We use new methods, specific for non-locally convex quasi-Banach spaces, to investigate when the quasi-greedy bases of a $p$-Banach space for $0< p<1$ are democratic. The novel techniques we obtain permit to show in particular that all quasi-greedy bases of the Hardy space $H_p({\mathbb {D}})$ for $0< p<1$ are democratic while, in contrast, no quasi-greedy basis of $H_p({\mathbb {D}}^d)$ for $d\ge 2$ is, solving thus a problem that was raised in [7]. Applications of our results to other spaces of interest both in functional analysis and approximation theory are also provided.

In this paper, we study intersection configurations – which describe the behaviour of multiple (finite) intersections of subgroups with respect to finite generability – in the realm of free and free times free-abelian (FTFA) groups. We say that a configuration is realizable in a group $G$ if there exist subgroups $H_1,\ldots, H_k \leqslant G$ realizing it. It is well known that free groups ${\mathbb {F}_{n}}$ satisfy the Howson property: the intersection of any two finitely generated subgroups is again finitely generated. We show that the Howson property is indeed the only obstruction for multiple intersection configurations to be realizable within nonabelian free groups. On the contrary, FTFA groups ${\mathbb {F}_{n}} \times \mathbb {Z}^m$ are well known to be non-Howson. We also study multiple intersections within FTFA groups, providing an algorithm to decide, given $k\geq 2$ finitely generated subgroups, whether their intersection is again finitely generated and, in the affirmative case, compute a ‘basis’ for it. We finally prove that any intersection configuration is realizable in an FTFA group ${\mathbb {F}_{n}} \times \mathbb {Z}^m$, for $n\geq 2$ and large enough $m$. As a consequence, we exhibit finitely presented groups where every intersection configuration is realizable.

In this paper, we study the behaviours of the commutators $[\vec b,\,T]$ generated by multilinear Calderón–Zygmund operators $T$ with $\vec b=(b_1,\,\ldots,\,b_m)\in L_{\rm loc}(\mathbb {R}^n)$ on weighted Hardy spaces. We show that for some $p_i\in (0,\,1]$ with $1/p=1/p_1+\cdots +1/p_m$, $\omega \in A_\infty$ and $b_i\in \mathcal {BMO}_{\omega,p_i}$ ($1\le i\le m$), which are a class of non-trivial subspaces of ${\rm BMO}$, the commutators $[\vec b,\,T]$ are bounded from $H^{p_1}(\omega )\times \cdots \times H^{p_m}(\omega )$ to $L^p(\omega )$. Meanwhile, we also establish the corresponding results for a class of maximal truncated multilinear commutators $T_{\vec b}^*$.

In the context of discrete nonautonomous dynamics, we prove that the homeomorphisms in the linearization theorem are $C^2$ diffeomorphisms. In contrast to other related works, our result does not involve non-resonance conditions or spectral gaps. Our approach is based on the interlacing of the properties of nonautonomous hyperbolicity of the linear part, and boundedness and Lipschitzness of the nonlinearities. Moreover, we propose a functional approach to find conditions for regularity of arbitrary degree.

This paper mainly considers the problem of generalizing a certain class of analytic functions by means of a class of difference operators. We consider some relations between starlike or convex functions and functions belonging to such classes. Some other useful properties of these classes are also considered.

We note that Gabber's rigidity theorem for the algebraic K-theory of henselian pairs also holds true for hermitian K-theory with respect to arbitrary form parameters.

We give a simple sufficient condition for Quinn’s ‘bordism-type’ spectra to be weakly equivalent to commutative symmetric ring spectra. We also show that the symmetric signature is (up to weak equivalence) a monoidal transformation between symmetric monoidal functors, which implies that the Sullivan–Ranicki orientation of topological bundles is represented by a ring map between commutative symmetric ring spectra. In the course of proving these statements, we give a new description of symmetric L theory which may be of independent interest.

In this paper we introduce and examine the differential subordination of the form

\[ p(z)+zp'(z)\varphi(p(z),zp'(z))\prec h(z),\quad z\in\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}, \]

$h$

$0\in h(\mathbb {D}).$

We first prove that the realization $A_{\mathrm {min}}$ of $A:={\operatorname {\mathrm {div}}}(Q\nabla )-V$ in $L^2({\mathbb {R}}^d)$ with unbounded coefficients generates a symmetric sub-Markovian and ultracontractive semigroup on $L^2({\mathbb {R}}^d)$ which coincides on $L^2({\mathbb {R}}^d)\cap C_b({\mathbb {R}}^d)$ with the minimal semigroup generated by a realization of $A$ on $C_b({\mathbb {R}}^d)$. Moreover, using time-dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernel of $A$ and deduce some spectral properties of $A_{\min }$ in the case of polynomially and exponentially growing diffusion and potential coefficients.

We study $L^p$-Sobolev regularity estimates for the restricted X-ray transforms generated by nondegenerate curves. Making use of the inductive strategy in the recent work by the authors, we establish the sharp $L^p$-regularity estimates for the restricted X-ray transforms in $\mathbb {R}^{d+1}$, $d\ge 3$. This extends the result due to Pramanik and Seeger in $\mathbb {R}^3$.

This paper concerns the monostable cooperative system with nonlocal diffusion and free boundaries, which has recently been discussed by Du and Ni [J. Differential equations 308(2021) 369-420 and arXiv:2010.01244]. We here aim at four aspects: the first is to give more accurate estimates for the longtime behaviours of the solution; the second is to discuss the limits of solution pair of a semi-wave problem; the third is to investigate the asymptotic behaviours of the corresponding Cauchy problem; the last is to study the limiting profiles of the solution as one of the expanding rates of free boundaries converges to $\infty$. Moreover, some epidemic models are given to illustrate their own rich longtime behaviours, which are quite different from those of the relevant existing works.

Let $\mathcal {M}$ be an Ahlfors $n$-regular Riemannian manifold such that either the Ricci curvature is non-negative or the Ricci curvature is bounded from below together with a bound on the gradient of the heat kernel. In the paper [IMRN, 2022, no. 2, 1245-1269] of Brazke–Schikorra–Sire, the authors characterised the BMO function $u : \mathcal {M} \to \mathbb {R}$ by a Carleson measure condition of its $\sigma$-harmonic extension $U:\mathcal {M}\times \mathbb {R}_+ \to \mathbb {R}$. This paper is concerned with the similar problem under a more general Dirichlet metric measure space setting, and the limiting behaviours of BMO & Carleson measure, where the heat kernel admits only the so-called diagonal upper estimate. More significantly, without the Ricci curvature condition, we relax the Ahlfors regularity to a doubling property, and remove the pointwise bound on the gradient of the heat kernel. Some similar results for the Lipschitz function are also given, and two open problems related to our main result are considered.

In this paper, we study the asymptotic profiles of positive solutions for diffusive logistic equations. The aim is to study the sharp effect of linear growth and nonlinear function. Both the classical reaction-diffusion equation and nonlocal dispersal equation are investigated. Our main results reveal that the linear and nonlinear parts of reaction term play quite different roles in the study of positive solutions.

Let $X$ and $Y$ be oriented topological manifolds of dimension $n\!+\!2$, and let $K\! \subset \! X$ and $J \! \subset \! Y$ be connected, locally-flat, oriented, $n$–dimensional submanifolds. We show that up to orientation preserving homeomorphism there is a well-defined connected sum $(X,K)\! \mathbin {\#}\! (Y,J)$. For $n = 1$, the proof is classical, relying on results of Rado and Moise. For dimensions $n=3$ and $n \ge 6$, results of Edwards-Kirby, Kirby, and Kirby-Siebenmann concerning higher dimensional topological manifolds are required. For $n = 2, 4,$ and $5$, Freedman and Quinn's work on topological four-manifolds is required along with the higher dimensional theory.

In this paper, we study existence of rotating periodic solutions for p-Laplacian differential systems. We first build a new continuation theorem by topological degree, and then obtain the existence of rotating periodic solutions for two kinds of p-Laplacian differential systems via this continuation theorem, extend some existing relevant results.

We investigate the existence of families of symmetric periodic solutions of second kind as continuation of the elliptical orbits of the two-dimensional Kepler problem for certain symmetric differentiable perturbations using Delaunay coordinates. More precisely, we characterize the sufficient conditions for its existence and its type of stability is studied. The estimate on the characteristic multipliers of the symmetric periodic solutions is the new contribution to the field of symmetric periodic solutions. In addition, we present some results about the relationship between our symmetric periodic solutions and those obtained by the averaging method for Hamiltonian systems. As applications of our main results, we get new families of periodic solutions for: the perturbed hydrogen atom with stark and quadratic Zeeman effect, for the anisotropic Seeligers two-body problem and to the planar generalized Størmer problem.

We compare two partitions of real bitangents to smooth plane quartics into sets of 4: one coming from the closures of connected components of the avoidance locus and another coming from tropical geometry. When both are defined, we use the Tarski principle for real closed fields in combination with the topology of real plane quartics and the tropical geometry of bitangents and theta characteristics to show that they coincide.

The main objective of this paper is to establish the convergence for the fractional $p$-Laplacian of sequences of nonnegative functions with $p>2$. Furthermore, we show the blow-up phenomena for solutions to the extended Nirenberg problem modelled by fractional $p$-Laplacian with the prescribed negative functions.

We extend the notion of regular coherence from rings to additive categories and show that well-known consequences of regular coherence for rings also apply to additive categories. For instance, the negative K-groups and all twisted Nil-groups vanish for an additive category, if it is regular coherent. This will be applied to nested sequences of additive categories, motivated by our ongoing project to determine the algebraic K-theory of the Hecke algebra of a reductive p-adic group.