Sharp weak bounds for the Hardy-Littlewood-Pólya operator and the weak bounds for the multilinear integral operator

TL;DR

The paper addresses sharp weak-type bounds for the $n$-dimensional Hardy-Littlewood-Pólya operator and extends the analysis to a general $m$-linear, radial-kernel integral operator in weighted spaces. By leveraging weighted $L^p$ spaces and radial symmetries, it derives explicit norm formulas under natural dimensional and weight constraints, including endpoint cases, and shows how particular kernels recover classical Hardy, HLP, and Hilbert bounds with computable constants. The results provide precise weak-type constants depending on the dimension $n$, the unit sphere measure $\omega_n$, and weights $(\beta,\gamma)$, advancing the understanding of sharp weighted weak-type estimates in harmonic analysis. This has implications for embedding properties of function spaces and for multilinear integral operators in high-dimensional settings.

Abstract

In this paper, we first obtain the operator norms of the $n$-dimensional Hardy-Littlewood-Pólya operator $\mathcal{H}$ from weighted Lebesgue spaces $L^p( \mathbb{R} ^n,| x |^β ) $ to weighted weak Lebesgue spaces $L^{q,\infty}(\mathbb{R} ^n,|x|^γ)$. Next, we obtain the weak bounds for the $m$-linear $n$-dimensional integral operator.

Theorems & Definitions (16)

• Definition 1.1: Hardydingyi
• Definition 1.2: mHLPnwei
• Definition 1.3: Hardydingyi
• Theorem 2.1
• Theorem 2.2
• Remark 1
• proof : Proof of Theorem \ref{['thm2.1']}:
• proof : Proof of Theorem \ref{['thm2.2']}:
• Theorem 3.1
• Corollary 3.1