Lower bounds for the weak-type constants of the operators $Λ_m$

TL;DR

This work analyzes weak-type $(1,1)$ constants for the family of Λ_m operators arising from the Beurling–Ahlfors transform on radial subspaces. It proves new, explicit lower bounds: w(Λ_0)=1/\ln(2), w(Λ_1)≥1.38, and liminf_m w(Λ_m)≥1.37, thereby disproving Gill's conjecture that these constants tend to 1. The approach constructs extremal function classes in $\mathcal{E}_m$ and an associated tractable subset $\mathcal{F}_m$ to obtain computable lower bounds via a supremum formula $W(b,d,m)$, with analogous results for the adjoint $Λ_m^*$. The results extend to adjoints, giving the same qualitative lower bounds for w$(Λ_m^*)$ and supporting a deeper relation between Λ_m and Λ_m^* in restricted settings; the paper also discusses conjectures about exact equalities among restricted weak-type constants and a potential Bellman-function route to sharp bounds. Overall, the paper advances understanding of how the Beurling–Ahlfors transform can behave in weighted radial contexts and shows that the weak-type constants do not collapse to 1 as the weight parameter grows, contradicting prior expectations.

Abstract

The operators $Λ_m$ ($m\in\mathbb{N}\cup \{0\}$) arise when one studies the action of the Beurling-Ahlfors transform on certain radial function subspaces. It is known that the weak-type $(1,1)$ constant of $Λ_0$ is equal to $1/\ln(2)\approx 1.44$. We construct examples showing that the weak-type $(1,1)$ constant of $Λ_1$ is larger than $1.38$ and that the weak-type $(1,1)$ constant of $Λ_m$ does not tend to $1$ when $m\to\infty$. This disproves a conjecture of Gill [Mich. Math. J. 59 (2010), No. 2, 353-363]. We also prove a companion result for the adjoint operators. This is the arXiv version of the paper - it includes some additional discussion in the appendices.

Figures (10)

• Figure 1: The light gray region is the set $\Omega_{m}$ consisting of parameters $b$, $d$ satisfying the constraints of the Definition \ref{['def:f-class-spec']}. Its boundary consists of the dashed lines described by: $b = b_{\min}(m)$, $d= d_{\min}(b,m)$, and $d=d_{\max}(b,m)$. The thick black line is the curve $d = d_{\mathrm{opt}}(b,m)$. Note that the axes have been scaled for readability.
• Figure 2: For a specific and appropriate choice of $d=d(b,m)$, $b_*=b_*(b,m)$, and $d_* = d_*(b_*(b,m),m)$ we have $W(b,d,m) = W_*(b_*,d_*,m)$.
• Figure 3: Claims \ref{['eq:W-vs-W-star-claim-0']}--\ref{['eq:W-vs-W-star-claim-3']} assert that for an appropriate choice of $d$, $b_*$, and $d_*$ we have $W(b,d,m) = W_*(b_*,d_*,m)$.
• Figure 4: Proof of Lemma \ref{['lem:asymptotic-push-case-0']}. The dashed lines, $x = \ln(3/2)$, $3e^{-y} = 2(2-e^x)$, and $e^{-y} = 2(2-e^x)$, divide the first quadrant into five sets according to the formulas for $U$ and $V$.
• Figure 5: Proof of Lemma \ref{['lem:asymptotic-push-case-0']} (cf. Figure \ref{['fig:asymptotic-case-0-push-the-cases']}). On the dotted lines $U(x,y,2(2-e^x))$ is constant; the arrow heads indicate in which direction $V(x,y,2(2-e^x))$ decreases and the thick lines indicate where the local minima can lie. Note that the axes have been scaled for readability.

Theorems & Definitions (90)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Lemma 2.1
• proof
• Definition 2.2
• Lemma 2.3
• proof
• Definition 2.4
• Lemma 2.5