We give a natural definition of the Morrey spaces for Radon measures which may be non-doubling but satisfy the growth condition. In these spaces we investigate the behavior of the maximal operator, the fractional integral operator, the singular integral operator and their vector-valued extensions.

Let A be a bounded self-adjoint operator on a separable Hilbert space H and H0 ⊂ H a closed invariant subspace of A. Assuming that H0 is of codimension 1, we study the variation of the invariant subspace H0 under bounded self-adjoint perturbations V of A that are off-diagonal with respect to the decomposition H = H0 ⊕ H1. In particular, we prove the existence of a one-parameter family of dense ...

We give partial answers to the following conjecture: the natural embedding of a rearrangement invariant space E into L1([0, 1]) is strictly singular if and only if G does not embed into E continuously, where G is the closure of the simple functions in the Orlicz space LΦ with Φ(x) = exp(x 2) − 1. In this paper we ask the following question. Given a rearrangement invariant space E on [0, 1], whe...

where T = M-'N. In this article we consider solutions of (1.1) where A is a singular operator and f is such that a solution exists, i.e., consistent singular systems. Singular systems of linear equations appear in many applications, such as when computing the stationary probability distributions of Markov chains [9] or Markov processes [2 ] . These particular applications appear in the analysis...

A family of selfadjoint operators of the Friedrichs model is considered. These symmetric type operators have one singular point, zero of order m. For every m > 3/2, we construct a rank 1 perturbation from the class Lip1 such that the corresponding operator has a sequence of eigenvalues converging to zero. Thus, near the singular point, there is no singular spectrum finiteness condition in terms...