Related papers: Holmstedt's formula for the $K$-functional: the li…

We consider K-interpolation methods involving slowly varying functions. Let $\overline{A}_{\theta,*}^{\mathcal{L}}$ and $\overline{A}_{\theta,*}^{\mathcal{R}}$ $(0\leq\theta\leq1)$ be the so called ${\mathcal{L}}$ or ${\mathcal{R}}$…

Let $(A_0, A_1)$ be a compatible couple of quasi-normed spaces, and let $\Phi_0$ and $\Phi_1$ be two general parameters of $K$-interpolation method. We compute $K$-functional for the couple $((A_0,A_1)_{\Phi_0}, (A_0, A_1)_{\Phi_1})$ in…

We consider the problem of complex interpolation of certain Hardy-type subspaces of K\"othe function spaces. For example, suppose $X_0$ and $X_1$ are K\"othe function spaces on the unit circle $\bold T,$ and let $H_{X_0}$ and $H_{X_1}$ be…

The paper provides a detailed study of crucial inequalities for smoothness and interpolation characteristics in rearrangement invariant Banach function spaces. We present a unified approach based on Holmstedt formulas to obtain these…

We consider a real interpolation method defined by means of slowly varying functions. We present some reiteration formulae including so called $L$ or $R$ limiting interpolation spaces. These spaces arise naturally in reiteration formulae…

In the paper Description of the $K$-spaces by means of $J$-spaces and the reverse problem, Math. Nachr. 296 (2023), no. 9, 4002--4031, we have establish conditions under which the limiting $K$-space $(X_0,X_1)_{0,q,b;K}$, involving a slowly…

We give a complete characterization of limiting interpolation spa\-ces for the real method of interpolation using extrapolation theory. For this purpose the usual tools (e.g., Boyd indices or the boundedness of Hardy type operators) are not…

Given $E_0, E_1, F_0, F_1, E$ rearrangement invariant function spaces, $a_0$, $a_1$, $b_0$, $b_1$, $b$ slowly varying functions and $0< \theta_0<\theta_1<1$, we characterize the interpolation spaces $$(\overline{X}^{\mathcal…

Our starting point is a lemma due to Varopoulos. We give a different proof of a generalized form this lemma, that yields an equivalent description of the $K$-functional for the interpolation couple $(X_0,X_1)$ where…

We obtain an explicit characterization of the $K$-functional of a pair of weighted classical Lorentz spaces of type $S$. We develop a method for obtaining such characterization based on a relation between the desired quantity and the…

We compute the K-functional related to some couple of spaces as small or classical Lebesgue space or Lorentz-Marcinkiewicz spaces completing the results of the previous works of the authors. This computation allows to determine the…

Let (A_0,A_1) be a compatible couple of Banach spaces in the interpolation theory sense. We give a formula for the K_t-functional of the interpolation couples (l_1(A_0),c_0(A_1)) or (l_1(A_0),l_infinity(A_1)) and (L_1(A_0),L_infinity(A_1)).

We study thin interpolating sequences $\{\lambda_n\}$ and their relationship to interpolation in the Hardy space $H^2$ and the model spaces $K_\Theta = H^2 \ominus \Theta H^2$, where $\Theta$ is an inner function. Our results, phrased in…

We present sharp interpolation theorems, including all limiting cases, for a class of quasilinear operators of joint weak type acting between Lorentz-Karamata spaces over $\sigma$-finite measure. This class contains many of the important…

Let K/F be a quadratic extension of number fields. After developing a theory of the Eisenstein series over F, we prove a formula which expresses a partial zeta function of K as a certain integral of the Eisenstein series. As an application,…

Let $(A_0, A_1)$ be an interpolation couple, and let $B_j$ be the closure of $A_0^\ast \cap A_1^\ast$ in $A_j^\ast$, $j = 0, 1$. For every $\theta \in \, ]0, 1[$, there exists a natural one to one contraction $R^\theta : A^\theta…

Given $E_0, E_1, E, F$ rearrangement invariant spaces, $a, b, b_0, b_1$ slowly varying functions and $0\leq \theta_0<\theta_1\leq 1$, we characterize the interpolation space $$(\overline{X}_{\theta_0,b_0,E_0}, \overline{X}^{\mathcal…

The paper is devoted to the construction of an optimal interpolation formula in $K_2(P_2)$ Hilbert space. Here the interpolation formula consists of a linear combination $\sum_{\beta=0}^NC_{\beta}(z)\varphi(x_\beta)$ of given values of a…

Interpolation inequalities in Triebel-Lizorkin-Lorentz spaces and Besov-Lorentz spaces are studied for both inhomogeneous and homogeneous cases. First we establish interpolation inequalities under quite general assumptions on the parameters…

Let $E, F, E_0, E_1$ be rearrangement invariant spaces; let $a, \mathrm{b}, \mathrm{b}_0, \mathrm{b}_1$ be slowly varying functions and $0< \theta_0,\theta_1<1$. We characterize the interpolation spaces $$\Big(\overline{X}^{\mathcal…