In 1993, N. Danikas and A. G. Siskakis showed that the Cesàro operator ${\mathcal{C}}$ is not bounded on  $H^{\infty }$ ; that is, ${\mathcal{C}}(H^{\infty })\nsubseteq H^{\infty }$ , but ${\mathcal{C}}(H^{\infty })$ is a subset of $BMOA$ . In 1997, M. Essén and J. Xiao gave that ${\mathcal{C}}(H^{\infty })\subsetneq {\mathcal{Q}}_{p}$ for every $0<p<1$ . In this paper, we characterize positive Borel measures $\unicode[STIX]{x1D707}$ such that ${\mathcal{C}}(H^{\infty })\subseteq M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ and show that ${\mathcal{C}}(H^{\infty })\subsetneq M({\mathcal{D}}_{\unicode[STIX]{x1D707}_{0}})\subsetneq \bigcap _{0<p<\infty }{\mathcal{Q}}_{p}$ by constructing some measures  $\unicode[STIX]{x1D707}_{0}$ . Here, $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ denotes the Möbius invariant function space generated by  ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$ , where ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$ is a Dirichlet space with superharmonic weight induced by a positive Borel measure $\unicode[STIX]{x1D707}$ on the open unit disk. Our conclusions improve results mentioned above.

1993年，N。Danikas和A. G. Siskakis证明了Cesàro运算符 $ {\ mathcal {C}} $ 不受  $ H ^ {\ infty} $的限制 ；即 $ {\ mathcal {C}}（H ^ {\ infty}）\ nsubseteq H ^ {\ infty} $ ，但是 $ {\ mathcal {C}}（H ^ {\ infty}）$ 是一个子集的 $ BMOA $ 。1997年，M。Essén和J. Xiao给出了 $ {\ mathcal {C}}（H ^ {\ infty}）\ subsetneq {\ mathcal {Q}} _ {p} $ 每 $ 0 <p <1 $ 。在本文中，我们表征了正的Borel度量 $ \ unicode [STIX] {x1D707} $ ，使得 $ {\ mathcal {C}}（H ^ {\ infty}）\ subseteq M（{\ mathcal {D}} _ { \ unicode [STIX] {x1D707}}）$ 并显示 $ {\ mathcal {C}}（H ^ {\ infty}）\ subsetneq M（{\ mathcal {D}} _ {\ unicode [STIX] {x1D707} _ {0}}）\ subsetneq \ bigcap _ {0 通过构造一些度量  $ \ unicode [STIX] {x1D707} _ {0} $来<p <\ infty} {\ mathcal {Q}} _ {p } $ 。在这里， $ M（{\ mathcal {D}} _ {\ unicode [STIX] {x1D707}}）$ 表示由  $ {\ mathcal {D}} _ {\ unicode [STIX] {x1D707 }} $ ，其中 $ {\ mathcal {D}} _ {\ unicode [STIX] {x1D707}} $是Dirichlet空间，其超谐调 权重由正Borel测度 $ \ unicode [STIX] {x1D707} $引起 。打开单元盘。我们的结论改善了上述结果。