538 research outputs found
A characterization of two weight norm inequality for Littlewood-Paley -function
Let and be the well-known high dimensional
Littlewood-Paley function which was defined and studied by E. M. Stein,
\begin{align*} g_{\lambda}^{*}(f)(x) =\bigg(\iint_{\mathbb R^{n+1}_{+}}
\Big(\frac{t}{t+|x-y|}\Big)^{n\lambda} |\nabla P_tf(y,t)|^2 \frac{dy
dt}{t^{n-1}}\bigg)^{1/2}, \ \quad \lambda > 1, \end{align*} where
, and ,
. In this paper, we give a characterization
of two-weight norm inequality for -function. We show that,
if and only if the two-weight Muchenhoupt condition
holds, and a testing condition holds : \begin{align*} \sup_{Q : cubes \ in
\mathbb R^n} \frac{1}{\sigma(Q)} \int_{\mathbb R^n} \iint_{\widehat{Q}}
\Big(\frac{t}{t+|x-y|}\Big)^{n\lambda}|\nabla P_t(\mathbf{1}_Q \sigma)(y,t)|^2
\frac{w dx dt}{t^{n-1}} dy < \infty, \end{align*} where is the
Carleson box over and is a pair of weights. We actually prove
this characterization for -function associated with more
general fractional Poisson kernel . Moreover, the corresponding results for
intrinsic -function are also presented.Comment: 21 pages, to appear in Journal of Geometric Analysi
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