Kernel theorems in the setting of mixed nonquasi-analytic classes

Let Ω₁⊂Rr and Ω₂⊂Rs be nonempty and open. We introduce the Beurling-Roumieu spaces D(ω₁,ω₂}(Ω₁×Ω₂), D(M,M^′}(Ω₁×Ω₂) and obtain tensor product representations of them. This leads for instance to kernel theorems of the following type: every continuous linear map from the Beurling space D(ω₁)(Ω₁) (respectively D(M)(Ω₁)) into the strong dual of the Roumieu space D{ω₂}(Ω₂) (respectively D{M^′}(Ω₂)) can be represented by a continuous linear functional on D(ω₁,ω₂}(Ω₁×Ω₂) (respectively D(M,M^′}(Ω₁×Ω₂)).     

Identifications in modular group algebras

Let G be a group, S a subgroup of G, and F a field of characteristic p. We denote the augmentation ideal of the group algebra FG by ω(G). The Zassenhaus-Jennings-Lazard series of G is defined by D_n(G)=G∩(1+ωⁿ(G)). We give a constructive proof of a theorem of Quillen stating that the graded algebra associated with FG is isomorphic as an algebra to the enveloping algebra of the restricted Lie algebra associated with the D_n(G). We then extend a theorem of Jennings that provides a basis for the quotient ωⁿ(G)/ωⁿ⁺¹(G) in terms of a basis of the restricted Lie algebra associated with the D_n(G). We shall use these theorems to prove the main results of this paper. For G a finite p-group and n a positive integer, we prove that G∩(1+ω(G)ωⁿ(S))=D_n+1(S) and G∩(1+ω²(G)ωⁿ(S))=D_n+2(S)D_n+1(S∩D₂(G)). The analogous results for integral group rings of free groups have been previously obtained by Gruenberg, Hurley, and Sehgal.     

The property O n for finite seminormal functors

We give a finite combinatorial test for finite seminormal functors to possess the property O _n and use it in establishing that in some cases this property leads to some well-known functors. For example, if some functor F possesses the property O ₂ then F ₂ coincides with either exp₂ or the squaring functor. Hence we conclude that if F(D ^ω ₁) and D ^ω ₁ are homeomorphic then F ₂ is either exp₂ or (·)².     

Harmonic measure on simply connected domains of fixed inradius

LetD?C be a simply connected domain that contains 0 and does not contain any disk of radius larger than 1. ForR>0, letω _D (R) denote the harmonic measure at 0 of the set {z:|z|?R}??D. Then it is shown thatthere exist β>0and C>0such that for each such D,ω _D (R)≤Ce ^?βR ,for every R>0. Thus a natural question is: What is the supremum of all β′s , call it β₀, for which the above inequality holds for every suchD? Another formulation of the problem involves hyperbolic metric instead of harmonic measure. Using this formulation a lower bound for β₀ is found. Upper bounds for β₀ can be obtained by constructing examples of domainsD. It is shown that a certain domain whose boundary consists of an infinite number of vertical half-lines, i.e. a comb domain, gives a good upper bound. This bound disproves a conjecture of C. Bishop which asserted that the strips of width 2 are extremal domains. Harmonic measures on comb domains are also studied.     

ω-hypoelliptic differential operators of constant strength

We study ω-hypoelliptic differential operators of constant strength. We show that any operator with constant strength and coefficients in which is homogeneous ω-hypoelliptic is also σ-hypoelliptic for any weight function σ=O(ω). We also present a sufficient condition in order to ensure that a differential operator admits a parametrix and, as a consequence, we obtain some conditions on the weights (ω,σ) to conclude that, for any operator P(x,D) with constant strength, the σ-hypoellipticity of the frozen operator P(x₀,D) implies the ω-hypoellipticity of P(x,D). This requires the use of pseudodifferential operators.     

Properties of derivations on some convolution algebras

For all convolution algebras L ¹[0, 1); L _loc ¹ and A(ω) = ∩ _n L ¹(ω _n ), the derivations are of the form D _μ f = Xf * μ for suitable measures μ, where (Xf)(t) = tf(t). We describe the (weakly) compact as well as the (weakly) Montel derivations on these algebras in terms of properties of the measure μ. Moreover, for all these algebras we show that the extension of D _μ to a natural dual space is weak-star continuous.     

Dynamical magnetic susceptibility in the spin-fermion model for cuprate superconductors

Using the method of diagram techniques for the spin and Fermi operators in the framework of the SU(2)-invariant spin-fermion model of the electron structure of the CuO₂plane of copper oxides, we obtain an exact representation of the Matsubara Green’s function D_⊥(k, iω _m ) of the subsystem of localized spins. This representation includes the Larkin mass operator Σ_L(k, iω _m ) and the strength and polarization operators P(k, iω _m ) and Π(k, iω _m ). The calculation in the one-loop approximation of the mass and strength operators for the Heisenberg spin system in the quantum spin-liquid state allows writing the Green’s function D_⊥(k, iω _m ) explicitly and establishing a relation to the result of Shimahara and Takada. An essential point in the developed approach is taking the spin-polaron nature of the Fermi quasiparticles in the spin-fermion model into account in finding the contribution of oxygen holes to the spin response in terms of the polarization operator Π(k, iω _m ).     

The trace-form of an algebraic number field

Let F be the rational field or a p-adic field, and let K an algebraic number field over F. If ω₁,…, ω_n is an integral basis for the ring $\text{D}$_L of integers in K, then the quadratic form Q whose matrix is (trace_K∥F(ω_iω_j)) has integral coefficients, and is called an integral trace-form. Q is determined by K up to integral equivalence. The purpose of this paper is to show that the genus of Q determines the ramification of primes in K.     

A note on D-spaces

We show that every regular T₁ submeta-Lindelöf space of cardinality ω₁ is D under MA+¬CH, which answers a question posed by Gruenhage (2011) [9]. Borges (1991) [5] asked if every monotonically normal paracompact space is a D-space, we give a characterization of paracompactness for monotonically normal spaces, which may be of some use in solving this problem.