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1.
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. 相似文献
2.
Let be the field of prime order It is known that for any integer one can construct a subset with such that One of the results of the present paper implies that if with then 相似文献
3.
Let be i.i.d. random variables with , and set . We prove that, for under the assumption that and Necessary and sufficient conditions for the convergence of the sum above were established by Lai (1974). 相似文献
4.
The main purpose of this paper is to obtain the well-known results of H.Fujita and K.Hayakawa on the nonexistence of nontrivial nonnegative global solutions for the Cauchy problem for the equation with on the half-space as a consequence of a new Liouville theorem of elliptic type for solutions of () on . This new result is in turn a consequence of other new phenomena established for nonlinear evolution problems. In particular, we prove that the inequality has no nontrivial solutions on when We also show that the inequality has no nontrivial nonnegative solutions for , and it has no solutions on bounded below by a positive constant for 1.$"> 相似文献
5.
Let , let and let be a bounded domain with a smooth boundary . Our purpose in this paper is to consider the existence of solutions of the problem: where 相似文献
6.
We say that the multinomial coefficient (m.c.) has order and power . Let be the number of m.c. that are not divisible by and have order with powers which are not larger than . If and then for any integer 相似文献
7.
We show that for any infinite set of unit vectors in the maximal operator defined by is not bounded in . 相似文献
8.
We prove that for Hilbert space operators and , it follows that ,\end{displaymath}"> where . Using the concept of -Gateaux derivative, we apply this result to characterize orthogonality in the sense of James in , and to give an easy proof of the characterization of smooth points in . 相似文献
9.
The following is proved: If is a function harmonic in the unit ball and if then the inequality holds, where is the nontangential maximal function of This improves a recent result of Stoll. This inequality holds for polyharmonic and hyperbolically harmonic functions as well. 相似文献
10.
Here is a particular case of the main result of this paper: Let be a bounded domain, with a boundary of class , and let be two continuous functions, , with 0$">, , with n$">. If and if the set of all global minima of the function has at least connected components, then, for each 0$"> small enough, the Neumann problem admits at least strong solutions in . 相似文献
11.
In this paper we give asymptotic estimates of the least energy solution of the functional as goes to infinity. Here is a smooth bounded domain of . Among other results we give a positive answer to a question raised by Chen, Ni, and Zhou (2000) by showing that . 相似文献
12.
In this note we consider the Sobolev inequality where is the best Sobolev constant and is the space obtained by taking the completion of with the norm . We prove here a refined version of this inequality, where is a positive constant, the distance is taken in the Sobolev space , and is the set of solutions which attain the Sobolev equality. This generalizes a result of Bianchi and Egnell (A note on the Sobolev inequality, J. Funct. Anal. 100 (1991), 18-24), which was posed by Brezis and Lieb (Sobolev inequalities with remainder terms, J. Funct. Anal. 62 (1985), 73-86). regarding the classical Sobolev inequality A key ingredient in our proof is the analysis of eigenvalues of the fourth order equation where and is the unique radial function in with . We will show that the eigenvalues of the above equation are discrete: and the corresponding eigenfunction spaces are 相似文献
13.
Let be a polar decomposition of an complex matrix . Then for every unitarily invariant norm , it is shown that where denotes the operator norm. This is a quantitative version of the well-known result that is normal if and only if . Related inequalities involving self-commutators are also obtained. 相似文献
14.
For a function defined on an interval let The principal result of this paper is the following Markov-type inequality for Müntz polynomials. Theorem. Let be an integer. Let be distinct real numbers. Let . Then where the supremum is taken for all (the span is the linear span over ). 相似文献
15.
Let denote the space of all real polynomials of degree at most . It is an old result of Stein and Wainger that for some constant depending only on . On the other hand, Carbery, Wainger and Wright claim that the true order of magnitude of the above principal value integral is . We prove that 相似文献
16.
Markov's inequality is for all polynomials . We prove a precise version of this inequality with an arbitrary continuum in the complex plane instead of the interval . 相似文献
17.
A very short proof is given of the inequality where and is the Poisson integral of 相似文献
18.
For a sparse polynomial , with and , we show that thus improving upon a bound of Mordell. Analogous results are obtained for Laurent polynomials and for mixed exponential sums. 相似文献
19.
Let be a polynomial with complex coefficients and roots , ..., , let denote its norm over the unit circle, and let denote Mahler's measure of . Gonçalves' inequality asserts that We prove that for , where is an explicit constant, and that for . We also establish additional lower bounds on the norms of a polynomial in terms of its coefficients. 相似文献
20.
A well-known multiplicative Weyl inequality states that the sequence of eigenvalues and the sequence of approximation numbers of any compact operator in a Banach space satisfy for all . We prove here that the constant is optimal, which solves a longstanding problem. 相似文献
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