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1.
Let and be relatively prime monic irreducible polynomials in (). In this paper, we give an elementary proof for the following law of quadratic reciprocity in : where is the Legendre symbol.
2.
Shinji Adachi Kazunaga Tanaka 《Proceedings of the American Mathematical Society》2000,128(7):2051-2057
We study Trudinger type inequalities in and their best exponents . We show for , ( is the surface area of the unit sphere in ), there exists a constant such that
for all . Here is defined by
It is also shown that with is false, which is different from the usual Trudinger's inequalities in bounded domains.
3.
Consecutive numbers with the same Legendre symbol 总被引:1,自引:0,他引:1
Zhi-Hong Sun 《Proceedings of the American Mathematical Society》2002,130(9):2503-2507
Let be an odd prime, and be a complete set of residues . The goal of the paper is to determine all the values of such that or , where is the Legendre symbol.
4.
Chunjie Wang 《Proceedings of the American Mathematical Society》2006,134(7):2061-2066
Let be the Bergman space over the open unit disk in the complex plane. Korenblum's maximum principle states that there is an absolute constant , such that whenever ( ) in the annulus , then . In this paper we prove that Korenblum's maximum principle holds with .
5.
A global compactness result for singular elliptic problems involving critical Sobolev exponent 总被引:11,自引:0,他引:11
Let be a bounded domain such that . Let be a (P.S.) sequence of the functional . We study the limit behaviour of and obtain a global compactness result.
6.
Philippe Poulin 《Proceedings of the American Mathematical Society》2007,135(1):77-85
It is well known that the Green function of the standard discrete Laplacian on , exhibits a pathological behavior in dimension . In particular, the estimate fails for . This fact complicates the study of the scattering theory of discrete Schrödinger operators. Molchanov and Vainberg suggested the following alternative to the standard discrete Laplacian, and conjectured that the estimate holds for all . In this paper we prove this conjecture.
7.
Raymond Mortini 《Proceedings of the American Mathematical Society》2007,135(6):1795-1801
Let be the Banach algebra of all bounded analytic functions in the unit disk . A function is said to be universal with respect to the sequence of noneuclidian translates, if the set is locally uniformly dense in the set of all holomorphic functions bounded by . We show that for any sequence of points in tending to the boundary there exists a closed subspace of , topologically generated by Blaschke products, and linear isometric to , such that all of its elements are universal with respect to noneuclidian translates. The proof is based on certain interpolation problems in the corona of . Results on cyclicity of composition operators in are deduced.
8.
Gangsong Leng Lin Si Qingsan Zhu 《Proceedings of the American Mathematical Society》2004,132(9):2655-2660
For let and denote the arithmetic mean and geometric mean of elements of , respectively. It is proved that if is an integer in , then
with equality if and only if . Furthermore, as a generalization of this inequality, a mixed power-mean inequality for subsets is established.
with equality if and only if . Furthermore, as a generalization of this inequality, a mixed power-mean inequality for subsets is established.
9.
Zinoviy Grinshpun 《Proceedings of the American Mathematical Society》2003,131(5):1591-1600
We prove the following theorem. Any isometric operator , that acts from the Hilbert space with nonnegative weight to the Hilbert space with nonnegative weight , allows for the integral representation
where the kernels and satisfy certain conditions that are necessary and sufficient for these kernels to generate the corresponding isometric operators.
where the kernels and satisfy certain conditions that are necessary and sufficient for these kernels to generate the corresponding isometric operators.
10.
K. Tanahashi A. Uchiyama M. Uchiyama 《Proceedings of the American Mathematical Society》2003,131(8):2549-2552
We show Schwarz type inequalities and consider their converses. A continuous function is said to be semi-operator monotone on if is operator monotone on . Let be a bounded linear operator on a complex Hilbert space and be the polar decomposition of . Let and for . (1) If a non-zero function is semi-operator monotone on , then for , where . (2) If are semi-operator monotone on , then for . Also, we show converses of these inequalities, which imply that semi-operator monotonicity is necessary.
11.
Zhangjian Hu 《Proceedings of the American Mathematical Society》2003,131(7):2171-2179
We define an extended Cesàro operator with holomorphic symbol in the unit ball of as
where is the radial derivative of . In this paper we characterize those for which is bounded (or compact) on the mixed norm space .
where is the radial derivative of . In this paper we characterize those for which is bounded (or compact) on the mixed norm space .
12.
G. A. Karagulyan 《Proceedings of the American Mathematical Society》2007,135(10):3133-3141
We show that for any infinite set of unit vectors in the maximal operator defined by is not bounded in .
13.
14.
Xavier Tolsa 《Proceedings of the American Mathematical Society》2000,128(7):2111-2119
We give a geometric characterization of those positive finite measures on with the upper density finite at -almost every , such that the principal value of the Cauchy integral of ,
{\varepsilon}} \frac{1}{\xi-z}\, d\mu(\xi),\end{displaymath}">
exists for -almost all . This characterization is given in terms of the curvature of the measure . In particular, we get that for , -measurable (where is the Hausdorff -dimensional measure) with , if the principal value of the Cauchy integral of exists -almost everywhere in , then is rectifiable.
15.
John R. Akeroyd 《Proceedings of the American Mathematical Society》2002,130(11):3349-3354
Let be a finite, positive Borel measure with support in such that - the closure of the polynomials in - is irreducible and each point in is a bounded point evaluation for . We show that if 0$">and there is a nontrivial subarc of such that
then for each nontrivial closed invariant subspace for the shift on .
-\infty,\end{displaymath}">
then for each nontrivial closed invariant subspace for the shift on .
16.
Roberto Camporesi Emmanuel Pedon 《Proceedings of the American Mathematical Society》2002,130(2):507-516
The continuous spectrum of the Dirac operator on the complex, quaternionic, and octonionic hyperbolic spaces is calculated using representation theory. It is proved that , except for the complex hyperbolic spaces with even, where .
17.
Sunggeum Hong 《Proceedings of the American Mathematical Society》2000,128(12):3529-3539
We consider operators associated with the Fourier multipliers and show that is of weak type on , , for the critical value .
18.
Heekyoung Hahn 《Proceedings of the American Mathematical Society》2007,135(8):2391-2401
Let SL be a genus zero Fuchsian group of the first kind with as a cusp, and let be the holomorphic Eisenstein series of weight on that is nonvanishing at and vanishes at all the other cusps (provided that such an Eisenstein series exists). Under certain assumptions on and on a choice of a fundamental domain , we prove that all but possibly of the nontrivial zeros of lie on a certain subset of . Here is a constant that does not depend on the weight, is the upper half-plane, and is the canonical hauptmodul for
19.
It is known that, given a Banach space , the modulus of convexity associated to this space is a non-negative function, non-decreasing, bounded above by the modulus of convexity of any Hilbert space and satisfies the equation for every , where is a constant. We show that, given a function satisfying these properties then, there exists a Banach space in such a way its modulus of convexity is equivalent to , in Figiel's sense. Moreover this Banach space can be taken to be two-dimensional.
20.
Wojciech Czaja Jacek Zienkiewicz 《Proceedings of the American Mathematical Society》2008,136(1):89-94
Given a Schrödinger operator on with nonnegative potential , we present an atomic characterization of the associated Hardy space .