In their 1976 paper, Nagel and Rudin characterize the closed unitarily and Möbius invariant spaces of continuous and $L^p$-functions on a sphere, for $1\leq p<\infty$. In this paper we provide an analogous characterization for the weak*-closed unitarily and Möbius invariant spaces of $L^\infty$-functions on a sphere. We also investigate the weak*-closed unitarily and Möbius invariant algebras of $L^\infty$-functions on a sphere.
In this paper we characterize spaces of continuous and $L^p$-functions on a compact Hausdorff space that are invariant under a transitive and continuous group action. This work generalizes Nagel and Rudin's 1976 results concerning unitarily and Möbius invariant spaces of continuous and measurable functions defined on the unit sphere in $\mathbb{C}^n$.
We establish functional analytic properties of the Stokes operator with bounded measurable coefficients on $L^p_σ (\mathbb{R}^d)$, $d \geq 2$, for $\lvert 1 / p - 1 / 2 \rvert < 1 / d$. These include optimal resolvent bounds and the property of maximal $L^q$-regularity. We further give regularity estimates on the gradient of the solution to the Stokes resolvent problem with bounded measurable coefficients. As a key to these results we establish the validity of a non-local Caccioppoli inequality to solutions of the Stokes resolvent problem.
In this paper we characterize spaces of $L^\infty$-functions on a compact Hausdorff space that are invariant under a transitive and continuous group action. This work generalizes the author's 2021 results concerning the specific case of unitarily and Möbius invariant spaces of $L^\infty$-functions defined on the unit sphere in $\mathbb{C}^n$.
We investigate off-diagonal decay properties of the generalized Stokes semigroup with bounded measurable coefficients on $\mathrm{L}^2_σ (\mathbb{R}^d)$. Such estimates are well-known for elliptic equations in the form of pointwise heat kernel bounds and for elliptic systems in the form of integrated off-diagonal estimates. On our way to unveil this off-diagonal behavior we prove resolvent estimates in Morrey spaces $\mathrm{L}^{2 , ν} (\mathbb{R}^d)$ with $0 \leq ν< 2$.
For a measurable space ($X,\mathcal{A}$), let $\mathcal{M}(X,\mathcal{A})$ be the corresponding ring of all real valued measurable functions and let $μ$ be a measure on ($X,\mathcal{A}$). In this paper, we generalize the so-called $m_μ$ and $U_μ$ topologies on $\mathcal{M}(X,\mathcal{A})$ via an ideal $I$ in the ring $\mathcal{M}(X,\mathcal{A})$. The generalized versions will be referred to as the $m_{μ_{I}}$ and $U_{μ_{I}}$ topology, respectively, throughout the paper. $L_{I}^{\infty} \left(μ\right)$ stands for the subring of $\mathcal{M}(X,\mathcal{A})$ consisting of all functions that are e…
Multiplication and composition operators between two Orlicz spaces
本文考虑复合算子$C_φ generated by nonsingular measurable transformation $ T $ and multiplication operator $ M_u $ generated by measurable function $ u $ between two different Orlicz spaces, then we investigate boundedness, compactness and essential norm of multiplication and composition operators in term of properties of the mapping $ φ $, the function $ u $ and the measure space $（X，X，μ）$.
We study real-valued valuations on the space of Lipschitz functions over the Euclidean unit sphere $S^{n-1}$. After introducing an appropriate notion of convergence, we show that continuous valuations are bounded on sets which are bounded with respect to the Lipschitz norm. This fact, in combination with measure theoretical arguments, will yield an integral representation for continuous and rotation invariant valuations on the space of Lipschitz functions over the 1-dimensional sphere.
