We prove several new results on the absolutely continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely continuous spectrum. Then we establish stability of the absolutely continuous spectrum in more general situations, where imbedded singular spectrum may occur. We present two kinds of optimal conditions for the stability of absolutely continuous spectrum: decay and smoothness. In the decay direction, we show that a sufficient (in the power scale) condition is |$q$($x$)|≤$C$(1+|$x$|)^{−1/4−ε}; in the smoothness direction, a sufficient condition in Hölder classes is$q$∈$C$^{1/2+ε}($R$). On the other hand, we show that there exist potentials which both satisfy |$q$($x$)|≤$C$(1+|$x$|)^{−1/4}and belong to$C$^{1/2}($R$) for which the spectrum becomes purely singular on the whole real axis, so that the above results are optimal within the scales considered.

We study the Möbius invariant spaces$Q$_{$p$}and$Q$_{$p, 0$}of analytic functions. These scales of spaces include BMOA=$Q$_{1}, VMOA=$Q$_{1, 0}and the Dirichlet space=$Q$_{0}. Using the Bergman metric, we establish decomposition theorems for these spaces. We obtain also a fractional derivative characterization for both$Q$_{$p$}and$Q$_{$p, 0$}.

A Sobolev space in several variables in an$L$^{1}-type norm is not complemented in its second dual. Hence it is not isomorphic as a Banach space to any complemented subspace of a Banach lattice.

We give a detailed description of nonconstant harmonic functions and their level sets on the Sierpiński gasket. We introduce a parameter, called$eccentricity$, which classifies these functions up to affine transformations$h→ah+b$. We describe three (presumably) distinct measures that describe how the eccentricities are distributed in the limit as we subdivide the gasket into smaller copies (cells) and restrict the harmonic function to the small cells. One measure simply counts the number of small cells with eccentricity in a specified range. One counts the contribution to the total energy coming from those cells. And one counts just those cells that intersect a fixed generic level set. The last measure yields a formula for the box dimension of a generic level set. All three measures are defined by invariance equations with respect to the same iterated function system, but with different weights. We also give a construction for a rectifiable curve containing a given level set. We exhibit examples where the curve has infinite winding number with respect to some points.

Let$r, s$∈ [0, 1], and let$X$be a Banach space satisfying the$M(r, s)$-inequality, that is, $$\parallel x^{***} \parallel \geqslant r\parallel \pi _X x^{***} \parallel + s\parallel x^{***} - \pi _X x^{***} \parallel for x^{***} \in X^{***} ,$$ where π_{$X$}is the canonical projection from$X$^{***}onto$X$^{*}. We show some examples of Banach spaces not containing$c$_{0}, having the point of continuity property and satisfying the above inequality for$r$not necessarily equal to one. On the other hand, we prove that a Banach space$X$satisfying the above inequality for$s$=1 admits an equivalent locally uniformly rotund norm whose dual norm is also locally uniformly rotund. If, in addition,$X$satisfies $$\mathop {\lim \sup }\limits_\alpha \parallel u^* + sx_\alpha ^* \parallel \leqslant \mathop {\lim \sup }\limits_\alpha \parallel v^* + x_\alpha ^* \parallel $$ whenever$u$^{*},$v$^{*}∈$X$^{*}with ‖$u$^{*}‖≤‖$v$^{*}‖ and ($x$_{α}^{*}) is a bounded weak^{*}null net in$X$^{*}, then$X$can be renormed to satisfy the,$M(r, 1)$and the$M(1, s)$-inequality such that$X$^{*}has the weak^{*}asymptotic-norming property I with respect to$B$_{$X$}.