JEDNOSTKA NAUKOWA KATEGORII A+

Weak type $(1,1)$ behavior for the Littlewood–Paley $\mathfrak g$-function

Xudong Lai Colloquium Mathematicum MSC: Primary 42B25; Secondary 42B99. DOI: 10.4064/cm8854-5-2022 Opublikowany online: 22 September 2022

Streszczenie

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 $\mathbb R^n$.

Autorzy

  • Xudong LaiInstitute for Advanced Study in Mathematics
    Harbin Institute of Technology
    Harbin, 150001, People’s Republic of China
    e-mail

Przeszukaj wydawnictwa IMPAN

Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki.

Przepisz kod z obrazka

Odśwież obrazek

Odśwież obrazek