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1.
We study the injectivity properties of the spherical mean value operators associated to the Gelfand pairs (H n,K), whereK is a compact subgroup ofU(n). We show that these spherical mean value operators are injective onL p Hn) for 1≤p<∞. Forp=∞, these operators are not injective. Nevertheless, if the spherical meansf*μ i overK-orbits of sufficiently many points (z i,t i) ∈H n vanish, we identify a necessary and sufficient condition on the points (z i,t i) which guaranteesf=0. ForK=U(n), this is equivalent to the condition for the two-radius theorem. Research supported by N.B.H.M. Research Grant, Govt. of India.  相似文献   

2.
We consider the Cauchy problem of Navier-Stokes equations in weak Morrey spaces. We first define a class of weak Morrey type spaces Mp*,λ(Rn) on the basis of Lorentz space Lp,∞ = Lp*(Rn)(in particular, Mp*,0(Rn) = Lp,∞, if p > 1), and study some fundamental properties of them; Second,bounded linear operators on weak Morrey spaces, and establish the bilinear estimate in weak Morrey spaces. Finally, by means of Kato's method and the contraction mapping principle, we prove that the Cauchy problem of Navier-Stokes equations in weak Morrey spaces Mp*,λ(Rn) (1<p≤n) is time-global well-posed, provided that the initial data are sufficiently small. Moreover, we also obtain the existence and uniqueness of the self-similar solution for Navier-Stokes equations in these spaces, because the weak Morrey space Mp*,n-p(Rn) can admit the singular initial data with a self-similar structure. Hence this paper generalizes Kato's results.  相似文献   

3.
By compatibly grading the p-part of the Hecke algebra associated to Sp n (ℤ) and the subring of ℚ[x 0±1,…,x n ±1] invariant under the associated Weyl group, we produce a matrix representation of the Satake isomorphism restricted to the corresponding finite dimensional components. In particular, using the elementary divisor theory of integral matrices, we show how to determine the entries of this matrix representation restricted to double cosets of a fixed similitude. The matrix representation is upper-triangular, and can be explicitly inverted. To address the specific question of characterizing families of Hecke operators whose generating series have “Euler” products, we define (n+1) families of polynomial Hecke operators t k n (p ) (in ℚ[x 0±1,…,x n ±1]) for Sp n whose generating series ∑t k n (p )v are rational functions of the form q k (v)−1, where q k is a polynomial in ℚ[x 0±1,…,x n ±1][v] of degree (2 n if k=0). For k=0 and k=1 the form of the polynomial is essentially that of the local factors in the spinor and standard zeta functions. For k>1, these appear to be new expressions. Taking advantage of the generating series and our ability to explicitly invert the Satake isomorphism, we explicitly compute the classical operators with the analogous properties in the case of genus 2. It is of interest to note that these operators lie in the full, but not generally the integral, Hecke algebra.   相似文献   

4.
Let ℛ n (t) denote the set of all reducible polynomials p(X) over ℤ with degree n ≥ 2 and height ≤ t. We determine the true order of magnitude of the cardinality |ℛ n (t)| of the set ℛ n (t) by showing that, as t → ∞, t 2 log t ≪ |ℛ2(t)| ≪ t 2 log t and t n ≪ |ℛ n (t)| ≪ t n for every fixed n ≥ 3. Further, for 1 < n/2 < k < n fixed let ℛ k,n (t) ⊂ ℛ n (t) such that p(X) ∈ ℛ k,n (t) if and only if p(X) has an irreducible factor in ℤ[X] of degree k. Then, as t → ∞, we always have t k+1 ≪ |ℛ k,n (t)| ≪ t k+1 and hence |ℛ n−1,n (t)| ≫ |ℛ n (t)| so that ℛ n−1,n (t) is the dominating subclass of ℛ n (t) since we can show that |ℛ n (t)∖ℛ n−1,n (t)| ≪ t n−1(log t)2.On the contrary, if R n s (t) is the total number of all polynomials in ℛ n (t) which split completely into linear factors over ℤ, then t 2(log t) n−1R n s (t) ≪ t 2 (log t) n−1 (t → ∞) for every fixed n ≥ 2.   相似文献   

5.
Saturation classes for the sequenceK n (f, x) = ∫f(xt) n (t) of linear operators whereK n(f, x) is of the limited oscillation type, that is,μ n (t) is monotonic fort ≠ [− n , n ],σ n =o(1),n → ∞ and ∫t 2m n (t), are obtained. Examples of applications to some sequences of non-positive operators are given.  相似文献   

6.
LetH n ≅ℝ2n ⋉ℝ be the Heisenberg group and letμ t be the normalized surface measure for the sphere of radiust in ℝ2n . Consider the maximal function defined byM f=sup t>0|f*μ t |. We prove forn≥2 thatM defines an operator bounded onL p (H n ) provided thatp>2n/(2n−1). This improves an earlier result by Nevo and Thangavelu, and the range forL p boundedness is optimal. We also extend the result to a more general class of surfaces and to groups satisfying a nondegeneracy condition; these include the groups of Heisenberg type. The second author was supported in part by the National Science Foundation.  相似文献   

7.
It is proved that, for any Lipschitz function f(t 1, ..., t n ) of n variables, the corresponding map f op: (A 1, ...,A n ) → f(A 1, ..., A n ) on the set of all commutative n-tuples of Hermitian operators on a Hilbert space is Lipschitz with respect to the norm of each Schatten ideal S p , p ∈ (1,∞). This result is applied to the functional calculus of normal operators and contractions. It is shown that Lipschitz functions of one variable preserve domains of closed derivations with values in S p . It is also proved that the map f op is Fréchet differentiable in the norm of S p if f is continuously differentiable.  相似文献   

8.
In this work, we study the continuity of pseudodifferential operators on local Hardy spaces h p (ℝ n ) and generalize the results due to Goldberg and Taylor by showing that operators with symbols in S 1,δ 0(ℝ n ), 0≤δ<1, and in some subclasses of S 1,10(ℝ n ) are bounded on h p (ℝ n ) (0<p≤1). As an application, we study the local solvability of the planar vector field L= t +ib(x,t) x , b(x,t)≥0, in spaces of mixed norm involving Hardy spaces. Work supported in part by CNPq, FINEP, and FAPESP.  相似文献   

9.
We are interested in stability/instability of the zero steady state of the superlinear parabolic equation u t + Δ2 u = |u| p-1 u in , where the exponent is considered in the “super-Fujita” range p > 1 + 4/n. We determine the corresponding limiting growth at infinity for the initial data giving rise to global bounded solutions. In the supercritical case p > (n + 4)/(n−4) this is related to the asymptotic behaviour of positive steady states, which the authors have recently studied. Moreover, it is shown that the solutions found for the parabolic problem decay to 0 at rate t −1/(p-1).  相似文献   

10.
In this paper, we consider problems of approximation of stochastic θ-integrals (θ) 0 t f(B(s))dB(s) with respect to a Brownian motion by sums of the form ∑ k=1 p fn(B n θ (tk-1))[B n θ (tk)-B n θ (tk-1], where the sequences {fn,n∈∕#x007D; and {[B n θ ,n∈∕} are convolution-type approximations of the functionf and Brownian motionB. Belorussian State University, F. Skoryna ave. 4, 220050 Minsk, Belorus. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 2, pp. 248–256, April–June, 1999. Translated by V. Mackevičius  相似文献   

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