimits }_{0}^{\infty }{\mathrm{e}}^{-\alpha ({x}^{2}+{r}^{2}) }\left ({\mathrm{e}}^{2r\alpha x} -{\mathrm{e}}^{-2r\alpha x}\right )\,\mathrm{d}x \\ & =& \frac{2\sqrt{\alpha }\|{f\|}_{\infty }} {\sqrt{\pi }} {\int \nolimits \nolimits }_{0}^{\infty }\left ({\mathrm{e}}^{-\alpha {(x-r)}^{2} } -{\mathrm{e}}^{-\alpha {(x+r)}^{2} }\right )\,\mathrm{d}x \\ & =& \frac{2\sqrt{\alpha }} {\sqrt{\pi }} \|{f\|}_{\infty }\left ({\int \nolimits \nolimits }_{-r}^{\infty }{\mathrm{e}}^{-\alpha {x}^{2} }\,\mathrm{d}x -{\int \nolimits \nolimits }_{r}^{\infty }{\mathrm{e}}^{-\alpha {x}^{2} }\,\mathrm{d}x\right ) \\ & =& \frac{2\sqrt{\alpha }} {\sqrt{\pi }} \|{f\|}_{\infty }{\int \nolimits \nolimits }_{-r}^{r}{\mathrm{e}}^{-\alpha {x}^{2}\,\mathrm{d}x } \\ & \leq & \frac{4r\sqrt{\alpha }} {\sqrt{\pi }} \|{f\|}_{\infty } = \frac{2\sqrt{\alpha }} {\sqrt{\pi }} \|{f\|}_{\infty }\vert z - w\vert.\end{array}$$

Thus, we have proved that

$$\vert {B}_{\alpha }f(z) - {B}_{\alpha }f(w)\vert \leq \frac{2\sqrt{\alpha }} {\sqrt{\pi }} \|{f\|}_{\infty }\vert z - w\vert $$

(3.14)

for all $f \in {L}^{\infty }(\mathbb{C})$ and all z and w in $\mathbb{C}$.

By Corollary 3.15, we have

$${B}_{\alpha }^{n}f(z) = {B}_{\frac{\alpha } {n} }f(z) = \frac{\alpha } {\pi n}\int \nolimits \nolimits \mathbb{C}f(w){\mathrm{e}}^{-\frac{\alpha } {n}\vert z-w{\vert }^{2} }\,\mathrm{d}A(w).$$

This, along with a simple change of variables, shows that

$${B}_{\alpha }^{n}f(z) = {B}_{ \alpha }g(z/\sqrt{n}),$$

where $g(z) = f(\sqrt{n}\,z)$. Combining this with the estimate in (3.14), we obtain the desired Lipschitz estimate in (3.13).

3.3 Fixed Points of the Berezin Transform

In the theory of Bergman spaces, it follows from a theorem of Ahern, Flores, and Rudin that a function is fixed by the Berezin transform in that context if and only if the function is harmonic, as long as the Berezin transform of the function is well defined. No other assumption on the function is necessary. See [1].

Therefore, it is natural to ask if the fixed points of the Berezin transform in our context here are exactly the harmonic functions as well. It turns out that the answer is negative in general, but positive under certain conditions.

Proposition 3.26.

Suppose f is a harmonic function on $\mathbb{C}$ satisfying condition (I ₁ ). Then $\widetilde{f} = f$.

Proof.

If f is harmonic, then f ∘ t _z is harmonic for every z. It follows from the mean value theorem for harmonic functions that

$$f \circ {t}_{z}(0) = \int \nolimits \nolimits \mathbb{C}f \circ {t}_{z}(w)\,\mathrm{d}{\lambda }_{\alpha }(w).$$

This shows that $f(z) =\widetilde{ f}(z)$ for every $z \in \mathbb{C}$.

The following result gives a partial converse to the proposition above.

Proposition 3.27.

If $f \in {L}^{\infty }(\mathbb{C})$ , then the following conditions are equivalent:

• (a) $\widetilde{f} = f$.

• (b) f is harmonic.

• (c) f is constant.

Proof.

Since f is bounded, the equivalence of (b) and (c) follows from the well-known maximum modulus principle for harmonic functions. If f is constant, then clearly $\widetilde{f} = f$. If $\widetilde{f} = f$, then ${\widetilde{f}}^{(n)} = f$ for all positive integers n. By Theorem 3.25, there exists a positive constant C such that

$$\vert f(z) - f(w)\vert \leq \frac{C} {\sqrt{n}}\vert z - w\vert $$

for all z and w in $\mathbb{C}$ with z ≠ w. Let n → ∞. We see that f must be constant.

Finally, in this section, we show by an example that there are more functions than the harmonic ones that are fixed by the Berezin transform.

Lemma 3.28.

For any complex ζ, let

$$I(\zeta ) = \frac{1} {\sqrt{\pi }}{\int \nolimits \nolimits }_{-\infty }^{\infty }{\mathrm{e}}^{\zeta t-{t}^{2} }\,\mathrm{d}t.$$

We have $I(\zeta ) ={ \mathrm{e}}^{{\zeta }^{2}/4 }$.

Proof.

It is clear that I(ζ) is an entire function of ζ. Differentiating under the integral sign, we obtain

$$\begin{array}{rcl} I^{\prime}(\zeta )& =& \frac{1} {\sqrt{\pi }}{\int \nolimits \nolimits }_{-\infty }^{\infty }t{\mathrm{e}}^{\zeta t-{t}^{2} }\,\mathrm{d}t \\ & =& \frac{1} {\sqrt{\pi }}{\int \nolimits \nolimits }_{-\infty }^{\infty }\left (t -\frac{\zeta } {2}\right ){\mathrm{e}}^{\zeta t-{t}^{2} }\,\mathrm{d}t + \frac{\zeta } {2}I(\zeta ) \\ & =& \frac{\zeta } {2}I(\zeta ).\end{array}$$

It follows that $I(\zeta ) = C{\mathrm{e}}^{{\zeta }^{2}/4 }$ for some constant C and all $\zeta \in \mathbb{C}$. It is well known that I(0) = 1. Thus, $I(\zeta ) ={ \mathrm{e}}^{{\zeta }^{2}/4 }$ for all $\zeta \in \mathbb{C}$.

Now fix two complex constants a and b such that a ² + b ² = 8απi and consider the function

$$f(z) ={ \mathrm{e}}^{ax+by},\quad z = x + \mathrm{i}y \in \mathbb{C},$$

which clearly satisfies condition (I ₁). A direct calculation shows that

$$\Delta f = ({a}^{2} + {b}^{2})f = 8\alpha \pi \mathrm{i}f,$$

so f is not harmonic. On the other hand,

$$\begin{array}{rcl} \widetilde{f}(z)& =& \int \nolimits \nolimits \mathbb{C}f(w + z)\,\mathrm{d}{\lambda }_{\alpha }(w) \\ & =& f(z)\int \nolimits \nolimits \mathbb{C}{\mathrm{e}}^{au+bv}\,\mathrm{d}{\lambda }_{ \alpha }(w), \\ \end{array}$$

where w = u + iv. Separating the variables, we obtain

$$\widetilde{f}(z) = f(z)I(a,\alpha )I(b,\alpha ),$$

where

$$I(\zeta, \alpha ) = \sqrt{\frac{\alpha } {\pi }}{\int \nolimits \nolimits }_{-\infty }^{\infty }{\mathrm{e}}^{\zeta t-\alpha {t}^{2} }\,\mathrm{d}t.$$

A simple change of variables gives

$$\widetilde{f}(z) = f(z)I(a/\sqrt{\alpha })I(b/\sqrt{\alpha }),$$

where I(ζ) is the function considered in Lemma 3.28 above. An application of Lemma 3.28 then gives

$$\widetilde{f}(z) = f(z){\mathrm{e}}^{({a}^{2}+{b}^{2})/(4\alpha ) } = f(z).$$

This shows that the function f is fixed by the Berezin transform, but it is not harmonic.

3.4 Fock–Carleson Measures

The main result of this section is the following:

Theorem 3.29.

Suppose μ is a positive Borel measure on $\mathbb{C}$ , 0 < p < ∞, and 0 < r < ∞. Then the following conditions are equivalent:

• (a) There exists a positive constant C such that

  $$\int \nolimits \nolimits \mathbb{C}\vert f(w){\mathrm{e}}^{-\frac{\alpha } {2} \vert w{\vert }^{2} }{\vert }^{p}\,\mathrm{d}\mu (w) \leq C\int \nolimits \nolimits \mathbb{C}\vert f(w){\mathrm{e}}^{-\frac{\alpha } {2} \vert w{\vert }^{2} }{\vert }^{p}\,\mathrm{d}A(w)$$

  for all entire functions f.

• (b) There exists a positive constant C such that

  $$\int \nolimits \nolimits \mathbb{C}{\mathrm{e}}^{-\frac{p\alpha } {2} \vert z-w{\vert }^{2} }\,\mathrm{d}\mu (w) \leq C$$

  for all $z \in \mathbb{C}$.

• (c) There exists a constant C > 0 such that μ(B(z,r)) ≤ C for all $z \in \mathbb{C}$.

Proof.

Fix a positive radius r and consider the lattice $r{\mathbb{Z}}^{2}$ in $\mathbb{C}$. Let {z _n } denote any fixed arrangement of this lattice into a sequence. For any entire function f, we set

$$I(f) = \int \nolimits \nolimits \mathbb{C}\vert f(w){\mathrm{e}}^{-\frac{\alpha } {2} \vert w{\vert }^{2} }{\vert }^{p}\,\mathrm{d}\mu (w).$$

Then

$$I(f) \leq \sum \limits_{n}{ \int \nolimits \nolimits }_{B({z}_{n},r)}\vert f(w){\mathrm{e}}^{-\frac{\alpha } {2} \vert w{\vert }^{2} }{\vert }^{p}\,\mathrm{d}\mu (w).$$

By Lemma 2.32 and the triangle inequality, there exists a constant C ₁ > 0 such that

$$\vert f(w){\mathrm{e}}^{-\frac{\alpha } {2} \vert w{\vert }^{2} }{\vert }^{p} \leq {C}_{ 1}{ \int \nolimits \nolimits }_{B({z}_{n},2r)}\vert f(u){\mathrm{e}}^{-\frac{\alpha } {2} \vert u{\vert }^{2} }{\vert }^{p}\,\mathrm{d}A(u)$$

for all w ∈ B(z _n , r). If condition (c) holds, then we can find a positive constant C ₂ (independent of f) such that

$$I(f) \leq {C}_{2}\sum \limits_{n}{ \int \nolimits \nolimits }_{B({z}_{n},2r)}\vert f(u){\mathrm{e}}^{-\frac{\alpha } {2} \vert u{\vert }^{2} }{\vert }^{p}\,\mathrm{d}A(u)$$

for all entire functions f. It is clear that there exists a positive integer N such that every point in the complex plane belongs to at most N of the disks B(z _n , 2r). Therefore,

$$I(f) \leq {C}_{2}N\int \nolimits \nolimits \mathbb{C}\vert f(u){\mathrm{e}}^{-\frac{\alpha } {2} \vert u{\vert }^{2} }{\vert }^{p}\,\mathrm{d}A(u).$$

This shows that condition (c) implies condition (a).

To show that condition (a) implies condition (b), simply take f = k _z and apply Lemma 2.33.

Finally, if condition (b) holds, then

$${\int \nolimits \nolimits }_{B(z,r)}{\mathrm{e}}^{-\frac{p\alpha } {2} \vert z-w{\vert }^{2} }\,\mathrm{d}\mu (w) \leq C$$

for all $z \in \mathbb{C}$. This clearly implies that

$$\mu (B(z,r)) \leq C{\mathrm{e}}^{\frac{p\alpha } {2} {r}^{2} }$$

for all $z \in \mathbb{C}$.

It is interesting to notice that condition (c) is independent of p and α. It follows that if condition (a) holds for some p > 0 and some α, then it holds for every p and every α (with the constant C dependent on p and α).

Similarly, condition (a) is independent of r. Therefore, if condition (c) holds for some r > 0, then it holds for every r > 0 (with the constant C dependent on r).

From now on, we will call any positive Borel measure μ that satisfies any of the equivalent conditions (a)–(c) above a Fock–Carleson measure. Similarly, we say that a positive Borel measure μ on $\mathbb{C}$ is a vanishing Fock–Carleson measure if

$${\lim }_{n\rightarrow \infty }\int \nolimits \nolimits \mathbb{C}\vert {f}_{n}(z){\mathrm{e}}^{-\frac{\alpha } {2} \vert z{\vert }^{2} }{\vert }^{p}\,\mathrm{d}\mu (z) = 0,$$

whenever {f _n } is a bounded sequence in F _α ^p that converges to 0 uniformly on compact subsets. We proceed to show that being a vanishing Fock–Carleson measure is also independent of p and α.

Theorem 3.30.

Suppose p > 0, α > 0, r > 0, and μ is a positive Borel measure on $\mathbb{C}$ . Then the following conditions are equivalent:

• (i) μ is a vanishing Fock–Carleson measure.

• (ii) $\int \nolimits \nolimits \mathbb{C}{\mathrm{e}}^{-\frac{p\alpha } {2} \vert z-w{\vert }^{2} }\,\mathrm{d}\mu (w) \rightarrow 0$ as z →∞.

• (iii) μ(B(z,r)) → 0 as z →∞.

Proof.

By the proof of Theorem 3.29, there exists a positive constant C (independent of z) such that

$$\mu (B(z,r)) \leq C\int \nolimits \nolimits \mathbb{C}{\mathrm{e}}^{-\frac{p\alpha } {2} \vert z-w{\vert }^{2} }\,\mathrm{d}\mu (w)$$

for all $z \in \mathbb{C}$. So condition (ii) implies (iii).

For any sequence z _n  → ∞, it is easy to see that the sequence of functions

$${f}_{n}(w) = {k}_{{z}_{n}}(w) = \frac{{\mathrm{e}}^{\alpha \bar{{z}}_{n}w}} {{\mathrm{e}}^{\alpha \vert {z}_{n}{\vert }^{2}/2}},\qquad w \in \mathbb{C},$$

satisfy $\|{f{}_{n}\|}_{p,\alpha } = 1$ and f _n (w) → 0 uniformly on compact sets. Therefore, condition (i) implies (ii).

On the other hand, carefully examining the proof of Theorem 3.29, we see that there is a positive constant C (independent of f) such that

$$\begin{array}{rcl} & & \int \nolimits \nolimits \mathbb{C}{\left \vert f(w){\mathrm{e}}^{-\alpha \vert w{\vert }^{2}/2 }\right \vert }^{p}\,\mathrm{d}\mu (w) \\ & & \quad \leq C\sum \limits_{k}\mu (B({z}_{k},r)){\int \nolimits \nolimits }_{B({z}_{k},2r)}{\left \vert f(w){\mathrm{e}}^{-\alpha \vert w{\vert }^{2}/2 }\right \vert }^{p}\,\mathrm{d}A(w), \\ \end{array}$$

(3.15)

where {z _k } is a fixed arrangement into a sequence of the lattice $r{\mathbb{Z}}^{2}$. If condition (iii) holds, then z↦μ(B(z, r)) is a bounded function, and for any ε > 0, there exists a positive integer N such that μ(B(z _k , r)) < ε whenever k > N. Thus, for any bounded sequence {f _n } in F _α ^p that converges to 0 uniformly on compact sets, we can estimate the sequence

$${I}_{n} = \int \nolimits \nolimits \mathbb{C}{\left \vert {f}_{n}(w){\mathrm{e}}^{-\alpha \vert w{\vert }^{2}/2 }\right \vert }^{p}\,\mathrm{d}\mu (w)$$

according to (3.15) as follows:

$$\begin{array}{rcl}{ I}_{n}& \leq & C\sum \limits_{k=1}^{N}{ \int \nolimits \nolimits }_{B({z}_{k},2r)}{\left \vert {f}_{n}(w){\mathrm{e}}^{-\alpha \vert w{\vert }^{2}/2 }\right \vert }^{p}\,\mathrm{d}A(w) \\ & & +C\epsilon \sum \limits_{k=N+1}^{\infty }{\int \nolimits \nolimits }_{B({z}_{k},2r)}{\left \vert {f}_{n}(w){\mathrm{e}}^{-\alpha \vert w{\vert }^{2}/2 }\right \vert }^{p}\,\mathrm{d}A(w), \\ \end{array}$$

(3.16)

where C is a positive constant independent of n. Since f _n (w) → 0 uniformly on compact sets in $\mathbb{C}$, we have

$${\lim }_{n\rightarrow \infty }\sum \limits_{k=1}^{N}{ \int \nolimits \nolimits }_{B({z}_{k},2r)}{\left \vert {f}_{n}(w){\mathrm{e}}^{-\alpha \vert w{\vert }^{2}/2 }\right \vert }^{p}\,\mathrm{d}A(w) = 0.$$

Let n → ∞ in (3.16). We obtain

$$\begin{array}{rcl} & & {limsup}_{n\rightarrow \infty }\int \nolimits \nolimits \mathbb{C}{\left \vert {f}_{n}(w){\mathrm{e}}^{-\alpha \vert w{\vert }^{2}/2 }\right \vert }^{p}\,\mathrm{d}\mu (w) \\ & & \quad \leq C\epsilon \sum \limits_{k=N+1}{ \int \nolimits \nolimits }_{B({z}_{k},2r)}{\left \vert {f}_{n}(w){\mathrm{e}}^{-\alpha \vert w{\vert }^{2}/2 }\right \vert }^{p}\,\mathrm{d}A(w).\end{array}$$

There is a positive integer m (depending on r only) such that every point in the complex plane belongs to at most m of the disks D(z _k , 2r). Therefore,

$$\begin{array}{rcl} & \sum \limits_{k=N+1}^{\infty }{\int \nolimits \nolimits }_{B({z}_{k},2r)}{\left \vert {f}_{n}(w){\mathrm{e}}^{-\alpha \vert w{\vert }^{2}/2 }\right \vert }^{p}\,\mathrm{d}A(w) \leq m\int \nolimits \nolimits \mathbb{C}{\left \vert {f}_{n}(w){\mathrm{e}}^{-\alpha \vert w{\vert }^{2}/2 }\right \vert }^{p}\,\mathrm{d}A(w) \leq C,& \\ \end{array}$$

where C is another positive constant independent of n (since {f _n } is a bounded sequence in F _α ^p). Therefore, we can find yet another positive constant C (independent of n and ε) such that

$${limsup}_{n\rightarrow \infty }\int \nolimits \nolimits \mathbb{C}{\left \vert {f}_{n}(w){\mathrm{e}}^{-\alpha \vert w{\vert }^{2}/2 }\right \vert }^{p}\,\mathrm{d}\mu (w) \leq C\epsilon.$$

Since ε is arbitrary, we have

$${\lim }_{n\rightarrow \infty }\int \nolimits \nolimits \mathbb{C}{\left \vert {f}_{n}(w){\mathrm{e}}^{-\alpha \vert w{\vert }^{2}/2 }\right \vert }^{p}\,\mathrm{d}\mu (w) = 0.$$

This shows that condition (iii) implies condition (i). The proof of the theorem is complete.

Carefully examining the proof of Theorems 3.29 and 3.30 above, we obtain the following characterization of Fock–Carleson and vanishing Fock–Carleson measures.

Corollary 3.31.

Suppose μ is a positive Borel measure on $\mathbb{C}$ , r > 0, and {z _n } is any arrangement into a sequence of the lattice $r{\mathbb{Z}}^{2}$ . Then

• (a) μ is a Fock–Carleson measure if and only if {μ(B(z _k ,r))} is in l ^∞.

• (b) μ is a vanishing Fock–Carleson measure if and only if the sequence {μ(B(z _k ,r))} is in c ₀.

Here, l ^∞ denotes the space of all bounded sequences, and c ₀ is the space of all sequences tending to 0.

Let μ be a complex, regular Borel measure μ on the complex plane. Define

$$\widetilde{\mu }(z) = \frac{\alpha } {\pi }\int \nolimits \nolimits \mathbb{C}\vert {k}_{z}(w){\vert }^{2}{\mathrm{e}}^{-\alpha \vert w{\vert }^{2} }\,\mathrm{d}\mu (w) = \frac{\alpha } {\pi }\int \nolimits \nolimits \mathbb{C}{\mathrm{e}}^{-\alpha \vert z-w{\vert }^{2} }\,\mathrm{d}\mu (w),$$

whenever these integrals converge. If dμ(z) = f(z)dA(z) and f satisfies condition (I ₁), it is clear that $\widetilde{\mu } =\widetilde{ f}$. Thus, we are going to call $\widetilde{\mu }$ the Berezin transform of the measure μ.

Taking p = 2 in Theorems 3.29 and 3.30, we see that a positive Borel measure μ on $\mathbb{C}$ is a Fock–Carleson measure if and only if $\widetilde{\mu } \in {L}^{\infty }(\mathbb{C})$, and μ is a vanishing Fock–Carleson measure if and only if $\widetilde{\mu } \in {C}_{0}(\mathbb{C})$.

We also note that when the radius r is fixed, the function z↦μ(B(z, r)) is a constant multiple of the averaging function

$$\widehat{{\mu }}_{r}(z) = \frac{\mu (B(z,r))} {\pi {r}^{2}}.$$

Thus, conditions on the function z↦μ(B(z, r)) can be replaced with the corresponding conditions on the averaging function $\widehat{{\mu }}_{r}$.

3.5 Functions of Bounded Mean Oscillation

For any positive radius r and every exponent p ∈ [1, ∞), we define BMO _r ^p to be the space of locally area-integrable functions f on $\mathbb{C}$ such that

$$\|{f\|}_{{\mathrm{BMO}}_{r}^{p}} {=\sup }_{z\in \mathbb{C}}M{O}_{p,r}(f)(z) < \infty, $$

where

$$M{O}_{p,r}(f)(z) ={ \left [ \frac{1} {\pi {r}^{2}}{ \int \nolimits \nolimits }_{B(z,r)}\vert f -\widehat{ {f}}_{r}(z){\vert }^{p}\,\mathrm{d}A\right ]}^{\frac{1} {p} }.$$

Here,

$$\widehat{{f}}_{r}(z) = \frac{1} {\pi {r}^{2}}{ \int \nolimits \nolimits }_{B(z,r)}f\,\mathrm{d}A$$

is the mean (average) of f over the Euclidean disk B(z, r). Clearly, BMO _r ^p is a linear space.

When p = 2, it is easy to see that

$$M{O}_{2,r}^{2}(f)(z) = \frac{1} {2{(\pi {r}^{2})}^{2}}{ \int \nolimits \nolimits }_{B(z,r)}{ \int \nolimits \nolimits }_{B(z,r)}\vert f(u) - f(v){\vert }^{2}\,\mathrm{d}A(u)\,\mathrm{d}A(v).$$

(3.17)

It is also easy to check that

$$M{O}_{2,r}^{2}(f)(z) =\widehat{ \vert f{\vert {}^{2}}}_{ r}(z) -\vert \widehat{{f}}_{r}(z){\vert }^{2}.$$

(3.18)

Lemma 3.32.

Let 1 ≤ p < ∞, r > 0, and f be a locally area-integrable function on $\mathbb{C}$ . Then f ∈BMO _r ^p if and only if there exists some C > 0 such that for any $z \in \mathbb{C}$ , there is a complex constant c _z with

$$\frac{1} {\pi {r}^{2}}{ \int \nolimits \nolimits }_{B(z,r)}\vert f(w) - {c}_{z}{\vert }^{p}\,\mathrm{d}A(w) \leq C.$$

(3.19)

Proof.

If f ∈ BMO _r ^p, then (3.19) holds with $C =\| {f\|}_{{\mathrm{BMO}}_{r}^{p}}^{p}$ and ${c}_{z} =\widehat{ {f}}_{r}(z)$.

On the other hand, if (3.19) holds, then by the triangle inequality for the L ^p integral,

$$\begin{array}{rcl} M{O}_{p,r}(f)(z)& =&{ \left [ \frac{1} {\pi {r}^{2}}{ \int \nolimits \nolimits }_{B(z,r)}\vert f -\widehat{ {f}}_{r}(z){\vert }^{p}\,\mathrm{d}A\right ]}^{\frac{1} {p} } \\ & \leq &{ \left [ \frac{1} {\pi {r}^{2}}{ \int \nolimits \nolimits }_{B(z,r)}\vert f - {c}_{z}{\vert }^{p}\,\mathrm{d}A\right ]}^{\frac{1} {p} } + \vert \widehat{{f}}_{r}(z) - {c}_{z}\vert. \end{array}$$

By Hölder’s inequality,

$$\begin{array}{rcl} & \vert \widehat{{f}}_{r}(z) - {c}_{z}\vert = \left \vert \frac{1} {\pi {r}^{2}}{ \int \nolimits \nolimits }_{B(z,r)}(f - {c}_{z})\,\mathrm{d}A\right \vert \leq {\left [ \frac{1} {\pi {r}^{2}}{ \int \nolimits \nolimits }_{B(z,r)}\vert f - {c}_{z}{\vert }^{p}\,\mathrm{d}A\right ]}^{\frac{1} {p} }.& \\ \end{array}$$

It follows that MO _p, r (f)(z) ≤ 2C for all $z \in \mathbb{C}$, so that f ∈ BMO _r ^p.

For any r > 0, we consider the space BO _r of continuous functions f on $\mathbb{C}$ such that the function

$${\omega }_{r}(f)(z) =\sup \{ \vert f(z) - f(w)\vert : w \in B(z,r)\}$$

is bounded on $\mathbb{C}$. We think of ω _r (f)(z) as the local oscillation of f at the point z.

Lemma 3.33.

The space BO _r is independent of r. Moreover, a continuous function f on the complex plane belongs to BO _r if and only if there exists a constant C > 0 such that

$$\vert f(z) - f(w)\vert \leq C(\vert z - w\vert + 1)$$

(3.20)

for all z and w in $\mathbb{C}$.

Proof.

If f satisfies the condition in (3.20), then clearly f ∈ BO _r .

To prove the other direction, assume that f ∈ BO _r . Thus, there exists a positive constant M such that

$$\vert f(u) - f(v)\vert \leq M,$$

(3.21)

whenever | u − v | ≤ r.

Let z and w be two arbitrary points in the complex plane. We are going to show that (3.20) holds for some positive constant C that is independent of z and w.

If | z − w | ≤ r, then (3.20) holds with C = M. If | z − w |  > r, we place points z ₀, …, z _n on the line segment from z to w in such a way that z ₀ = z, z _n  = w, | z _k  − z _k + 1 |  = r for 0 ≤ k < n − 1, and | z _n − 1 − z _n  | ≤ r. By the triangle inequality and (3.21),

$$\vert f(z) - f(w)\vert \leq \sum \limits_{k=0}^{n-1}\vert f({z}_{ k}) - f({z}_{k+1})\vert \leq nM.$$

Since (n − 1)r ≤ | z − w | ≤ nr, we have

$$nr \leq \vert z - w\vert + r \leq \max (1,1/r)(\vert z - w\vert + 1).$$

With C = max(M, 1, 1 ∕ r), we obtain the desired estimate in (3.20).

Since BO _r is actually independent of the radius r, we will write BO for BO _r . The initials in BO stand for bounded oscillation. It is clear that

$$\|{f\|}_{\mathrm{BO}} =\sup \{ \vert f(z) - f(w)\vert : \vert z - w\vert \leq 1\}$$

defines a complete seminorm on BO.

We will make the connection between BMO _r ^p and the weighted Gaussian measures dλ_α with the help of Fock–Carleson measures. More specifically, for any 1 ≤ p < ∞ and r > 0, we use BA _r ^p to denote the space of Lebesgue measurable functions f on $\mathbb{C}$ such that $\widehat{\vert f{\vert {}^{p}}}_{r}(z)$ is bounded. By the characterization of Fock–Carleson measures in Sect. 3.4, the space BA _r ^p is independent of r. Therefore, we will write BA^p for BA _r ^p. More specifically, a Lebesgue measurable function f on $\mathbb{C}$ belongs to BA _r ^p if and only if

$$\|{f\|}_{{\mathrm{BA}}^{p}}^{p} {=\sup }_{ z\in \mathbb{C}}\widetilde{\vert f{\vert }^{p}}(z) < \infty, $$

where $\widetilde{\vert f{\vert }^{p}}$ is the Berezin transform of | f | ^p with respect to the Gaussian measure dλ_α. Although the weight parameter α appears in the definition of the norm above, the space BA^p is independent of α.

The space BA^p depends on p. In fact, if 1 ≤ p < q < ∞, then BA^q ⊂ BA^p and the containment is strict.

We now describe the structure of BMO _r ^p in terms of the relatively simple spaces BO and BA^p. Recall that φ _z (w) = z − w.

Theorem 3.34.

Let α > 0, r > 0, and 1 ≤ p < ∞. Suppose f is a locally area-integrable function on $\mathbb{C}$ . Then the following conditions are equivalent:

• (a) f ∈BMO _r ^p.

• (b) f ∈BO + BA ^p.

• (c) f satisfies condition (I ₁ ), and there exists a positive constant C such that

  $$\int \nolimits \nolimits \mathbb{C}\vert f \circ {\varphi }_{z}(w) -\widetilde{ f}(z){\vert }^{p}\,\mathrm{d}{\lambda }_{ \alpha }(w) \leq C$$

  (3.22)

  for all $z \in \mathbb{C}$.

• (d) There exists a positive constant C such that for any $z \in \mathbb{C}$ , there is some complex number c _z with

  $$\int \nolimits \nolimits \mathbb{C}\vert f \circ {\varphi }_{z}(w) - {c}_{z}{\vert }^{p}\,\mathrm{d}{\lambda }_{ \alpha }(w) \leq C.$$

  (3.23)

Proof.

Let f ∈ BMO_2r ^p and | z − w | ≤ r. We have

$$\begin{array}{rcl} \vert \widehat{{f}}_{r}(z) -\widehat{ {f}}_{r}(w)\vert & \leq & \vert \widehat{{f}}_{r}(z) -\widehat{ {f}}_{2r}(z)\vert + \vert \widehat{{f}}_{2r}(z) -\widehat{ {f}}_{r}(w)\vert \\ &\leq & \frac{1} {\pi {r}^{2}}{ \int \nolimits \nolimits }_{B(z,r)}\vert f(u) -\widehat{ {f}}_{2r}(z)\vert \,\mathrm{d}A(u) \\ & & + \frac{1} {\pi {r}^{2}}{ \int \nolimits \nolimits }_{B(w,r)}\vert f(u) -\widehat{ {f}}_{2r}(z)\vert \,\mathrm{d}A(u).\end{array}$$

Since B(z, r) and B(w, r) are both contained in B(z, 2r), it follows from Hölder’s inequality that the two integral summands above are both bounded by a constant that is independent of z and w. This proves that $\widehat{{f}}_{r}$ belongs to BO _r  = BO.

On the other hand, we can show that the function $g = f -\widehat{ {f}}_{r}$ belongs to BA^p whenever f ∈ BMO_2r ^p. In fact, it follows from (3.17) that f ∈ BMO_2r ^p implies that f ∈ BMO _r ^p, and it follows from the triangle inequality for L ^p integrals that

$$\begin{array}{rcl}{ \left [\widehat{\vert g{\vert {}^{p}}}_{ r}(z)\right ]}^{\frac{1} {p} }& =&{ \left [ \frac{1} {\pi {r}^{2}}{ \int \nolimits \nolimits }_{B(z,r)}\vert f(u) -\widehat{ {f}}_{r}(u){\vert }^{p}\,\mathrm{d}A(u)\right ]}^{\frac{1} {p} } \\ & \leq &{ \left [ \frac{1} {\pi {r}^{2}}{ \int \nolimits \nolimits }_{B(z,r)}\vert f(u) -\widehat{ {f}}_{r}(z){\vert }^{p}\,\mathrm{d}A(u)\right ]}^{\frac{1} {p} } \\ & & +{\left [ \frac{1} {\pi {r}^{2}}{ \int \nolimits \nolimits }_{B(z,r)}\vert \widehat{{f}}_{r}(u) -\widehat{ {f}}_{r}(z){\vert }^{p}\,\mathrm{d}A(u)\right ]}^{\frac{1} {p} } \\ & \leq & \|{f\|}_{{\mathrm{BMO}}_{r}^{p}} + {\omega }_{r}(\widehat{{f}}_{r})(z).\end{array}$$

Since $\widehat{{f}}_{r} \in {\mathrm{BO}}_{r}$ and f ∈ BMO _r ^p, we have g ∈ BA^p.

Thus, we have proved that f ∈ BMO_2r ^p implies

$$f =\widehat{ {f}}_{r} + (f -\widehat{ {f}}_{r}) \in \mathrm{BO} +{ \mathrm{BA}}^{p}.$$

Since r is arbitrary, we conclude that BMO _r ^p ⊂ BO + BA^p, which proves that condition (a) implies condition (b).

It is clear that every function in BO satisfies condition (I _p ). Also, every function in BA^p satisfies condition (I _p ). Therefore, condition (b) implies that f satisfies condition (I _p ). Since p ≥ 1, f also satisfies condition (I ₁). In particular, condition (b) implies that the Berezin transform of f is well defined.

By the triangle inequality and Hölder’s inequality,

$$\|f \circ {\varphi }_{z} -\widetilde{ f}{(z)\|}_{{L}^{p}(\mathrm{d}{\lambda }_{\alpha })} \leq \| f \circ {\varphi {}_{z}\|}_{{L}^{p}(\mathrm{d}{\lambda }_{\alpha })} + \vert \widetilde{f}(z)\vert \leq 2\,\widetilde{\vert f{\vert }^{p}}(z).$$

We see that condition (3.22) holds whenever f ∈ BA^p. On the other hand, it follows from Hölder’s inequality that

$$\begin{array}{rcl} \|f \circ {\varphi }_{z} -\widetilde{ f}{(z)\|}_{{L}^{p}(\mathrm{d}{\lambda }_{\alpha })}^{p}\!\!\!& \!\!\!& = \int \nolimits \nolimits \mathbb{C}\vert f(z\! -\! w) -\widetilde{ f}(z){\vert }^{p}\,\mathrm{d}{\lambda }_{ \alpha }(w) \\ \!\!\!& \!\!\!& \leq \int \nolimits \nolimits \mathbb{C} \int \nolimits \nolimits \mathbb{C}\vert f(z\! -\! w) - f(z\! -\! u){\vert }^{p}\,\mathrm{d}{\lambda }_{ \alpha }(w)\mathrm{d}{\lambda }_{\alpha }(u).\end{array}$$

This together with Lemma 3.33 shows that for any f ∈ BO,

$$\|f \circ {\varphi }_{z} -\widetilde{ f}{(z)\|}_{{L}^{p}(\mathrm{d}{\lambda }_{\alpha })}^{p} \leq {C}^{p} \int \nolimits \nolimits \mathbb{C} \int \nolimits \nolimits \mathbb{C}{\left [\vert u - w\vert + 1\right ]}^{p}\,\mathrm{d}{\lambda }_{ \alpha }(w)\,\mathrm{d}{\lambda }_{\alpha }(u).$$

The integral on the right-hand side above converges. Thus, condition (3.22) holds for all f ∈ BO as well, and we have proved that condition (b) implies condition (c).

Mimicking the proof of Lemma 3.32, we easily obtain the equivalence of conditions (c) and (d).

Finally, if condition (3.22) holds, we can find a positive constant C such that

$$\begin{array}{rcl} & & \frac{C} {\pi {r}^{2}}{ \int \nolimits \nolimits }_{B(z,r)}\vert f(w) -\widetilde{ f}(z){\vert }^{p}\,\mathrm{d}A(w) \\ & & \quad \leq \int \nolimits \nolimits \mathbb{C}\vert f(w) -\widetilde{ f}(z){\vert }^{p}\vert {k}_{ z}(w){\vert }^{2}\,\mathrm{d}{\lambda }_{ \alpha }(w) \\ & & \quad = \int \nolimits \nolimits \mathbb{C}\vert f \circ {\varphi }_{z}(w) -\widetilde{ f}(z){\vert }^{p}\,\mathrm{d}{\lambda }_{ \alpha }(w).\end{array}$$

This, along with Lemma 3.32, then shows that condition (c) implies condition (a).

As a consequence of Theorem 3.34, we see that the space BMO _r ^p is independent of r and the Berezin transform of every function in BMO _r ^p is well defined. Thus, we will write BMO^p for BMO _r ^p and define a complete seminorm on BMO^p by

$$\|{f\|}_{{\mathrm{BMO}}^{p}} {=\sup }_{z\in \mathbb{C}}\|f \circ {\varphi }_{z} -\widetilde{ f}{(z)\|}_{{L}^{p}(\mathrm{d}{\lambda }_{\alpha })} {=\sup }_{z\in \mathbb{C}}\|f \circ {t}_{z} -\widetilde{ f}{(z)\|}_{{L}^{p}(\mathrm{d}{\lambda }_{\alpha })}.$$

One of the nice features of this seminorm is that it is invariant under the actions of t _a , τ _a , and φ _a .

The proof of Theorem 3.34 also shows that every function in BMO^p satisfies condition (I _p ). In particular, ${\mathrm{BMO}}^{p} \subset {L}^{p}(\mathbb{C},\mathrm{d}{\lambda }_{\alpha })$.

Theorem 3.35.

If 1 < p < ∞, then there exists a positive constant C = C(p,α) such that

$$\vert \widetilde{f}(z) -\widetilde{ f}(w)\vert \leq C\|{f\|}_{{\mathrm{BMO}}^{p}}\vert z - w\vert $$

for all z and w in $\mathbb{C}$ and all f ∈BMO ^p.

Proof.

Fix any $z \in \mathbb{C}$ and fix any directional parameter θ. Consider the curve γ(t) = z + e^iθ t, which is traced out by a particle that starts at z, with unit speed, and in the θ-direction. Recall that

$$\widetilde{f}(\gamma (t)) = \frac{\alpha } {\pi }\int \nolimits \nolimits \mathbb{C}f(u){\mathrm{e}}^{-\alpha \vert \gamma (t)-u{\vert }^{2} }\,\mathrm{d}A(u).$$

Differentiating under the integral sign gives

$$\frac{\mathrm{d}} {\mathrm{d}t}\widetilde{f}(\gamma (t)) = -\frac{2{\alpha }^{2}} {\pi } \int \nolimits \nolimits \mathbb{C}f(u){\mathrm{e}}^{-\alpha \vert \gamma (t)-u{\vert }^{2} }\mathrm{Re}\,\left [\gamma ^{\prime}(t)(\overline{\gamma (t)} -\overline{u})\right ]\,\mathrm{d}A(u).$$

For any fixed t, the function

$$h(u) = \mathrm{Re}\,\left [\gamma ^{\prime}(t)(\overline{\gamma (t)} - u)\right ]$$

is harmonic, so it is fixed by the Berezin transform. It follows that

$$\frac{\alpha } {\pi }\int \nolimits \nolimits \mathbb{C}{\mathrm{e}}^{-\alpha \vert \gamma (t)-u{\vert }^{2} }\mathrm{Re}\,\left [\gamma ^{\prime}(t)(\overline{\gamma (t)} -\overline{u})\right ]\,\mathrm{d}A(u) =\widetilde{ h}(\gamma (t)) = 0.$$

Therefore, $\mathrm{d}\widetilde{f}(\gamma (t))/\mathrm{d}t$ is equal to

$$-\frac{2{\alpha }^{2}} {\pi } \int \nolimits \nolimits \mathbb{C}(f(u) -\widetilde{ f}(\gamma (t))){\mathrm{e}}^{-\alpha \vert \gamma (t)-u{\vert }^{2} }\mathrm{Re}\,\left [\gamma ^{\prime}(t)(\overline{\gamma (t)} -\overline{u})\right ]\,\mathrm{d}A(u).$$

Let q be the conjugate exponent, 1 ∕ p + 1 ∕ q = 1. Then by Hölder’s inequality, $\vert \mathrm{d}\widetilde{f}(\gamma (t))/\mathrm{d}t\vert$ is less than or equal to

$$\frac{2{\alpha }^{2}} {\pi }{ \left [\int \nolimits \nolimits \mathbb{C}\vert f(u) -\widetilde{ f}(\gamma (t)){\vert }^{p}{\mathrm{e}}^{-\alpha \vert \gamma (t)-u{\vert }^{2} }\,\mathrm{d}A(u)\right ]}^{\frac{1} {p} }$$

times

$${ \left [\int \nolimits \nolimits \mathbb{C}\vert \gamma (t) - u{\vert }^{q}{\mathrm{e}}^{-\alpha \vert \gamma (t)-u{\vert }^{2} }\,\mathrm{d}A(u)\right ]}^{\frac{1} {q} }.$$

(3.24)

The integral in (3.24) is, via a simple change of variables, equal to

$$\int \nolimits \nolimits \mathbb{C}\vert u{\vert }^{q}{\mathrm{e}}^{-\alpha \vert u{\vert }^{2} }\,\mathrm{d}A(u),$$

which is clearly convergent. Therefore, there exists a positive constant C = C(α, p) such that

$$\left \vert \frac{\mathrm{d}} {\mathrm{d}t}\widetilde{f}(\gamma (t))\right \vert \leq CM{O}_{p}(f)(\gamma (t)) \leq C\|{f\|}_{{\mathrm{BMO}}^{p}}$$

for all t, where

$$\|{f\|}_{{\mathrm{BMO}}^{p}} {=\sup }_{z\in \mathbb{C}}M{O}_{p}(f)(z) {=\sup }_{z\in \mathbb{C}}\|f \circ {\varphi }_{z} -\widetilde{ f}{(z)\|}_{{L}^{p}(\mathrm{d}{\lambda }_{\alpha })}.$$

Integrating with respect to t, we obtain

$$\vert \widetilde{f}(z) -\widetilde{ f}(w)\vert \leq C\|{f\|}_{{\mathrm{BMO}}^{p}}\vert z - w\vert $$

for all z and w in $\mathbb{C}$.

The following result gives another way to split the space BMO^p into the sum of two simpler spaces: a space of “smooth” functions and a space of “small” functions.

Theorem 3.36.

Suppose f ∈BMO ^p and 1 ≤ p < ∞. Then $\widetilde{f} \in \mathrm{BO}$ and $f -\widetilde{ f} \in {\mathrm{BA}}^{p}$.

Proof.

It is easy to see that there is a positive constant C such that

$$\begin{array}{rcl} \vert \widetilde{f}(z) -\widehat{ {f}}_{r}(z)\vert & \leq & \frac{1} {\pi {r}^{2}}{ \int \nolimits \nolimits }_{B(z,r)}\vert f(w) -\widetilde{ f}(z)\vert \,\mathrm{d}A(w) \\ & \leq & C{\int \nolimits \nolimits }_{B(z,r)}\vert f(w) -\widetilde{ f}(z)\vert \vert {k}_{z}(w){\vert }^{2}\,\mathrm{d}{\lambda }_{ \alpha }(w) \\ & \leq & C\int \nolimits \nolimits \mathbb{C}\vert f \circ {\varphi }_{z}(w) -\widetilde{ f}(z)\vert \,\mathrm{d}{\lambda }_{\alpha }(w) \\ & \leq & C\|f \circ {\varphi }_{z} -\widetilde{ f}{(z)\|}_{{L}^{p}(\mathrm{d}{\lambda }_{\alpha })}, \\ \end{array}$$

where the last step follows from Hölder’s inequality. This shows that $\widetilde{f} -\widehat{ {f}}_{r}$ is a bounded function. Since a bounded continuous function belongs to both BO and BA^p, we have $\widetilde{f} -\widehat{ {f}}_{r} \in \mathrm{BO} \cap {\mathrm{BA}}^{p}$.

Write

$$f -\widetilde{ f} = (f -\widehat{ {f}}_{r}) - (\widetilde{f} -\widehat{ {f}}_{r}),$$

and recall from Theorem 3.34 that $f -\widehat{ {f}}_{r}$ is in BA^p. We conclude that $f -\widetilde{ f}$ belongs to BA^p. Similarly, we can write

$$\widetilde{f} =\widehat{ {f}}_{r} + (\widetilde{f} -\widehat{ {f}}_{r})$$

and infer that $\widetilde{f} \in \mathrm{BO}$.

Corollary 3.37.

If 1 < p < ∞, then

$${\mathrm{BMO}}^{p} = \mathrm{LIP} +{ \mathrm{BA}}^{p},$$

where LIP is the space of all Lipschitz functions on $\mathbb{C}$ . Moreover, a canonical decomposition is given by $f =\widetilde{ f} + (f -\widetilde{ f})$.

The next result characterizes entire functions in BMO^p.

Proposition 3.38.

Suppose 1 ≤ p < ∞ and f is an entire function. Then f ∈BMO ^p if and only if f is a linear polynomial.

Proof.

When f is entire, we have $\widehat{{f}}_{r} = f$ because of the mean value theorem. It follows from Theorem 3.34 (and its proof) that $f =\widehat{ {f}}_{r} \in \mathrm{BO}$ whenever f ∈ BMO^p. Thus, there exists a positive constant C such that

$$\vert f(z) - f(w)\vert \leq C(\vert z - w\vert + 1)$$

for all z and w. Let w = 0 and use Cauchy’s estimate. We conclude that f must be a linear polynomial.

Conversely, if f is a linear polynomial, then f is Lipschitz in the Euclidean metric. In particular, f ∈ BO, and so f ∈ BMO^p.

Let VMO _r ^p denote the space of locally area-integrable functions f such that

$${\lim }_{z\rightarrow \infty }M{O}_{p,r}(f)(z) = 0.$$

It is clear that VMO _r ^p is a subspace of BMO _r ^p. Just like BMO _r ^p, the space VMO _r ^p is also independent of r, and we will write VMO^p for VMO _r ^p.

Similarly, we consider the space VO _r consisting of continuous functions f such that

$${\lim }_{z\rightarrow \infty }{\omega }_{r}(f)(z) = 0.$$

It can be shown that VO _r is independent of r, and we will write VO for VO _r . The initials in VO stand for “vanishing oscillation.”

We also consider the space VA _r ^p consisting of functions such that

$${\lim }_{z\rightarrow \infty } \frac{1} {\pi {r}^{2}}{ \int \nolimits \nolimits }_{B(z,r)}\vert f(w){\vert }^{p}\,\mathrm{d}A(w) = 0.$$

According to the characterizations of vanishing Fock–Carleson measures in Sect. 3.4, the space VA _r ^p is independent of r and consists of functions f such that $\widetilde{\vert f{\vert }^{p}}(z) \rightarrow 0$ as z → ∞. We will write VA^p for VA _r ^p. The initials in VA^p stand for “vanishing average.” The following theorem describes the structure of VMO^p.

Theorem 3.39.

Suppose 1 ≤ p < ∞, r > 0, and f is locally area integrable. Then the following conditions are equivalent:

• (i) f ∈VMO ^p = VMO _r ^p.

• (ii) MO _p (f)(z) → 0 as z →∞.

• (iii) f ∈VO + VA ^p.

Moreover, there are two canonical decompositions for condition (iii) above:

$$f =\widetilde{ f} + (f -\widetilde{ f}),\quad f =\widehat{ {f}}_{r} + (f -\widehat{ {f}}_{r}).$$

We omit the proof.

Corollary 3.40.

Suppose f is an entire function. Then f ∈VMO ^p if and only if f is constant.

3.6 Notes

The Berezin transform was introduced in [23] and then studied systematically in [23,  24,  25,  26,  27] for a number of reproducing Hilbert spaces. It has become an indispensable tool in the study of operators on function spaces, including Hankel operators, Toeplitz operators, and composition operators. See [250] for applications of the Berezin transform in the theory of Bergman spaces. In particular, the proofs of Propositions 3.3–3.6 were adapted from the corresponding ones in [250].

In the setting of Fock spaces and when parametrized appropriately, the Berezin transform is nothing but the heat transform. This connection with the heat equation makes the Berezin transform on Fock spaces particularly useful. The semigroup property of the heat transforms was first observed in [30].

The Lipschitz estimate for the Berezin transform of a bounded linear operator on the Fock space is due to Coburn. See [54,  55]. Propositions 3.9–3.11 and Corollary 3.12 are taken from [55], and these results will be needed in Chap. 6 when we study Toeplitz operators on the Fock space.

Theorem 3.25, the Lipschitz estimate for the Berezin transform of a bounded function, was first proved in [29]. Together with the semigroup property, this result shows that the Berezin transform is a rapidly smoothing operation on bounded functions, and consequently, a bounded function that is fixed by the Berezin transform must be constant. On the other hand, there exist unbounded functions fixed by the Berezin transform that are not harmonic. The example in Sect. 3.3 was taken from [84]. This example shows the sharp contrast with the Bergman space theory, where the fixed points of the Berezin transform are exactly the harmonic functions; see [1].

The characterization of Fock–Carleson measures is analogous to the characterization of Carleson measures for Bergman spaces. The material in Sect. 3.4 is taken from [132]. See [250] for the corresponding results in the Bergman space theory. Note that the notion of Carleson measures was initially introduced in the Hardy space setting, where a geometric characterization is much more difficult. See [76].

The notion of BMO and VMO using a fixed Euclidean radius was first introduced in [257,  32]. This idea was then generalized to the setting of bounded symmetric domains in [21] and to the case of strongly pseudoconvex domains in [149], with the Euclidean metric replaced by the Bergman metric. The resulting spaces are independent of the particular radius used, but the dependence on the exponent p was observed and studied in [248] in the context of Bergman spaces on the unit ball. The extension to the Fock space setting is straightforward.

The Lipschitz estimate for the Berezin transform of a function in BMO was first proved in [21] in the context of Bergman spaces on bounded symmetric domains. The extension to the Fock space, Theorem 3.35, was first carried out in [13].

3.7 Exercises

• 1. Show that the Lipschitz constant $2\sqrt{\alpha }$ in Corollary 3.8 is best possible.

• 2. Show that the spaces BMO^p and VMO^p are complete under the norm

  $$\|f\| =\| {f\|}_{{\mathrm{BMO}}^{p}} + \vert \widetilde{f}(0)\vert.$$

• 3. Characterize the multipliers of the spaces BMO^p and VMO^p.

• 4. Show that the function | z | belongs to BMO^p but the function | z | ² does not belong to BMO^p.

• 5. Show that the function $\sqrt{\vert z\vert }$ belongs to VMO^p.

• 6. Show that the function ${\mathrm{e}}^{\mathrm{i}\sqrt{\vert z\vert }}$ belongs to VMO^p.

• 7. Study the behavior of the Berezin transform of the function ln | z | , which is harmonic everywhere except the origin.

• 8. If $f \in {L}^{\infty }(\mathbb{C})$, show that the sequence $\{\widetilde{{f}}^{(n)}\}$ converges to a constant function as n → ∞. Moreover, the convergence is uniform on any compact subset of $\mathbb{C}$.

• 9. If f is locally L ^p-integrable and

  $${\lim }_{z\rightarrow \infty }f(z) = L$$

  exists, then f ∈ VMO^p.

• 10. A function f is “eventually slowly varying” if, for any ε > 0, there exist positive numbers R and δ such that | f(z) − f(w) |  < ε whenever | z |  > R, | w |  > R, and | z − w |  < δ. Show that every eventually slowly varying function is in VMO^p.

• 11. Characterize harmonic functions in BMO^p.

• 12. Suppose α, β, and γ are positive parameters. Show that for 1 ≤ p ≤ ∞, we have Q _α L _β ^p ⊂ L _γ ^p if and only if α² ∕ γ ≤ 2α − β.

• 13. Show that the Berezin transform B _α is never bounded on L _β ^p, where α and β are positive weight parameters.

• 14. If f ∈ BMO¹, show that B _α( | f | ) − | B _α f | is bounded for α > 0.

• 15. Does the boundedness of B _α( | f | ) − | B _α f | imply f ∈ BMO¹?

• 16. Consider the previous two problems for 1 < p < ∞.

• 17. Show that ${B}_{\alpha }{f}_{r}(z) = {B}_{\alpha /{r}^{2}}f(rz)$, where f _r (z) = f(rz).

• 18. Show that B _α is a bounded and self-adjoint operator on ${L}^{2}(\mathbb{C},\mathrm{d}A)$.

• 19. Show that BA^q ⊂ BA^p whenever 1 ≤ p ≤ q < ∞. Furthermore, the inclusion is strict if p < q.

• 20. If f ∈ BMO^p, then | f | ∈ BMO^p. Similarly, if f ∈ VMO^p, then | f | ∈ VMO^p.