Carleson's decomposition of functions in $BMO_L$
Prof. Peng Chen
2023-08-16 16:30-17:30
MCM110
Speaker: Prof. Peng Chen (Sun Yat-sen University)Time: 16:30-17:30 August 16, 2023 (Wednesday)
Place: MCM110
Title: Carleson's decomposition of functions in $BMO_L$
Abstract: Let $X$ be a metric space with doubling measure, and $L$ be a nonnegative self-adjoint operator on $L^2(X)$ whose heat kernel satisfies the Gaussian upper bound. Suppose $ f$ is in the space $ {\rm BMO}_L(X)$ associated with the operator $L$ and has compact support. We show that there exist $g\in L^\infty(X)$ and a finite Carleson measure $\sigma$ such that $$ f(x)=g(x)+\int_{X\times [0,\infty)} K_{e^{-t^2L}}(x,y)d\sigma(y,t) $$ with $\|g\|_{L^\infty}+\|\sigma\|_{\mathcal C}\leq C\|f\|_{BMO_L}$. This extends the result for the classical John-Nirenberg BMO space by Carleson (1976) (see also Garnett and Jones (1982), Uchiyama (1980) and Wilson (1988)) to the BMO setting associated with operators.The theory of multilinear multipliers originated in the work of Coifman and Meyer in 1970s, which had fruitful applications including the famous Bony decomposition. In past decades, various multilinear multipliers were introduced with different backgrounds. In this talk, I will discuss several closely related multipliers including multilinear multipliers without decay, multilinear lattice bump multipliers, and multilinear Hormander multiliers. Finally, I will present our new sharp result on multilinear multi-parameter Hormander multipliers, which is joint work with J. Chen, G. Lu, B. Park, and L. Zhang.
