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A direct proof of Open image in new window

  • Akihito Uchiyama
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

$$ \left| {\smallint _{{\text{R}}^n } f\left( x \right)g\left( x \right)dx} \right| \leqslant C\left( n \right)\left\| {S_1 \left( {tD_t u} \right)} \right\|_{L^1 } \left\| g \right\|_{{\text{BMO,}}} $$
(12.1)
where
$$ f,g \in L^2 , $$
(12.2)
$$ u\left( {x,t} \right) = \smallint _{{\text{R}}^n } P\left( {x - y,t} \right)f\left( y \right)dy. $$
(12.3)
(In (12.3), P (x, t) denotes the Poisson kernel.) Since \( \left\| {S_1 \left( {tD_t u} \right)} \right\|_{L^1 } \) dominates \( \left\| f \right\|_{H^1 } \) by (10.19), (12.1) follows from Lemma 2.2 (with p = 1) and Theorem 2.2. In this section, we explain C. Fefferman’s direct proof of (12.1), which is one of the oldest proofs of his H 1-BMO duality theorem. (Another one of the oldest proofs will be explained in Section 19.)

Keywords

Euclidean Space Harmonic Function Measurable Function Number Theory Algebraic Geometry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Japan 2001

Authors and Affiliations

  • Akihito Uchiyama

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