A direct proof of Open image in new window

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


$$ \left| {\smallint _{{\text{R}}^n } f\left( x \right)g\left( x \right)dx} \right| \leqslant C\left( n \right)\left\| {S_1 \left( {tD_t u} \right)} \right\|_{L^1 } \left\| g \right\|_{{\text{BMO,}}} $$
$$ f,g \in L^2 , $$
$$ u\left( {x,t} \right) = \smallint _{{\text{R}}^n } P\left( {x - y,t} \right)f\left( y \right)dy. $$
(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.)


Euclidean Space Harmonic Function Measurable Function Number Theory Algebraic Geometry 
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© Springer Japan 2001

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  • Akihito Uchiyama

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