A Littlewood-Paley type inequality

Abstract.

In this note we prove the following theorem: Let u be a harmonic function in the unit ball $B \subset {\mathbf{R}}^{n}$ and $p \in {\left[ {\frac{{n - 2}} {{n - 1}},1} \right]}$. Then there is a constant C = C(p, n) such that

$$ {\mathop {\sup }\limits_{0 \leqslant r < 1} }{\kern 1pt} {\kern 1pt} {\int_S {{\left| {u{\left( {r\zeta } \right)}} \right|}^{p} d\sigma {\left( \zeta \right)} \leqslant C{\left( {{\left| {u{\left( 0 \right)}} \right|}^{p} + {\int_B {{\left| {\nabla u{\left( x \right)}} \right|}^{p} {\left( {1 - {\left| x \right|}} \right)}^{{p - 1}} dV{\left( x \right)}} }} \right)}} } $$

.