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

Abstract

Let $n\in \mathbb{N}\cap [2,\infty )$ and $\Omega \in L^1({\mathbb{S}}^{n-1})$ with $\Omega \not\equiv 0$. In this article, we introduce a new family of lifted rough maximal operators $\{{\mathcal{M}}_{\theta }^{\Omega }{\}}_{\theta \in (0,\infty )}$ 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 $\gamma \in \mathbb{R}\setminus \{0\}$. For the endpoint case $p=1$ and $\Omega \in L(\log L)({\mathbb{S}}^{n-1})$, we prove that the above estimate holds if and only if $\gamma \in (-\infty ,-n)\cup (0,\infty )$. 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 $M_{\Omega }^{\ast }$, 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.