Poisson area and non-tangential maximal functions on manifolds with ends and applications

Abstract

Let $M=\mathbb R^m\sharp(\mathbb R^n\times\mathbb S^{m-n})$, where $m>n\geq3$. The full-cone Poisson area and non-tangential maximal functions fail to be of weak type $(p,p)$ for $1<p<2$. Let $K$ be the compact part joining the two ends and put $ρ(x)=1+d(x,K)$. For fixed small $c>0$, we restrict the horizontal distance from the vertex to $cρ(x)$ and retain every positive time. The resulting maximal function is of weak type $(1,1)$ and bounded on $L^p$ for $1<p\leq\infty$. The corresponding area function uses the full spatial and time gradient. It satisfies a Fefferman--Stein distribution inequality. Its $L^p$ norm is equivalent to that of $f$ for $1<p\leq2$. This area function defines a global Hardy space. We prove its $Δ$-atomic description, its injective embedding into $L^1$, its characterization by the restricted maximal function, and its real interpolation identity with $L^2$. Direct estimates for spectral-band kernels then give a Mihlin multiplier theorem from this Hardy space to $L^1$. Interpolation and duality give $L^p$ boundedness for every $1<p<\infty$.