Weak type $(1,1)$ behavior for the Littlewood–Paley $\mathfrak g$-function
Volume 171 / 2023
Abstract
For $f\in L^p(\mathbb R^n)\ (1\le p \lt \infty )$, the classical Littlewood–Paley $\mathfrak g$-function is defined by $$\mathfrak g(f)(x)=\Bigl (\int _0^\infty |\nabla u(x,t)|^2t\,dt\Big )^{1/2}, $$ where $u(x,t)$ denotes the Poisson integral of $f$. The following two weak type $(1,1)$ behaviors for the operator $\mathfrak {g}$ are established: $$ \lambda m(\{x\in \mathbb R ^n:\mathfrak g(f)(x) \gt \lambda \})\lesssim n^3\|f\|_1,$$ $$\lim _{\lambda \to 0_+}\lambda m(\{x\in \mathbb R ^n:\mathfrak g(f)(x) \gt \lambda \})=\frac {\sqrt {2}\,c_n\omega_{n-1}}{2n}\Bigl |\int _{\mathbb R ^n}f(x)\,dx\Big |,$$ for any $f\in L^1(\mathbb R^n)$, where $c_n$ is the constant in the Poisson kernel and $\omega _{n-1}$ is the area of the unit sphere in $\mathbb R^n$.
