共查询到20条相似文献,搜索用时 31 毫秒
1.
Trieu Le 《Complex Analysis and Operator Theory》2010,4(2):391-399
For a bounded measurable function f on the open unit disk \({\mathbb{D}}\) , let T f denote the corresponding Toeplitz operator on the Bergman space \({A^2(\mathbb{D})}\) . A recent result of Luecking shows that if T f has finite rank, then f must be the zero function. Using a refined version of this result, we show that if all, except possibly one, of the functions f 1,..., f m are radial and \({T_{f_1}\cdots T_{f_m}}\) has finite rank, then one of these functions must be zero. 相似文献
2.
We prove that a deformation of a hypersurface in an (n + 1)-dimensional real space form \({{\mathbb S}^{n+1}_{p,1}}\) induces a Hamiltonian variation of the normal congruence in the space \({{\mathbb L}({\mathbb S}^{n+1}_{p,1})}\) of oriented geodesics. As an application, we show that every Hamiltonian minimal submanifold in \({{\mathbb L}({\mathbb S}^{n+1})}\) (resp. \({{\mathbb L}({\mathbb H}^{n+1})}\)) with respect to the (para-)Kähler Einstein structure is locally the normal congruence of a hypersurface \({\Sigma}\) in \({{\mathbb S}^{n+1}}\) (resp. \({{\mathbb H}^{n+1}}\)) that is a critical point of the functional \({{\mathcal W}(\Sigma) = \int_\Sigma\left(\Pi_{i=1}^n|\epsilon+k_i^2|\right)^{1/2}}\), where ki denote the principal curvatures of \({\Sigma}\) and \({\epsilon \in \{-1, 1\}}\). In addition, for \({n = 2}\), we prove that every Hamiltonian minimal surface in \({{\mathbb L}({\mathbb S}^{3})}\) (resp. \({{\mathbb L}({\mathbb H}^{3})}\)), with respect to the (para-)Kähler conformally flat structure, is the normal congruence of a surface in \({{\mathbb S}^{3}}\) (resp. \({{\mathbb H}^{3}}\)) that is a critical point of the functional \({{\mathcal W}\prime(\Sigma) = \int_\Sigma\sqrt{H^2-K+1}}\) (resp. \({{\mathcal W}\prime(\Sigma) = \int_\Sigma\sqrt{H^2-K-1}}\)), where H and K denote, respectively, the mean and Gaussian curvature of \({\Sigma}\). 相似文献
3.
In this paper we consider the Schrödinger operator ?Δ + V on \({\mathbb R^d}\), where the nonnegative potential V belongs to the reverse Hölder class \({B_{q_{_1}}}\) for some \({q_{_1}\geq \frac{d}{2}}\) with d ≥ 3. Let \({H^1_L(\mathbb R^d)}\) denote the Hardy space related to the Schrödinger operator L = ?Δ + V and \({BMO_L(\mathbb R^d)}\) be the dual space of \({H^1_L(\mathbb R^d)}\). We show that the Schrödinger type operator \({\nabla(-\Delta +V)^{-\beta}}\) is bounded from \({H^1_L(\mathbb R^d)}\) into \({L^p(\mathbb R^d)}\) for \({p=\frac{d}{d-(2\beta-1)}}\) with \({ \frac{1}{2}<\beta<\frac{3}{2} }\) and that it is also bounded from \({L^p(\mathbb R^d)}\) into \({BMO_L(\mathbb R^d)}\) for \({p=\frac{d}{2\beta-1}}\) with \({ \frac{1}{2}<\beta< 2}\). 相似文献
4.
In this paper, we study the existence and multiplicity of homoclinic solutions for the following second-order p(t)-Laplacian–Hamiltonian systemswhere \({t \in \mathbb{R}}\), \({u \in \mathbb{R}^n}\), \({p \in C(\mathbb{R},\mathbb{R})}\) with p(t) > 1, \({a \in C(\mathbb{R},\mathbb{R})}\), \({W\in C^1(\mathbb{R}\times\mathbb{R}^n,\mathbb{R})}\) and \({\nabla W(t,u)}\) is the gradient of W(t, u) in u. The point is that, assuming that a(t) is bounded in the sense that there are constants \({0<\tau_1<\tau_2<\infty}\) such that \({\tau_1\leq a(t)\leq \tau_2 }\) for all \({t \in \mathbb{R}}\) and W(t, u) is of super-p(t) growth or sub-p(t) growth as \({|u|\rightarrow \infty}\), we provide two new criteria to ensure the existence and multiplicity of homoclinic solutions, respectively. Recent results in the literature are extended and significantly improved.
相似文献
$$\frac{{\rm d}}{{\rm d}t}(|\dot{u}(t)|^{p(t)-2}\dot{u}(t))-a(t)|u(t)|^{p(t)-2}u(t)+\nabla W(t,u(t))=0,$$
5.
The Gamma semigroup with parameter \(b>0\) on \(L^p(\mathbb R^+)\) is defined by Let S denote the multiplication operator \(f(x)\rightarrow xf(x)\) with maximal domain D(S) in \(L^p(\mathbb R^+)\). The bounded operator V on \(L^p(\mathbb R^+)\) is S-Volterra if D(S) is V-invariant and \([S,V]=V^2\) on D(S). For \(1<p<\infty \), we characterize the Gamma semigroup as the unique regular semigroup \(V(\cdot )\) on \(L^p(\mathbb R^+)\) with imaginary type less than \(\pi \), such that V(1) is S-Volterra and \(V(1)u^b=Su^b\), where \(u^b(x):=e^{-bx}\).
相似文献
$$\begin{aligned} W_b(t)f(x)=\frac{1}{\Gamma (t)}\int _0^x(x-y)^{t-1}e^{-b(x-y)}f(y)\,dy. \end{aligned}$$
6.
We show that for every sequence \({(p_n)_{n\in\mathbb{N}}}\) with 1 ≤ p n ≤ 2 there exists an \({\mathcal{L}_1}\) -space with the Radon-Nikodým containing an isomorphic copy of \({\ell_1(\ell_{p_n})}\) . 相似文献
7.
Let \(\mathcal Lf(x)=-\Delta f (x)+V(x)f(x)\), V?≥?0, \(V\in L^1_{loc}(\mathbb R^d)\), be a non-negative self-adjoint Schrödinger operator on \(\mathbb R^d\). We say that an L 1-function f is an element of the Hardy space \(H^1_{\mathcal L}\) if the maximal functionbelongs to \(L^1(\mathbb R^d)\). We prove that under certain assumptions on V the space \(H^1_{\mathcal L}\) is also characterized by the Riesz transforms \(R_j=\frac{\partial}{\partial x_j}\mathcal L^{-1\slash 2}\), j?=?1,...,d, associated with \(\mathcal L\). As an example of such a potential V one can take any V?≥?0, \(V\in L^1_{loc}\), in one dimension.
相似文献
$ \mathcal M_{\mathcal L} f(x)=\sup\limits_{t>0}|e^{-t\mathcal L} f(x)| $
8.
Miloš S. Kurilić 《Order》2017,34(2):235-251
For a partial order \(\mathbb {P}\) having infinite antichains by \(\mathfrak {a}(\mathbb {P})\) we denote the minimal cardinality of an infinite maximal antichain in \(\mathbb {P}\) and investigate how does this cardinal invariant of posets behave in finite products. In particular we show that \(\min \{ \mathfrak {a}(\mathbb {P}),\mathfrak {p} (\text {sq} \mathbb {P}) \} \leq \mathfrak {a} (\mathbb {P}^{n} ) \leq \mathfrak {a} (\mathbb {P})\), for all \(n\in \mathbb {N}\), where \(\mathfrak {p} (\text {sq} \mathbb {P})\) is the minimal size of a centered family without a lower bound in the separative quotient of the poset \(\mathbb {P}\), or \(\mathfrak {p} (\text {sq} \mathbb {P})=\infty \), if there is no such family. So we have \(\mathfrak {a} (\mathbb {P} \times \mathbb {P})=\mathfrak {a} (\mathbb {P})\) whenever \(\mathfrak {p} (\text {sq} \mathbb {P})\geq \mathfrak {a} (\mathbb {P})\) and we show that, in addition, this equality holds for all posets obtained from infinite Boolean algebras of size ≤ø 1 by removing zero, all reversed trees, all atomic posets and, in particular, for all posets of the form \(\langle \mathcal {C} ,\subset \rangle \), where \(\mathcal {C}\) is a family of nonempty closed sets in a compact T 1-space containing all singletons. As a by-product we obtain the following combinatorial statement: If X is an infinite set and {A i ×B i :i∈I} an infinite partition of the square X 2, then at least one of the families {A i :i∈I} and {B i :i∈I} contains an infinite partition of X. 相似文献
9.
Any continuous strictly monotonic function \({F : {{\mathbb R}^{{\geq}0} \to {\mathbb R}}}\) with F(0) = 0 and F(t) → ∞ for t → ∞ gives rise to a topological rotational spread of \({{\rm PG}\,(3,{\mathbb R})}\); this spread is non-regular, if F is not linear. The action of the group \({SO_3({\mathbb R})}\) on this spread yields a topological parallelism of \({{\rm PG}\,(3,{\mathbb R})}\). The article also contains a short investigation on rotational spreads. Moreover, we construct a parallelism P 72 of \({{\rm PG}\,(3,{\mathbb R})}\) which is composed of piecewise regular spreads each consisting of two segments which are tacked together along a common regulus. Using Klein’s correspondence of line geometry and the Thas–Walker construction we represent every parallel class of P 72 via two parallel half-lines being non-interior to a given sphere in \({{\mathbb R}^3}\). The parallelism P 72 contains exactly one regular spread, all other members of P 72 are piecewise regular spreads with two segments. However, P 72 is not topological. 相似文献
10.
Let M be a smooth compact oriented Riemannian manifold, and let Δ M be the Laplace–Beltrami operator on M. Say \({0 \neq f \in \mathcal{S}(\mathbb {R}^+)}\) , and that f (0) = 0. For t > 0, let K t (x, y) denote the kernel of f (t 2 Δ M ). We show that K t is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator f (t 2Δ) on \({\mathbb {R}^n}\) . We define continuous \({\mathcal {S}}\)-wavelets on M, in such a manner that K t (x, y) satisfies this definition, because of its localization near the diagonal. Continuous \({\mathcal {S}}\)-wavelets on M are analogous to continuous wavelets on \({\mathbb {R}^n}\) in \({\mathcal {S}}\) (\({\mathbb {R}^n}\)). In particular, we are able to characterize the Hölder continuous functions on M by the size of their continuous \({\mathcal {S}}\)-wavelet transforms, for Hölder exponents strictly between 0 and 1. If M is the torus \({\mathbb T^2}\) or the sphere S 2, and f (s) = se ?s (the “Mexican hat” situation), we obtain two explicit approximate formulas for K t , one to be used when t is large, and one to be used when t is small. 相似文献
11.
We study the asymptotic distribution of the critical values of f n as n goes to infinity. Let X be a closed manifold and \({f:X \times X \to \mathbb{R}}\) be a smooth function. Define \({f_n : X^{n+1} \to \mathbb{R}}\) by \({f_n(x_1, \ldots, x_{n+1}) := \frac{1}{n}\sum f(x_i, x_{i+1})}\) . 相似文献
12.
Jose Franco 《Central European Journal of Mathematics》2012,10(3):927-941
We study the representation theory of the solution space of the one-dimensional Schrödinger equation with singular potential V λ (x) = λx ?2 as a representation of \(\widetilde{SL(2,\mathbb{R})}\). The subspace of solutions for which the action globalizes is constructed via nonstandard induction outside the semisimple category. By studying the subspace of K-finite vectors in this space, a distinguished family of potentials, parametrized by the triangular numbers is shown to generate a global representation of \(\widetilde{SL(2,\mathbb{R})}\) ? H 3, where H 3 is the three-dimensional Heisenberg group. 相似文献
13.
Let M be a left module for the Schur algebra S(n, r), and let \({s \in \mathbb{Z}^+}\) . Then \({M^{\otimes s}}\) is a \({(S(n,\,rs), F{\mathfrak{S}_{s}})}\) -bimodule, where the symmetric group \({{\mathfrak{S}_s}}\) on s letters acts on the right by place permutations. We show that the Schur functor f rs sends \({M^{\otimes s}}\) to the \({(F{\mathfrak{S}_{rs}},F{\mathfrak{S}_s})}\) -bimodule \({F\mathfrak{S}_{rs}\otimes_{F(\mathfrak{S}_{r}\wr{\mathfrak{S}_s})} ((f_rM)^{\otimes s}\otimes_{F} F{\mathfrak{S}_s})}\) . As a corollary, we obtain the image under the Schur functor of the Lie power L s (M), exterior power \({\bigwedge^s(M)}\) of M and symmetric power S s (M). 相似文献
14.
We prove the existence of infinitely many solutions for where V(x) satisfies \(\lim _{|x| \rightarrow \infty } V(x) = V_\infty >0\) and some conditions. We require conditions on f(u) only around 0 and at \(\infty \).
相似文献
$$\begin{aligned} - \Delta u + V(x) u = f(u) \quad \text { in } \mathbb {R}^N, \quad u \in H^1(\mathbb {R}^N), \end{aligned}$$
15.
We prove that the maximal dimension of a p-central subspace of the generic symbol p-algebra of prime degree p is \({p+1}\). We do it by proving the following number theoretic fact: let \({\{s_1,\dots,s_{p+1}\}}\) be \({p+1}\) distinct nonzero elements in the additive group \({G=(\mathbb{Z}/p \mathbb{Z}) \times (\mathbb{Z}/p \mathbb{Z})}\), then every nonzero element \({g \in G}\) can be expressed as \({d_1 s_1+\dots+d_{p+1} s_{p+1}}\) for some non-negative integers \({d_1,\dots,d_{p+1}}\) with \({d_1+\dots+d_{p+1}\leq p-1}\). 相似文献
16.
Banach spaces which are Grothendieck spaces with the Dunford–Pettis property (briefly, GDP) are classical. A systematic treatment of GDP-Fréchet spaces occurs in Bonet and Ricker (Positivity 11:77–93, 2007). This investigation is continued here for locally convex Hausdorff spaces. The product and (most) inductive limits of GDP-space are again GDP-spaces. Also, every complete injective space is a GDP-space. For \({p\in \{0\}\cup[1,\infty)}\) it is shown that the classical co-echelon spaces k p (V) and \({K_p(\overline{V})}\) are GDP-spaces if and only if they are Montel. On the other hand, \({K_\infty(\overline{V})}\) is always a GDP-space and k ∞(V) is a GDP-space whenever its (Fréchet) predual, i.e., the Köthe echelon space λ 1(A), is distinguished. 相似文献
17.
Dražen Adamović Victor G. Kac Pierluigi Möseneder Frajria Paolo Papi Ozren Perše 《Japanese Journal of Mathematics》2017,12(2):261-315
We present methods for computing the explicit decomposition of the minimal simple affine W-algebra \({W_k(\mathfrak{g}, \theta)}\) as a module for its maximal affine subalgebra \({\mathscr{V}_k(\mathfrak{g}^{\natural})}\) at a conformal level k, that is, whenever the Virasoro vectors of \({W_k(\mathfrak{g}, \theta)}\) and \({\mathscr{V}_k(\mathfrak{g}^\natural)}\) coincide. A particular emphasis is given on the application of affine fusion rules to the determination of branching rules. In almost all cases when \({\mathfrak{g}^{\natural}}\) is a semisimple Lie algebra, we show that, for a suitable conformal level k, \({W_k(\mathfrak{g}, \theta)}\) is isomorphic to an extension of \({\mathscr{V}_k(\mathfrak{g}^{\natural})}\) by its simple module. We are able to prove that in certain cases \({W_k(\mathfrak{g}, \theta)}\) is a simple current extension of \({\mathscr{V}_k(\mathfrak{g}^{\natural})}\). In order to analyze more complicated non simple current extensions at conformal levels, we present an explicit realization of the simple W-algebra \({W_{k}(\mathit{sl}(4), \theta)}\) at k = ?8/3. We prove, as conjectured in [3], that \({W_{k}(\mathit{sl}(4), \theta)}\) is isomorphic to the vertex algebra \({\mathscr{R}^{(3)}}\), and construct infinitely many singular vectors using screening operators. We also construct a new family of simple current modules for the vertex algebra \({V_k (\mathit{sl}(n))}\) at certain admissible levels and for \({V_k (\mathit{sl}(m \vert n)), m\ne n, m,n\geq 1}\) at arbitrary levels. 相似文献
18.
Let n and s be integers such that \(1\le s<\frac{n}{2}\), and let \(M_n(\mathbb {K})\) be the ring of all \(n\times n\) matrices over a field \(\mathbb {K}\). Denote by \([\frac{n}{s}]\) the least integer m with \(m\ge \frac{n}{s}\). In this short note, it is proved that if \(g:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\) is a map such that \(g\left( \sum _{i=1}^{[\frac{n}{s}]}A_i\right) =\sum _{i=1}^{[\frac{n}{s}]}g(A_i)\) holds for any \([\frac{n}{s}]\) rank-s matrices \(A_1,\ldots ,A_{[\frac{n}{s}]}\in M_n(\mathbb {K})\), then \(g(x)=f(x)+g(0)\), \(x\in M_n(\mathbb {K})\), for some additive map \(f:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\). Particularly, g is additive if \(char\mathbb {K}\not \mid \left( [\frac{n}{s}]-1\right) \). 相似文献
19.
Victor Cuauhtemoc García 《The Ramanujan Journal》2018,47(1):85-98
Let p be a large prime number and f(x) be an integer-valued function defined in \({\mathbb F}_p\). The Littlewood problem in \({{\mathbb {F}}}_p\) is to establish non-trivial lower bounds for the \(\ell _1\) norm of exponential sums involving f(x). In the present paper, we establish new lower bounds for exponential sums including polynomials, powers of any primitive root and subgroups of \(\mathbb {F}_p^*.\) 相似文献
20.
Pérez-Izquierdo and Shestakov recently extended the PBW theorem to Malcev algebras. It follows from their construction that for any Malcev algebra M over a field of characteristic ≠ 2, 3 there is a representation of the universal nonassociative enveloping algebra U(M) by linear operators on the polynomial algebra P(M). For the nilpotent non-Lie Malcev algebra \(\mathbb{M}\) of dimension 5, we use this representation to determine explicit structure constants for \(U(\mathbb{M})\); from this it follows that \(U(\mathbb{M})\) is not power-associative. We obtain a finite set of generators for the alternator ideal \(I(\mathbb{M}) \subset U(\mathbb{M})\) and derive structure constants for the universal alternative enveloping algebra \(A(\mathbb{M}) = U(\mathbb{M})/I(\mathbb{M})\), a new infinite dimensional alternative algebra. We verify that the map \(\iota\colon \mathbb{M} \to A(\mathbb{M})\) is injective, and so \(\mathbb{M}\) is special. 相似文献