It is shown for the non-centered Hardy–Littlewood maximal operator M that ∥DMf∥1≤Cn∥Df∥1 for all radial functions in W1,1(Rn).

The partial entanglement entropy (PEE) $s_{\mathcal{A}}(\mathcal{A}_i)$ characterizes how much the subset $\mathcal{A}_i$ of $\mathcal{A}$ contribute to the entanglement entropy $S_{\mathcal{A}}$. We find one additional physical requirement for $s_{\mathcal{A}}(\mathcal{A}_i)$, which is the invariance under a permutation. The partial entanglement entropy proposal satisfies all the physical requirements. We show that for Poincar\'e invariant theories the physical requirements are enough to uniquely determine the PEE (or the entanglement contour) to satisfy a general formula. This is the first time we find the PEE can be uniquely determined. Since the solution of the requirements is unique and the \textit{PEE proposal} is a solution, the \textit{PEE proposal} is justified for Poincar\'e invariant theories.

In this paper, we consider the Cauchy problem to the planar non-resistive magnetohydrodynamic equations without heat conductivity, and establish the global well-posedness of strong solutions with large initial data. The key ingredient of the proof is to establish the a priori estimates on the effective viscous flux and a newly introduced ``transverse effective viscous flux" vector field inducted by the transverse magnetic field. The initial density is assumed only to be uniformly bounded and of finite mass and, in particular, the vacuum and discontinuities of the density are allowed.

We study property ($T$) and the fixed-point property for actions on$L$^{$p$}and other Banach spaces. We show that property ($T$) holds when$L$^{2}is replaced by$L$^{$p$}(and even a subspace/quotient of$L$^{$p$}), and that in fact it is independent of 1≤$p$<∞. We show that the fixed-point property for$L$^{$p$}follows from property ($T$) when 1<$p$< 2+$ε$. For simple Lie groups and their lattices, we prove that the fixed-point property for$L$^{$p$}holds for any 1<$p$<∞ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive spaces.

A similar formula to the one established by Ansemil and Floret for symmetric tensor products of direct sums is proved for alternating and Jacobian tensor products. It is then applied to stable spaces where a number of isomorphisms between spaces of tensors or multilinear forms are unveiled. A connection between these problems and irreducible group representations is made.