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

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 $f$, $$\sup_{\theta,\lambda\in(0,\infty)}\lambda^p \underset{{\mathcal M}^\Omega_\theta(f)(x,t) > \lambda t^\frac{\gamma}{p}} {\int_{\mathbb R^n}\int_0^\infty} t^{\gamma-1}\,dt\,dx \sim \|f\|_{L^p(\mathbb{R}^n)}^p$$ 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 Sj\"ogren and Soria in page 228 of [Israel J. Math. 95 (1996)]. Moreover, although the operator $M^\ast_\Omega$, arising from the method of rotation of Calder\'on 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.