On representation of the solution of the Neumann problem in a domain with peak by the harmonic simple layer potential

The problem of finding a solution of the Neumann problem for the Laplacian in the form of a simple layer potential Vρ with unknown density ρ is known to be reducible to a boundary integral equation of the second kind to be solved for density. The Neumann problem is examined in a bounded n-dimensional domain Ω⁺ (n > 2) with a cusp of an outward isolated peak either on its boundary or in its complement Ω⁻ = R ⁿ \Ω⁺. Let Γ be the common boundary of the domains Ω^±, Tr(Γ) be the space of traces on Γ of functions with finite Dirichlet integral over R ⁿ , and Tr(Γ)* be the dual space to Tr(Γ). We show that the solution of the Neumann problem for a domain Ω⁻ with a cusp of an inward peak may be represented as Vρ⁻, where ρ⁻ ∈ Tr(Γ)* is uniquely determined for all Ψ⁻ ∈ Tr(Γ)*. If Ω⁺ is a domain with an inward peak and if Ψ⁺ ∈ Tr(Γ)*, Ψ⁺ ⊥ 1, then the solution of the Neumann problem for Ω⁺ has the representation u ⁺ = Vρ⁺ for some ρ⁺ ∈ Tr(Γ)* which is unique up to an additive constant ρ₀, ρ₀ = V ⁻¹(1). These results do not hold for domains with outward peak.     

Semi-primary hereditary rings

A semi-primary hereditary ring Σ, with radicalM and residue ring Γ=Σ/M, is uniquely determined by Γ and a Γ-bimoduleA=M/M ², whenever Σ admits a splitting Σ=Γ+A+M ².     

On the homotopy of the stable mapping class group

By considering all surfaces and their mapping class groups at once, it is shown that the classifying space of the stable mapping class group after plus construction, BΓ_∞ ⁺, has the homotopy type of an infinite loop space. The main new tool is a generalized group completion theorem for simplicial categories. The first deloop of BΓ_∞ ⁺ coincides with that of Miller [M] induced by the pairs of pants multiplication. The classical representation of the mapping class group onto Siegel's modular group is shown to induce a map of infinite loop spaces from BΓ_∞ ⁺ to K-theory. It is then a direct consequence of a theorem by Charney and Cohen [CC] that there is a space Y such that BΓ_∞ ⁺≃Im J _(1/2)×Y, where Im J _(1/2) is the image of J localized away from the prime 2. Oblatum 23-X-1995 &19-XI-1996     

Twisted traces of singular values

Let p be a prime for which the congruence group Γ₀(p)^* is of genus zero, and j _p ^* be the corresponding Hauptmodul. We investigate the twisted traces of singular values of j _p ^* and construct infinite products related to them. This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-331-C00006).     

Resolvent of the Laplacian on geometrically finite hyperbolic manifolds

For geometrically finite hyperbolic manifolds Γ\ℍ ⁿ⁺¹, we prove the meromorphic extension of the resolvent of Laplacian, Poincaré series, Einsenstein series and scattering operator to the whole complex plane. We also deduce the asymptotics of lattice points of Γ in large balls of ℍ ⁿ⁺¹ in terms of the Hausdorff dimension of the limit set of Γ.     

Solvability of nonlocal elliptic problems in weight spaces

In this paper we consider elliptic equations of order 2m in a bounded domainQ є R ⁿ with boundaryδQ and nonlocal conditions relating the traces of the solution and its derivatives on (n − 1)-dimensional smooth manifolds Γ _i (∪ _i =∂δQ) to their values on some compact setF ⊂Q, whereF ∩δQ ≠ Φ. The Fredholm solvability of these problems in the weight spacesV _{p, a} ^/l+2m (Q) is proved for arbitrary 1<p <∞. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 882–898, June, 2000. This research was supported by the Russian Foundation for Basic Research under grant No. 99-01-00028.     

Comparison of Twisted P-Form Spectra for Flat Manifolds with Diagonal Holonomy

We give an explicit formula for the multiplicities of the eigenvalues ofthe Laplacian acting on sections of natural vector bundles over acompact flat Riemannian manifold M = \ ⁿ , a Bieberbach group. In the case of the Laplacian acting onp-forms, twisted by a unitary character of , when hasdiagonal holonomy group F ₂ ^k , these multiplicities have acombinatorial expression in terms of integral values of Krawtchoukpolynomials and the so called Sunada numbers. If the Krawtchoukpolynomial K _p ⁿ (x)does not have an integral root, this expressioncan be inverted and conversely, the presence of such roots allows toproduce many examples of p-isospectral manifolds that are notisospectral on functions. We compare the notions of twistedp-isospectrality, twisted Sunada isospectrality and twisted finitep-isospectrality, a condition having to do with a finite part of thespectrum, proving several implications among them and getting a converseto Sunada's theorem in our context, for n 8. Furthermore, a finitepart of the spectrum determines the full spectrum. We give new pairs ofnonhomeomorphic flat manifolds satisfying some kind of isospectralityand not another. For instance: (a) manifolds which are isospectral onp-forms for only a few values of p 0, (b) manifolds which aretwisted isospectral for every , a nontrivial character of F, and(c) large twisted isospectral sets.     

A canonical subspace of modular forms of half-integral weight

We characterise the space of newforms of weight k + 1/2 on Γ₀(4N), N odd and square-free (studied by the second and third authors with Vasudevan) under the Atkin-Lehner W(4) operator. As an application, we show that the (±1)-eigensubspaces of the W(4) operator on the space of modular forms of weight k + 1/2 on Γ₀(4N) is mapped to modular forms of weight 2k on Γ₀(N), under a class of Shimura maps. The existence of such subspaces having this mapping property was conjectured by Zagier in a private communication. One of the special features of the (±1)-eigensubspaces is that the (2k + 1)-th power of the classical theta series of weight 1/2 belongs to the +  eigensubspace and hence this gives interesting congruences for r _2k+1(p ²).     

On the outradius of the teichmüller space

Let Γ be a fuchsian group which preserves the unit disc Δ and hence also its complement Δ^* in the Riemann sphere . The Bers embedding represents the Teichm=:uller space T(Γ) of Γ in the space (B (Δ^*, Γ) of bounded quadratic differentials for Γ in Δ^*. Then, T(Γ) is included in the closed ball centred at the origin of radius 6 inB(Δ^*, Γ) with respect to the norm employed in a paper by Nehari [The Schwarzian derivative and Schlicht functions; Bull. Amer. Math. Soc. 55 (1949), 545–551]. In other words the outradiuso(Γ) ofT(Γ) is not greater than 6. The purpose of this paper is to give a complete characterization of a fuchsian group Γ for which the outradiuso(Γ) ofT(Γ) attains this extremal value 6. The main theorem is: Let Γ be a fuchsian group preserving Δ^*. Then the outradiuso(Γ) of the Teichmüller spaceT(Γ) equals 6 if and only if for any positive numberd, either (i) there exists a hyperbolic disc of radiusd precisely invariant under the trivial subgroup, or (ii) there exists the collar of widthd about the axis of a hyperbolic element of Γ. Dedicated to Professor K?taro Oikawa on his 60th birthday     

Unique ergodicity of flows on homogeneous spaces

LetG be a unimodular Lie group, Γ a co-compact discrete subgroup ofG and ‘a’ a semisimple element ofG. LetT _a be the mapgΓ →ag Γ:G/Γ →G/Γ. The following statements are pairwise equivalent: (1) (T _a, G/Γ,θ) is weak-mixing. (2) (T _a, G/Γ) is topologically weak-mixing. (3) (G ^u, G/Γ) is uniquely ergodic. (4) (G ^u, G/Γ,θ) is ergodic. (5) (G ^u, G/Γ) is point transitive. (6) (G ^u, G/Γ) is minimal. If in additionG is semisimple with finite center and no compact factors, then the statement “(T _a, G/Γ,θ) is ergodic” may be added to the above list. The authors were partially supported by NSF grant MCS 75-05250.     

Mutual isomorphisms of translations of a homogeneous flow

Ergodic one-parameter flows (G/Γ,g _ℝ) induced by the left action of a subgroupg _ℝ ⊂G on homogeneous spaces of finite volume are considered. Letℳ ⊂ ℝ⁺ be the set of allt>0 such that the cascade (G/Γ,g _t ℤ) is metrically isomorphic to the cascade (G/Γ,g _ℤ). We prove that eitherℳ is at most countable or the subgroupg _ℝ is horocyclic andℳ=ℝ⁺. We prove that a metric isomorphism of ergodic quasi-unipotent cascades (or flows) is affine on almost all fibers of a certain natural bundle. The result generalizes Witte's theorem on the affinity of such isomorphisms of cascades with the mixing property; this is applied to the study of the structure of the setℳ ⊂ ℝ⁺. The proof is based on the fundamental Ratner theorem stating that the ergodic measures of unipotent cascades are algebraic. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 98–110, July, 1995. This research was partially supported by the Russian Foundation for Basic Research under grant No. 94-01-00082a and by the International Science Foundation.     

Pseudodifferential Operators Approach to Singular Integral Operators in Weighted Variable Exponent Lebesgue Spaces on Carleson Curves

The main results of the paper are: (1) The boundedness of singular integral operators in the variable exponent Lebesgue spaces L ^p(·)(Γ, w) on a class of composed Carleson curves Γ where the weights w have a finite set of oscillating singularities. The proof of this result is based on the boundedness of Mellin pseudodifferential operators on the spaces L^p(·)(\mathbbR ₊,dm){L^{p(\cdot )}(\mathbb{R} _{+},d\mu)} where dμ is an invariant measure on multiplicative group ${\mathbb{R}_{+}=\left\{r\in \mathbb{R}:r >0 \right\}}${\mathbb{R}_{+}=\left\{r\in \mathbb{R}:r >0 \right\}}. (2) Criterion of local invertibility of singular integral operators with piecewise slowly oscillating coefficients acting on L ^p(·)(Γ, w) spaces. We obtain this criterion from the corresponding criteria of local invertibility at the point 0 of Mellin pseudodifferential operators on \mathbbR₊{\mathbb{R}_{+}} and local invertibility of singular integral operators on \mathbbR{\mathbb{R}}. (3) Criterion of Fredholmness of singular integral operators in the variable exponent Lebesgue spaces L ^p(·)(Γ, w) where Γ belongs to a class of composed Carleson curves slowly oscillating at the nodes, and the weight w has a finite set of slowly oscillating singularities.     

On semigroups generated by restrictions of elliptic operators to invariant subspaces

Let A be the closed unbounded operator inL _p(G) that is associated with an elliptic boundary value problem for a bounded domainG. We prove the existence of a spectral projectionE determined by the set Γ = {λ;θ ₁≦argλ≦θ ₂} and show thatAE is the infinitesimal generator of an analytic semigroup provided that the following conditions hold: 1<p<∞; the boundary ϖΓ of Γ is contained in the resolvent setp(A) ofA;π/2≦θ<θ ₂≦3π/2 ; and there exists a constantc such that (I)││(λ-A)^-1││≦c/│λ│ for λ∈ϖΓ. The following consequence is obtained: Suppose that there exist constantsM andc such that λ∈p(A) and estimate (I) holds provided that |λ|≧M and Re λ=0. Then there exist bounded projectionE ⁻ andE ⁺ such thatA is completely reduced by the direct sum decompositionL _p(G)=E⁻L_p (G) ⊕E⁺L_p (G) and each of the operatorsAE ⁻ and—AE ⁺ is the infinitestimal generator of an analytic semigroup.     

Approximation by Faber-Laurent rational functions in the mean of functions of callsL p(Γ) with 1

Çavu§  A.  Israfilov  D. M. 《分析论及其应用》1995,11(1):105-118

Nonholonomic connections with extended Galilean groups of transformations

We continue the investigation of regular representations of the extended Galilean group {ie463-01} acting on the smooth canonical fiber bundle π: R ⁿ⁺¹ → I R ⁿ . First-and second-order nonholonomic affine connections Γ ₁, Γ ₂, Γ _1,2 are constructed using the results presented in papers [3, 4, 5, 6].     

On the abelianizations of congruence subgroups of Aut(<Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript>)

Let F _n be the free group of rank n, and let Aut⁺(F _n ) be its special automorphism group. For an epimorphism π : F _n → G of the free group F _n onto a finite group G we call the standard congruence subgroup of Aut⁺(F _n ) associated to G and π. In the case n = 2 we fully describe the abelianization of Γ⁺(G, π) for finite abelian groups G. Moreover, we show that if G is a finite non-perfect group, then Γ⁺(G, π) ≤ Aut⁺(F ₂) has infinite abelianization.     

A congruence for the Fourier coefficients of a modular form and its application to quadratic forms

Let F(z)=∑ _n=1^∞ A(n)q ⁿ denote the unique weight 6 normalized cuspidal eigenform on Γ₀(4). We prove that A(p)≡0,2,−1(mod 11) when p≠11 is a prime. We then use this congruence to give an application to the number of representations of an integer by quadratic form of level 4.      

On inhomogeneous diophantine approximation and Hausdorff dimension

Let Γ = Z A + Z ⁿ  ⊂ R ⁿ be a dense subgroup of rank n + 1 and let [^(w)] \hat{w} (A) denote the exponent of uniform simultaneous rational approximation to the generating point A. For any real number v ≥  [^(w)] \hat{w} (A), the Hausdorff dimension of the set B _v of points in R ⁿ that are v-approximable with respect to Γ is shown to be equal to 1/v.     

Deformations of complex structures on Γ/SL 2(C)

LetG be a connected complex semisimple Lie group. Let Γ be a cocompact lattice inG. In this paper, we show that whenG isSL ₂(C), nontrivial deformations of the canonical complex structure onX exist if and only if the first Betti number of the lattice Γ is non-zero. It may be remarked that for a wide class of arithmetic groups Γ, one can find a subgroup Γ′ of finite index in Γ, such that Γ′/[Γ′,Γ′] is finite (it is a conjecture of Thurston that this is true for all cocompact lattices inSL(2, C)). We also show thatG acts trivially on the coherent cohomology groupsH ⁱ(Γ/G, O) for anyi≥0.     

L^p-gradient Estimates of Symmetric Markov Semigroups for 1 〈 p ≤ 2

For 1 〈 p ≤2, an L^p-gradient estimate for a symmetric Markov semigroup is derived in a general framework, i.e. ‖Γ^/2（Ttf）‖p≤Cp/√t‖p, where F is a carre du champ operator. As a simple application we prove that F1/2（（I- L） ^-α） is a bounded operator from L^p to L^v provided that 1 〈 p 〈 2 and 1/2〈α〈1. For any 1 〈 p 〈 2, q 〉 2 and 1/2 〈α 〈 1, there exist two positive constants cq,α,Cp,α such that ‖Df‖p≤ Cp,α‖（I - L）^αf‖p,Cq,α（I-L）^（1-α）‖Df‖q＋‖f‖q, where D is the Malliavin gradient （[2]） and L the Ornstein-Uhlenbeck operator.