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

Let $n\in \mathbb{N}\cap [2,\mathrm{\infty })$ and $\mathrm{\Omega }\in L^1({\mathbb{S}}^{n-1})$ with $\mathrm{\Omega }\equiv ̸0$. In this article, we introduce a new family of lifted rough maximal operators $\{{\mathcal{M}}_{\theta }^{\mathrm{\Omega }}{\}}_{\theta \in (0,\mathrm{\infty })}$ in the upper-half plane and establish their optimal weak-type estimates. Specifically, we prove that, for any $p\in (1,\mathrm{\infty })$, the estimate, with the positive equivalence constants independent of $f$, $$\underset{\theta ,\lambda \in (0,\mathrm{\infty })}{\sup }{\lambda }^p{t}^{\gamma -1}dtdx\sim \mathrm{\parallel }f{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}^p$$ 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 $\mathrm{\Omega }\in L(\log L)({\mathbb{S}}^{n-1})$, we prove that the above estimate holds if and only if $\gamma \in (-\mathrm{\infty },-n)\cup (0,\mathrm{\infty })$. 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 $M_{\mathrm{\Omega }}^{\ast }$, 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.