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

• Autorzy: Adam Osękowski
• Języki publikacji: EN

Abstrakty

EN

Let $(h_{k})_{k≥0}$ be the Haar system on [0,1]. We show that for any vectors $a_{k}$ from a separable Hilbert space 𝓗 and any $ε_{k} ∈ [-1,1]$, k = 0,1,2,..., we have the sharp inequality
$||∑_{k=0}^{n} ε_{k}a_{k}h_{k}||_{W([0,1])} ≤ 2||∑_{k=0}^{n} a_{k}h_{k}||_{L^{∞}([0,1])}$, n = 0,1,2,...,
where W([0,1]) is the weak-$L^{∞}$ space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound
$||Y||_{W(Ω)} ≤ 2||X||_{L^{∞}(Ω)}$,
where X and Y stand for 𝓗-valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.

Kategorie tematyczne

• 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
• 31B05: Harmonic, subharmonic, superharmonic functions
• 60G42: Martingales with discrete parameter
• 60G44: Martingales with continuous parameter

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2014
• Tom: 62
• Numer: 2
• Strony: 187-196

Daty

wydano

2014

Twórcy

autor

Adam Osękowski

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

Identyfikatory

DOI

10.4064/ba62-2-7