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No abstract uploaded!

We prove that if$T$is a strictly singular one-to-one operator defined on an infinite dimensional Banach space$X$, then for every infinite dimensional subspace$Y$of$X$there exists an infinite dimensional subspace$Z$of$X$such that$Z∩Y$is infinite dimensional,$Z$contains orbits of$T$of every finite length and the restriction of$T$to$Z$is a compact operator.

Let ($A$_{$0, A$}_{$1$}) be a compatible pair of quasi-Banach spaces and 1et$A$be a corresponding space of real interpolation type such that$A$_{$0$}∩$A$_{$1$}is not dense in$A$. Upper and lower estimates are obtained for the distance of any element$f$of$A$from$A$_{$0$}∩$A$_{$1$}. These lead to formulae for the distance in a large number of concrete situations, such as when$A$_{$0$}∩$A$_{$1$}=$L$^{$∞$}and$A$is either weak-$L$^{q}, a ‘grand’ Lebesgue space or an Orlicz space of exponential type.

In this note, we consider the regularity of solutions of the nonlinear elliptic systems of$n$-Laplacian type involving measures, and prove that the gradients of the solutions are in the weak Lebesgue space$L$^{$n$,∞}. We also obtain the$a priori$global and local estimates for the$L$^{$n$,∞}-norm of the gradients of the solutions without using BMO-estimates. The proofs are based on a new lemma on the higher integrability of functions.