Some estimates for the Schrödinger type operators with nonnegative potentials

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 \({ \frac{1}{2}<\beta< 2}\).     

Sharp weighted bounds for the Hardy–Littlewood maximal operators on Musielak–Orlicz spaces

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].     

Interpolation of sum and intersection spaces of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>-type and applications to the Stokes problem in general unbounded domains

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.     

Conformal embeddings of affine vertex algebras in minimal <Emphasis Type="Italic">W</Emphasis>-algebras II: decompositions

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.     

Locally algebraic vectors in the Breuil–Herzig ordinary part

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.     

A Beurling-Blecher-Labuschagne Theorem for Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras

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 \({0 < p \leq\infty}\).     

Some remarks on energy inequalities for harmonic maps with potential

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 \({i^*(\mathcal{P},\ell+1)\neq i^*(\mathcal{Q},\ell+1)}\).     

Weyl Type Asymptotics and Bounds for the Eigenvalues of Functional-Difference Operators for Mirror Curves

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.     

Behaviour at the non-positive integers of Dirichlet series associated to polynomials of several variables

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.     

The generalized Korteweg-de Vries equation with time oscillating nonlinearity in scale critical Sobolev space

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 \({|\omega| \to \infty}\).

     

Interpolation Between $$H^{p(\cdot )}({\mathbb {R}}^n)$$ and $$L^\infty ({\mathbb {R}}^n)$$L∞(Rn): Real Method

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 \(WL^{p(\cdot )}({\mathbb {R}}^n)\).     

{{C^m}} Positive Eigenvectors for Linear Operators Arising in the Computation of Hausdorff Dimension

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 \({s_0}\).     

Generalized Joseph–Lundgren exponent and intersection properties for supercritical quasilinear elliptic equations

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.

     

Continuity of the first eigenvalue for a family of degenerate eigenvalue problems

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 \({[0,2)}\).     

Closed Convex Hulls of Unitary Orbits in {C^{*}_{r}(\mathbb{F}_{\infty})}

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 \({C^*_r(\mathbb{F}_{\infty})}\).     

About the Uniform Hölder Continuity of Generalized Riemann Function

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.

     

Orthogonal Polynomials for Modified Chebyshev Measure of the First Kind

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.     

An inverse spectral theorem for Kreĭn strings with a negative eigenvalue

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.

     

Number fields in fibers: the geometrically abelian case with rational critical values

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.     

Layered solutions for a fractional inhomogeneous Allen–Cahn equation

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.