On estimate of operator for $0<p<\infty $

Abstract: Operators such as Carleson operator are known to be bounded on $L^p$ for all $1<p<\infty$, but not from $L^1$ to weak-$L^1$ and from $H^p$ to $L^p$ for each $0<p\leq 1$, the object of this article is to give a estimate for all $0<p<\infty$. For the weights $w$ satisfying the doubling condition of order $q$ with $0<q<p$ and the reverse H\"{o}lder condition, by using some new functions spaces, we prove that: $\bullet$ some sublinear operators are bounded from some subspaces of $L^p_w$ to $L^p_w$ and to themselves for all $0<p< \infty$; in particular, these imply the endpoint estimates from $H^p_w$ to $L^p_w$ and from $H^p_w$ to itself for all $0<p\leq 1$; these results are applied to many operators, such as Hardy-Littlewood maximal operator, singular integral operators with rough kernels, Calder\'{o}n commutators, Carleson operator, the polynomial Carleson operator, et al, and give the endpoint versions of classical theorems such as Carleson-Hunt theorem and a conjecture of Stein; $\bullet$ $H^p_w$ with $0<p\leq 1$ is characterized by blocks without vanishing moment conditions; $\bullet$ $H^p_w$ with $0<p\leq 1$ is characterized by a convolution maximal function with a non-smooth kernel.