Estimates of Weighted Hardy Operators and Commutators on Total Morrey Spaces

We determine the exact operator norms of the weighted Hardy operator $U_{\psi }$ and the weighted Cesàro operator $V_{\psi }$ on Total Morrey spaces $L_{p,\lambda ,\mu }\left({\mathbb{R}}^n\right)$. We also study their commutators with symbols $b\in \mathrm{BMO}\left({\mathbb{R}}^n\right)$ satisfying the additional dilation condition ${\sup }_{0<t<1}{\displaystyle \frac{{\parallel b-b(t·)\parallel }_{L^{\infty }\left({\mathbb{R}}^n\right)}}{\log (2/t)}}<\infty .$ Assuming in addition that the corresponding underlying weighted Hardy or Cesàro operator is bounded, we obtain necessary and sufficient integral conditions on the weight function ensuring the boundedness of the first-order and higher-order commutators for every symbol satisfying this dilation condition. The results reflect the two-scale structure of Total Morrey spaces: the weighted Hardy operator is governed by the large-scale parameter $\mu$, whereas the weighted Cesàro operator is governed by the small-scale parameter $\lambda$. The exact operator norm formulas extend their classical Morrey-space counterparts, while the commutator criteria are obtained for $\mathrm{BMO}$ symbols satisfying the stated additional dilation condition.