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1.
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   

2.
Let \({\varphi}\) be a Musielak–Orlicz function satisfying that, for any \({(x,\,t)\in{\mathbb R}^n \times [0, \infty)}\), \({\varphi(\cdot,\,t)}\) belongs to the Muckenhoupt weight class \({A_\infty({\mathbb R}^n)}\) with the critical weight exponent \({q(\varphi) \in [1,\,\infty)}\) and \({\varphi(x,\,\cdot)}\) is an Orlicz function with uniformly lower type \({p^{-}_{\varphi}}\) and uniformly upper type \({p^+_\varphi}\) satisfying \({q(\varphi) < p^{-}_{\varphi}\le p^{+}_{\varphi} < \infty}\). In this paper, the author obtains a sharp weighted bound involving \({A_\infty}\) constant for the Hardy–Littlewood maximal operator on the Musielak–Orlicz space \({L^{\varphi}}\). This result recovers the known sharp weighted estimate established by Hytönen et al. in [J. Funct. Anal. 263:3883–3899, 2012].  相似文献   

3.
In a general unbounded uniform C 2-domain \({\Omega \subset \mathbb{R}^n, n \geq 3}\) , and \({1\leq q\leq \infty}\) consider the spaces \({\tilde{L}^q(\Omega)}\) defined by \({\tilde{L^q}(\Omega) := \left\{\begin{array}{ll}L^q(\Omega)+L^2(\Omega),\quad q < 2, \\ L^q(\Omega)\cap L^2(\Omega),\quad q\geq 2, \end{array}\right.}\) and corresponding subspaces of solenoidal vector fields, \({\tilde{L}^q_\sigma(\Omega)}\) . By studying the complex and real interpolation spaces of these we derive embedding properties for fractional order spaces related to the Stokes problem and L p ? L q -type estimates for the corresponding semigroup.  相似文献   

4.
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   

5.
For a fairly general reductive group \({G_{/\mathbb{Q}_p}}\), we explicitly compute the space of locally algebraic vectors in the Breuil–Herzig construction \({\Pi(\rho)^{ord}}\), for a potentially semistable Borel-valued representation \({\rho}\) of \({Gal(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)}\). The point being we deal with the whole representation, not just its socle—and we go beyond \({GL_n(\mathbb{Q}_p)}\). In the case of \({GL_2(\mathbb{Q}_p)}\), this relation is one of the key properties of the \({p}\)-adic local Langlands correspondence. We give an application to \({p}\)-adic local-global compatibility for \({\Pi(\rho)^{ord}}\) for modular representations, but with no indecomposability assumptions.  相似文献   

6.
We prove a Beurling-Blecher-Labuschagne theorem for \({H^\infty}\)-invariant spaces of \({L^p(\mathcal{M},\tau)}\) when \({0 < p \leq\infty}\), using Arveson’s non-commutative Hardy space \({H^\infty}\) in relation to a von Neumann algebra \({\mathcal{M}}\) with a semifinite, faithful, normal tracial weight \({\tau}\). Using the main result, we are able to completely characterize all \({H^\infty}\)-invariant subspaces of \({L^p(\mathcal{M} \rtimes_\alpha \mathbb{Z},\tau)}\), where \({\mathcal{M} \rtimes_\alpha \mathbb{Z} }\) is a crossed product of a semifinite von Neumann algebra \({\mathcal{M}}\) by the integer group \({\mathbb{Z}}\), and \({H^\infty}\) is a non-selfadjoint crossed product of \({\mathcal{M}}\) by \({\mathbb{Z}^+}\). As an example, we characterize all \({H^\infty}\)-invariant subspaces of the Schatten p-class \({S^p(\mathcal{H})}\), where \({H^\infty}\) is the lower triangular subalgebra of \({B(\mathcal{H})}\), for each \({0 < p \leq\infty}\).  相似文献   

7.
We call the \({\delta}\)-vector of an integral convex polytope of dimension d flat if the \({\delta}\)-vector is of the form \({(1,0,\ldots,0,a,\ldots,a,0,\ldots,0)}\), where \({a \geq 1}\). In this paper, we give the complete characterization of possible flat \({\delta}\)-vectors. Moreover, for an integral convex polytope \({\mathcal{P}\subset \mathbb{R}^N}\) of dimension d, we let \({i(\mathcal{P},n)=|n\mathcal{P}\cap \mathbb{Z}^N|}\) and \({i^*(\mathcal{P},n)=|n(\mathcal{P} {\setminus}\partial \mathcal{P})\cap \mathbb{Z}^N|}\). By this characterization, we show that for any \({d \geq 1}\) and for any \({k,\ell \geq 0}\) with \({k+\ell \leq d-1}\), there exist integral convex polytopes \({\mathcal{P}}\) and \({\mathcal{Q}}\) of dimension d such that (i) For \({t=1,\ldots,k}\), we have \({i(\mathcal{P},t)=i(\mathcal{Q},t),}\) (ii) For \({t=1,\ldots,\ell}\), we have \({i^*(\mathcal{P},t)=i^*(\mathcal{Q},t)}\), and (iii) \({i(\mathcal{P},k+1) \neq i(\mathcal{Q},k+1)}\) and \({i^*(\mathcal{P},\ell+1)\neq i^*(\mathcal{Q},\ell+1)}\).  相似文献   

8.
We investigate Weyl type asymptotics of functional-difference operators associated to mirror curves of special del Pezzo Calabi-Yau threefolds. These operators are \({H(\zeta) = U + U^{-1} + V + \zeta V^{-1}}\) and \({H_{m,n} = U + V + q^{-mn}U^{-m}V^{-n}}\), where \({U}\) and \({V}\) are self-adjoint Weyl operators satisfying \({UV = q^{2}VU}\) with \({q = {\rm e}^{{\rm i}\pi b^{2}}}\), \({b > 0}\) and \({\zeta > 0}\), \({m, n \in \mathbb{N}}\). We prove that \({H(\zeta)}\) and \({H_{m,n}}\) are self-adjoint operators with purely discrete spectrum on \({L^{2}(\mathbb{R})}\). Using the coherent state transform we find the asymptotical behaviour for the Riesz mean \({\sum_{j\ge 1}(\lambda - \lambda_{j})_{+}}\) as \({\lambda \to \infty}\) and prove the Weyl law for the eigenvalue counting function \({N(\lambda)}\) for these operators, which imply that their inverses are of trace class.  相似文献   

9.
In this paper, we relate the special values at a non-positive integer \({\underline{\mathbf{s}}=(s_{1},\ldots, s_{r})= -\underline{\mathbf{N}}= (-N_{1},\ldots, -N_{r})}\) obtained by meromorphic continuation of the multiple Dirichlet series \({{Z(\underline{\mathbf{P}}, \underline{\mathbf{s}})=\sum_{\underline{m}\in {\mathbb{N}}^{*n}}{\frac{1}{\prod_{i=1}^{r}{P_{i}^{ s_{i}}(\underline{m})}}}}}\) to special values of the function \({Y(\underline{\mathbf{P}}, \underline{\mathbf{s}})=\int_{[1, +\infty[^{n}} {\prod_{i=1}^{r}{P_{i}^{- s_{i}}(\underline{\mathbf{x}})}\; d{\underline{\mathbf{x}}}}}\) where \({\underline{\mathbf{P}}=(P_{1},..., P_{r}),\; (r\geq 1)}\) are elliptic polynomials in “\({n}\) ” variables. We prove a simple relation between \({Z(\underline{\mathbf{P}}_{\underline{\mathbf{a}}}, -\underline{\mathbf{N}})}\) and \({Y(\underline{\mathbf{P}}_{\underline{\mathbf{a}}}, -\underline{\mathbf{N}})}\), such that for all \({\underline{\mathbf{a}} \in {\mathbb{R}}^{n}_{+}}\), we denote \({\underline{\mathbf{P}}_{\underline{\mathbf{a}}}:=(P_{1 \underline{\mathbf{a}}},\ldots, P_{r \underline{\mathbf{a}}})}\), where \({P_{i\;\underline{\mathbf{a}}}(\underline{\mathbf{x}}):= P_i(\underline{\mathbf{x}}+ \underline{\mathbf{a}})\; (1\leq i\leq r)}\) is the shifted polynomial.  相似文献   

10.
We consider the generalized Korteweg-de Vries (gKdV) equation with the time oscillating nonlinearity:
$${\partial}_t u+{\partial}_x^3 u+ g(\omega t) {\partial}_x (|u|^{p-1}u)= 0, \quad (t, x) \in \mathbb{R} \times \mathbb{R}.$$
Under the suitable assumption on g, we show that if the nonlinear term is mass critical or supercritical i.e., \({p \geq 5}\) and \({u(0) \in \dot{H}^{s_{p}}}\), where \({s_{p} = 1/2 - 2/(p-1)}\) is a scale critical exponent, then there exists a unique global solution to (gKdV) provided that \({|\omega|}\) is sufficiently large. We also obtain the behavior of the solution to (gKdV) as \({|\omega| \to \infty}\).
  相似文献   

11.
Let \(p(\cdot ):\ {\mathbb {R}}^n\rightarrow (0,\infty )\) be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into “good” and “bad” parts and then prove the following real interpolation theorem between the variable Hardy space \(H^{p(\cdot )}({\mathbb {R}}^n)\) and the space \(L^{\infty }({\mathbb {R}}^n)\): \((H^{p(\cdot )}(\mathbb R^n),L^{\infty }({\mathbb {R}}^n))_{\theta ,\infty }= WH^{p(\cdot )/(1-\theta )}({\mathbb {R}}^n),\quad \mathrm{where}~\theta \in (0,1), \mathrm{and}\) \(WH^{p(\cdot )/(1-\theta )}({\mathbb {R}}^n)\) denotes the variable weak Hardy space. As an application, the variable weak Hardy space \(WH^{p(\cdot )}({\mathbb {R}}^n)\) with \(p_-:=\mathop {\text {ess inf}}\limits _{x\in {{{\mathbb {R}}}^n}}p(x)\in (1,\infty )\) is proved to coincide with the variable Lebesgue space \(WL^{p(\cdot )}({\mathbb {R}}^n)\).  相似文献   

12.
We consider a broad class of linear Perron–Frobenius operators \({\Lambda:X \rightarrow X}\), where \({X}\) is a real Banach space of \({C^m}\) functions. We prove the existence of a strictly positive \({C^m}\) eigenvector \({v}\) with eigenvalue \({r=r(\Lambda) =}\) the spectral radius of \({\Lambda}\). We prove (see Theorem 6.5 in Sect. 6 of this paper) that \({r(\Lambda)}\) is an algebraically simple eigenvalue and that, if \({\sigma(\Lambda)}\) denotes the spectrum of the complexification of \({\Lambda,\sigma(\Lambda) \backslash \{r(\Lambda)\}\subseteq \{\zeta \in \mathbb{C} \big| |\zeta| \le r_*\}}\), where \({r_* < r(\Lambda)}\). Furthermore, if \({u \in X}\) is any strictly positive function, \({(\frac 1r \Lambda)^k(u) \rightarrow s_u v}\) as \({k \rightarrow \infty}\), where \({s_u > 0}\) and convergence is in the norm topology on \({X}\). In applications to the computation of Hausdorff dimension, one is given a parametrized family \({\Lambda_s,s > s_*}\), of such operators and one wants to determine the (unique) value \({s_0}\) such that \({r(\Lambda_{s_0})=1}\). In another paper (Falk and Nussbaum in C\({^{\rm m}}\) Eigenfunctions of Perron–Frobenius operators and a new approach to numerical computation of Hausdorff dimension, submitted) we prove that explicit estimates on the partial derivatives of the positive eigenvector \({v_s}\) of \({\Lambda_s}\) can be obtained and that this information can be used to give rigorous, sharp upper and lower bounds for \({s_0}\).  相似文献   

13.
We study the solution \({u(r,\rho)}\) of the quasilinear elliptic problem
$$\begin{cases}r^{-(\gamma-1)}(r^{\alpha}|u'|^{\beta-1}u')'+|u|^{p-1}u=0, & 0 < r < \infty, \\u(0)=\rho > 0,\ u'(0)=0.\end{cases}$$
The usual Laplace, \({m}\)-Laplace, and \({k}\)-Hessian operators are included in the differential operator \({r^{-(\gamma-1)}(r^{\alpha}|u'|^{\beta-1}u')'}\). Under certain conditions on \({\alpha}\), \({\beta}\), \({\gamma}\), and \({p}\), the equation has a singular positive solution \({u^*(r)}\) and the solution \({u(r,\rho)}\) is positive for \({r\ge 0}\). We study the intersection numbers between \({u(r,\rho)}\) and \({u^*(r)}\) and between \({u(r,\rho_0)}\) and \({u(r,\rho_1)}\). A generalized Joseph–Lundgren exponent \({p^*_{JL}}\) plays a crucial role. The main technique is a phase plane analysis. In particular, we use two changes of variables which transform the equation into two autonomous systems.
  相似文献   

14.
For each \({\alpha\in[0,2)}\) we consider the eigenvalue problem \({-{\rm div}(|x|^\alpha \nabla u)=\lambda u}\) in a bounded domain \({\Omega\subset \mathbb{R}^N}\) (\({N\geq 2}\)) with smooth boundary and \({0\in \Omega}\) subject to the homogeneous Dirichlet boundary condition. Denote by \({\lambda_1(\alpha)}\) the first eigenvalue of this problem. Using \({\Gamma}\)-convergence arguments we prove the continuity of the function \({\lambda_1}\) with respect to \({\alpha}\) on the interval \({[0,2)}\).  相似文献   

15.
Let \({C^*_r(\mathbb{F}_{\infty})}\) be the reduced C*-algebra of the free group on infinitely many generators. Say that \({a, b \in C^*_r(\mathbb{F}_{\infty})_{SA}}\). Then \({a}\) is majorized by \({b}\) if and only if \({a \in \overline{Conv(U(b))}.}\) In particular, \({\tau(b)1 \in \overline{Conv(U(b))}.}\) Moreover, in the above results, we provide uniform bounds for the number of unitary conjugates needed for a given approximation. In the above, \({Conv(U(b))}\) is the convex hull of the unitary orbit of \({b}\) in \({C^*_r(\mathbb{F}_{\infty})}\).  相似文献   

16.
In this paper, we study the uniform Hölder continuity of the generalized Riemann function \({R_{\alpha,\beta} \,\,{\rm (with}\,\, \alpha > 1 \,\,{\rm and}\,\, \beta > 0}\)) defined by
$$R_{\alpha,\beta}(x) = \sum_{n=1}^{+\infty} \frac{\sin(\pi n^\beta x)}{n^\alpha},\quad x \in \mathbb{R},$$
using its continuous wavelet transform. In particular, we show that the exponent we find is optimal. We also analyse the behaviour of \({R_{\alpha,\beta} \,\,{\rm as}\,\, \beta}\) tends to infinity.
  相似文献   

17.
A string is a pair \({(L, \mathfrak{m})}\) where \({L \in[0, \infty]}\) and \({\mathfrak{m}}\) is a positive, possibly unbounded, Borel measure supported on [0, L]; we think of L as the length of the string and of \({\mathfrak{m}}\) as its mass density. To each string a differential operator acting in the space \({L^2(\mathfrak{m})}\) is associated. Namely, the Kre?n–Feller differential operator \({-D_{\mathfrak{m}}D_x}\) ; its eigenvalue equation can be written, e.g., as
$$f^{\prime}(x) + z \int_0^L f(y)\,d\mathfrak{m}(y) = 0,\quad x \in\mathbb R,\ f^{\prime}(0-) = 0.$$
A positive Borel measure τ on \({\mathbb R}\) is called a (canonical) spectral measure of the string \({\textsc S[L, \mathfrak{m}]}\) , if there exists an appropriately normalized Fourier transform of \({L^2(\mathfrak{m})}\) onto L 2(τ). In order that a given positive Borel measure τ is a spectral measure of some string, it is necessary that: (1) \({\int_{\mathbb R} \frac{d\tau(\lambda)}{1+|\lambda|} < \infty}\) . (2) Either \({{\rm supp} \tau \subseteq [0, \infty)}\) , or τ is discrete and has exactly one point mass in (?∞, 0). It is a deep result, going back to Kre?n in the 1950’s, that each measure with \({\int_{\mathbb R}\frac{d\tau(\lambda)}{1+|\lambda|} < \infty}\) and \({{\rm supp} \tau \subseteq [0, \infty)}\) is a spectral measure of some string, and that this string is uniquely determined by τ. The question remained open, which conditions characterize whether a measure τ with \({{\rm supp} \tau \not\subseteq [0, \infty)}\) is a spectral measure of some string. In the present paper, we answer this question. Interestingly, the solution is much more involved than the first guess might suggest.
  相似文献   

18.
Given numbers \({n,s \in \mathbb{N}}\), \({n \geq 2}\), and the \({n}\)th-degree monic Chebyshev polynomial of the first kind \({\widehat T_n(x)}\), the polynomial system “induced” by \({\widehat T_n(x)}\) is the system of orthogonal polynomials \({\{p_{k}^{n,s} \}}\) corresponding to the modified measure \({d \sigma^{n,s}(x)=\widehat T^{2s}_n(x) d\sigma(x)}\), where \({d\sigma(x)=1/\sqrt{1-x^{2}}dx}\) is the Chebyshev measure of the first kind. Here we are concerned with the problem of determining the coefficients in the three-term recurrence relation for the polynomials \({p^{n,s}_{k}}\). The desired coefficients are obtained analytically in a closed form.  相似文献   

19.
Let X be an algebraic curve over \({\mathbb {Q}}\) and \({t\in {\mathbb {Q}}(X)}\) a non-constant rational function such that \({{\mathbb {Q}}(X)\ne {\mathbb {Q}}(t)}\). For every \({ n \in {\mathbb {Z}}}\) pick \({P_ n \in X(\bar{{\mathbb {Q}}})}\) such that \({t(P_n)=n}\). We conjecture that, for large N, among the number fields \({\mathbb {Q}}(P_1), \ldots , {\mathbb {Q}}(P_N)\) there are at least cN distinct. We prove this conjecture in the special case when \(\bar{{\mathbb {Q}}}(X)/\bar{{\mathbb {Q}}}(t)\) is an abelian field extension and the critical values of t are all rational. This implies, in particular, that our conjecture follows from a more famous conjecture of Schinzel.  相似文献   

20.
We consider the problem
$$\varepsilon^{2s} (-\partial_{xx})^s \tilde{u}(\tilde{x}) -V(\tilde{x})\tilde{u}(\tilde{x})(1-\tilde{u}^2(\tilde{x}))=0 \quad{\rm in} \mathbb{R},$$
where \({(-\partial_{xx})^s}\) denotes the usual fractional Laplace operator, \({\varepsilon > 0}\) is a small parameter and the smooth bounded function V satisfies \({{\rm inf}_{\tilde{x} \in \mathbb{R}}V(\tilde{x}) > 0}\). For \({s\in(\frac{1}{2},1)}\), we prove the existence of separate multi-layered solutions for any small \({\varepsilon}\), where the layers are located near any non-degenerate local maximal points and non-degenerate local minimal points of function V. We also prove the existence of clustering-layered solutions, and these clustering layers appear within a very small neighborhood of a local maximum point of V.
  相似文献   

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