Optimal Weak-Type Estimates and Their Applications of Lifted Rough Maximal Operators

Abstract

Let $n\in\mathbb N\cap[2,\infty)$ and $Ω\in L^1(\mathbb S^{n-1})$ with $Ω\not\equiv 0$. In this article, we introduce a new family of lifted rough maximal operators $\{\mathcal{M}_θ^Ω\}_{θ\in(0,\infty)}$ in the upper-half plane and establish their optimal weak-type estimates. Specifically, we prove that, for any $p \in (1, \infty)$, the estimate, with the positive equivalence constants independent of $f$, \[ \sup_{θ,λ\in(0,\infty)}λ^p \underset{{\mathcal M}^Ω_θ(f)(x,t) > λt^\fracγ{p}} {\int_{\mathbb R^n}\int_0^\infty} t^{γ-1}\,dt\,dx \sim \|f\|_{L^p(\mathbb{R}^n)}^p \] holds for all $f\in L^p(\mathbb R^n)$ if and only if $γ\in\mathbb R\setminus\{0\}$. For the endpoint case $p=1$ and $Ω\in L(\log L)(\mathbb{S}^{n-1})$, we prove that the above estimate holds if and only if $γ\in (-\infty, -n) \cup (0, \infty)$. As applications, we obtain weak-type estimates for generalized Poisson integrals without any logarithmic integrability assumptions, which gives an affirmative answer to the question posed by Sjögren and Soria in page 228 of [Israel J. Math. 95 (1996)]. Moreover, although the operator $M^\ast_Ω$, arising from the method of rotation of Calderón and Zygmund, is not of weak type $(1,1)$, we find that its lifted variant is weak type $(1,1)$. In addition, we establish a new characterization of Hardy spaces in terms of truncated rough singular integrals.