Volume 34, Issue 6

In this paper, we are concerned with the limit theory of stochastic McKean–Vlasov equations. First, we prove the optimal L p {L^{p}} ( p ⩾ 2 {p\geqslant 2} ) strong convergence rate of the Wong–Zakai approximation for stochastic McKean–Vlasov equations. Then we show the support theorem for stochastic McKean–Vlasov equations.

The article is devoted to studying the Singer transfer. The image of the Singer transfer Tr * ℝ ⁢ ℙ ∞ {\operatorname{Tr}_{*}^{\mathbb{R}\mathbb{P}^{\infty}}} for the infinite real projective space is proved to be a module over the image of the transfer Tr * {\operatorname{Tr}_{*}} for the sphere. Further, the algebraic Kahn–Priddy homomorphism is shown to be an epimorphism from Im ⁡ Tr * ℝ ⁢ ℙ ∞ {\operatorname{Im}\operatorname{Tr}_{*}^{\mathbb{R}\mathbb{P}^{\infty}}} onto Im ⁡ Tr * {\operatorname{Im}\operatorname{Tr}_{*}} in positive stems. The indecomposable elements h ^ i {\widehat{h}_{i}} for i ≥ 1 {i\geq 1} and c ^ i , d ^ i , e ^ i , f ^ i , p ^ i {\widehat{c}_{i},\widehat{d}_{i},\widehat{e}_{i},\widehat{f}_{i},\widehat{p}_{% i}} for i ≥ 0 {i\geq 0} are in the image of the Singer transfer Tr * ℝ ⁢ ℙ ∞ {\operatorname{Tr}_{*}^{\mathbb{R}\mathbb{P}^{\infty}}} , whereas the ones g ^ i {\widehat{g}_{i}} for i ≥ 1 {i\geq 1} and D ^ 3 ⁢ ( i ) , p ^ i ′ {\widehat{D}_{3}(i),\widehat{p}^{\prime}_{i}} for i ≥ 0 {i\geq 0} are not in its image. This transfer is shown to be not monomorphic in every positive homological degree. The transfer behavior is also investigated near “critical elements”. The squaring operation on the domain of Tr * ℝ ⁢ ℙ ∞ {\operatorname{Tr}_{*}^{\mathbb{R}\mathbb{P}^{\infty}}} is proved to be eventually isomorphic. This phenomenon leads to the so-called “stability” of the Singer transfer for the infinite real projective space under the iterated squaring operation.

We prove that the Stiefel–Whitney classes of a moment-angle manifold, not necessarily smooth, are trivial. We also consider Stiefel–Whitney classes of the partial quotient of a moment-angle manifold.

Let p be a prime and let G be a finite group. A complex character of G is called almost p -rational if its values belong to a cyclotomic field ℚ ⁢ ( e 2 ⁢ π ⁢ i / n ) {{\mathbb{Q}}(e^{2\pi i/n})} for some n ∈ ℤ + {n\in{\mathbb{Z}}^{+}} not divisible by p 2 {p^{2}} . We prove that, in contrast to usual p -rational characters, there are “many” almost p -rational irreducible characters in finite groups. We obtain both explicit and asymptotic bounds for the number of almost p -rational irreducible characters of G in terms of p . In fact, motivated by the McKay–Navarro conjecture, we obtain the same bound for the number of such characters of p ′ {p^{\prime}} -degree and prove that, in the minimal situation, the number of almost p -rational irreducible p ′ {p^{\prime}} -characters of G coincides with that of 𝐍 G ⁢ ( P ) {{\mathbf{N}}_{G}(P)} for P ∈ Syl p ⁡ ( G ) {P\in{\operatorname{Syl}}_{p}(G)} . Lastly, we propose a new way to detect the cyclicity of Sylow p -subgroups of a finite group G from its character table, using almost p -rational irreducible p ′ {p^{\prime}} -characters and the blockwise refinement of the McKay–Navarro conjecture.

We show that a four-dimensional homogeneous manifold with half-harmonic Weyl curvature tensor is symmetric or homothetic either to the only nonsymmetric anti-self-dual homogeneous manifold or to the 3-symmetric space.

Let the iterated function systems { S i } i = 1 N \{S_{i}\}_{i=1}^{N} be defined by S i ⁢ ( x ) = ( - 1 ) i - 1 ⁢ ρ ⁢ ( x + d i ) S_{i}(x)=(-1)^{i-1}\rho(x+d_{i}) , x ∈ R x\in\mathbb{R} , d i ∈ D d_{i}\in D , where 0 < ρ < 1 0<\rho<1 , and 𝐷 is a finite subset of ℤ. Let the measure μ ρ , P \mu_{\rho,P} be generated by the IFS and the probability weight P = { p i } i = 1 N P=\{p_{i}\}_{i=1}^{N} . In this paper, we introduce a new way to consider the spectrality of μ ρ , P \mu_{\rho,P} , and obtain a sufficient and necessary condition for N = 2 N=2 , and construct two examples of spectral measures for N = 2 ⁢ L ≥ 4 N=2L\geq 4 and N = 2 ⁢ L + 1 N=2L+1 , respectively.

In this article, we derive lower bounds for the number of distinct prime divisors of families of non-zero Fourier coefficients of non-CM primitive cusp forms and more generally of non-CM primitive Hilbert cusp forms. In particular, for the Ramanujan Δ-function, we show that, for any ϵ > 0 \epsilon>0 , there exist infinitely many natural numbers 𝑛 such that τ ⁢ ( p n ) \tau(p^{n}) has at least 2 ( 1 - ϵ ) ⁢ log ⁡ n log ⁡ log ⁡ n 2^{(1-\epsilon)\frac{\log n}{\log\log n}} distinct prime factors for almost all primes 𝑝. This improves and refines the existing bounds. We also study lower bounds for absolute norms of radicals of non-zero Fourier coefficients of modular forms alluded to above.

Let ( 𝕏 , d , μ ) {(\mathbb{X},d,\mu)} be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and let X ⁢ ( 𝕏 ) {X(\mathbb{X})} be a ball quasi-Banach function space on 𝕏 {\mathbb{X}} . In this article, the authors introduce the weak Hardy space W ⁢ H ~ X ⁢ ( 𝕏 ) {\widetilde{WH}_{X}(\mathbb{X})} associated with X ⁢ ( 𝕏 ) {X(\mathbb{X})} via the Lusin area function. Then the authors characterize W ⁢ H ~ X ⁢ ( 𝕏 ) {\widetilde{WH}_{X}(\mathbb{X})} by the molecule, the grand maximal function, and the Littlewood–Paley g -function and g λ * {g^{*}_{\lambda}} -function. Moreover, all these results have a wide generality. Particularly, the results of this article are also new even when they are applied, respectively, to weighted Lebesgue spaces, Orlicz spaces, and variable Lebesgue spaces, which actually are new even on RD-spaces (that is, spaces of homogeneous type with additional reverse doubling condition). The main novelties of this article exist in that the authors take full advantage of the geometrical properties of 𝕏 {\mathbb{X}} expressed by both the dyadic cubes and the exponential decay of the approximations of the identity to overcome the difficulties caused by the deficiencies of both the explicit expression of the quasi-norm of X ⁢ ( 𝕏 ) {X(\mathbb{X})} and the reverse doubling condition of μ, and that the authors use the tent space on 𝕏 × ℤ {\mathbb{X}\times\mathbb{Z}} to characterize W ⁢ H ~ X ⁢ ( 𝕏 ) {\widetilde{WH}_{X}(\mathbb{X})} by the Littlewood–Paley g λ * {g^{*}_{\lambda}} -function, where the range of λ might be best possible in some cases.

In this paper, we study Jacobi forms associated with the Leech lattice Λ which are invariant under the Conway group Co 0 {\operatorname{Co}_{0}} . We determine and construct generators of modules of both weak and holomorphic Jacobi forms of integral weight and fixed index t ≤ 3 {t\leq 3} . As applications, (i) we find the modular linear differential equations satisfied by the holomorphic generators; (ii) we determine the decompositions of many products of orbits of Leech vectors; (iii) we calculate the intersections between orbits and Leech vectors; (iv) we derive some conjugate relations among orbits modulo t ⁢ Λ {t\Lambda} .

In this paper, we investigate Hölder continuity of quasiconformal mappings in ℝ n {\mathbb{R}^{n}} from the points of view of quasihyperbolic geometry and the theory of Gromov hyperbolic spaces. We establish several characterizations of Gromov hyperbolic domains satisfying the Gehring–Martio-type quasihyperbolic boundary conditions. As applications, we generalize certain results concerning Hölder continuity of conformal mappings, establishing counterparts of results of Becker and Pommerenke, Smith and Stegenga, and Näkki and Palka in higher-dimensional Euclidean spaces.

In this article, we investigate the quantitative unique continuation properties of complex-valued solutions to drift equations in the plane. We consider equations of the form Δ ⁢ u + W ⋅ ∇ ⁡ u = 0 {\Delta u+W\cdot\nabla u=0} in ℝ 2 {\mathbb{R}^{2}} , where W = W 1 + i ⁢ W 2 {W=W_{1}+iW_{2}} with each W j {W_{j}} being real-valued. Under the assumptions that W j ∈ L q j {W_{j}\in L^{q_{j}}} for some q 1 ∈ [ 2 , ∞ ] {q_{1}\in[2,\infty]} , q 2 ∈ ( 2 , ∞ ] {q_{2}\in(2,\infty]} and that W 2 {W_{2}} exhibits rapid decay at infinity, we prove new global unique continuation estimates. This improvement is accomplished by reducing our equations to vector-valued Beltrami systems. Our results rely on a novel order of vanishing estimate combined with a finite iteration scheme.

Let 𝔤 {\mathfrak{g}} be a complex semisimple Lie algebra and let θ be a finite-order automorphism of 𝔤 {\mathfrak{g}} . Let 𝔤 0 {\mathfrak{g}_{0}} be the subalgebra { X ∈ 𝔤 : θ ⁢ ( X ) = X } {\{X\in\mathfrak{g}:\theta(X)=X\}} . In this article, we study for which pairs ( V 1 , V 2 ) {(V_{1},V_{2})} , consisting of two irreducible finite-dimensional representations of 𝔤 {\mathfrak{g}} , we have res 𝔤 0 ⁡ V 1 ≃ res 𝔤 0 ⁡ V 2 . \operatorname{res}_{\mathfrak{g}_{0}}V_{1}\simeq\operatorname{res}_{\mathfrak{% g}_{0}}V_{2}. In many cases, we show that V 1 {V_{1}} and V 2 {V_{2}} have isomorphic restrictions to 𝔤 0 {\mathfrak{g}_{0}} if and only if V 1 {V_{1}} is isomorphic to V 2 σ {V_{2}^{\sigma}} for some outer automorphism σ of 𝔤 {\mathfrak{g}} .

To study the boundedness of bi-parameter singular integral operators of non-convolution type in the Journé class, Fefferman discovered a boundedness criterion on bi-parameter Hardy spaces H p ⁢ ( ℝ n 1 × ℝ n 2 ) {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} by considering the action of the operators on rectangle atoms. More recently, the theory of multiparameter local Hardy spaces has been developed by the authors. In this paper, we establish this type of boundedness criterion on weighted bi-parameter local Hardy spaces h ω p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}_{\omega}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} . In comparison with the unweighted case, the uniform boundedness of rectangle atoms on weighted local bi-parameter Hardy spaces, which is crucial to establish the atomic decomposition on bi-parameter weighted local Hardy spaces, is considerably more involved. As an application, we establish the boundedness of bi-parameter pseudodifferential operators, including h ω p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}_{\omega}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} to L ω p ⁢ ( ℝ n 1 + n 2 ) {L_{\omega}^{p}(\mathbb{R}^{n_{1}+n_{2}})} and h ω p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}_{\omega}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} to h ω p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}_{\omega}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for all 0 < p ≤ 1 {0<p\leq 1} , which sharpens our earlier result even in the unweighted case requiring max ⁡ { n 1 n 1 + 1 , n 2 n 2 + 1 } < p ≤ 1 . \max\Bigl{\{}\frac{n_{1}}{n_{1}+1},\frac{n_{2}}{n_{2}+1}\Bigr{\}}<p\leq 1.