Good λ inequalities for nontangential maximal functions and S-functions of harmonic functions

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


For a harmonic function u(x, t) on R+n+1, let Nδu be as in (0.2) and let
$$ \begin{gathered} S_\delta \left( {t\left| {\nabla _u } \right|} \right)\left( x \right) \hfill \\ = \left\{ {\Gamma \left( {\frac{{n + 2}} {2}} \right)\pi ^{ - n/2} \smallint \smallint _{\Gamma \left( {x,\delta } \right)} t^2 \left| {\nabla u\left( {y,t} \right)} \right|^2 \left( {\delta t} \right)^{ - n} dydt/t} \right\}^{1/2} , \hfill \\ \end{gathered} $$
where ∇ = (Dt,D X1,…,D Xn). We will show the following precise relations between Nδu and Sδ (t|∇u|).


Distribution Function Euclidean Space Harmonic Function Number Theory Geometric Property 
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© Springer Japan 2001

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  • Akihito Uchiyama

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