Fock projections on vector-valued $L^p$-spaces with matrix weights

TL;DR

The paper addresses bounding the Fock projection $P_{\alpha}$ on vector-valued $L^p$-spaces with matrix weights over $\C^n$ equipped with Gaussian measures. It develops a matrix-weighted theory via a restricted $\mathcal{A}_{p,r}$-condition, proving that $P_{\alpha}$ is bounded on $L^p_{\alpha,W}(\C^n;\C^d)$ if and only if $W$ satisfies this condition for some (hence all) $r>0$, with quantitative norm estimates. A key methodological contribution is the use of integral operators tied to normalized reproducing kernels and a matrix-weighted maximal Fock projection, together with duality, to bridge boundedness of $P_{\alpha}$ and the $\mathcal{A}_{p,r}$-property. The endpoint $p=\infty$ yields new results even in the scalar case and confirms the robustness of the $\mathcal{A}_{p,r}$ framework in the Fock space setting, extending classical matrix-weighted harmonic analysis to several complex variables.

Abstract

In this paper, we characterize the $d\times d$ matrix weights $W$ on $\mathbb{C}^n$ such that the Fock projection $P_α$ is bounded on the vector-valued spaces $L^p_{α,W}(\mathbb{C}^n;\mathbb{C}^d)$ induced by $W$ and the Gaussian measures. It is proved that for $1\leq p\leq\infty$, the Fock projection $P_α$ is bounded on $L^p_{α,W}(\mathbb{C}^n;\mathbb{C}^d)$ if and only if $W$ satisfies a restricted $\mathcal{A}_p$-condition. Our result is new even in the scalar setting at the endpoint $p=\infty$.

Theorems & Definitions (16)

• Lemma 2.1
• Remark 2.2
• Theorem 3.1
• Proposition 3.2
• proof
• Lemma 3.3
• Lemma 3.4
• proof
• proof : Proof of Theorem \ref{['main']}. (a)$\Longrightarrow$(c)
• Lemma 3.5