Existence theorems of nonnegative entire solutions of stationary critical $p$-fractional Hardy Schrödinger Kirchhoff equations are presented in this paper. The equations we treat deal with Hardy terms and critical nonlinearities and the main theorems extend several recent results on the topic. The paper contains also some open problems.

In this article we characterize all possible cases that may occur in the rela- tions between the sets of $p$ for which weak type $(p; p)$ and strong type $(p; p)$ inequalities for the Hardy-Littlewood maximal operators, both centered and non-centered, hold in the context of general metric measure spaces.

We introduce a version of Farber's topological complexity suitable for investigating mechanical systems whose configuration spaces exhibit symmetries. Our invariant has vastly different properties to the previous approaches of Colman-Grant, Dranishnikov, and Lubawski-Marzantowicz. In particular, it is bounded from above by Farber's topological complexity.

For an integrable function $f$ on $[0,1)^d$, let $S(f)$ and $Mf$ denote the corresponding dyadic square function and the dyadic maximal function of $f$, respectively. The paper contains the proofs of the following statements.

(i) If $w$ is a dyadic $A_1$ weight on $[0,1)^d$, then $$ ||S(f)||_{L^1(w)}\leq \sqrt{5}[w]_{A_1}^{1/2}||Mf||_{L^1(w)}. $$ The exponent $1/2$ is shown to be the best possible.

(ii) For any $p>1$, there are no constants $c_p$, $\alpha_p$ depending only on $p$ such that for all dyadic $A_p$ weights $w$ on $[0,1)^d$, $$ ||S(f)||_{L^1(w)}\leq c_p[w]_{A_p}^{\alpha_p}||Mf||_{L^1(w)}. $$

Given a positive function $F$ defined on the unit Euclidean sphere and satisfying a suitable convexity condition, we consider, for hypersurfaces $M^n$ immersed in the Euclidean space $\mathbb R^{n+1}$, the so-called $k$-th anisotropic mean curvatures $H_k^F$, $0\leq k\leq n$. For fixed $0\leq r\leq s\leq n$, a hypersurface $M^n$ of $\mathbb{R}^{n+1}$ is said to be $(r,s,F)$-linear Weingarten when its $k$-th anisotropic mean curvatures $H_k^F$, $r\leq k\leq s$, are linearly related. In this setting, we establish the concept of stability concerning closed $(r,s,F)$-linear Weingarten hypersurfaces immersed in $\mathbb R^{n+1}$ and, afterwards, we prove that such a hypersurface is stable if, and only if, up to translations and homotheties, it is the Wulff shape of $F$. For $r=s$ and $F\equiv1$, our results amount to the standard stability studied, for instance, by Alencar-do Carmo-Rosenberg [1].

In this paper some qualitative and geometric aspects of nonsmooth vector fields theory are discussed. A Poincaré-Bendixson Theorem for a class of nonsmooth systems is presented. In addition, a minimal set in planar Filippov systems not predicted in classical Poincaré-Bendixson theory and whose interior is non-empty is exhibited. The concepts of limit sets, recurrence, and minimal sets for nonsmooth systems are defined and compared with the classical ones. Moreover some differences between them are pointed out.

Let $\mathsf M_{\mathsf{S}}$ denote the strong maximal operator on $\mathbb{R}^n$ and let $w$ be a non-negative, locally integrable function. For $\alpha\in(0,1)$ we define the weighted Tauberian constant $\mathsf C_{\mathsf {S},w}$ associated with $\mathsf M_{\mathsf{S}}$ by \[ \mathsf C_{\mathsf{S},w}(\alpha) := \sup_{\begin{subarray}{c} E\subset \mathbb{R}^n \\ 0\lt w(E) \lt+\infty\end{subarray}}\frac{1}{w(E)}w(\{x\in\mathbb{R}^n: \mathsf M_{\mathsf{S}}( {\mathbf 1}_E)(x)>\alpha\}). \] We show that $\lim_{\alpha\to 1^-} \mathsf C_{\mathsf {S},w}(\alpha)=1$ if and only if $w\in A_\infty^*$, that is if and only if $w$ is a strong Muckenhoupt weight. This is quantified by the estimate $\mathsf C_{\mathsf {S},w}(\alpha)-1\lesssim_{n} (1-\alpha)^{ (cn [w]_{A_\infty^*})^{-1}}$ as $\alpha\to 1^-$, where $c>0$ is a numerical constant independent of $n$; this estimate is sharp in the sense that the exponent $1/(cn[w]_{A_\infty^*})$ can not be improved in terms of $[w]_{A_\infty^*}$. As corollaries, we obtain a sharp reverse Hölder inequality for strong Muckenhoupt weights in $\mathbb{R}^n$ as well as a quantitative imbedding of $A_\infty^*$ into $A_{p}^*$. We also consider the strong maximal operator on $\mathbb{R}^n$ associated with the weight $w$ and denoted by $\mathsf M_{\mathsf{S}} ^{w}$. In this case the corresponding Tauberian constant $\mathsf C_{\mathsf{S}} ^w$ is defined by \[ \mathsf C _{\mathsf{S}}^w(\alpha) := \sup_{\begin{subarray}{c} E\subset \mathbb{R}^n \\ 0\lt w(E) \lt +\infty\end{subarray}}\frac{1}{w(E)}w(\{x\in\mathbb{R}^n: \mathsf M_{\mathsf{S}}^{w}({\mathbf 1}_E)(x)>\alpha\}). \] We show that there exists some constant $c_{w,n}>0$ depending only on $w$ and the dimension $n$ such that $\mathsf C_{\mathsf{S}} ^w(\alpha)-1 \lesssim_{w,n} (1-\alpha)^{ c_{w,n} }$ as $\alpha\to 1^-$ whenever $w\in A_\infty^*$ is a strong Muckenhoupt weight.

In a recent paper, Csörnyei and Wilson prove that curves in Euclidean space of $\sigma$-finite length have tangents on a set of positive $\mathscr{H}^{1}$-measure. They also show that a higher dimensional analogue of this result is not possible without some additional assumptions. In this note, we show that if $\Sigma\subseteq \mathbb{R}^{d+1}$ has the property that each ball centered on $\Sigma$ contains two large balls in different components of $\Sigma^{c}$ and $\Sigma$ has $\sigma$-finite $\mathscr{H}^{d}$-measure, then it has $d$-dimensional tangent points in a set of positive $\mathscr{H}^{d}$-measure. As an application, we show that if the dimension of harmonic measure for an NTA domain in $\mathbb{R}^{d+1}$ is less than $d$, then the boundary domain does not have $\sigma$-finite $\mathscr{H}^{d}$-measure.

We also give shorter proofs that Semmes surfaces are uniformly rectifiable and, if $\Omega\subseteq \mathbb{R}^{d+1}$ is an exterior corkscrew domain whose boundary has locally finite $\mathscr{H}^{d}$-measure, one can find a Lipschitz subdomain intersecting a large portion of the boundary.

In this paper we prove that for every complete $n$-dimensional Riemannian manifold without Green's function and with its sectional curvatures satisfying $K \le -1$, the exponent of convergence is greater than or equal to $n-1$. Furthermore, we show that this inequality is sharp. This result is well known for manifolds with constant sectional curvatures $K = -1$.

The aim of this paper is to prove a trace theorem for Besov functions in the metric setting, generalizing a known result from A. Jonsson and H. Wallin in the Euclidean case. We show that the trace of a Besov space defined in a ‘big set’ $X$ is another Besov space defined in the ‘small set’ $F\subset X$. The proof is divided in three parts. First we see that Besov functions in $F$ are restrictions of functions of the same type (but greater regularity) in $X$, that is we prove an extension theorem and mention examples where this theorem holds. Next, as an auxiliary result that can also be interesting on its own, we show that the interpolation between certain potential spaces gives a Besov space. Finally, to obtain that Besov functions in $X$ can in fact be restricted to $F$, a restriction theorem, we first prove that this result holds for functions in the potential space, and then by the interpolation result previously shown, it must hold in the Besov case. For the interpolation and restriction theorems, we make additional assumptions on the spaces $X$ and $F$, and on the order of regularity of the functions involved. We include an interesting example of our trace theorem, not covered by the classical one.

We present the theory of the Dirichlet problem for nonlocal operators which are the generators of general pure-jump symmetric Lévy processes whose Lévy measures need not be absolutely continuous. We establish basic facts about the Sobolev spaces for such operators, in particular we prove the existence and uniqueness of weak solutions. We present strong and weak variants of maximum principle, and $L^\infty$ bounds for solutions. We also discuss the related extension problem in $C^{1,1}$ domains.

For $V$ a vector space over a field, or more generally, over a division ring, it is well-known that every $x\in\operatorname{End}(V)$ has an inner inverse; that is, that there exists $y\in\operatorname{End}(V)$ satisfying $xyx=x$. We show here that a large class of such $x$ have inner inverses $y$ that satisfy with $x$ an infinite family of additional monoid relations, making the monoid generated by $x$ and $y$ what is known as an inverse monoid (definition recalled). We obtain consequences of these relations, and related results.

P. Nielsen and J. Šter [16] show that a much larger class of elements $x$ of rings $R$, including all elements of von Neumann regular rings, have inner inverses satisfying arbitrarily large finite subsets of the abovementioned set of relations. But we show by example that the endomorphism ring of any infinite-dimensional vector space contains elements having no inner inverse that simultaneously satisfies all those relations.

A tangential result gives a condition on an endomap $x$ of a set $S$ that is necessary and sufficient for $x$ to have a strong inner inverse in the monoid of all endomaps of $S$.

By means of the modified Abel lemma on summation by parts, we investigate infinite series involving quadratic and cubic harmonic numbers. Several infinite series identities are established for $\pi^2$ and $\zeta(3)$ as consequences.