Communications in Contemporary Mathematics

Recently, both the bilinear decompositions $h^1({ℝ}^n)\times \mathrm{bmo}({ℝ}^n)\subset L^1({ℝ}^n)+h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)$ and $h^1({ℝ}^n)\times \mathrm{bmo}({ℝ}^n)\subset L^1({ℝ}^n)+h^{log}({ℝ}^n)$ were established. In this paper, the authors prove in some sense that the former is sharp, while the latter is not. To this end, the authors first introduce the local Orlicz-slice Hardy space which contains $h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)$, a variant of the local Orlicz Hardy space, introduced by Bonami and Feuto as a special case, and obtain its dual space by establishing its characterizations via atoms, finite atoms, and various maximal functions, which are new even for $h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)$. The relationship $h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)⊊h^{log}({ℝ}^n)$ is also clarified.

In this paper, we are concerned with the following Dirichlet problem for nonlinear equations involving the fractional $p$-Laplacian:

We extend to the spatial case a technique of integration of the close encounters formulated by Tullio Levi-Civita for the planar restricted three-body problem. We consider the Hamiltonian introduced in the Kustaanheimo–Stiefel regularization and construct a complete integral of the related Hamilton–Jacobi equation by means of a series convergent in a neighborhood of the collisions with the primary or secondary body.

We study images of equilibrium (Gibbs) states for a class of non-invertible transformations associated to conformal iterated function systems (IFSs) with overlaps ${𝒮}$. We prove exact dimensionality for these image measures, and find a dimension formula using their overlap numbers. In particular, we obtain a geometric formula for the dimension of self-conformal measures for IFSs with overlaps, in terms of the overlap numbers. This implies a necessary and sufficient condition for dimension drop. If $\nu ={\pi }_{\ast }\mu$ is a self-conformal measure, then $\mathrm{\text{HD}}(\nu )<\frac{h(\mu )}{|\chi (\mu )|}$ if and only if the overlap number $o({𝒮},\mu )>1$. Examples are also discussed.

In this paper, we obtain local derivative estimates of Shi-type for the heat equation coupled to the Ricci flow. As applications, in part combining with Kuang’s work, we extend some results of Zhang and Bamler–Zhang including distance distortion estimates and a backward pseudolocality theorem for Ricci flow on compact manifolds to the noncompact case.

We investigate a Maclaurin inequality for vectors and its connection to an Aleksandrov-type inequality for parallelepipeds.

A virtual $n$-string $(M,\alpha )$ consists of a closed, oriented surface $M$ and a collection $\alpha$ of $n$ oriented, closed curves immersed in $M$. We consider virtual $n$-strings up to virtual homotopy, i.e. stabilizations, destabilizations, stable homeomorphism, and homotopy.

Recently, Cahn proved that any virtual $1$-string can be virtually homotoped to a minimally filling and crossing-minimal representative by monotonically decreasing both genus and the number of self-intersections. We generalize her result to the case of non-parallel virtual $n$-strings.

Cahn also proved that any two crossing-irreducible representatives of a virtual $1$-string are related by isotopy, Type 3 moves, stabilizations, destabilizations, and stable homeomorphism. Kadokami claimed that this held for virtual $n$-strings in general, but Gibson found a counterexample for $5$-strings. We show that Kadokami’s statement holds for non-parallel $n$-strings and exhibit a counterexample for general virtual $3$-strings.

We obtain Margulis-type asymptotic estimates for the number of free homotopy classes of closed geodesics on certain manifolds without conjugate points. Our results cover all compact surfaces of genus at least 2 without conjugate points.

We introduce a very general extension of the monomorphism category as studied by Ringel and Schmidmeier which in particular covers generalized species over locally bounded quivers. We prove that analogues of the kernel and cokernel functor send almost split sequences over the path algebra and the preprojective algebra to split or almost split sequences in the monomorphism category. We derive this from a general result on preservation of almost split morphisms under adjoint functors whose counit is a monomorphism. Despite of its generality, our monomorphism categories still allow for explicit computations as in the case of Ringel and Schmidmeier.