Estimates at or beyond endpoint in harmonic analysis: Bochner-Riesz means and spherical means

We introduce some new functions spaces to investigate some problems at or beyond endpoint. First, we prove that Bochner-Riesz means $B_R^\lambda$ are bounded from some subspaces of $L^p_{|x|^\alpha}$ to $L^p_{|x|^\alpha}$ for $ \frac{n-1}{2(n+1)}<\lambda \leq \frac{n-1}{2}, 0 < p\leq p'_\lambda=\frac{2n}{n+1+2\lambda}, n(\frac{p}{p_\lambda}-1)< \alpha<n(\frac{p}{p'_\lambda}-1)$, and $0<R<\infty,$ and so are the maximal Bochner-Riesz means $B_*^\lambda$ for $ \frac{n-1}{2}\leq \lambda < \infty, 0 < p\leq 1$ and $-n< \alpha<n(p-1)$. From these we obtain the $L^p_{|x|^\alpha}$-norm convergent property of $B_R^\lambda $ for these $\lambda,p,$ and $\alpha$. Second, let $n\geq 3,$ we prove that the maximal spherical means are bounded from some subspaces of $L^p_{|x|^\alpha}$ to $L^p_{|x|^\alpha}$ for $0<p\leq \frac{n}{n-1}$ and $ -n(1-\frac{p}{2})<\alpha<n(p-1)-n$. We also obtain a $L^p_{|x|^\alpha}$-norm convergent property of the spherical means for such $p$ and $\alpha$. Finally, we prove that some new types of $|x|^\alpha$-weighted estimates hold at or beyond endpoint for many operators, such as Hardy-Littlewood maximal operator, some maximal and truncated singular integral operators, the maximal Carleson operator, etc. The new estimates can be regarded as some substitutes for the $(H^p,H^p)$ and $(H^p,L^p)$ estimates for the operators which fail to be of types $(H^p,H^p)$ and $(H^p,L^p)$.