On weighted compactness of commutators of bilinear maximal Calderón–Zygmund singular integral operators

Let T be a bilinear Calderón–Zygmund singular integral operator and let T * {T^{*}} be its corresponding truncated maximal operator. For any b ∈ BMO ⁡ ( ℝ n ) {b\in\operatorname{BMO}({\mathbb{R}^{n}})} and b → = ( b 1 , b 2 ) ∈ BMO ⁡ ( ℝ n ) × BMO ⁡ ( ℝ n ) {\vec{b}=(b_{1},b_{2})\in\operatorname{BMO}({\mathbb{R}^{n}})\times% \operatorname{BMO}({\mathbb{R}^{n}})} , let T b , j * {T^{*}_{b,j}} ( j = 1 , 2 {j=1,2} ) and T b → * {T^{*}_{\vec{b}}} be the commutators in the j-th entry and the iterated commutators of T * {T^{*}} , respectively. In this paper, for all 1 < p 1 , p 2 < ∞ {1<p_{1},p_{2}<\infty} , 1 p = 1 p 1 + 1 p 2 {\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}} , we show that T b , j * {T^{*}_{b,j}} and T b → * {T^{*}_{\vec{b}}} are compact operators from L p 1 ⁢ ( w 1 ) × L p 2 ⁢ ( w 2 ) {L^{p_{1}}(w_{1})\times L^{p_{2}}(w_{2})} to L p ⁢ ( v w → ) {L^{p}(v_{\vec{w}})} if b , b 1 , b 2 ∈ CMO ⁡ ( ℝ n ) {b,b_{1},b_{2}\in\operatorname{CMO}(\mathbb{R}^{n})} and w → = ( w 1 , w 2 ) ∈ A p → {\vec{w}=(w_{1},w_{2})\in A_{\vec{p}}} , v w → = w 1 p / p 1 ⁢ w 2 p / p 2 {v_{\vec{w}}=w_{1}^{p/p_{1}}w_{2}^{p/p_{2}}} . Here CMO ⁡ ( ℝ n ) {\operatorname{CMO}(\mathbb{R}^{n})} denotes the closure of 𝒞 c ∞ ⁢ ( ℝ n ) {\mathcal{C}_{c}^{\infty}(\mathbb{R}^{n})} in the BMO ⁡ ( ℝ n ) {\operatorname{BMO}(\mathbb{R}^{n})} topology and A p → {A_{\vec{p}}} is the multiple weights class.

Upper estimates of heat kernels for non-local Dirichlet forms on doubling spaces

In this paper, we present a new approach to obtaining the off-diagonal upper estimate of the heat kernel for any regular Dirichlet form without a killing part on the doubling space. One of the novelties is that we have obtained the weighted L 2 {L^{2}} -norm estimate of the survival function 1 - P t B ⁢ 1 B {1-P_{t}^{B}1_{B}} for any metric ball B, which yields a nice tail estimate of the heat semigroup associated with the Dirichlet form. The parabolic L 2 {L^{2}} mean-value inequality is borrowed to use.

Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities

The Levi theorem tells us that every finite-dimensional Lie algebra is the semi-direct product of a semi-simple Lie algebra and a solvable Lie algebra. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. Therefore, it is important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this paper, we give a new connection between nilpotent Lie algebras and nilradicals of derivation Lie algebras of isolated complete intersection singularities. As an application, we obtain the correspondence between the nilpotent Lie algebras of dimension less than or equal to 7 and the nilradicals of derivation Lie algebras of isolated complete intersection singularities with modality less than or equal to 1. Moreover, we give a new characterization theorem for zero-dimensional simple complete intersection singularities.

On cutting blocking sets and their codes

Let PG ⁡ ( r , q ) {\operatorname{PG}(r,q)} be the r-dimensional projective space over the finite field GF ⁡ ( q ) {\operatorname{GF}(q)} . A set 𝒳 {\mathcal{X}} of points of PG ⁡ ( r , q ) {\operatorname{PG}(r,q)} is a cutting blocking set if for each hyperplane Π of PG ⁡ ( r , q ) {\operatorname{PG}(r,q)} the set Π ∩ 𝒳 {\Pi\cap\mathcal{X}} spans Π. Cutting blocking sets give rise to saturating sets and minimal linear codes, and those having size as small as possible are of particular interest. We observe that from a cutting blocking set obtained in [20], by using a set of pairwise disjoint lines, there arises a minimal linear code whose length grows linearly with respect to its dimension. We also provide two distinct constructions: a cutting blocking set of PG ⁡ ( 3 , q 3 ) {\operatorname{PG}(3,q^{3})} of size 3 ⁢ ( q + 1 ) ⁢ ( q 2 + 1 ) {3(q+1)(q^{2}+1)} as a union of three pairwise disjoint q-order subgeometries, and a cutting blocking set of PG ⁡ ( 5 , q ) {\operatorname{PG}(5,q)} of size 7 ⁢ ( q + 1 ) {7(q+1)} from seven lines of a Desarguesian line spread of PG ⁡ ( 5 , q ) {\operatorname{PG}(5,q)} . In both cases, the cutting blocking sets obtained are smaller than the known ones. As a byproduct, we further improve on the upper bound of the smallest size of certain saturating sets and on the minimum length of a minimal q-ary linear code having dimension 4 and 6.

Inertia groups and smooth structures on quaternionic projective spaces

This paper deals with certain results on the number of smooth structures on quaternionic projective spaces, obtained through the computation of inertia groups and their analogues, which in turn are computed using techniques from stable homotopy theory. We show that the concordance inertia group is trivial in dimension 20, but there are many examples in high dimensions where the concordance inertia group is non-trivial. We extend these to computations of concordance classes of smooth structures. These have applications to 3-sphere actions on homotopy spheres and tangential homotopy structures.

Embeddings of locally compact abelian p-groups in Hawaiian groups

We show that locally compact abelian p-groups can be embedded in the first Hawaiian group on a compact path connected subspace of the Euclidean space of dimension four. This result gives a new geometric interpretation for the classification of locally compact abelian groups which are rich in commuting closed subgroups. It is then possible to introduce the idea of an algebraic topology for topologically modular locally compact groups via the geometry of the Hawaiian earring. Among other things, we find applications for locally compact groups which are just noncompact.

A theorem of Roe and Strichartz on homogeneous trees

Let 𝔛 {\mathfrak{X}} be a homogeneous tree and let ℒ {\mathcal{L}} be the Laplace operator on 𝔛 {\mathfrak{X}} . In this paper, we address problems of the following form: Suppose that { f k } k ∈ ℤ {\{f_{k}\}_{k\in\mathbb{Z}}} is a doubly infinite sequence of functions in 𝔛 {\mathfrak{X}} such that for all k ∈ ℤ {k\in\mathbb{Z}} one has ℒ ⁢ f k = A ⁢ f k + 1 {\mathcal{L}f_{k}=Af_{k+1}} and ∥ f k ∥ ≤ M {\lVert f_{k}\rVert\leq M} for some constants A ∈ ℂ {A\in\mathbb{C}} , M > 0 {M>0} and a suitable norm ∥ ⋅ ∥ {\lVert\,\cdot\,\rVert} . From this hypothesis, we try to infer that f 0 {f_{0}} , and hence every f k {f_{k}} , is an eigenfunction of ℒ {\mathcal{L}} . Moreover, we express f 0 {f_{0}} as the Poisson transform of functions defined on the boundary of 𝔛 {\mathfrak{X}} .

On length densities

For a commutative cancellative monoid M, we introduce the notion of the length density of both a nonunit x ∈ M {x\in M} , denoted LD ⁡ ( x ) {\operatorname{LD}(x)} , and the entire monoid M, denoted LD ⁡ ( M ) {\operatorname{LD}(M)} . This invariant is related to three widely studied invariants in the theory of nonunit factorizations, L ⁢ ( x ) {L(x)} , ℓ ⁢ ( x ) {\ell(x)} , and ρ ⁢ ( x ) {\rho(x)} . We consider some general properties of LD ⁡ ( x ) {\operatorname{LD}(x)} and LD ⁡ ( M ) {\operatorname{LD}(M)} and give a wide variety of examples using numerical semigroups, Puiseux monoids, and Krull monoids. While we give an example of a monoid M with irrational length density, we show that if M is finitely generated, then LD ⁡ ( M ) {\operatorname{LD}(M)} is rational and there is a nonunit element x ∈ M {x\in M} with LD ⁡ ( M ) = LD ⁡ ( x ) {\operatorname{LD}(M)=\operatorname{LD}(x)} (such a monoid is said to have accepted length density). While it is well known that the much studied asymptotic versions of L ⁢ ( x ) {L(x)} , ℓ ⁢ ( x ) {\ell(x)} , and ρ ⁢ ( x ) {\rho(x)} (denoted L ¯ ⁢ ( x ) {\overline{L}(x)} , ℓ ¯ ⁢ ( x ) {\overline{\ell}(x)} , and ρ ¯ ⁢ ( x ) {\overline{\rho}(x)} ) always exist, we show the somewhat surprising result that LD ¯ ⁢ ( x ) = lim n → ∞ ⁡ LD ⁡ ( x n ) {\overline{\operatorname{LD}}(x)=\lim_{n\rightarrow\infty}\operatorname{LD}(x^% {n})} may not exist. We also give some finiteness conditions on M that force the existence of LD ¯ ⁢ ( x ) {\overline{\operatorname{LD}}(x)} .

Resistance scaling on 4N-carpets

The 4 ⁢ N {4N} -carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a 4 ⁢ N {4N} -carpet F, let { F n } n ≥ 0 {\{F_{n}\}_{n\geq 0}} be the natural decreasing sequence of compact pre-fractal approximations with ⋂ n F n = F {\bigcap_{n}F_{n}=F} . On each F n {F_{n}} , let ℰ ⁢ ( u , v ) = ∫ F N ∇ ⁡ u ⋅ ∇ ⁡ v ⁢ d ⁢ x {\mathcal{E}(u,v)=\int_{F_{N}}\nabla u\cdot\nabla v\,dx} be the classical Dirichlet form and u n {u_{n}} be the unique harmonic function on F n {F_{n}} satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by [M. T. Barlow and R. F. Bass, On the resistance of the Sierpiński carpet, Proc. Roy. Soc. Lond. Ser. A 431 (1990), no. 1882, 345–360], we prove a resistance estimate of the following form: there is ρ = ρ ⁢ ( N ) > 1 {\rho=\rho(N)>1} such that ℰ ⁢ ( u n , u n ) ⁢ ρ n {\mathcal{E}(u_{n},u_{n})\rho^{n}} is bounded above and below by constants independent of n. Such estimates have implications for the existence and scaling properties of Brownian motion on F.