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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*}where $\Delta _{\Phi }u=\text {div}(\varphi (x,|\nabla u|)\nabla u)$
for a generalized N-function $\Phi (x,t)=\int _{0}^{|t|}\varphi (x,s)s\,ds$
. We consider $\Omega \subset \mathbb {R}^{N}$
to be a smooth bounded domain that contains two disjoint open regions $\Omega _N$
and $\Omega _p$
such that $\overline {\Omega _N}\cap \overline {\Omega _p}=\emptyset$
. The main feature of the problem $(P)$
is that the operator $-\Delta _{\Phi }$
behaves like $-\Delta _N$
on $\Omega _N$
and $-\Delta _p$
on $\Omega _p$
. We assume the nonlinearity $f:\Omega \times \mathbb {R}\to \mathbb {R}$
of two different types, but both behave like $e^{\alpha |t|^{\frac {N}{N-1}}}$
on $\Omega _N$
and $|t|^{p^{*}-2}t$
on $\Omega _p$
as $|t|$
is large enough, for some $\alpha >0$
and $p^{*}=\frac {Np}{N-p}$
being the critical Sobolev exponent for $1< p< N$
. In this context, for one type of nonlinearity $f$
, we provide a multiplicity of solutions in a general smooth bounded domain and for another type of nonlinearity $f$
, in an annular domain $\Omega$
, we establish existence of multiple solutions for the problem $(P)$
that are non-radial and rotationally non-equivalent.
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 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 $\Omega \in L^{r}(\Sigma )$
, $1 < r < \infty$
.
$W^{m,\frac {n}{m}}(\mathbb {R}^n)$
: a simple approach
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,
for any
$$ \begin{align*}\sup_{\|\nabla^mu\|_{\frac{n}{m}}^{\frac{n}{m}}+\|u\|_{\frac{n}{m}}^{\frac{n}{m}}\leq1}\int_{\mathbb{R}^n}\Phi\Big(\beta_{n,m} (1+\alpha \|u\|_{\frac{n}{m}}^{\frac{n}{m}} )^{\frac{m}{n-m}}|u|^{\frac{n}{n-m}}\Big)dx<\infty, \end{align*} $$
$0<\alpha <1$
. Furthermore, we construct a proper test function sequence to derive the sharpness of the exponent
$\alpha $
of the above Adams inequalities. Namely, we will show that if
$\alpha \ge 1$
, then the above supremum is infinite.
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
where 
\begin{equation*} \mathcal{F}({\bf u}) \colon =\int_{\Omega} f(x,{\bf u},D{\bf u})\,\textrm{d} x, \end{equation*}
$f$ satisfies a uniform VMO condition with respect to the 
$x$-variable and is continuous with respect to 
${\bf u}$. The growth condition with respect to the gradient variable is assumed a general one.
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.
