We show that the unit ball of the subspace $M_W^0$ of ordered continuous elements of $M_W$ has no extreme points, where $M_W$ is the Marcinkiewicz function space generated by a decreasing weight function $w$ over the interval $(0,\infty)$ and $W(t) = \int_0^tw$, $t\in(0,\infty)$. We also present here a proof of the fact that a function $f$ in the unit ball of $M_W$ is an extreme point if and only if $f^*=w$.

After a short expose of the history of the Beckenbach--Dresher inequality, general result and the Aczel type inequality are given and super(sub)additivity of the function $G_{p,q,u}(f,g;A,B) :=\frac{A^{\frac up}(f^p)}{B^{\frac{u-1}{q}}(g^q)}$ is proved. Also, a difference which is inspired by one integral analogue of the Beckenbach--Dresher inequality is considered.

The classical Bohr inequality says that $|a+b|^2\leq p|a|^2+q|b|^2$ for all scalars $a, \ b$ and positive $p,q$ with $\frac 1p + \frac 1q =1.$ The equality holds if and only if $(p-1)a=b.$ Several authors discussed operator version of Bohr inequality. In this paper, we give a unified proof to operator generalizations of Bohr inequality. One viewpoint of ours is a matrix inequality, and the other is a generalized parallelogram law for absolute value of operators, i.e., for operators $A$ and $B$ on a Hilbert space and $t\neq0$, $$|A-B|^2+\frac{1}{t}|tA+B|^2=(1+t)|A|^2+(1+\frac{1}{t})|B|^2.$$

$L_p-L_q$--boundedness of the map $f\to w(x)\int_{a(x)}^{b(x)}k(x,y)f(y)v(y)dy$ is described by different types of criteria expressed in terms of given parameters $p,q \in (0,\infty)$, strictly increasing boundaries $a(x)$ and $b(x)$, locally integrable weight functions $v,w$ and a positive continuous kernel $k(x,y)$ satisfying some growth conditions.

We discuss the proofs of the existence and uniqueness of solutions of the Navier--Stokes equation driven with additive noise in three dimensions, in the presence of a strong uni-directional mean flow with some rotation. We also discuss how the existence of a unique invariant measure is established and the properties of this measure are described. The invariant measure is used to prove Kolmogorov's scaling in 3-dimensional turbulence including the celebrated $-5/3$ power law for the decay of the power spectrum of a turbulent 3-dimensional flow. Then we briefly describe the mathematical proof of Kolmogorov's statistical theory of turbulence.

In this paper, we show a complement of Ando--Hiai inequality: Let $A$ and $B$ be positive invertible operators on a Hilbert space $H$ and $\alpha\in [0,1]$. If $A\ \sharp_{\alpha}\ B \leq I$, then $$A^r \, \sharp_\alpha \, B^r \leq \|(A \,\sharp_\alpha \, B)^{-1}\|^{1-r} I \quad \text{for all }\, 0 <r \leq 1, $$ where $I$ is the identity operator and the symbol $\| \cdot \|$ stands for the operator norm.

We study new conditions on a radial function $f$ in order to have the almost everywhere convergence of the spherical partial Fourier integrals.

Multiscale stochastic homogenization is studied for quasilinear hyperbolic problems. We consider the asymptotic behaviour of a sequence of realizations of the form ${\frac{\partial^2 u^\omega_{\varepsilon}}{\partial t^2}} - \mathrm{div}\left(a\left(T_1(\frac{x}{\varepsilon_1})\omega_1, T_2(\frac{x}{\varepsilon_2})\omega_2 ,t, D u^\omega_{\varepsilon}\right)\right)-\Delta(\frac{\partial u^\omega_{\varepsilon}}{\partial t}) +G\left(T_3(\frac{x}{\varepsilon_3})\omega_3 ,t,\frac{\partial u^\omega_{\varepsilon}}{\partial t}\right)=f$. It is shown, under certain structure assumptions on the random maps $a\left(\omega_1,\omega_2,t,\xi\right)$ and $G\left(\omega_3,t,\eta\right)$, that the sequence $\{u^\omega_\epsilon\}$ of solutions converges weakly in $L^p(0,T;W^{1,p}_0(\Omega))$ to the solution $u$ of the homogenized problem ${\frac{\partial^2 u}{\partial t^2}} - \mathrm{div}\left( b \left( t,D u \right)\right)-\Delta(\frac{\partial u}{\partial t})+{\overline G}(t,\frac{\partial u}{\partial t}) = f$.

It is shown that the conditions of the validity of the Hardy inequality coincide with the conditions on the spectrum of some (nonlinear) differential operators to be bounded from below and discrete.

In this paper, we state, prove and discuss a new refined general weighted discrete Hardy-type inequality with a non-negative kernel, related to an arbitrary non-negative convex (or positive concave) function on a real interval and to a positive real parameter. As its consequences, obtained by rewriting it for various suitably chosen parameters, kernels, weights and convex (or concave) functions, we derive new weighted and unweighted generalizations and refinements of some well-known inequalities such as Carleman's inequality and the so-called Godunova's inequality. Finally, by employing exponential and logarithmic convexity, as special cases of the usual convexity, we obtain some further refinements of the inequalities mentioned above.

We consider the Edmunds--Triebel logarithmic spaces $A_\theta (logA)_{b,q} $ produced by a Banach couple $\overline{A}=(A_0, \; A_1),$ as special cases of extrapolation spaces and get estimates of a measure of weak noncompactness of the unit balls of these spaces in terms of the measures of weak noncompactness of the unit balls of the spaces $A_0$ and $A_1.$ We obtain also estimates of the $n$-th Jordan--von Neumann constant $C^n_{NJ}$ and the $n$-th James constant $J_n$ of the spaces $A_\theta (logA)_{b,q} $ in terms of the corresponding constants of the spaces $A_0$ and $A_1.$

In this paper, improvements for superquadratic functions of Jensen--Steffensen's and related inequalities are discussed. For superquadratic functions which are not convex we get inequalities analog to Jensen--Steffensen's inequality for convex functions. For superquadratic functions which are convex (including many useful functions), we get improvements and extensions of Jensen--Steffensen's inequality and related inequalities.

Some generalizations of Ostrowski inequality are given by using biparametric Euler identities involving real Borel measures and harmonic sequences of functions.

On the occasion of Lars-Erik Persson's 65th birthday there was organized a conference Analysis, Inequalities and Homogenization Theory (AIHT), which was held at Lulea} University of Technology in June 8-11, 2009, in Lulea-Sweden. The first plenary lecture had the title above and here I present an abbreviated form of this lecture.

On the occasion of his 65th birthday, we had an interview with Professor Lars-Erik Persson. We present several questions in this interview together with his answers. Some of the answers are indeed surprising also to us who know him well.