Given $p\in[1,\infty)$ and a bounded open set $\Omega\subset\mathbb{R}^d$ with Lipschitz boundary, we study the $\Gamma$-convergence of the weighted fractional seminorm

\begin{equation*}[u]_{s,p,f}^p=\int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{|\tilde u(x)- \tilde u(y)|^p}{\|x-y\|^{d+sp}}\,f(x)\,f(y)\,\mathrm{d} x\,\mathrm{d} y,\end{equation*}

as $s\to1^-$ for $u\in L^p(\Omega)$, where $\tilde u=u$ on $\Omega$ and $\tilde u=0$ on $\mathbb{R}^d\setminus\Omega$. Assuming that $(f_s)_{s\in(0,1)}\subset L^\infty(\mathbb{R}^d;[0,\infty))$ and $f\in\mathrm{Lip}_b(\mathbb{R}^d;(0,\infty))$ are such that $f_s\to f$ in $L^\infty(\mathbb{R}^d)$ as $s\to1^-$, we show that $(1-s)[u]_{s,p,f_s}^p$ $\Gamma$-converges to the Dirichlet $p$-energy weighted by $f^2$. In the case $p=2$, we also prove the convergence of the corresponding gradient flows.

Structural changes of the pore space and clogging phenomena are inherent to many porous media applications. However, related analytical investigations remain challenging due to potentially vanishing coefficients in the respective systems of partial differential equations. In this research, we apply an appropriate scaling of the unknowns and work with porosity-weighted function spaces. This enables us to prove existence, uniqueness and non-negativity of weak solutions to a combined flow and transport problem with vanishing, but prescribed porosity field, permeability and diffusion.

In this article, we consider diffusive transport of a reactive substance in a saturated porous medium including variable porosity. Thereby, the evolution of the microstructure is caused by precipitation of the transported substance. We are particularly interested in analysing the model when the equations degenerate due to clogging. Introducing an appropriate weighted function space, we are able to handle the degeneracy and obtain analytical results for the transport equation. Also the decay behaviour of this solution with respect to the porosity is investigated. There a restriction on the decay order is assumed, that is, besides low initial concentration also dense precipitation leads to possible high decay. We obtain nonnegativity and boundedness for the weak solution to the transport equation. Moreover, we study an ordinary differential equation (ODE) describing the change of porosity. Thereby, the control of an appropriate weighted norm of the gradient of the porosity is crucial for the analysis of the transport equation. In order to obtain global in time solutions to the overall coupled system, we apply a fixed point argument. The problem is solved for substantially degenerating hydrodynamic parameters.

New transference results for Fourier multiplier operators defined by regulated symbols are presented. We prove restriction and extension of multipliers between weighted Lebesgue spaces with two different weights, which belong to a class more general than periodic weights, and two different exponents of integrability that can be below one.

We also develop some ad-hoc methods that apply to weights defined by the product of periodic weights with functions of power type. Our vector-valued approach allows us to extend our results to transference of maximal multipliers and provide transference of Littlewood–Paley inequalities.

This paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s. It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original contributions include the extension of the fast component-by-component (CBC) construction of lattice rules that achieve the optimal convergence order (a rate of almost 1/N, where N is the number of points, independently of dimension) to so-called “product and order dependent” (POD) weights, as seen in some recent applications. Although the paper has a strong focus on lattice rules, the function space settings are applicable to all QMC methods. Furthermore, the error analysis and construction of lattice rules can be adapted to polynomial lattice rules from the family of digital nets.

Representation theorems are proved for Banach ideal spaces with the Fatou property which are built by the Calderón–Lozanovskiĭ construction. Factorization theorems for operators in spaces more general than the Lebesgue ${{L}^{p}}$ spaces are investigated. It is natural to extend the Gagliardo theorem on the Schur test and the Rubio de Francia theorem on factorization of the Muckenhoupt ${{A}_{p}}$ weights to reflexive Orlicz spaces. However, it turns out that for the scales far from ${{L}^{p}}$ -spaces this is impossible. For the concrete integral operators it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces are not valid. Representation theorems for the Calderón–Lozanovskiĭ construction are involved in the proofs.

Averages in weighted spaces $L_{\phi }^{p}[-1,1]$ defined by additions on $[-1,\,1]$ will be shown to satisfy strong converse inequalities of type $\text{A}$ and $\text{B}$ with appropriate $K$ -functionals. Results for higher levels of smoothness are achieved by combinations of averages. This yields, in particular, strong converse inequalities of type $\text{D}$ between $K$ -functionals and suitable difference operators.