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Let$f=g$_{t}+$h$_{t}be the optimal decomposition for calculating the exact value of the$K$-functional$K(t, f$; $$\bar X$$ ) of an element$f$with respect to a couple $$\bar X$$ =($X$_{0},$X$_{1}) of Banach lattices of measurable functions. It is shown that this decomposition has a rather simple form in many cases where one of the spaces$X$_{0}and$X$_{1}is either$L$^{∞}or$L$^{1}. Many examples are given of couples of lattices $$\bar X$$ for which |$g$_{t}| increases monotonically a.e. with respect to$t$. It is shown that this property implies a sharpened estimate from above for the Brudnyi-Krugljak$K$-divisibility constant γ( $$\bar X$$ ) for the couple. But it is also shown that certain couples $$\bar X$$ do not have this property. These also provide examples of couples of lattices for which γ( $$\bar X$$ ).

Let $A=-(\nabla-i\vec{a})^2+V$ be a magnetic Schrödinger operator acting on$L$^{2}($R$^{$n$}),$n$≥1, where $\vec{a}=(a_1,\cdots,a_n)\in L^2_{\rm loc}$ and 0≤$V$∈$L$^{1}_{loc}. Following [1], we define, by means of the area integral function, a Hardy space$H$^{1}_{$A$}associated with$A$. We show that Riesz transforms (∂/∂$x$_{$k$}-$i$$a$_{$k$})$A$^{-1/2}associated with$A$, $k=1,\cdots,n$ , are bounded from the Hardy space$H$^{1}_{$A$}into$L$^{1}. By interpolation, the Riesz transforms are bounded on$L$^{$p$}for all 1<$p$≤2.

We study spectral and scattering properties of the Laplacian$H$^{(σ)}=-Δ in $L_2(\mathbf{R}^{d+1}_+)$ corresponding to the boundary condition $\frac{\partial u}{\partial\nu} + \sigma u = 0$ with a periodic function σ. For non-negative σ we prove that$H$^{(σ)}is unitarily equivalent to the Neumann Laplacian$H$^{(0)}. In general, there appear additional channels of scattering due to surface states. We prove absolute continuity of the spectrum of$H$^{(σ)}under mild assumptions on σ.