We verify a conjecture on Triebel-Lizorkin-Lorentz spaces F_q^s L_p,r(ℝ^n) proving that they are multiplication algebras if s> n/p . We also prove the corresponding conjecture on Besov-Lorentz spaces B_q^s L_p,r(ℝ^n) . Moreover, we show that in a certain range of parameters the condition s>n/p is necessary for these spaces to be multiplication algebras.

We investigate the interpolation properties of compact multilinear operators by the real method between quasi-Banach spaces. As an application we establish a reinforced version of a multilinear Marcinkiewicz theorem.

We work with Besov spaces with Lorentz smoothness B^s _q L_p,r (ℝ^n ) . Here -∞

Let \mathcal{I} be a surjective closed operator ideal with associated outer measure \gamma_{{\mathcal{I}}} , and let \gamma_{{\mathfrak{I}}} be the extension of \gamma_{{\mathcal{I}}} to bilinear operators. We establish a convexity inequality for the measure \gamma_{{\mathfrak{I}}} of a bilinear operator interpolated by the real method. Our arguments do not require that \mathcal{I} be symmetric.

We describe a procedure for extending the inner measure β __ℐ associated to an operator ideal ℐ to a measure β __𝔍 for bounded bilinear operators T. When ℐ is injective and closed, we show that β __𝔍(T)=0 if and only if T=RS for some bounded bilinear operator S and R∈ℐ . If ℐ satisfies the Σ _r -condition, then we establish a convexity inequality for the measure β __𝔍 of a bilinear operator interpolated by the real method.

We extend the (outer) measure γ_I associated to an operator ideal I to a measure γ_J for bounded bilinear operators. If I is surjective and closed, and J is the class of those bilinear operators such that γ_J(T)=0 , we prove that J coincides with the composition bideal I∘B . If I satisfies the Σr-condition, we establish a simple necessary and sufficient condition for an interpolated operator by the real method to belong to J . Furthermore, if in addition I is symmetric, we prove a formula for the measure γ_J of an operator interpolated by the real method. In particular, results apply to weakly compact operators.

We give estimates for the measure of non-compactness of an operator interpolated by the limiting methods involving slowly varying functions. As applications we establish estimates for the measure of non-compactness of operators acting between Lorentz–Karamata spaces.

We prove that, under a mild condition on a couple (A(0), A(1)) of quasi-Banach spaces, all real interpolation spaces (A(0), A(1))(theta,p) with 0 < theta < 1 and 0 < p < infinity are different from each other. In the Banach case and for 1 <= p <= infinity this was shown by Janson, Nilsson, Peetre and Zafran, thus solving an old problem posed by J. -L. Lions. Moreover, we give an application to certain spaces which are important objects in Operator Theory and which consist of bounded linear operators whose approximation numbers belong to Lorentz sequence spaces. (C) 2022 The Author(s). Published by Elsevier Inc.

We prove the following dichotomy for the spaces ℒ ,, () (X, Y) of all operators T ∈ ℒ(X, Y) whose approximation numbers belong to the Lorentz-Zygmund sequence spaces ℓp,q(log ℓ)α: If X and Y are infinite-dimensional Banach spaces, then the spaces ℒ ,, () (X, Y) with 0 < p < ∞, 0 < q ≤ ∞ and α ∈ ℝ are all different from each other, but otherwise, if X or Y are finite-dimensional, they are all equal (to ℒ(X, Y)). Moreover we show that the scale { L_∞ ,q^(a)(X,Y)} _0 < q < ∞ is strictly increasing in q, where ℒ ∈, () (X, Y) is the space of all operators in ℒ(X, Y) whose approximation numbers are in the limiting Lorentz sequence space ∓∈,q.

We establish atomic decompositions and characterizations in terms of wavelets for Besov-Lorentz spaces BqsLp,r(Rn) and for Triebel-Lizorkin-Lorentz spaces FqsLp,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 BqsLp,∞(Rn). Finally, we describe BqsLp,r(Rn) as an approximation space, which allows us to show new sufficient conditions on parameters for BqsLp,r(Rn) to be a multiplication algebra.

We show a necessary and sufficient condition for weak compactness of bilinear operators interpolated by the real method. This characterization does not hold for interpolated operators by the complex method.

We study the interpolation properties of weakly compact bilinear operators by the real method and also by the complex method. We also study the factorization property of weakly compact bilinear operators through reflexive Banach spaces.

We establish compactness interpolation results for bilinear operators of convolution type and for operators of product type among quasi-Banach spaces. We do not assume any auxiliary condition on the spaces.

We establish a compactness interpolation result for bilinear operators of the type proved by Janson for bounded bilinear operators. We also give an application to compactness of convolution operators. (C) 2020 Elsevier Inc. All rights reserved.

. We study the description by means of the J -functional of logarithmic interpolation spaces ( A 0 ; A 1 ) 1;q;A in the category of the p -normed quasi-Banach couples (0 < p (cid:20) 1). When ( A 0 ; A 1 ) 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.

We work with Triebel–Lizorkin spaces $$F_{q}^{s}L_{p,r}({\mathbb {R}}^{n})$$ and Besov spaces $$B_{q}^{s} L_{p,r}({\mathbb {R}}^{n})$$ 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 $$F_{q}^{s}L_{p,r}({\mathbb {R}}^{n})$$ and $$B_{q}^{s}L_{p,r}({\mathbb {R}}^{n})$$ to be multiplication algebras.

We complete the range of the parameters in the interpolation formula established by Mastyło and Silva for the measure of non-compactness of a bilinear operator interpolated by the real method.

Let $$d\in {\mathbb {N}}$$ and let $$\Omega $$ be a bounded Lipschitz domain in $${\mathbb {R}}^d$$. We prove that the embedding $$I_d{:}B^d _{p,q}(\Omega ) \longrightarrow L_p (\log L)_a (\Omega )$$ is nuclear if $$a<-1$$ and $$1\le p,q\le \infty $$, while if $$-1<a<0$$, $$2<p<\infty $$ and $$p\le q \le \infty $$ the embedding $$I_d$$ fails to be nuclear. Furthermore, if $$a=-1$$, the embedding $$I_d{:}B^d _{\infty ,\infty }(\Omega ) \longrightarrow L_\infty (\log L)_{-1} (\Omega )$$ is not nuclear.

We investigate the interpolation spaces $$\left( A_{0}, A_{1}\right) _{1,\infty , (0, \alpha _{\infty })}$$ formed by all $$ a \in A_{0}+A_{1}$$, having a finite norm: $$\begin{aligned} {}\left\Vert a \right\Vert _{\left( A_{0}, A_{1}\right) _{1,\infty , (0, \alpha _{\infty })}} = \max \left\{ \sup _{0< t< 1} \frac{K(t,a)}{t}, \sup _{1< t < \infty } \frac{(1+\log t)^{\alpha _{\infty }}K(t,a)}{t}\right\} , \end{aligned}$$where K(t, a) is the K-functional. We show that they have a description in terms of the J-functional which is of another nature than the description of the other logarithmic interpolation spaces. We also determine the associate space of $$\left( E_{0}, E_{1}\right) _{1,\infty , (0, \alpha _{\infty })}$$ when $$E_{0}$$ and $$E_{1}$$ are Banach function spaces, and the dual space of $$(A_0,A_1)_{1,\infty , (0, \alpha _\infty )}^{\circ }$$.

We derive interpolation formulae for the measure of non-compactness of operators interpolated by logarithmic methods with $\theta = 0,1$ between quasi-Banach spaces. Applications are given to operators between Lorentz–Zygmund spaces.