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1.
For a complex vector space , let be the algebra of polynomial functions on . In this paper, we construct bases for the algebra of all highest weight vectors in , where and for all , and the algebra of highest weight vectors in .
2.
Matthias Lesch Markus J. Pflaum 《Transactions of the American Mathematical Society》2000,352(11):4911-4936
We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space . For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone , we construct a unique ``symbol valued trace', which extends the -trace on operators of small order. This construction is in the spirit of a trace due to Kontsevich and Vishik in the nonparametric case. Our trace allows us to construct various trace functionals in a systematic way. Furthermore, we study the higher-dimensional eta-invariants on algebras with parameter space . Using Clifford representations we construct for each first order elliptic differential operator a natural family of parametric pseudodifferential operators over . The eta-invariant of this family coincides with the spectral eta-invariant of the operator.
3.
Shulim Kaliman Sté phane Vé né reau Mikhail Zaidenberg 《Transactions of the American Mathematical Society》2004,356(2):509-555
The Abhyankar-Sathaye Problem asks whether any biregular embedding can be rectified, that is, whether there exists an automorphism such that is a linear embedding. Here we study this problem for the embeddings whose image is given in by an equation , where and . Under certain additional assumptions we show that, indeed, the polynomial is a variable of the polynomial ring (i.e., a coordinate of a polynomial automorphism of ). This is an analog of a theorem due to Sathaye (1976) which concerns the case of embeddings . Besides, we generalize a theorem of Miyanishi (1984) giving, for a polynomial as above, a criterion for when .
4.
Rü diger W. Braun Reinhold Meise B. A. Taylor 《Transactions of the American Mathematical Society》2004,356(4):1315-1383
The local Phragmén-Lindelöf condition for analytic subvarieties of at real points plays a crucial role in complex analysis and in the theory of constant coefficient partial differential operators, as Hörmander has shown. Here, necessary geometric conditions for this Phragmén-Lindelöf condition are derived. They are shown to be sufficient in the case of curves in arbitrary dimension and of surfaces in . The latter result leads to a geometric characterization of those constant coefficient partial differential operators which are surjective on the space of all real analytic functions on .
5.
Albert Jeu-Liang Sheu 《Proceedings of the American Mathematical Society》2001,129(11):3307-3311
We show that the C*-algebra of a quantum sphere , 1$">, consists of continuous fields of operators in a C*-algebra , which contains the algebra of compact operators with , such that is a constant function of , where is the quotient map and is the unit circle.
6.
Cyrus P. Aryana Kevin F. Clancey 《Proceedings of the American Mathematical Society》2004,132(10):3007-3018
A study is made of the eigenvalues of self-adjoint Toeplitz operators on multiply connected planar regions having holes. The presence of eigenvalues is detected through an analysis of the zeros of translations of theta functions restricted to in .
7.
Michal Feckan 《Proceedings of the American Mathematical Society》2008,136(2):453-459
We investigate the solvability of polynomial equations on the -algebra of arithmetic functions .
8.
Georg Schneider 《Proceedings of the American Mathematical Society》2004,132(8):2399-2409
We consider Hankel operators of the form . Here . We show that in the case of one complex dimension the Hankel operators are compact but not Hilbert-Schmidt if 2k$">.
9.
Simon Gindikin Bernhard Krö tz 《Transactions of the American Mathematical Society》2002,354(8):3299-3327
In this paper we define a distinguished boundary for the complex crowns of non-compact Riemannian symmetric spaces . The basic result is that affine symmetric spaces of can appear as a component of this boundary if and only if they are non-compactly causal symmetric spaces.
10.
C. Denson Hill Mauro Nacinovich 《Proceedings of the American Mathematical Society》2005,133(6):1843-1850
There is an intrinsic notion of what it means for a contact manifold to be the smooth boundary of a Stein manifold. The same concept has another more extrinsic formulation, which is often used as a convenient working hypothesis. We give a simple proof that the two are equivalent. Moreover it is shown that, even though a border always exists, its germ is not unique; nevertheless the germ of the Dolbeault cohomology of any border is unique. We also point out that any Stein fillable compact contact -manifold has a geometric realization in via an embedding, or in via an immersion.
11.
We study the problem when an infinite system of linear functional equations
has a real analytic solution on for every right-hand side and give a complete characterization of such sequences of analytic functionals . We also show that every open set has a complex neighbourhood such that the positive answer is equivalent to the positive answer for the analogous question with solutions holomorphic on .
has a real analytic solution on for every right-hand side and give a complete characterization of such sequences of analytic functionals . We also show that every open set has a complex neighbourhood such that the positive answer is equivalent to the positive answer for the analogous question with solutions holomorphic on .
12.
Laura Felicia Matusevich Ezra Miller Uli Walther 《Journal of the American Mathematical Society》2005,18(4):919-941
We analyze the behavior of the holonomic rank in families of holonomic systems over complex algebraic varieties by providing homological criteria for rank-jumps in this general setting. Then we investigate rank-jump behavior for hypergeometric systems arising from a integer matrix and a parameter . To do so we introduce an Euler-Koszul functor for hypergeometric families over , whose homology generalizes the notion of a hypergeometric system, and we prove a homology isomorphism with our general homological construction above. We show that a parameter is rank-jumping for if and only if lies in the Zariski closure of the set of -graded degrees where the local cohomology of the semigroup ring supported at its maximal graded ideal is nonzero. Consequently, has no rank-jumps over if and only if is Cohen-Macaulay of dimension .
13.
Piotr Kot 《Proceedings of the American Mathematical Society》2007,135(12):3895-3903
We consider a circular, bounded, strictly convex domain with boundary of class . For any compact subset of we construct a sequence of homogeneous polynomials on which are big at each point of . As an application for any circular subset of type we construct a holomorphic function which is square integrable on and such that where denotes unit disc in .
14.
Krzysztof Plotka 《Proceedings of the American Mathematical Society》2003,131(4):1031-1041
We say that a function is a Hamel function ( ) if , considered as a subset of , is a Hamel basis for . We prove that every function from into can be represented as a pointwise sum of two Hamel functions. The latter is equivalent to the statement: for all there is a such that . We show that this fails for infinitely many functions.
15.
Steve Hofmann 《Proceedings of the American Mathematical Society》2008,136(12):4223-4233
We consider divergence form elliptic operators , defined in , where the coefficient matrix is , uniformly elliptic, complex and -independent. Using recently obtained results concerning the boundedness and invertibility of layer potentials associated to such operators, we show that if in , then for any vector-valued we have the bilinear estimate where and where is the usual non-tangential maximal operator. The result is new even in the case of real symmetric coefficients and generalizes an analogous result of Dahlberg for harmonic functions on Lipschitz graph domains. We also identify the domain of the generator of the Poisson semigroup for the equation in
16.
Francesca Astengo Bianca Di Blasio 《Proceedings of the American Mathematical Society》2006,134(5):1319-1329
The generalised Cayley transform from an Iwasawa -group into the corresponding real unit sphere induces isomorphisms between suitable Sobolev spaces and . We study the differential of , and we obtain a criterion for a function to be in .
17.
Alexander Brudnyi 《Transactions of the American Mathematical Society》2005,357(5):1859-1875
Let be a family of holomorphic functions in a domain depending holomorphically on . We study the distribution of zeros of in a subdomain whose boundary is a closed non-singular analytic curve. As an application, we obtain several results about distributions of zeros of families of generalized exponential polynomials and displacement maps related to certain ODE's.
18.
Alexandru D. Ionescu Stephen Wainger 《Journal of the American Mathematical Society》2006,19(2):357-383
We prove that if is a Calderón-Zygmund kernel and is a polynomial of degree with real coefficients, then the discrete singular Radon transform operator extends to a bounded operator on , . This gives a positive answer to an earlier conjecture of E. M. Stein and S. Wainger.
19.
El Maati Ouhabaz 《Proceedings of the American Mathematical Society》2006,134(12):3567-3575
We prove sharp large time Gaussian estimates for heat kernels of elliptic and Schrödinger operators, including Schrödinger operators with magnetic fields. Our estimates are then used to prove that for general (magnetic) Schrödinger operators , we have the -estimate (for large ): where is the spectral bound of The same estimate holds for elliptic and Schrödinger operators on general domains.
20.
Directional derivative estimates for Berezin symbols of bounded operators on Bergman spaces of arbitrary bounded domains in are obtained. These estimates also hold in the setting of the Segal-Bargmann space on . It is also shown that our estimates are sharp at every point of by exhibiting the optimizers explicitly.