A characterization of two weight norm inequality for Littlewood-Paley gλ∗g_{\lambda}^{*}-function

Let $n\ge 2$ and $g_{\lambda}^{*}$ 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 $P_tf(y,t)=p_t*f(y)$, $p_t(y)=t^{-n}p(y/t)$ and $p(x) = (1+|x|^2)^{-(n+1)/2}$, $\nabla =(\frac{\partial}{\partial y_1},\ldots,\frac{\partial}{\partial y_n},\frac{\partial}{\partial t})$. In this paper, we give a characterization of two-weight norm inequality for $g_{\lambda}^{*}$-function. We show that, $\big\| g_{\lambda}^{*}(f \sigma) \big\|_{L^2(w)} \lesssim \big\| f \big\|_{L^2(\sigma)}$ if and only if the two-weight Muchenhoupt $A_2$ 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 $\widehat{Q}$ is the Carleson box over $Q$ and $(w, \sigma)$ is a pair of weights. We actually prove this characterization for $g_{\lambda}^{*}$-function associated with more general fractional Poisson kernel $p^\alpha(x) = (1+|x|^2)^{-{(n+\alpha)}/{2}}$. Moreover, the corresponding results for intrinsic $g_{\lambda}^*$-function are also presented.Comment: 21 pages, to appear in Journal of Geometric Analysi

The CMO\mathrm{CMO}-Dirichlet problem for elliptic systems in the upper half-space

We prove that for any second-order, homogeneous, $N \times N$ elliptic system $L$ with constant complex coefficients in $\mathbb{R}^n$, the Dirichlet problem in $\mathbb{R}^n_+$ with boundary data in $\mathrm{CMO}(\mathbb{R}^{n-1}, \mathbb{C}^N)$ is well-posed under the assumption that $d\mu(x', t) := |\nabla u(x)|^2\, t \, dx' dt$ is a strong vanishing Carleson measure in $\mathbb{R}^n_+$ in some sense. This solves an open question posed by Martell et al. The proof relies on a quantitative Fatou-type theorem, which not only guarantees the existence of the pointwise nontangential boundary trace for smooth null-solutions satisfying a strong vanishing Carleson measure condition, but also includes a Poisson integral representation formula of solutions along with a characterization of $\mathrm{CMO}(\mathbb{R}^{n-1}, \mathbb{C}^N)$ in terms of the traces of solutions of elliptic systems. Moreover, we are able to establish the well-posedness of the Dirichlet problem in $\mathbb{R}^n_+$ for a system $L$ as above in the case when the boundary data belongs to $\mathrm{XMO}(\mathbb{R}^{n-1}, \mathbb{C}^N)$, which lines in between $\mathrm{CMO}(\mathbb{R}^{n-1}, \mathbb{C}^N)$ and $\mathrm{VMO}(\mathbb{R}^{n-1}, \mathbb{C}^N)$. Analogously, we formulate a new brand of strong Carleson measure conditions and a characterization of $\mathrm{XMO}(\mathbb{R}^{n-1}, \mathbb{C}^N)$ in terms of the traces of solutions of elliptic systems