共查询到20条相似文献,搜索用时 879 毫秒
1.
Following the lines of [13] we introduce the classes of mixed smoothness αp(Rn), αp(Zn) for a multi-index α = (α1,…, αn). Such classes are naturally tied up with the study of semi-elliptic differential and difference equations.Besides a brief presentation of such classes, we concentrate our research on the study of mixed homogeneous multipliers with homogeneity β = (β1, …, βn) and their preservation of mixed homogeneous Hölder classes α∞, for a different multi-index α.In the last paragraph we apply the results to produce various improvements of the classical Schauder's estimates, for differential and difference equations, in the parabolic and elliptic case. 相似文献
2.
Lisa A Mantini 《Journal of Functional Analysis》1985,60(2):211-242
Starting from the realization of the Fock space as L2-cohomology of p + q, 0,p(p + q) = ⊕m?m0,p(p + q), an integral transform is constructed which is a direct-image mapping from m0,p(p + q) into the space of holomorphic sections of some vector bundle Em over M ≈ U(p, q)/(U(q) × U(p)), m ? 0. The transform intertwines the natural actions of U(p, q) and is injective if m ? 0, so it provides a geometric realization of the ladder representations of U(p, q). The sections in the image of the transform satisfy certain linear differential equations, which are explicitly described. For example, Maxwell's equations are of this form if p = q = 2 and m = 2. Thus, this transform is analogous to the Penrose correspondence. 相似文献
3.
The set of all rearrangement invariant function spaces on [0,1] having the p-Banach–Saks property has a unique maximal element for all p∈(1,2]. For p=2 this is L2, for p∈(1,2) this is Lp,∞0. We compute the Banach–Saks index for the families of Lorentz spaces , 1?q?∞, and Lorentz–Zygmund spaces L(p,α), , extending the classical results of Banach–Saks and Kadec–Pelczynski for Lp-spaces. Our results show that the set of rearrangement invariant spaces with Banach–Saks index p∈(1,2] is not stable with respect to the real and complex interpoltaion methods. To cite this article: E.M. Semenov, F.A. Sukochev, C. R. Acad. Sci. Paris, Ser. I 337 (2003). 相似文献
4.
Let and denote respectively the space of n×n complex matrices and the real space of n×n hermitian matrices. Let p,q,n be positive integers such that p?q?n. For , the (p,q)-numerical range of A is the set , where Cp(X) is the pth compound matrix of X, and Jq is the matrix Iq?On-q. Let denote n or . The problem of determining all linear operators T: → such that is treated in this paper. 相似文献
5.
René Carmona 《Journal of Functional Analysis》1979,33(3):259-296
First we compute Brownian motion expectations of some Kac's functionals. This allows a complete study of the semigroups generated by the formal differential operator on the various Lebesgue's spaces Lq=Lqn, dx, whenever the negative part of V is in L∞ + Lp for some . Our approach is probabilistic and some of the proofs are surprisingly elementary. The negative infinitesimal generators of our semigroups are shown to be reasonable self-adjoint extensions of H. Under mild assumptions on V, H is unitary equivalent to the Dirichlet operator, say D, associated to its groundstate measure. We study regularity of the semigroups generated by D. We concentrate on hyper and supercontractivity and we give, using probabilistic techniques, new examples of potential functions V which give rise to hyper and supercontractive Dirichlet semigroups. 相似文献
6.
Helmut Abels 《Mathematische Nachrichten》2006,279(4):351-367
In this paper we prove unique solvability of the generalized Stokes resolvent equations in an infinite layer Ω0 = ℝn –1 × (–1, 1), n ≥ 2, in Lq ‐Sobolev spaces, 1 < q < ∞, with slip boundary condition of on the “upper boundary” ∂Ω+0 = ℝn –1 × {1} and non‐slip boundary condition on the “lower boundary” ∂Ω–0 = ℝn –1 × {–1}. The solution operator to the Stokes system will be expressed with the aid of the solution operators of the Laplace resolvent equation and a Mikhlin multiplier operator acting on the boundary. The present result is the first step to establish an Lq ‐theory for the free boundary value problem studied by Beale [9] and Sylvester [22] in L 2‐spaces. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
7.
Zhongli Wei 《Journal of Mathematical Analysis and Applications》2007,328(2):1255-1267
This paper investigates the existence of positive solutions of singular Dirichlet boundary value problems for second order differential system. A necessary and sufficient condition for the existence of C[0,1]×C[0,1] positive solutions as well as C1[0,1]×C1[0,1] positive solutions is given by means of the method of lower and upper solutions and the fixed point theorems. Our nonlinearity fi(t,x1,x2) may be singular at x1=0, x2=0, t=0 and/or t=1, i=1,2. 相似文献
8.
《Nonlinear Analysis: Theory, Methods & Applications》2004,56(2):185-199
In this paper we are concerned with positive solutions of the doubly nonlinear parabolic equation ut=div(um−1|∇u|p−2∇u)+Vum+p−2 in a cylinder , with initial condition u(·,0)=u0(·)⩾0 and vanishing on the parabolic boundary . Here (resp. ) is a bounded domain with smooth boundary, , , 1<p<N and m+p−2>0. The critical exponents are found and the nonexistence results are proved for . 相似文献
9.
Stephen C Milne 《Journal of Combinatorial Theory, Series A》1982,33(1):36-47
Let Vn(q) denote the n-dimensional vector space over the finite field with q elements, and Ln(q) be the lattice of subspaces of Vn(q). Two rank- and order-preserving maps from Ln(q) onto the lattice of subsets of an n-set are constructed. Three equivalent formulations of these maps are given: an inductive procedure based on an elementary combinatorial interpretation of a well-known pair of difference equations satisfied by the Gaussian coefficients [], a direct set-theoretical definition, and, a direct definition involving a certain pair of modular chains in Ln(q). The direct set-theoretical definition of one of these maps has already been given by Knuth. Knuth's map, however, may be systematically discovered by means of the inductive procedure and the direct lattice-theoretic definition shows how it can be generalized. As a further application of the pair of difference equations satisfied by [], a direct-combinatorial proof of an identity of Carlitz that expands Gaussian coefficients in terms of binomial coefficients has been formulated. 相似文献
10.
In the space L 2[0, π], we consider the operators $$ L = L_0 + V, L_0 = - y'' + (\nu ^2 - 1/4)r^{ - 2} y (\nu \geqslant 1/2) $$ with the Dirichlet boundary conditions. The potential V is the operator of multiplication by a function (in general, complex-valued) in L 2[0, π] satisfying the condition $$ \int\limits_0^\pi {r^\varepsilon } (\pi - r)^\varepsilon |V(r)|dr < \infty , \varepsilon \in [0,1] $$ . We prove the trace formula Σ n=1 ∞ [µ n ? λ n ? Σ k=1 m α k (n) ] = 0. 相似文献
11.
Hermann König 《Journal of Functional Analysis》1977,24(1):32-51
For an open set Ω ? N, 1 ? p ? ∞ and λ ∈ +, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators , 1 ? p, q ? ∞ and a quasibounded domain Ω ? N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map exists and belongs to the given Banach ideal : Assume the quasibounded domain fulfills condition Ckl for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any to the boundary ?Ω tends to zero as for , and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ , μ > λ S(; p,q:N) and v > N/l · λD(;p,q), one has that belongs to the Banach ideal . Here λD(;p,q;N)∈+ and λS(;p,q;N)∈+ are the D-limit order and S-limit order of the ideal , introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings lpn → lqn for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω fulfills condition C1l.For an open set Ω in N, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in N and give sufficient conditions on λ such that the Sobolev imbedding operator exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω is a quasibounded open set in N. 相似文献
12.
《Nonlinear Analysis: Theory, Methods & Applications》2004,57(3):349-362
We show that in a smooth bounded domain , n⩾2, all global nonnegative solutions of ut−Δum=up with zero boundary data are uniformly bounded in by a constant depending on and τ but not on u0, provided that 1<m<p<[(n+1)/(n−1)]m. Furthermore, we prove an a priori bound in depending on under the optimal condition 1<m<p<[(n+2)/(n−2)]m. 相似文献
13.
E. D. Nursultanov 《Proceedings of the Steklov Institute of Mathematics》2006,255(1):185-202
Let (X, Y) be a pair of normed spaces such that X ? Y ? L 1[0, 1] n and {e k } k be an expanding sequence of finite sets in ? n with respect to a scalar or vector parameter k, k ∈ ? or k ∈ ? n . The properties of the sequence of norms $\{ \left\| {S_{e_k } (f)} \right\|x\} _k $ of the Fourier sums of a fixed function f ∈ Y are studied. As the spaces X and Y, the Lebesgue spaces L p [0, 1], the Lorentz spaces L p,q [0, 1], L p,q [0, 1] n , and the anisotropic Lorentz spaces L p,q*[0, 1] n are considered. In the one-dimensional case, the sequence {e k } k consists of segments, and in the multidimensional case, it is a sequence of hyperbolic crosses or parallelepipeds in ? n . For trigonometric polynomials with the spectrum given by step hyperbolic crosses and parallelepipeds, various types of inequalities for different metrics in the Lorentz spaces L p,q [0, 1] n and L p,q*[0, 1] n are obtained. 相似文献
14.
We consider nonlinear boundary value problems of the type for the existence of solutions. It is assumed that L is a 2nth-order linear differential operator in the real Hilbert space S = L2[a, b] which admits a decomposition of the form where T is an nth-order linear differential operator and N is a nonlinear operator defined on a subspace of S. The decomposition of L induces a natural decomposition of the generalized inverse of L. Using the method of “alternative problems,” we split the boundary value problem into an equivalent system of two equations. The theory of monotone operators and the theory of nonlinear Hammerstein equations are then utilized to consider the solvability of the equivalent system. 相似文献
15.
Structure is developed on the set of real-valued stochastic processes in terms of the authors recently defined statistical measures making explicit an -calculus over the structure. This proves that the stochastic-differential equation y=x, where x is a stochastic process and is an nth order linear-stochastic differential operator with up to n ? 1 stochastic-process coefficients, is solved by Adomian's series, and finally, establishes the existence and uniqueness of the statistical measures of the solution process. 相似文献
16.
Allan J. Sieradski 《Topology》1978,17(1):85-93
THIS PAPER investigates the structure of the semigroup Σ generated by a set S of non-cancellation examples in the homotopy category. The featured spaces are the 3-dimensional Lens spaces Lp.q. Their products Lp.q × S3 with the 3-sphere S3 are shown to have the same simple-homotopy type, while their own products are shown to determine a unique-division semigroup. 相似文献
17.
Nonlinear partial differential operators having the form G(u) = g(u, D1u,…, DNu), with g?C(R × RN), are here shown to be precisely those operators which are local, (locally) uniformly continuous on, , and (roughly speaking) translation invariant. It is also shown that all such partial differential operators are necessarily bounded and continuous with respect to the norm topologies of . 相似文献
18.
Allan C. Peterson 《Journal of Mathematical Analysis and Applications》1975,52(3):573-582
In this paper we are concerned with proving comparison theorems, under various assumptions, for the (p, q)-boundary value problem for the nth order nonlinear differential equation and the linear differential equation lny = (?1)qk(t)y, where ln is the classical nth order linear operator with leading coefficient one. 相似文献
19.
J. V. Brawley 《Linear algebra and its applications》1975,10(3):199-217
Let F=GF(q) denote the finite field of order q, and let . Then f(x) defines, via substitution, a function from Fn×n, the n×n matrices over F, to itself. Any function which can be represented by a polynomialf(x)?F[x] is called a scalar polynomial function on Fn×n. After first determining the number of scalar polynomial functions on Fn×n, the authors find necessary and sufficient conditions on a polynomial in order that it defines a permutation of (i) n, the diagonalizable matrices in Fn×n, (ii)n, the matrices in Fn×n all of whose roots are in F, and (iii) the matric ring Fn×n itself. The results for (i) and (ii) are valid for an arbitrary field F. 相似文献
20.
In this paper, we discuss properties of the ω,q-Bernstein polynomials introduced by S. Lewanowicz and P. Woźny in [S. Lewanowicz, P. Woźny, Generalized Bernstein polynomials, BIT 44 (1) (2004) 63–78], where fC[0,1], ω,q>0, ω≠1,q−1,…,q−n+1. When ω=0, we recover the q-Bernstein polynomials introduced by [G.M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511–518]; when q=1, we recover the classical Bernstein polynomials. We compute the second moment of , and demonstrate that if f is convex and ω,q(0,1) or (1,∞), then are monotonically decreasing in n for all x[0,1]. We prove that for ω(0,1), qn(0,1], the sequence converges to f uniformly on [0,1] for each fC[0,1] if and only if limn→∞qn=1. For fixed ω,q(0,1), we prove that the sequence converges for each fC[0,1] and obtain the estimates for the rate of convergence of by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions. 相似文献