We characterize the functions with ‘small’ bounded mean oscillation (BMO) norm by establishing the precise connection between the space BMO and class $A_\infty$ of Muckenhoupt weights. We prove that there exists a universal constant $c^*_2$ such that $\Vert f \Vert_{BMO} \lt c^*_2$ if and only if $\exp f \in A_2$, where $c^*_2$ is the sharp constant in the John and Nirenberg inequality. Similarly, in dimension one, we prove that $\Vert f \Vert_{BLO} \lt 1$ if and only if $\exp f \in A_1$. As application we introduce a structure of metric space in $A_\infty$ and prove that the closed unit ball of $A_\infty$ is a Banach space.

In this short note, we show that every convex, order-bounded above functional on a Fréchet lattice is automatically continuous. This improves a result in Ruszczyński and Shapiro ((2006) Mathematics of Operations Research 31(3), 433–452.) and applies to many deviation and variability measures. We also show that an order-continuous, law-invariant functional on an Orlicz space is strongly consistent everywhere, extending a result in Krätschmer et al. ((2017) Finance and Stochastics 18(2), 271–295.).

We develop a new method suitable for establishing lower bounds on the ball measure of noncompactness of operators acting between considerably general quasinormed function spaces. This new method removes some of the restrictions oft-presented in the previous work. Most notably, the target function space need not be disjointly superadditive nor equipped with a norm. Instead, a property that is far more often at our disposal is exploited—namely the absolute continuity of the target quasinorm.

We use this new method to prove that limiting Sobolev embeddings into spaces of Brezis–Wainger type are so-called maximally noncompact, i.e. their ball measure of noncompactness is the worst possible.

The article introduces and studies Hausdorff–Berezin operators on the unit ball in a complex space. These operators are a natural generalization of the Berezin transform. In addition, the class of such operators contains, for example, the invariant Green potential, and some other operators of complex analysis. Sufficient and necessary conditions for boundedness in the space of p – integrable functions with Haar measure (invariant with respect to involutive automorphisms of the unit ball) are given. We also provide results on compactness of Hausdorff–Berezin operators in Lebesgue spaces on the unit ball. Such operators have previously been introduced and studied in the context of the unit disc in the complex plane. Present work is a natural continuation of these studies.

We prove some results on weakly almost square Banach spaces and their relatives. On the one hand, we discuss weak almost squareness in the setting of Banach function spaces. More precisely, let $(\Omega,\Sigma)$ be a measurable space, let E be a Banach lattice and let $\nu:\Sigma \to E^+$ be a non-atomic countably additive measure having relatively norm compact range. Then the space $L_1(\nu)$ is weakly almost square. This result applies to some abstract Cesàro function spaces. Similar arguments show that the Lebesgue–Bochner space $L_1(\mu,Y)$ is weakly almost square for any Banach space Y and for any non-atomic finite measure µ. On the other hand, we make some progress on the open question of whether there exists a locally almost square Banach space, which fails the diameter two property. In this line, we prove that if X is any Banach space containing a complemented isomorphic copy of c0, then for every $0 \lt \varepsilon \lt 1$, there exists an equivalent norm $|\cdot|$ on X satisfying the following: (i) every slice of the unit ball $B_{(X,|\cdot|)}$ has diameter 2; (ii) $B_{(X,|\cdot|)}$ contains non-empty relatively weakly open subsets of arbitrarily small diameter and (iii) $(X,|\cdot|)$ is (r, s)-SQ for all $0 \lt r,s \lt \frac{1-\varepsilon}{1+\varepsilon}$.

Given a $\sigma $-finite measure space $(X,\mu )$, a Young function $\Phi $, and a one-parameter family of Young functions $\{\Psi _q\}$, we find necessary and sufficient conditions for the associated Orlicz norms of any function $f\in L^\Phi (X,\mu )$ to satisfy

$$\begin{align*}\lim_{q\rightarrow \infty}\|f\|_{L^{\Psi_q}(X,\mu)}=C\|f\|_{L^\infty(X,\mu)}. \end{align*}$$

The constant C is independent of f and depends only on the family $\{\Psi _q\}$. Several examples of one-parameter families of Young functions satisfying our conditions are given, along with counterexamples when our conditions fail.

Answering a question by Chatterji–Druţu–Haglund, we prove that, for every locally compact group $G$, there exists a critical constant $p_G \in [0,\infty ]$ such that $G$ admits a continuous affine isometric action on an $L_p$ space ($0< p<\infty$) with unbounded orbits if and only if $p \geq p_G$. A similar result holds for the existence of proper continuous affine isometric actions on $L_p$ spaces. Using a representation of cohomology by harmonic cocycles, we also show that such unbounded orbits cannot occur when the linear part comes from a measure-preserving action, or more generally a state-preserving action on a von Neumann algebra and $p>2$. We also prove the stability of this critical constant $p_G$ under $L_p$ measure equivalence, answering a question of Fisher.

We establish multiplicity results for the following class of quasilinear problems P

\begin{equation*} \left\{ \begin{array}{@{}l} -\Delta_{\Phi}u=f(x,u) \quad \mbox{in} \quad \Omega, \\ u=0 \quad \mbox{on} \quad \partial \Omega, \end{array} \right. \end{equation*}

$\Delta _{\Phi }u=\text {div}(\varphi (x,|\nabla u|)\nabla u)$

$\Phi (x,t)=\int _{0}^{|t|}\varphi (x,s)s\,ds$

$\Omega \subset \mathbb {R}^{N}$

$\Omega _N$

$\Omega _p$

$\overline {\Omega _N}\cap \overline {\Omega _p}=\emptyset$

$(P)$

$-\Delta _{\Phi }$

$-\Delta _N$

$\Omega _N$

$-\Delta _p$

$\Omega _p$

$f:\Omega \times \mathbb {R}\to \mathbb {R}$

$e^{\alpha |t|^{\frac {N}{N-1}}}$

$\Omega _N$

$|t|^{p^{*}-2}t$

$\Omega _p$

$|t|$

$\alpha >0$

$p^{*}=\frac {Np}{N-p}$

$1< p< N$

$f$

$f$

$\Omega$

$(P)$

We consider a class of generalized nonlocal $p$-Laplacian equations. We find some proper structural conditions to establish a version of nonlocal Harnack inequalities of weak solutions to such nonlocal problems by using the expansion of positivity and energy estimates.

We study nonlinear measure data elliptic problems involving the operator of generalized Orlicz growth. Our framework embraces reflexive Orlicz spaces, as well as natural variants of variable exponent and double-phase spaces. Approximable and renormalized solutions are proven to exist and coincide for arbitrary measure datum and to be unique when for a class of data being diffuse with respect to a relevant nonstandard capacity. A capacitary characterization of diffuse measures is provided.

We present new estimates in the setting of weighted Lorentz spaces of operators satisfying a limited Rubio de Francia condition; namely $T$ is bounded on $L^{p}(v)$ for every $v$ in an strictly smaller class of weights than the Muckenhoupt class $A_p$. Important examples will be the Bochner–Riesz operators $BR_\lambda$ with $0<\lambda <{(n-1)}/2$, sparse operators, Hörmander multipliers with a limited regularity condition and rough operators with $\Omega \in L^{r}(\Sigma )$, $1 < r < \infty$.

Our aim in this paper is to establish Trudinger’s exponential integrability for Riesz potentials in weighted Morrey spaces on the half space. As an application, we obtain Trudinger’s inequality for Riesz potentials in the framework of double phase functionals.

In this paper, we develop an extremely simple method to establish the sharpened Adams-type inequalities on higher-order Sobolev spaces $W^{m,\frac {n}{m}}(\mathbb {R}^n)$ in the entire space $\mathbb {R}^n$ , which can be stated as follows: Given $\Phi \left ( t\right ) =e^{t}-\underset {j=0}{\overset {n-2}{\sum }} \frac {t^{j}}{j!}$ and the Adams sharp constant $\beta _{n,m}$ . Then,

$$ \begin{align*}\sup_{\|\nabla^mu\|_{\frac{n}{m}}^{\frac{n}{m}}+\|u\|_{\frac{n}{m}}^{\frac{n}{m}}\leq1}\int_{\mathbb{R}^n}\Phi\Big(\beta_{n,m} (1+\alpha \|u\|_{\frac{n}{m}}^{\frac{n}{m}} )^{\frac{m}{n-m}}|u|^{\frac{n}{n-m}}\Big)dx<\infty, \end{align*} $$

$0<\alpha <1$

$\alpha $

$\alpha \ge 1$

Our argument avoids applying the complicated blow-up analysis often used in the literature to deal with such sharpened inequalities.

We prove the partial Hölder continuity for minimizers of quasiconvex functionals

\begin{equation*} \mathcal{F}({\bf u}) \colon =\int_{\Omega} f(x,{\bf u},D{\bf u})\,\textrm{d} x, \end{equation*}

$f$

$x$

${\bf u}$

This paper establishes the mapping properties of pseudo-differential operators and the Fourier integral operators on the weighted Morrey spaces with variable exponents and the weighted Triebel–Lizorkin–Morrey spaces with variable exponents. We obtain these results by extending the extrapolation theory to the weighted Morrey spaces with variable exponents. This extension also gives the mapping properties of Calderón–Zygmund operators on the weighted Hardy–Morrey spaces with variable exponents and the wavelet characterizations of the weighted Hardy–Morrey spaces with variable exponents.

Based on the Gale–Ryser theorem [2, 6], for the existence of suitable $(0,1)$-matrices for different partitions of a natural number, we revisit the classical result of Lorentz [4] regarding the characterization of a plane measurable set, in terms of its cross-sections, and extend it to general measure spaces.

We investigate the boundedness, compactness, invertibility and Fredholmness of weighted composition operators between Lorentz spaces. It is also shown that the classes of Fredholm and invertible weighted composition maps between Lorentz spaces coincide when the underlying measure space is nonatomic.

In this paper we give sufficient conditions to obtain continuity results of solutions for the so called ϕ-Laplacian Δϕ with respect to domain perturbations. We point out that this kind of results can be extended to a more general class of operators including, for instance, nonlocal nonstandard growth type operators.

The asymptotic behavior of solutions to a family of Dirichlet boundary value problems, involving differential operators in divergence form, on a domain equipped with a Finsler metric is investigated. Solutions are shown to converge uniformly to the distance function to the boundary of the domain, which takes into account the Finsler norm involved in the equation. This implies that a well-known result in the analysis of problems modeling torsional creep continues to hold in this more general setting.

We study the stability of the differential process of Rochberg and Weiss associated with an analytic family of Banach spaces obtained using the complex interpolation method for families. In the context of Köthe function spaces, we complete earlier results of Kalton (who showed that there is global bounded stability for pairs of Köthe spaces) by showing that there is global (bounded) stability for families of up to three Köthe spaces distributed in arcs on the unit circle while there is no (bounded) stability for families of four or more Köthe spaces. In the context of arbitrary pairs of Banach spaces, we present some local stability results and some global isometric stability results.