Coorbits for projective representations with an application to Bergman spaces

Abstract

Representation theory of locally compact topological groups is a powerful tool to analyze Banach spaces of functions and distributions. It provides a unified framework for constructing function spaces and to study several generalizations of the wavelet transform. Recently representation theory has been used to provide atomic decompositions for a large collection of classical Banach spaces. But in some natural situations, including Bergman spaces on bounded domains, representations are too restrictive. The proper tools are projective representations. In this paper we extend known techniques from representation theory to also include projective representations. This leads naturally to twisted convolution on groups avoiding the usual central extension of the group. As our main application we obtain atomic decompositions of Bergman spaces on the unit ball through the holomorphic discrete series for the group of isometries of the ball.

Decompositions of reproducing kernel spaces for twisted convolution

From now on we let $\phi (x)=W^\rho _u(u)$ for some fixed $u\in \mathcal {S}.$ Then $B_u^\sigma = \{ f\in B \mid f=f\#\phi \}$. Given a compact neighbourhood U of the identity in G, a U-dense and well-spread sequence $\{ x_i\}\subseteq G$ and a U-BUPU $\{\psi _i \}$, we formally define the operators

$$\begin{aligned} T_1 f&= \sum _i f(x_i) \sigma (x,x^{-1}x_i)\psi _i\#\phi \\ T_2 f&= \sum _i \lambda _i(f) \ell _{x_i}^\sigma \phi \\ T_3 f&= \sum _i c_i f(x_i) \ell _{x_i}^\sigma \phi \end{aligned}$$

where $\lambda _i(f) = \int f(y)\psi _i(y)\overline{\sigma (y,y^{-1}x_i)}\,dy$ and $c_i = \int \psi _i(y)\,dy$. The following results will establish when these operators are well defined on $B_u^\sigma$.

Define the local oscillations

$$\begin{aligned} \mathrm {osc}_{U}^{r^\sigma } f(x) = \sup _{y\in U} | r^\sigma _y f(x)-f(x)| \quad \text {and} \quad \mathrm {osc}_{U}^{\ell ^\sigma } f(x) = \sup _{y\in U} | \ell ^\sigma _y f(x)-f(x)|. \end{aligned}$$

Proposition 2

If $f\in B_u^\sigma$ then

$$\begin{aligned} |T_1f(x)-f(x)|&\le |f|*\mathrm {osc}^{r^\sigma }_{U^{-1}} \phi (x) \\ |T_2f(x)-f(x)|&\le |f|*\mathrm {osc}^{\ell ^\sigma }_{U^{-1}} \phi (x) \\ |T_3f(x)-f(x)|&\le |f|*\mathrm {osc}^{r^\sigma }_{U^{-1}} \phi *(|\phi | +\mathrm {osc}^{\ell ^\sigma }_{U^{-1}} \phi )(x) + |f|*\mathrm {osc}_{U^{-1}}^{\ell ^\sigma }\phi (x). \end{aligned}$$

Proof

We see that

$$\begin{aligned} \big |\sum _i f(x_i) \sigma (x,x^{-1}x_i)\psi _i(x) -f(x)\big | \le \sum _i \big |f(x_i)\sigma (x,x^{-1}x_i)-f(x)\big |\psi _i(x) \end{aligned}$$

and for $x\in x_iU$ we get that $x_i \in xU^{-1}$, so

$$\begin{aligned} \big |f(x_i)\sigma (x,x^{-1}x_i)-f(x)\big | \le \sup _{y\in U^{-1}} \big | f(xy)\sigma (x,y) - f(x)\big | = \mathrm {osc}_{U^{-1}}^{r^\sigma } f(x). \end{aligned}$$

Next, if $f=f\#\phi$ we get

$$\begin{aligned} \big |r^\sigma _y f(x) - f(x)\big | \le \int |f(z)||\phi (z^{-1}xy)\overline{\sigma (z,z^{-1}xy)}\sigma (x,y) - \phi (z^{-1}x)\overline{\sigma (z,z^{-1})}|\,dz. \end{aligned}$$

Since $\sigma (z,z^{-1}xy) = \sigma (z,z^{-1}x)\sigma (x,y)\overline{\sigma (z^{-1}x,y)}$ we get that the integral above reduces to

$$\begin{aligned} \int |f(z)||r^\sigma _y \phi (z^{-1}x) - \phi (z^{-1}x)|\,dz. \end{aligned}$$

Taking supremum we get the desired result.

We have that

$$\begin{aligned} f(x)-T_2f(x)&=\int f(y) \ell ^\sigma _y\phi (x)\,dy - \int \sum _i f(y)\psi _i(y)\overline{\sigma (y,y^{-1}x_i)}\,dy \ell ^\sigma _{x_i} \phi (x) \\&= \int \sum _i f(y)\psi _i(y)\big [\ell ^\sigma _y\phi (x)- \overline{\sigma (y,y^{-1}x_i)} \ell ^\sigma _{x_i} \phi (x)\big ]\,dy \\&= \int \sum _i f(y)\psi _i(y)\ell ^\sigma _y \big [\phi (x)-\ell ^\sigma _{y^{-1}x_i} \phi (x)\big ]\,dy. \end{aligned}$$

When $y\in x_iU$ for a compact neighbourhood U of the identity, then $y^{-1}x_i \in U^{-1}$. Thus we get

$$\begin{aligned} |f(x)-T_2f(x)|&\le \int \sum _i |f(y)| \psi _i(y)|\ell ^\sigma _y \phi (x)-\ell ^\sigma _{y^{-1}x_i} \phi (x)|\,dy \\&\le \int \sum _i |f(y)| \psi _i(y)|\ell ^\sigma _y \mathrm {osc}_{U^{-1}}^{\ell ^\sigma } \phi |\,dy \\&\le \int |f(y)| \ell _y \mathrm {osc}_{U^{-1}}^{\ell ^\sigma } \phi \,dy. \end{aligned}$$

This shows the claim.

$$\begin{aligned} |T_3f(x) - f(x)| \le \int \sum _i \psi _i(y) |f(x_i)\ell ^\sigma _{x_i}\phi (x) - f(y)\ell ^\sigma _y\phi (x)|\,dy \end{aligned}$$

Let us rewrite part of the integrand when $y\in x_iU$

$$\begin{aligned}&|f(x_i)\ell ^\sigma _{x_i}\phi (x) - f(y)\ell ^\sigma _y\phi (x)| \\&\quad = |f(x_i)\sigma (y,y^{-1}x_i) \ell ^\sigma _y\ell ^\sigma _{y^{-1}x_i}\phi (x) - f(y)\ell ^\sigma _y\phi (x)| \\&\quad \le |[f(x_i)\sigma (y,y^{-1}x_i)-f(y)] \ell ^\sigma _y\ell ^\sigma _{y^{-1}x_i}\phi (x)| + |f(y)||\ell ^\sigma _y\ell ^\sigma _{y^{-1}x_i}\phi (x)\ - \ell ^\sigma _y\phi (x)| \\&\quad = |[r^\sigma _{y^{-1}x_i}f(y)-f(y)] \ell ^\sigma _y\ell ^\sigma _{y^{-1}x_i}\phi (x)| + |f(y)||\ell ^\sigma _y\ell ^\sigma _{y^{-1}x_i}\phi (x)\ - \ell ^\sigma _y\phi (x)| \\&\quad \le \mathrm {osc}^{r^\sigma }_{U^{-1}}f(y) [|\ell ^\sigma _y\ell ^\sigma _{y^{-1}x_i}\phi (x) - \ell ^\sigma _y \phi (x)|+|\ell ^\sigma _y\phi (x)|] + |f(y)||\ell ^\sigma _y \mathrm {osc}^{\ell ^\sigma }_{U^{-1}} \phi (x)| \\&\quad \le \mathrm {osc}^{r^\sigma }_{U^{-1}}f(y) [|\ell ^\sigma _y\mathrm {osc}^{\ell ^\sigma _{U^{-1}}} \phi (x)| +|\ell ^\sigma _y\phi (x)|] + |f(y)||\ell ^\sigma _y \mathrm {osc}^{\ell ^\sigma }_{U^{-1}} \phi (x)|. \end{aligned}$$

As before the oscillation of f can be transferred onto the kernel, and the final result is obtained. $\square$

From this we obtain

Corollary 1

Let B be a solid BF-space and assume that $f\mapsto f*|\phi |$, $f\mapsto f*\mathrm {osc}^{\ell ^\sigma }_{U^{-1}}\phi$ and $f\mapsto f*\mathrm {osc}^{r^\sigma }_{U^{-1}}\phi$ are bounded on B, then $T_1,T_2,T_3$ are well-defined bounded operators on $B^\#_u$.

Moreover, if there are constants $C_U$ for which

$$\begin{aligned} \Vert f*\mathrm {osc}^{\ell ^\sigma }_{U^{-1}}\phi \Vert \le C_U \Vert f\Vert \text { and } \Vert f*\mathrm {osc}^{r^\sigma }_{U^{-1}}\phi \Vert \le C_U \Vert f\Vert \end{aligned}$$

and $\lim _{U\rightarrow \{ e\}} C_U =0$, then there is a U small enough as well as U-dense $\{ x_i\}$ such that the operators are invertible on $B^\#_u$.

We will now use the special form of $\phi$ to find oscillation estimates via derivatives. We have defined $\phi (x) = \langle u, \rho (x)u\rangle$, and from this and Remark 1 we get

$$\begin{aligned} \mathrm {osc}_U^{r^\sigma } \phi (x) = \sup _{y\in U} |\langle \rho ^*(x^{-1})u,\rho (y)u-u\rangle |, \end{aligned}$$

and

$$\begin{aligned} \mathrm {osc}_U^{\ell ^\sigma } \phi (x) = \sup _{y\in U} |\langle \rho ^*(y)u-u, \rho (x)u\rangle |. \end{aligned}$$

In light of this is seems possible to evaluate the oscillation by a certain level of smoothness of the vector u, and this is exactly the approach we will take. We let $v\in \mathcal {S}$ and $\lambda \in \mathcal {S}^*$ be arbitrary elements and define the functions $H(y) = \langle \lambda , \rho (y)u\rangle$ and $K(y) = \langle \rho ^*(y)u,v\rangle .$ We will now investigate the local oscillations of H and K in terms of derivatives, but first we need to introduce some notation.

If f is a function on G and X is in $\mathfrak {g}$, then define

$$\begin{aligned} Xf(y) = \frac{d}{d t}\Big |_{t=0} f(y\exp (tX)). \end{aligned}$$

We now fix a basis $X_1,\dots ,X_n$ for the Lie algebra $\mathfrak {g}$, and for a multi-index $\alpha$ we define

$$\begin{aligned} X^\alpha f = X_1^{\alpha _1}\cdots X_n^{\alpha _n} f. \end{aligned}$$

We will investigate oscillations of H and K on the specific neighbourhood

$$\begin{aligned} U_\epsilon = \{ \exp (t_1X_1)\cdots \exp (t_nX_n)\mid -\epsilon \le t_k\le \epsilon \}. \end{aligned}$$

Remember that we choose the cocycle $\sigma$ and $\epsilon >0$ such that $\sigma$ is $C^\infty$ on a neighbourhood containing $U_\epsilon \times U_\epsilon$. According to Lemma 2.5 in [4] there is a constant $C_\epsilon$ such that

$$\begin{aligned} \sup _{y\in U_\epsilon } |H(y)-H(e)|&\le C_\epsilon \sum _{{\mathop {|\delta |=|\alpha |}\limits ^{1\le |\alpha |\le n}}} \int _{[-\epsilon ,\epsilon ]^{|\delta |}} |X^\alpha H (\tau _\delta (t_1,\ldots ,t_n))| (dt_1)^{\delta _1} \dots (dt_n)^{\delta _n}, \end{aligned}$$

and

$$\begin{aligned} \sup _{y\in U_\epsilon } |K(y)-K(e)|&\le C_\epsilon \sum _{{\mathop {|\delta |=|\alpha |}\limits ^{1\le |\alpha |\le n}}} \int _{[-\epsilon ,\epsilon ]^{|\delta |}} |X^\alpha K (\tau _\delta (t_1,\ldots ,t_n))| (dt_1)^{\delta _1} \dots (dt_n)^{\delta _n}, \end{aligned}$$

where $\tau _\delta (t_1,\ldots ,t_n) = \exp (\delta _1t_1X_1)\cdots \exp (\delta _nt_nX_n)$ for a multi-index $\delta$. Due to the special form of H

$$\begin{aligned} XH(y) = \frac{d}{dt}\Big |_{t=0} \langle \lambda , \rho (y\exp (tX)) u\rangle = \frac{d}{dt}\Big |_{t=0} \langle \lambda ,\rho (y)\rho (\exp (tX))u\rangle \overline{\sigma (y,\exp (tX))}. \end{aligned}$$

Therefore $X^\delta H(y)$ can be expressed as a sum

$$\begin{aligned} \sum _{|\gamma |\le |\delta |} \langle \lambda , \rho (y) \rho (X_n)^{\gamma _n}\cdots \rho (X_1)^{\gamma _1}u \rangle g_\gamma (y), \end{aligned}$$

where $g_\gamma$ is an appropriate derivative of the cocycle $\sigma$ of order $|\gamma |$. Notice, that the $g_\gamma$’s do not depend on the vectors v and $\lambda$ used to define H and K. If y is in the compact set $U_\epsilon$ the functions $g_\gamma$ are uniformly bounded, and therefore there is a constant $D_\epsilon$ such that

$$\begin{aligned} \sup _{y\in U_\epsilon }&|H(y)-H(e)| \\&\le D_\epsilon \sum _{{\mathop {|\delta |=|\alpha |}\limits ^{1\le |\alpha |\le n}}} \int _{[-\epsilon ,\epsilon ]^{|\delta |}} |\langle \lambda ,\rho (\tau _\delta (\mathbf {t})) \rho (X_n)^{\delta _n}\cdots \rho (X_1)^{\delta _1} u)| (dt_1)^{\delta _1} \dots (dt_n)^{\delta _n}, \end{aligned}$$

when we write $\mathbf {t}=(t_1,t_2,\ldots ,t_n)$. From this we get that

$$\begin{aligned}&\mathrm {osc}^{r^\sigma }_U \phi (x) \\&\quad \le D_\epsilon \sum _{{\mathop {|\delta |=|\alpha |}\limits ^{1\le |\alpha |\le n}}} \int _{[-\epsilon ,\epsilon ]^{|\delta |}} |\langle u,\rho (x\tau _\delta (\mathbf {t})) \rho (X_n)^{\delta _n}\cdots \rho (X_1)^{\delta _1} u)| (dt_1)^{\delta _1} \dots (dt_n)^{\delta _n}. \end{aligned}$$

Treating K the same way we get

$$\begin{aligned}&\mathrm {osc}^{\ell ^\sigma }_U \phi (x) \\&\quad \le D_\epsilon \sum _{{\mathop {|\delta |=|\alpha |}\limits ^{1\le |\alpha |\le n}}} \int _{[-\epsilon ,\epsilon ]^{|\delta |}} |\langle \rho ^*(X_n)^{\delta _n} \cdots \rho ^*(X_1)^{\delta _1} u ,\rho (\tau _\delta (\mathbf {t})^{-1}x)u )| (dt_1)^{\delta _1} \dots (dt_n)^{\delta _n}. \end{aligned}$$

Lemma 8

Assume that B is a solid BF-space on which left and right translations are continuous. If $f\mapsto f*|\langle u,\rho (\cdot )\rho (X_n)^{\delta _n}\cdots \rho (X_1)^{\delta _1} u\rangle |$ and $f\mapsto f*|\langle \rho ^*(X_n)^{\delta _n}\cdots \rho ^*(X_1)^{\delta _1} u,\rho (\cdot )u\rangle |$ are bounded on B for all $|\delta |\le \dim (G)$, then

$$\begin{aligned} \Vert f*\mathrm {osc}_U^{\ell ^\sigma } \phi \Vert _B \le C_\epsilon \Vert f\Vert _B \end{aligned}$$

and

$$\begin{aligned} \Vert f*\mathrm {osc}_U^{r^\sigma } \phi \Vert _B \le C_\epsilon \Vert f\Vert _B. \end{aligned}$$

Moreover, $\lim _{\epsilon \rightarrow 0} C_\epsilon =0$.

Proof

Write $\mathbf {t}=(t_1,\ldots ,t_n)$. Notice that

$$\begin{aligned} |f*&\mathrm {osc}_U^{\ell ^\sigma } \phi (x) | \\&\le D_\epsilon \sum _{{\mathop {|\delta |=|\alpha |}\limits ^{1\le |\alpha |\le n}} } \int _{[-\epsilon ,\epsilon ]^{|\delta |}} |\ell _{\tau _\delta (\mathbf {t})} f|*|W_{\rho ^*(X_n)^{\delta _n} \cdots \rho ^*(X_1)^{\delta _1} u}(u)(x) )| (dt_1)^{\delta _1} \dots (dt_n)^{\delta _n}. \end{aligned}$$

Since left translation is continuous on B the right hand side defines a function in B by Theorem 3.29 in [37], and by solidity $f*\mathrm {osc}_U^{\ell ^\sigma } \phi (x)$ is also in B. Moreover, $\Vert f*\mathrm {osc}_U^{\ell ^\sigma } \phi \Vert \le C_\epsilon \Vert f\Vert$, where $C_\epsilon$ is equal to $D_\epsilon$ multiplied by a polynomial in $\epsilon$ with no constant term. Since $D_\epsilon$ is uniform in $\epsilon$ we see that $\lim _{\epsilon \rightarrow 0} C_\epsilon =0$.

The proof for convolution with right oscillations follows in a similar manner. $\square$