The equivalence theorem for logarithmic interpolation spaces in the quasi-Banach case

We study the description by means of the J-functional of logarithmic interpolation spaces (A0, A1) 1, q, A in the category of the p-normed quasi-Banach couples (0 < p ≤ 1). When (A0, A1) is a Banach couple, it is known that the description changes depending on the relationship between q and A. In our more general setting, the parameter p also has an important role as the results show

On compactness theorems for logarithmic interpolation methods

Let (A0;A1) be a Banach couple, (B0;B1) a quasi-Banach couple, 0 B0 is bounded and T : A1 -> B1 is compact, then the interpolated operator by the logarithmic method T : (A0,A1)1;q;A -> (B0;B1)1;q;A is compact too. This result allows the extension of some limit variants of Krasnosel'skii's compact interpolation theorem

Logarithmic interpolation methods and measure of non-compactness

We derive interpolation formulae for the measure of non-compactness of operators interpolated by logarithmic methods with [θ] = 0; 1 between quasi-Banach spaces. Applications are given to operators between Lorentz-Zygmund spaces

Traces of some weighted function spaces and related non‐standard real interpolation of Besov spaces

We study traces of weighted Triebel–Lizorkin spaces F p , q s ( R n , w ) $F^s_{p,q}(\mathbb {R}^n,w)$ on hyperplanes R n − k $\mathbb {R}^{n-k}$ , where the weight is of Muckenhoupt type. We concentrate on the example weight w α ( x ) = | x n | α $w_\alpha (x) = {\big\vert x_n\big\vert }^\alpha$ when | x n | ≤ 1 $\big\vert x_n\big\vert \le 1$ , x ∈ R n $x\in \mathbb {R}^n$ , and w α ( x ) = 1 $w_\alpha (x)=1$ otherwise, where α > − 1 $\alpha >-1$ . Here we use some refined atomic decomposition argument as well as an appropriate wavelet representation in corresponding (unweighted) Besov spaces. The second main outcome is the description of the real interpolation space ( B p 1 , p 1 s 1 ( R n − k ) , B p 2 , p 2 s 2 ( R n − k ) ) θ , r $\big (B^{s_1}_{p_1,p_1}\big (\mathbb {R}^{n-k}\big ), B^{s_2}_{p_2,p_2}{\big (\mathbb {R}^{n-k}\big )\big )}_{\theta ,r}$ , 0 0$sufficiently large, 0 < θ < 1$0<\theta <1$, 0 < r ≤ ∞$

Function spaces of Lorentz-Sobolev type: Atomic decompositions, characterizations in terms of wavelets, interpolation and multiplications

We establish atomic decompositions and characterizations in terms of wavelets for Besov-Lorentz spaces Bsq Lp,r (Rn) and for Triebel-Lizorkin-Lorentz spaces Fsq Lp,r (Rn) in the whole range of parameters. As application we obtain new interpolation formulae between spaces of Lorentz-Sobolev type. We also remove the restrictions on the parameters in a result of Peetre on optimal embeddings of Besov spaces. Moreover, we derive results on diffeomorphisms, extension operators and multipliers for Bsq Lp,∞ (Rn). Finally, we describe Bsq Lp,r (Rn) as an approximation space, which allows us to show new sufficient conditions on parameters for Bsq Lp,r (Rn) to be a multiplication algebra

Interpolation of the measure of non-compactness of bilinear operators among quasi-Banach spaces

Working in the setting of quasi-Banach couples, we establish a formula for the measure of non-compactness of bilinear operators interpolated by the general real method. The result applies to the real method and to the real method with a function parameter

Associate spaces of logarithmic interpolation spaces and generalized Lorentz-Zygmund spaces

We determine the associate space of the logarithmic interpolation space (X0, X1)1,q,A where X0 and X1 are Banach function spaces over a σ-finite measure space (Ω, µ). Particularizing the results for the case of the couple (L1, L∞) over a non-atomic measure space, we recover results of Opic and Pick on associate spaces of generalized Lorentz-Zygmund spaces L(∞,q;A). We also establish the corresponding results for sequence spaces

On function spaces of Lorentz–Sobolev type

We work with Triebel–Lizorkin spaces FsqLp,r(Rn) and Besov spaces BsqLp,r(Rn) with Lorentz smoothness. Using their characterizations by real interpolation we show how to transfer a number of properties of the usual Triebel–Lizorkin and Besov spaces to the spaces with Lorentz smoothness. In particular, we give results on diffeomorphisms, extension operators, multipliers and we also show sufficient conditions on parameters for FsqLp,r(Rn) and BsqLp,r(Rn) to be multiplication algebras