  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.
