A complete description of the real interpolation space $L={(L_{p₀}\left(\omega ₀\right),...,L_{pₙ}\left(\omega ₙ\right))}_{\theta ⃗,q}$ is given. An interesting feature of the result is that the whole measure space (Ω,μ) can be divided into disjoint pieces ${\Omega }_i$ (i ∈ I) such that L is an $l_q$ sum of the restrictions of L to ${\Omega }_i$, and L on each ${\Omega }_i$ is a result of interpolation of just two weighted $L_p$ spaces. The proof is based on a generalization of some recent results of the first two authors concerning real interpolation of vector-valued spaces.

Let $\theta \in (0,1)$, $\lambda \in [0,1)$ and $p,p_0,p_1\in (1,\infty ]$ be such that $(1-\theta )/p_0+\theta /p_1=1/p$, and let $\varphi ,{\varphi }_0,{\varphi }_1$ be some admissible functions such that $\varphi ,{\varphi }_0^{p/p_0}$ and ${\varphi }_1^{p/p_1}$ are equivalent. We first prove that, via the $\pm$ interpolation method, the interpolation $\langle L_{{\varphi }_0}^{p_0),\lambda }\left(𝒳\right),L_{{\varphi }_1}^{p_1),\lambda }\left(𝒳\right),\theta \rangle$ of two generalized grand Morrey spaces on a quasi-metric measure space $𝒳$ is the generalized grand Morrey space $L_{\varphi }^{p),\lambda }\left(𝒳\right)$. Then, by using block functions, we also find a predual space of the generalized grand Morrey space. These results are new even for generalized grand Lebesgue spaces.

Geometric structure of Cesàro function spaces $Ce{s}_p\left(I\right)$, where I = [0,1] and [0,∞), is investigated. Among other matters we present a description of their dual spaces, characterize the sets of all q ∈ [1,∞] such that $Ce{s}_p[0,1]$ contains isomorphic and complemented copies of $l_q$-spaces, show that Cesàro function spaces fail the fixed point property, give a description of subspaces generated by Rademacher functions in spaces $Ce{s}_p[0,1]$.

We prove that for a normed linear space $X$, if $f_1:X\to ℝ$ is continuous and semiconvex with modulus $\omega$, $f_2:X\to ℝ$ is continuous and semiconcave with modulus $\omega$ and $f_1\le f_2$, then there exists $f\in C^{1,\omega }\left(X\right)$ such that $f_1\le f\le f_2$. Using this result we prove a generalization of Ilmanen lemma (which deals with the case $\omega \left(t\right)=t$) to the case of an arbitrary nontrivial modulus $\omega$. This generalization (where a $C_{loc}^{1,\omega }$ function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.

Let $(A_0,A_1)$ be a regular interpolation couple. Under several different assumptions on a fixed $A^{\beta }$, we show that $A^{\theta }=A_{\theta }$ for every $\theta \in (0,1)$. We also deal with assumptions on ${\overline{A}}^{\beta }$, the closure of $A^{\beta }$ in the dual of ${(A_0^{*},A_1^{*})}_{\beta }$.

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...

Let $T$ be a tree with $n$ vertices. To each edge of $T$ we assign a weight which is a positive definite matrix of some fixed order, say, $s$. Let $D_{ij}$ denote the sum of all the weights lying in the path connecting the vertices $i$ and $j$ of $T$. We now say that $D_{ij}$ is the distance between $i$ and $j$. Define $D:=\left[D_{ij}\right]$, where $D_{ii}$ is the $s\times s$ null matrix and for $i\ne j$, $D_{ij}$ is the distance between $i$ and $j$. Let $G$ be an arbitrary connected weighted graph with $n$ vertices, where each weight is a positive definite matrix of order...

CONTENTSPREFACE..........................................................................................................................................................................3INTRODUCTION............................................................................................................................................................. 41. Notation. 2. Subject of the paper.Chapter I. DECOMPOSITION OF Σ INTO ${\Sigma }_1$, ${\Sigma }_2$, ${\Sigma }_3$, ${\Sigma }_4$ INESSENTIAL RESTRICTIONOF GENERALITY ...............................................................................................................................................................

The aim of this paper is to characterize the $L^p-L^q$ boundedness of two classes of integral operators from $L^p(𝒰,\mathrm{d}{V}_{\alpha })$ to $L^q(𝒰,\mathrm{d}{V}_{\beta })$ in terms of the parameters $a$, $b$, $c$, $p$, $q$ and $\alpha$, $\beta$, where $𝒰$ is the Siegel upper half-space. The results in the presented paper generalize a corresponding result given in C. Liu, Y. Liu, P. Hu, L. Zhou (2019).