In this paper, we obtain a fractional Hardy inequality in the case $Q<sp$ on homogeneous Lie groups, and as an application we show the corresponding uncertainty principle. Also, we show a fractional Hardy–Sobolev-type inequality on homogeneous Lie groups. In addition, we prove fractional logarithmic Hardy–Sobolev and fractional Nash-type inequalities on homogeneous Lie groups. We note that the case $Q>sp$ was extensively studied in the literature, while here we are dealing with the complementary range $Q<sp$.

Let G be a linear Lie group with Lie algebra $\mathfrak{g}$. An element $X\in \mathfrak{g}$ is called ${\mathrm{Ad}}_G$-real if $-X=gX{g}^{-1}$ for some $g\in G$. Moreover, if $-X=gX{g}^{-1}$ holds for some involution $g\in G$, then X is called strongly ${\mathrm{Ad}}_G$-real. We have classified the ${\mathrm{Ad}}_G$-real and strongly ${\mathrm{Ad}}_G$-real orbits in the special linear Lie algebra $\mathfrak{s}\mathfrak{l}(n,\mathbb{F})$ for $\mathbb{F}=\mathbb{C}$ or $\mathbb{H}$.

We prove new sufficient bump conditions for general iterated commutators of Calderón–Zygmund operators and fractional integral operators. As an application of our results, we derive two weight compactness theorems for higher-order iterated commutators with a $CMO$ function.

In this note, we provide a new partial solution to the Hurwitz existence problem for surface branched covers. Namely, we consider candidate branch data with base surface the sphere and one partition of the degree having length 2, and we fully determine which of them are realizable and which are exceptional. The case where the covering surface is also the sphere was solved somewhat recently by Pakovich, and we deal here with the case of positive genus. We show that the only other exceptional candidate data, besides those of Pakovich (five infinite families and one sporadic case), are a well-known, very specific infinite family in degree 4 (indexed by the genus of the candidate covering surface, which can attain any value), five sporadic cases (four in genus 1 and one in genus 2), and another infinite family in genus 1 also already known. Since the degree is a composite number for all these exceptional data, our findings provide more evidence for the prime-degree conjecture. Our argument proceeds by induction on the genus and on the number of branching points, so our results logically depend on those of Pakovich, and we do not employ the technology of constellations on which his proof is based.

We improve the multiparameter version of Rota’s theorem obtained by Frangos and Sucheston for bistochastic operators to the case of positive linear contractions. As a special case, we also obtain the multiparamter convergence of Burkholder–Chow type for positive self-adjoint contractions on $L_2$ spaces. The idea behind our approach can be applied to obtain a ratio multiparameter version of Doob’s theorem. We show that the weak-type maximal inequality approach to proving almost everywhere convergence makes it possible to give a positive answer to the question raised by Burkholder about positive definite self-adjoint contractions on $L_1$.

Let $P^{(\frac{1}{2})}(n)$ denote the middle prime factor of n (taking into account multiplicity). More generally, one can consider, for any $\mathit{\alpha }\in (0,1)$, the α-positioned prime factor of n, $P^{(\mathit{\alpha })}(n)$. It has previously been shown that $loglog{P}^{(\mathit{\alpha })}(n)$ has normal order $\mathit{\alpha }loglogx$, and its values follow a Gaussian distribution around this value. We extend this work by obtaining an asymptotic formula for the count of $n\le x$ for which $P^{(\mathit{\alpha })}(n)=p$, for primes p in a wide range up to x. We give several applications of these results, including an exploration of the geometric mean of the middle prime factors, for which we find that $\frac{1}{x}{\sum }_{1<n\le x}log{P}^{(\frac{1}{2})}(n)\sim A{(logx)}^{\mathit{\phi }-1}$, where φ is the golden ratio, and A is an explicit constant. Along the way, we obtain an extension of Lichtman’s recent work on the “dissected” Mertens’ theorem sums ${\sum }_{P^{+}(n)\le y,\mathrm{\Omega }(n)=k}\frac{1}{n}$ for large values of k.

We consider the conditional control problem introduced by Lions in his lectures at the Collège de France in November 2016. In his lectures, Lions emphasized some of the major differences with the analysis of classical stochastic optimal control problems, and in so doing, raised the question of the possible differences between the value functions resulting from optimization over the class of Markovian controls as opposed to the general family of open-loop controls. The goal of the paper is to elucidate this quandary and settle Lions’ original conjecture in the case of a relaxed version of the problem. First, we justify the mathematical formulation of the conditional control problem by the description of a practical model from evolutionary biology. Next, we relax the original formulation by the introduction of soft as opposed to hard killing, and using a mimicking argument, we reduce the open-loop optimization problem to an optimization over a specific class of feedback controls. After proving existence of optimal feedback control functions, we prove a superposition principle allowing us to recast the original stochastic control problems as deterministic control problems for dynamical systems of probability Gibbs measures. Next, we characterize the solutions by forward–backward systems of coupled nonlinear and nonlocal partial differential equations (PDEs), very much in the spirit of some of the mean field game (MFG) systems. From there, we identify a common optimizer, proving the conjecture of equality of the value functions. Finally, we illustrate the results with convincing numerical experiments.