This paper deals with Besov spaces of logarithmic smoothness $B_{p,r}^{0,b}$ formed by periodic functions. We study embeddings of $B_{p,r}^{0,b}$ into Lorentz-Zygmund spaces $L_{p,q}(log L)_{β}$. Our techniques rely on the approximation structure of $B_{p,r}^{0,b}$, Nikol'skiĭ type inequalities, extrapolation properties of $L_{p,q}(log L)_{β}$ and interpolation.
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We prove a reiteration theorem for interpolationmethods defined by means of polygons, and a Wolff theorem for the case when the polygon has 3 or 4 vertices. In particular, we establish a Wolff theorem for Fernandez' spaces, which settles a problem left over in [5].
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We review several results on interpolation of Banach algebras and factorization of weakly compact homomorphisms. We also establish a new result on interpolation of multilinear operators.
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We investigate the behaviour of bilinear operators under limiting real methods. As an application, we show an interpolation formula for spaces of linear operators. Some results on norm estimates for bounded linear operators are also established.
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We study limiting K- and J-methods for arbitrary Banach couples. They are related by duality and they extend the methods already known in the ordered case. We investigate the behaviour of compact operators and we also discuss the representation of the methods by means of the corresponding dual functional. Finally, some examples of limiting function spaces are given.
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The paper establishes necessary and sufficient conditions for compactness of operators acting between general K-spaces, general J-spaces and operators acting from a J-space into a K-space. Applications to interpolation of compact operators are also given.
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We investigate compact operators between approximation spaces, paying special attention to the limit case. Applications are given to embeddings between Besov spaces.
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We characterize all the extreme points of the unit ball in the space of trilinear forms on the Hilbert space $ℂ^2$. This answers a question posed by R. Grząślewicz and K. John [7], who solved the corresponding problem for the real Hilbert space $ℝ^2$. As an application we determine the best constant in the inequality between the Hilbert-Schmidt norm and the norm of trilinear forms.
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We investigate the behaviour of the measure of non-compactness of an operator under real interpolation. Our results refer to general Banach couples. An application to the essential spectral radius of interpolated operators is also given.
We determine the exact dependence on \(\theta,q,p\) of the constants in the equivalence theorem for the real interpolation method \((A_0,A_1)_{\theta,q}\) with pairs of \(p\)-normed spaces.
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