Sharp Weak-Type Inequality for the Haar System, Harmonic Functions and Martingales

Adam Osękowski¹

¹ Department of Mathematics, Informatics and Mechanics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 62 (2014) no. 2, pp. 187-196

Résumé

Let $(h_k)_{k\geq 0}$ be the Haar system on $[0,1]$. We show that for any vectors $a_k$ from a separable Hilbert space $\mathcal{H}$ and any $\varepsilon_k\in [-1,1]$, $k=0,1,2,\ldots,$ we have the sharp inequality $$ \Bigl\|\sum_{k=0}^n \varepsilon_ka_kh_k\Big\|_{W([0,1])}\leq 2\Bigl\|\sum_{k=0}^n a_kh_k\Big\|_{L^\infty([0,1])},\quad\ n=0,1,2,\ldots,$$ where $W([0,1])$ is the weak-$L^\infty$ space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound $$ \|Y\|_{W(\varOmega)}\leq 2\|X\|_{L^\infty(\varOmega)},$$ where $X$ and $Y$ stand for $\mathcal{H}$-valued martingales such that $Y$ is differentially subordinate to $X$. An application to harmonic functions on Euclidean domains is presented.

Détail

DOI : 10.4064/ba62-2-7

Keywords: geq haar system vectors separable hilbert space mathcal varepsilon ldots have sharp inequality bigl sum varepsilon leq bigl sum infty quad ldots where weak l infty space introduced bennett devore sharpley above estimate generalized sharp weak type bound varomega leq infty varomega where stand mathcal valued martingales differentially subordinate nbsp application harmonic functions euclidean domains presented