| Abstract: | A measure $\\mu \]$ is called Carleson measure, iff the condition of Carleson type $\\mu ({Q^*}) \le C{\left| Q \right|^\alpha }(\alpha \ge 1)\]$ is satisfied, where C is a constant independent of the cube Q with edge length
$\q > 0\]$ in $\{R^n}\]$ and $\{Q^*} = \{ (y,t) \in R_ + ^{n + 1}|y \in Q,0 < t < q\} \]$. In this paper the following theorem is established:"Suppose that $\\mu \]$ is a Carleson measure,$\\phi (y,t)\]$ is continuous in $\R_ + ^{n + 1}\]$ and $\{\phi ^*}(x) = \mathop {\sup }\limits_{\left| {y - x} \right| < t,(y,t) \in R_ + ^{n + 1}} \left| {\phi (y,t)} \right|\]$. Then the following inequalities hold: (1)$\\mu (\{ \left| {\phi (y,t)} \right| > s\} ) \le C{\left| {\{ {\phi ^*}(x) > s\} } \right|^\alpha }(\forall s > 0)\]$, (2)$\\int_{R_ + ^{n + 1}} {{{\left| {\phi (y,t)} \right|}^\alpha }} \alpha \mu \le C{\left {\int_{{R^n}} {{\phi ^*}(x)dx} } \right]^\alpha }\]$, (3)$\\int_{{Q^n}} {{{\left| {\phi (y,t)} \right|}^\alpha }} d\mu \le C{\left {\int_{3Q} {{\phi ^*}(x)dx} } \right]^\alpha }\]$ where 3Q denotes the cube with the same center as Q but of edge length 3q,
In virtue of this theorem, the proof of three propositions in the paper of C. Feffer-man and E. M. Stein (Acta Math., 1972) is simplified. |
