Endpoint Boundedness of Toeplitz Operators

• The criterion characterizes boundedness by requiring weights to meet u-adapted Bekollè–Bonami conditions coupled with reverse Hölder regularity.
• The approach uses sparse domination and dyadic decompositions to obtain weak-type (1,1) estimates for Toeplitz operators on weighted Bergman spaces.
• Illustrative examples with radial and conformal Jacobian weights demonstrate both the sharpness of the result and its limitations in broader settings.

The endpoint boundedness criterion for Toeplitz operators describes necessary and sufficient conditions for extending these operators as bounded linear maps at critical values of the involved function space exponents, particularly in weighted Bergman spaces on the unit ball $\mathbb{C}^n$. Central to this theory is the control exerted by the symbol $u \in L^{\infty}(\mathbb{B}_n)$ and the weighted classes $\sigma$ drawn from the adapted Bekollè–Bonami weights, denoted $B_p$ and related families, along with further regularity encoded by reverse Hölder conditions. Endpoint refers specifically to the weak-type $(1,1)$ result, addressing the most singular $L^1$ range where classical Calderón–Zygmund methods require significant modification.

1. Toeplitz Operators and Weighted Bergman Spaces

Toeplitz operators $T_u$ on the Bergman space are defined via integral kernels,

$$T_u f(z) = \int_{\mathbb{B}_n} K(z, w) u(w) f(w)\, dV(w), \qquad K(z, w) = \frac{1}{(1 - z \cdot \overline{w})^{n+1}},$$

where $u\in L^\infty(\mathbb{B}_n)$ is the symbol and $dV$ denotes normalized Lebesgue measure on the ball. Weighted theory situates this operator as acting naturally on spaces $L^1_\sigma = L^1(\mathbb{B}_n, d\mu)$ with $d\mu = \sigma\,dV$, and one seeks the boundedness from $L^1_\sigma$ to weak-$L^1_\sigma$ (the endpoint $L^{1,\infty}_\sigma$) (Stockdale et al., 2021).

2. Endpoint Weak-Type $(1,1)$ Theorem

Let $u\in L^\infty(\mathbb{B}_n)$ and $\sigma$ be a nonnegative locally integrable weight. The endpoint theorem states: $$T_u : L^1_\sigma \to L^{1, \infty}_\sigma \text{ is bounded}$$ if and only if $\sigma$ belongs to the $u$-adapted Bekollè–Bonami class $uB_1$ and, in practice, satisfies a reverse Hölder property $RH_r$ for some $r>1$.

More precisely, there exists $C=C(n)$ such that

$$\|T_u\|_{L^1_\sigma \to L^{1, \infty}_\sigma} \leq C\,[\sigma]_{uB_1}\,[\sigma]_{RH_r}\,(1+\log r'), \qquad r' = \frac{r}{r-1},$$

and if $\sigma$ further satisfies a dyadic doubling condition, one obtains

$$\|T_u\|_{L^1_\sigma \to L^{1, \infty}_\sigma} \lesssim ([\sigma]_{uB_1} + 1) \log (e + [\sigma]_{B_1}).$$

This result is sharp in all known cases; the logarithmic term is generally unavoidable when tracking dyadic decompositions (Stockdale et al., 2021).

3. Bekollè–Bonami and Related Weight Classes

Bekollè–Bonami classes $B_p$ encode the compatibility of a weight $\omega$ with the Bergman kernel on dyadic cube systems. Explicitly,

$$\omega \in B_p \; \Longleftrightarrow \; [\omega]_{B_p} := \sup_{e, K\in\mathcal{D}^e} \langle \omega \rangle_K \, \langle \omega^{1-p'}\rangle_K^{p-1} < \infty$$

with $p' = \frac{p}{p-1}$ and $$\langle \omega \rangle_K = \frac{1}{|K|} \int_K \omega\, dV$$. For the endpoint $p=1$,

$[\omega]_{B_1} := \sup_{e, K} \langle \omega \rangle_K \, \esssup_K(\omega^{-1}) < \infty.$

The class $uB_1$ is defined by incorporating the supremum norm of $u$ over dyadic tents, and similar expressions hold for $uB_p$.

Reverse Hölder classes $RH_r$ are given by

$$[\sigma]_{RH_r} := \sup_{e,\,K} \left\langle \sigma^r \right\rangle_K^{1/r} \left\langle \sigma \right\rangle_K^{-1} < \infty,$$

with weights in $B_1$ (under mild doubling) automatically in $RH_r$ for some $r>1$ (Stockdale et al., 2021).

4. Sparse Domination and Calderón–Zygmund Methods

The proof utilizes sparse domination: for any $f$,

$$|T_u f(z)| \lesssim \sum_{e=1}^M \sum_{K\in\mathcal{D}^e} \|u\|_{L^\infty(R)}\, \frac{1}{|R|}\int_R |f|\, dV \, \mathbf{1}_K(z).$$

A Calderón–Zygmund decomposition on the dyadic systems, together with weak-type $(1,1)$ control of the Hardy–Littlewood maximal operator, enables

$$\mu\left\{ |T_u f| > 2\lambda \right\} \lesssim [\sigma]_{uB_1}\,\mu\left\{ M(uf)>\lambda \right\} + \sum_{k\ge 0} C_k [\sigma]_{uB_1}\, \mu\left\{ |T_u f| > 2^{-k} \lambda \right\}$$

with $C_k$ decaying as a negative power of $k$ up to logarithms. Iteration and the choice of a Young function $\Phi(t)=t^r$ systematically produce the stated reverse Hölder dependence and weak–$(1,1)$ bound (Stockdale et al., 2021).

5. Illustrative Examples and Boundary Cases

• Radial weights: For $\sigma_b(z)=(1 - |z|^2)^b$ with $b\in(-1, 0]$, $\sigma_b$ satisfies $B_1$ with $[\sigma_b]_{B_1} \sim \frac{1}{1+b}$ and doubling, giving

$$P:L^1_{\sigma_b} \to L^{1,\infty}_{\sigma_b},\quad \|P\| \lesssim \frac{1}{1+b} \log\left( e + \frac{1}{1+b} \right).$$

• Conformal Jacobian weights: For $g$ univalent in $\mathbb{D}$ and $\sigma = |g'|^\alpha$, $\sigma \in B_1 \cap RH_r$ via Koebe distortion.
• Limitations: Outside $uB_1$ (even if $\sigma\in B_1$), no general weak–$(1,1)$ bound is known and necessity remains open (Stockdale et al., 2021).

6. Summary and Significance

The endpoint boundedness theory for Toeplitz operators on weighted Bergman spaces characterizes the admissible weights for which the operator extends as weak-type $(1,1)$ from $L^1_\sigma$ to $L^{1,\infty}_\sigma$. This is achieved via the quantitative $uB_1$ condition, reverse Hölder regularity, and compatibility with dyadic decomposition structures. All criteria are quantitative and checkable through Carleson-tent or dyadic-cube averages. The theory extends, with analogous structure, to related spaces and further general symbols, and sharp quantitative dependence in operator norms is obtained. This advances a rigorous framework for endpoint analysis in Bergman and related function spaces, elucidating the boundary between boundedness and failure in weighted singular integral settings (Stockdale et al., 2021).