Brownian Martingales and Some New Results on Interpolation of Hardy Spaces

Quanhua Xu

doi:10.1023/a:1008604106453

Brownian Martingales and Some New Results on Interpolation of Hardy Spaces

Quanhua Xu^*

^*Corresponding author for this work

Sorbonne Université
UMR de Mathématiques

Research output: Contribution to journal › Article › peer-review

Abstract

Let H₁(ℝⁿ) be the usual Hardy space on ℝⁿ. We show that the couple (H₁(ℝⁿ), L_∞(ℝⁿ)) is a Calderón couple. This result immediately follows from the following stronger one: Given any f ∈ H₁(ℝⁿ) + L_∞(ℝⁿ) there exist two linear operators U and V satisfying the properties: (i) U f = Nf (Nf being the nontangential maximal function of f) and U is contractive from H₁(ℝⁿ) to L₁(ℝⁿ) and also from L_∞(ℝⁿ) to L_∞(ℝⁿ); (ii) V (Nf) = f, V is similtaneously bounded from L1(ℝⁿ) to H₁(ℝⁿ) and from L_∞(ℝⁿ) to L_∞(ℝⁿ) and the norms of V on these spaces are controlled by a universal constant. We also have similar results on the couple (L_p(ℝⁿ), BMO(ℝⁿ)) for every 1 < p < ∞. Our approach to these results is via Brownian motion.

Original language	English
Pages (from-to)	257-277
Number of pages	21
Journal	Potential Analysis
Volume	11
Issue number	3
DOIs	https://doi.org/10.1023/a:1008604106453
State	Published - 1999
Externally published	Yes

Keywords

Brownian motion
Calderón couple
Hardy space
Interpolation

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title = "Brownian Martingales and Some New Results on Interpolation of Hardy Spaces",

abstract = "Let H1(ℝn) be the usual Hardy space on ℝn. We show that the couple (H1(ℝn), L∞(ℝn)) is a Calder{\'o}n couple. This result immediately follows from the following stronger one: Given any f ∈ H1(ℝn) + L∞(ℝn) there exist two linear operators U and V satisfying the properties: (i) U f = Nf (Nf being the nontangential maximal function of f) and U is contractive from H1(ℝn) to L1(ℝn) and also from L∞(ℝn) to L∞(ℝn); (ii) V (Nf) = f, V is similtaneously bounded from L1(ℝn) to H1(ℝn) and from L∞(ℝn) to L∞(ℝn) and the norms of V on these spaces are controlled by a universal constant. We also have similar results on the couple (Lp(ℝn), BMO(ℝn)) for every 1 < p < ∞. Our approach to these results is via Brownian motion.",

keywords = "Brownian motion, Calder{\'o}n couple, Hardy space, Interpolation",

author = "Quanhua Xu",

year = "1999",

doi = "10.1023/a:1008604106453",

language = "英语",

volume = "11",

pages = "257--277",

journal = "Potential Analysis",

issn = "0926-2601",

publisher = "Springer Netherlands",

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T1 - Brownian Martingales and Some New Results on Interpolation of Hardy Spaces

AU - Xu, Quanhua

PY - 1999

Y1 - 1999

N2 - Let H1(ℝn) be the usual Hardy space on ℝn. We show that the couple (H1(ℝn), L∞(ℝn)) is a Calderón couple. This result immediately follows from the following stronger one: Given any f ∈ H1(ℝn) + L∞(ℝn) there exist two linear operators U and V satisfying the properties: (i) U f = Nf (Nf being the nontangential maximal function of f) and U is contractive from H1(ℝn) to L1(ℝn) and also from L∞(ℝn) to L∞(ℝn); (ii) V (Nf) = f, V is similtaneously bounded from L1(ℝn) to H1(ℝn) and from L∞(ℝn) to L∞(ℝn) and the norms of V on these spaces are controlled by a universal constant. We also have similar results on the couple (Lp(ℝn), BMO(ℝn)) for every 1 < p < ∞. Our approach to these results is via Brownian motion.

AB - Let H1(ℝn) be the usual Hardy space on ℝn. We show that the couple (H1(ℝn), L∞(ℝn)) is a Calderón couple. This result immediately follows from the following stronger one: Given any f ∈ H1(ℝn) + L∞(ℝn) there exist two linear operators U and V satisfying the properties: (i) U f = Nf (Nf being the nontangential maximal function of f) and U is contractive from H1(ℝn) to L1(ℝn) and also from L∞(ℝn) to L∞(ℝn); (ii) V (Nf) = f, V is similtaneously bounded from L1(ℝn) to H1(ℝn) and from L∞(ℝn) to L∞(ℝn) and the norms of V on these spaces are controlled by a universal constant. We also have similar results on the couple (Lp(ℝn), BMO(ℝn)) for every 1 < p < ∞. Our approach to these results is via Brownian motion.

KW - Brownian motion

KW - Calderón couple

KW - Hardy space

KW - Interpolation

UR - https://www.scopus.com/pages/publications/0033458201

U2 - 10.1023/a:1008604106453

DO - 10.1023/a:1008604106453

M3 - 文章

AN - SCOPUS:0033458201

SN - 0926-2601

VL - 11

SP - 257

EP - 277

JO - Potential Analysis

JF - Potential Analysis

IS - 3

ER -

Brownian Martingales and Some New Results on Interpolation of Hardy Spaces

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Keywords

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