The Transition Density of Brownian Motion Killed on a Bounded Set

We study the transition density of a standard two-dimensional Brownian motion killed when hitting a bounded Borel set A. We derive the asymptotic form of the density, say \(p^A_t(\mathbf{x},\mathbf{y})\), for large times t and for \(\mathbf{x}\) and \(\mathbf{y}\) in the exterior of A valid uniformly under the constraint \(|\mathbf{x}|\vee |\mathbf{y}| =O(t)\). Within the parabolic regime \(|\mathbf{x}|\vee |\mathbf{y}| = O(\sqrt{t})\) in particular \(p^A_t(\mathbf{x},\mathbf{y})\) is shown to behave like \(4e_A(\mathbf{x})e_A(\mathbf{y}) (\lg t)^{-2} p_t(\mathbf{y}-\mathbf{x})\) for large t, where \(p_t(\mathbf{y}-\mathbf{x})\) is the transition kernel of the Brownian motion (without killing) and \(e_A\) is the Green function for the ‘exterior of A’ with a pole at infinity normalized so that \(e_A(\mathbf{x}) \sim \lg |\mathbf{x}|\). We also provide fairly accurate upper and lower bounds of \(p^A_t(\mathbf{x},\mathbf{y})\) for the case \(|\mathbf{x}|\vee |\mathbf{y}|>t\) as well as corresponding results for the higher dimensions.     

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.     

On the global convergence rate of the gradient descent method for functions with Hölder continuous gradients

The gradient descent method minimizes an unconstrained nonlinear optimization problem with \({\mathcal {O}}(1/\sqrt{K})\), where K is the number of iterations performed by the gradient method. Traditionally, this analysis is obtained for smooth objective functions having Lipschitz continuous gradients. This paper aims to consider a more general class of nonlinear programming problems in which functions have Hölder continuous gradients. More precisely, for any function f in this class, denoted by \({{\mathcal {C}}}^{1,\nu }_L\), there is a \(\nu \in (0,1]\) and \(L>0\) such that for all \(\mathbf{x,y}\in {{\mathbb {R}}}^n\) the relation \(\Vert \nabla f(\mathbf{x})-\nabla f(\mathbf{y})\Vert \le L \Vert \mathbf{x}-\mathbf{y}\Vert ^{\nu }\) holds. We prove that the gradient descent method converges globally to a stationary point and exhibits a convergence rate of \({\mathcal {O}}(1/K^{\frac{\nu }{\nu +1}})\) when the step-size is chosen properly, i.e., less than \([\frac{\nu +1}{L}]^{\frac{1}{\nu }}\Vert \nabla f(\mathbf{x}_k)\Vert ^{\frac{1}{\nu }-1}\). Moreover, the algorithm employs \({\mathcal {O}}(1/\epsilon ^{\frac{1}{\nu }+1})\) number of calls to an oracle to find \({\bar{\mathbf{x}}}\) such that \(\Vert \nabla f({{\bar{\mathbf{x}}}})\Vert <\epsilon \).     

$${\sigma_2}$$-Energy as a polyconvex functional on a space of self-maps of annuli in the multi-dimensional calculus of variations

Let \({\mathbb{X} \subset \mathbb {R}^n}\) be a bounded Lipschitz domain and consider the energy functional

$${{\mathbb F}_{\sigma_2}}[u; \mathbb{X}] := \int_\mathbb{X} {\mathbf F}(\nabla u) \, dx,$$

over the space of admissible maps

$${{\mathcal {A}_\varphi}(\mathbb{X}) :=\{u \in W^{1,4}(\mathbb{X}, {\mathbb{R}^n}) : {\rm det}\, \nabla u > 0\, {\rm for}\, {\mathcal {L}^n}{\rm -a.e. in}\, \mathbb{X}, u|_{\partial \mathbb{X}} =\varphi \}},$$

where the integrand \({{\mathbf F}\colon \mathbb M_{n\times n}\to \mathbb{R}}\) is quasiconvex and sufficiently regular. Here our attention is paid to the prototypical case when \({{\mathbf F}(\xi):=\frac{1}{2}\sigma_2(\xi)+\Phi(\det\xi)}\). The aim of this paper is to discuss the question of multiplicity versus uniqueness for extremals and strong local minimizers of \({\mathbb F_{\sigma_2}}\) and the relation it bares to the domain topology. In contrast, for constructing explicitly and directly solutions to the system of Euler–Lagrange equations associated to \({{\mathbb F}_{\sigma_2}}\), we use a topological class of maps referred to as generalised twists and relate the problem to extremising an associated energy on the compact Lie group \({\mathbf {SO}(n)}\). The main result is a surprising discrepancy between even and odd dimensions. In even dimensions the latter system of equations admits infinitely many smooth solutions amongst such maps whereas in odd dimensions this number reduces to one.

     

Sharp MSE Bounds for Proximal Denoising

Denoising has to do with estimating a signal \(\mathbf {x}_0\) from its noisy observations \(\mathbf {y}=\mathbf {x}_0+\mathbf {z}\). In this paper, we focus on the “structured denoising problem,” where the signal \(\mathbf {x}_0\) possesses a certain structure and \(\mathbf {z}\) has independent normally distributed entries with mean zero and variance \(\sigma ^2\). We employ a structure-inducing convex function \(f(\cdot )\) and solve \(\min _\mathbf {x}\{\frac{1}{2}\Vert \mathbf {y}-\mathbf {x}\Vert _2^2+\sigma {\lambda }f(\mathbf {x})\}\) to estimate \(\mathbf {x}_0\), for some \(\lambda >0\). Common choices for \(f(\cdot )\) include the \(\ell _1\) norm for sparse vectors, the \(\ell _1-\ell _2\) norm for block-sparse signals and the nuclear norm for low-rank matrices. The metric we use to evaluate the performance of an estimate \(\mathbf {x}^*\) is the normalized mean-squared error \(\text {NMSE}(\sigma )=\frac{{\mathbb {E}}\Vert \mathbf {x}^*-\mathbf {x}_0\Vert _2^2}{\sigma ^2}\). We show that NMSE is maximized as \(\sigma \rightarrow 0\) and we find the exact worst-case NMSE, which has a simple geometric interpretation: the mean-squared distance of a standard normal vector to the \({\lambda }\)-scaled subdifferential \({\lambda }\partial f(\mathbf {x}_0)\). When \({\lambda }\) is optimally tuned to minimize the worst-case NMSE, our results can be related to the constrained denoising problem \(\min _{f(\mathbf {x})\le f(\mathbf {x}_0)}\{\Vert \mathbf {y}-\mathbf {x}\Vert _2\}\). The paper also connects these results to the generalized LASSO problem, in which one solves \(\min _{f(\mathbf {x})\le f(\mathbf {x}_0)}\{\Vert \mathbf {y}-{\mathbf {A}}\mathbf {x}\Vert _2\}\) to estimate \(\mathbf {x}_0\) from noisy linear observations \(\mathbf {y}={\mathbf {A}}\mathbf {x}_0+\mathbf {z}\). We show that certain properties of the LASSO problem are closely related to the denoising problem. In particular, we characterize the normalized LASSO cost and show that it exhibits a “phase transition” as a function of number of observations. We also provide an order-optimal bound for the LASSO error in terms of the mean-squared distance. Our results are significant in two ways. First, we find a simple formula for the performance of a general convex estimator. Secondly, we establish a connection between the denoising and linear inverse problems.     

Transcendence of zeros of Jacobi forms

A special case of a fundamental theorem of Schneider asserts that if \(j(\tau )\) is algebraic (where j is the classical modular invariant), then any zero z not in \(\mathbf{Q}.L_\tau := \mathbf{Q}\oplus \mathbf{Q}\tau \) of the Weierstrass function \(\wp (\tau ,\cdot )\) attached to the lattice \(L_\tau =\mathbf{Z}\oplus \mathbf{Z}\tau \) is transcendental. In this note we generalize this result to holomorphic Jacobi forms of weight k and index \(m\in \mathbf{N}\) with algebraic Fourier coefficients.     

Arithmetic Groups,Base Change,and Representation Growth

Consider an arithmetic group \({\mathbf{G}(O_S)}\), where \({\mathbf{G}}\) is an affine group scheme with connected, simply connected absolutely almost simple generic fiber, defined over the ring of S-integers O _S of a number field K with respect to a finite set of places S. For each \({n \in \mathbb{N}}\), let \({R_n(\mathbf{G}(O_S))}\) denote the number of irreducible complex representations of \({\mathbf{G}(O_S)}\) of dimension at most n. The degree of representation growth \({\alpha(\mathbf{G}(O_S)) = \lim_{n \rightarrow\infty}\log R_n(\mathbf{G}(O_S)) / \log n}\) is finite if and only if \({\mathbf{G}(O_S)}\) has the weak Congruence Subgroup Property. We establish that for every \({\mathbf{G}(O_S)}\) with the weak Congruence Subgroup Property the invariant \({\alpha(\mathbf{G}(O_S))}\) is already determined by the absolute root system of \({\mathbf{G}}\). To show this we demonstrate that the abscissae of convergence of the representation zeta functions of such groups are invariant under base extensions \({K{\subset}L}\). We deduce from our result a variant of a conjecture of Larsen and Lubotzky regarding the representation growth of irreducible lattices in higher rank semi-simple groups. In particular, this reduces Larsen and Lubotzky’s conjecture to Serre’s conjecture on the weak Congruence Subgroup Property, which it refines.     

Helicoidal surfaces satisfying {\Delta ^{II}\mathbf{G}=f(\mathbf{G}+C)}

In this paper, we study helicoidal surfaces without parabolic points in Euclidean 3-space \({\mathbb{R} ^{3}}\), satisfying the condition \({\Delta ^{II}\mathbf{G}=f(\mathbf{G}+C)}\), where \({\Delta ^{II}}\) is the Laplace operator with respect to the second fundamental form, f is a smooth function on the surface and C is a constant vector. Our main results state that helicoidal surfaces without parabolic points in \({ \mathbb{R} ^{3}}\) which satisfy the condition \({\Delta ^{II} \mathbf{G}=f(\mathbf{G}+C)}\), coincide with helicoidal surfaces with non-zero constant Gaussian curvature.     

Equilibrium Diffusion on the Cone of Discrete Radon Measures

Let \({\mathbb {K}(\mathbb {R}^{d})}\) denote the cone of discrete Radon measures on \(\mathbb {R}^{d}\). There is a natural differentiation on \(\mathbb {K}(\mathbb {R}^{d})\): for a differentiable function \(F:\mathbb {K}(\mathbb {R}^{d})\to \mathbb {R}\), one defines its gradient \(\nabla ^{\mathbb {K}}F\) as a vector field which assigns to each \(\eta \in \mathbb {K}(\mathbb {R}^{d})\) an element of a tangent space \(T_{\eta }(\mathbb {K}(\mathbb {R}^{d}))\) to \(\mathbb {K}(\mathbb {R}^{d})\) at point η. Let \(\phi :\mathbb {R}^{d}\times \mathbb {R}^{d}\to \mathbb {R}\) be a potential of pair interaction, and let μ be a corresponding Gibbs perturbation of (the distribution of) a completely random measure on \(\mathbb {R}^{d}\). In particular, μ is a probability measure on \(\mathbb {K}(\mathbb {R}^{d})\) such that the set of atoms of a discrete measure \(\eta \in \mathbb {K}(\mathbb {R}^{d})\) is μ-a.s. dense in \(\mathbb {R}^{d}\). We consider the corresponding Dirichlet form

$$\mathcal{E}^{\mathbb{K}}(F,G)={\int}_{\mathbb K(\mathbb{R}^{d})}\langle\nabla^{\mathbb{K}} F(\eta), \nabla^{\mathbb{K}} G(\eta)\rangle_{T_{\eta}(\mathbb{K})}\,d\mu(\eta). $$

Integrating by parts with respect to the measure μ, we explicitly find the generator of this Dirichlet form. By using the theory of Dirichlet forms, we prove the main result of the paper: If d ≥ 2, there exists a conservative diffusion process on \(\mathbb {K}(\mathbb {R}^{d})\) which is properly associated with the Dirichlet form \(\mathcal {E}^{\mathbb {K}}\).

     

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

Transformation formulae and asymptotic expansions for double holomorphic Eisenstein series of two complex variables

The main object of study in this paper is the double holomorphic Eisenstein series \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) having two complex variables \(\mathbf{s}=(s_1,s_2)\) and two parameters \(\mathbf{z}= (z_1,z_2)\) which satisfies either \(\mathbf{z}\in (\mathfrak {H}^+)^2\) or \(\mathbf{z}\in (\mathfrak {H}^-)^2\), where \(\mathfrak {H}^{\pm }\) denotes the complex upper and lower half-planes, respectively. For \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\), its transformation properties and asymptotic aspects are studied when the distance \(|z_2-z_1|\) becomes both small and large under certain natural settings on the movement of \(\mathbf{z}\in (\mathfrak {H}^{\pm })^2\). Prior to the proofs our main results, a new parameter \(\eta \), which plays a pivotal role in describing our results, is introduced in connection with the difference \(z_2-z_1\). We then establish complete asymptotic expansions for \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) when \(\mathbf{z}\) moves within the poly-sector either \((\mathfrak {H}^+)^2\) or \((\mathfrak {H}^-)^2\), so as to \(\eta \rightarrow 0\) through \(|\arg \eta |<\pi /2\) in the ascending order of \(\eta \) (Theorem 1). This further leads us to show that counterpart expansions exist for \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) in the descending order of \(\eta \) as \(\eta \rightarrow \infty \) through \(|\arg \eta |<\pi /2\) (Theorem 2). Our second main formula in Theorem 2 yields a functional equation for \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) (Corollaries 2.1, 2.2), and also reduces naturally to various expressions of \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) in closed forms for integer lattice point \(\mathbf{s}\in \mathbb {Z}^2\) (Corollaries 2.3–2.17). Most of these results reveal that the particular values of \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) at \(\mathbf{s}\in \mathbb {Z}^2\) are closely linked to Weierstraß’ elliptic function, the classical Eisenstein series reformulated by Ramanujan, and the Jordan–Kronecker type functions, each associated with the bases \(2\pi (1, z_j)\), \(j=1,2\). The latter two functions were extensively utilized by Ramanujan in the course of developing his theories of Eisenstein series, elliptic functions, and theta functions. As for the methods used, crucial roles in the proofs are played by the Mellin–Barnes type integrals, manipulated with several properties of hypergeometric functions; the transference from Theorem 1 to Theorem 2 is, for instance, achieved by a connection formula for Kummer’s confluent hypergeometric functions.     

Few associative triples,isotopisms and groups

Let Q be a quasigroup. For \(\alpha ,\beta \in S_Q\) let \(Q_{\alpha ,\beta }\) be the principal isotope \(x*y = \alpha (x)\beta (y)\). Put \(\mathbf a(Q)= |\{(x,y,z)\in Q^3;\) \(x(yz)) = (xy)z\}|\) and assume that \(|Q|=n\). Then \(\sum _{\alpha ,\beta }\mathbf a(Q_{\alpha ,\beta })/(n!)^2 = n^2(1+(n-1)^{-1})\), and for every \(\alpha \in S_Q\) there is \(\sum _\beta \mathbf a(Q_{\alpha ,\beta })/n! = n(n-1)^{-1}\sum _x(f_x^2-2f_x+n)\ge n^2\), where \(f_x=|\{y\in Q;\) \( y = \alpha (y)x\}|\). If G is a group and \(\alpha \) is an orthomorphism, then \(\mathbf a(G_{\alpha ,\beta })=n^2\) for every \(\beta \in S_Q\). A detailed case study of \(\mathbf a(G_{\alpha ,\beta })\) is made for the situation when \(G = \mathbb Z_{2d}\), and both \(\alpha \) and \(\beta \) are “natural” near-orthomorphisms. Asymptotically, \(\mathbf a(G_{\alpha ,\beta })>3n\) if G is an abelian group of order n. Computational results: \(\mathbf a(7) = 17\) and \(\mathbf a(8) \le 21\), where \(\mathbf a(n) = \min \{\mathbf a(Q);\) \( |Q|=n\}\). There are also determined minimum values for \(\mathbf a(G_{\alpha ,\beta })\), G a group of order \(\le 8\).     

Lower order eigenvalues of a system of equations of the drifting Laplacian on the metric measure spaces

Let \(\Omega \) be a bounded domain with smooth boundary in an n-dimensional metric measure space \((\mathbb {R}^n, \langle ,\rangle , e^{-\phi }dv)\) and let \(\mathbf {u}=(u^1, \ldots , u^n)\) be a vector-valued function from \(\Omega \) to \(\mathbb {R}^n\). In this paper, we investigate the Dirichlet eigenvalue problem of a system of equations of the drifting Laplacian: \(\mathbb {L}_{\phi } \mathbf {u} + \alpha [ \nabla (\mathrm {div}\mathbf { u}) -\nabla \phi \mathrm {div} \mathbf {u}]= - \widetilde{\sigma } \mathbf {u}\), in \( \Omega \), and \(u|_{\partial \Omega }=0,\) where \(\mathbb {L}_{\phi } = \Delta - \nabla \phi \cdot \nabla \) is the drifting Laplacian and \(\alpha \) is a nonnegative constant. We establish some universal inequalities for lower order eigenvalues of this problem on the metric measure space \((\mathbb {R}^n, \langle ,\rangle , e^{-\phi }dv)\) and the Gaussian shrinking soliton \((\mathbb {R}^n, \langle ,\rangle _{\mathrm {can}}, e^{-\frac{|x|^2}{4}}dv, \frac{1}{2})\). Moreover, we give an estimate for the upper bound of the second eigenvalue of this problem in terms of its first eigenvalue on the gradient product Ricci soliton \((\Sigma \times \mathbb {R}, \langle ,\rangle , e^{-\frac{\kappa t^2}{2}}dv, \kappa )\), where \( \Sigma \) is an Einstein manifold with constant Ricci curvature \(\kappa \).     

Homoclinic Solutions for <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)-Laplacian–Hamiltonian Systems Without Coercive Conditions

In this paper, we study the existence and multiplicity of homoclinic solutions for the following second-order p(t)-Laplacian–Hamiltonian systems

$$\frac{{\rm d}}{{\rm d}t}(|\dot{u}(t)|^{p(t)-2}\dot{u}(t))-a(t)|u(t)|^{p(t)-2}u(t)+\nabla W(t,u(t))=0,$$

where \({t \in \mathbb{R}}\), \({u \in \mathbb{R}^n}\), \({p \in C(\mathbb{R},\mathbb{R})}\) 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.

     

A Characterization of Two-Weight Norm Inequality for Littlewood–Paley $$g_{\lambda }^{*}$$-Function

Let \(n\ge 2\) and \(g_{\lambda }^{*}\) be the well-known high-dimensional Littlewood–Paley function which was defined and studied by E. M. Stein,

$$\begin{aligned} g_{\lambda }^{*}(f)(x) =\bigg (\iint _{\mathbb {R}^{n+1}_{+}} \Big (\frac{t}{t+|x-y|}\Big )^{n\lambda } |\nabla P_tf(y,t)|^2 \frac{\mathrm{d}y \mathrm{d}t}{t^{n-1}}\bigg )^{1/2}, \ \quad \lambda > 1, \end{aligned}$$

where \(P_tf(y,t)=p_t*f(y)\), \(p_t(y)=t^{-n}p(y/t)\), and \(p(x) = (1+|x|^2)^{-(n+1)/2}\), \(\nabla =(\frac{\partial }{\partial y_1},\ldots ,\frac{\partial }{\partial y_n},\frac{\partial }{\partial t})\). In this paper, we give a characterization of two-weight norm inequality for \(g_{\lambda }^{*}\)-function. We show that \(\big \Vert g_{\lambda }^{*}(f \sigma ) \big \Vert _{L^2(w)} \lesssim \big \Vert f \big \Vert _{L^2(\sigma )}\) if and only if the two-weight Muckenhoupt \(A_2\) condition holds, and a testing condition holds:

$$\begin{aligned} \sup _{Q : \text {cubes}~\mathrm{in} \ {\mathbb {R}^n}} \frac{1}{\sigma (Q)} \int _{{\mathbb {R}^n}} \iint _{\widehat{Q}} \Big (\frac{t}{t+|x-y|}\Big )^{n\lambda }|\nabla P_t(\mathbf {1}_Q \sigma )(y,t)|^2 \frac{w \mathrm{d}x \mathrm{d}t}{t^{n-1}} \mathrm{d}y < \infty , \end{aligned}$$

where \(\widehat{Q}\) is the Carleson box over Q and \((w, \sigma )\) is a pair of weights. We actually prove this characterization for \(g_{\lambda }^{*}\)-function associated with more general fractional Poisson kernel \(p^\alpha (x) = (1+|x|^2)^{-{(n+\alpha )}/{2}}\). Moreover, the corresponding results for intrinsic \(g_{\lambda }^*\)-function are also presented.

     

Conformally Covariant Differential Operators for the Diagonal Action of $$O(p,\,q)$$ on Real Quadrics

Let \(X=G/P\) be a real projective quadric, where \(G=O(p,\,q)\) and P is a parabolic subgroup of G. Let \((\pi _{\lambda ,\epsilon },\, \mathcal H_{\lambda ,\epsilon })_{ (\lambda ,\epsilon )\in {\mathbb {C}}\times \{\pm \}}\) be the family of (smooth) representations of G induced from the characters of P. For \((\lambda ,\, \epsilon ),\, (\mu ,\, \eta )\in {\mathbb {C}}\times \{\pm \},\) a differential operator \(\mathbf D_{(\mu ,\eta )}^\mathrm{reg}\) on \(X\times X,\) acting G-covariantly from \({\mathcal {H}}_{\lambda ,\epsilon } \otimes {\mathcal {H}}_{\mu , \eta }\) into \({\mathcal {H}}_{\lambda +1,-\epsilon } \otimes {\mathcal {H}}_{\mu +1, -\eta }\) is constructed.     

Frame completions with prescribed norms: local minimizers and applications

Let \(\mathcal {F}_{0}=\{f_{i}\}_{i\in \mathbb {I}_{n_{0}}}\) be a finite sequence of vectors in \(\mathbb {C}^{d}\) and let \(\mathbf {a}=(a_{i})_{i\in \mathbb {I}_{k}}\) be a finite sequence of positive numbers, where \(\mathbb {I}_{n}=\{1,\ldots , n\}\) for \(n\in \mathbb {N}\). We consider the completions of \(\mathcal {F}_{0}\) of the form \(\mathcal {F}=(\mathcal {F}_{0},\mathcal {G})\) obtained by appending a sequence \(\mathcal {G}=\{g_{i}\}_{i\in \mathbb {I}_{k}}\) of vectors in \(\mathbb {C}^{d}\) such that ∥g _i ∥² = a _i for \(i\in \mathbb {I}_{k}\), and endow the set of completions with the metric \(d(\mathcal {F},\tilde {\mathcal {F}}) =\max \{ \,\|g_{i}-\tilde {g}_{i}\|: \ i\in \mathbb {I}_{k}\}\) where \(\tilde {\mathcal {F}}=(\mathcal {F}_{0},\,\tilde {\mathcal {G}})\). In this context we show that local minimizers on the set of completions of a convex potential P _φ , induced by a strictly convex function φ, are also global minimizers. In case that φ(x) = x ² then P _φ is the so-called frame potential introduced by Benedetto and Fickus, and our work generalizes several well known results for this potential. We show that there is an intimate connection between frame completion problems with prescribed norms and frame operator distance (FOD) problems. We use this connection and our results to settle in the affirmative a generalized version of Strawn’s conjecture on the FOD.     

Absolute continuity and singularity of Palm measures of the Ginibre point process

We prove a dichotomy between absolute continuity and singularity of the Ginibre point process \(\mathsf {G}\) and its reduced Palm measures \(\{\mathsf {G}_{\mathbf {x}}, \mathbf {x} \in \mathbb {C}^{\ell }, \ell = 0,1,2\ldots \}\), namely, reduced Palm measures \(\mathsf {G}_{\mathbf {x}}\) and \(\mathsf {G}_{\mathbf {y}}\) for \(\mathbf {x} \in \mathbb {C}^{\ell }\) and \(\mathbf {y} \in \mathbb {C}^{n}\) are mutually absolutely continuous if and only if \(\ell = n\); they are singular each other if and only if \(\ell \not = n\). Furthermore, we give an explicit expression of the Radon–Nikodym density \(d\mathsf {G}_{\mathbf {x}}/d \mathsf {G}_{\mathbf {y}}\) for \(\mathbf {x}, \mathbf {y} \in \mathbb {C}^{\ell }\).     

Conflict free colorings of (strongly) almost disjoint set-systems

\(f\: \cup {\mathcal {A}}\to {\rho}\) is called a conflict free coloring of the set-system\({\mathcal {A}}\)(withρcolors) if

$\forall A\in {\mathcal {A}}\ \exists\, {\zeta}<{\rho} (|A\cap f^{-1}\{{\zeta}\}|=1).$

The conflict free chromatic number\(\operatorname {\chi _{\rm CF}}\, ({\mathcal {A}})\) of \({\mathcal {A}}\) is the smallest ρ for which \({\mathcal {A}}\) admits a conflict free coloring with ρ colors.

\({\mathcal {A}}\) is a (λ,κ,μ)-system if \(|{\mathcal {A}}| = \lambda\), |A|=κ for all \(A \in {\mathcal {A}}\), and \({\mathcal {A}}\) is μ-almost disjoint, i.e. |A∩A′|<μ for distinct \(A, A'\in {\mathcal {A}}\). Our aim here is to study

$\operatorname {\chi _{\rm CF}}\, (\lambda,\kappa,\mu) = \sup \{\operatorname {\chi _{\rm CF}}\, ({\mathcal {A}})\: {\mathcal {A}}\mbox{ is a } (\lambda,\kappa,\mu)\mbox{-system}\}$

for λ≧κ≧μ, actually restricting ourselves to λ≧ω and μ≦ω.

For instance, we prove that

? for any limit cardinal κ (or κ=ω) and integers n≧0, k>0, GCH implies

$\operatorname {\chi _{\rm CF}}\, (\kappa^{+n},t,k+1) =\begin{cases}\kappa^{+(n+1-i)}&; \text{if \ } i\cdot k < t \le (i+1)\cdot k,\ i =1,\dots,n;\\[2pt]\kappa&; \text{if \ } (n+1)\cdot k < t;\end{cases}$

? if λ≧κ≧ω>d>1, then λ<κ ^+ω implies \(\operatorname {\chi _{\rm CF}}\, (\lambda,\kappa,d) <\omega\) and λ≧? _ω (κ) implies \(\operatorname {\chi _{\rm CF}}\, (\lambda,\kappa,d) = \omega\);? GCH implies \(\operatorname {\chi _{\rm CF}}\, (\lambda,\kappa,\omega) \le \omega_{2}\) for λ≧κ≧ω ₂ and V=L implies \(\operatorname {\chi _{\rm CF}}\, (\lambda,\kappa,\omega) \le \omega_{1}\) for λ≧κ≧ω ₁;? the existence of a supercompact cardinal implies the consistency of GCH plus \(\operatorname {\chi _{\rm CF}}\,(\aleph_{\omega+1},\omega_{1},\omega)= \aleph_{\omega+1}\) and \(\operatorname {\chi _{\rm CF}}\, (\aleph_{\omega+1},\omega_{n},\omega) = \omega_{2}\) for 2≦n≦ω;? CH implies \(\operatorname {\chi _{\rm CF}}\, (\omega_{1},\omega,\omega) = \operatorname {\chi _{\rm CF}}\, (\omega_{1},\omega_{1},\omega) = \omega_{1}\), while \(MA_{\omega_{1}}\) implies \(\operatorname {\chi _{\rm CF}}\, (\omega_{1},\omega,\omega) = \operatorname {\chi _{\rm CF}}\, (\omega_{1},\omega_{1},\omega) = \omega\).     

Least energy sign-changing solutions for a class of fourth order Kirchhoff-type equations in $$\mathbb {R}^{N}$$

In this article we study the problem

$$\begin{aligned} \Delta ^{2}u-\left( a+b\int _{\mathbb {R}^{N}}\left| \nabla u\right| ^{2}dx\right) \Delta u+V(x)u=\left| u\right| ^{p-2}u\ \text { in }\mathbb {R}^{N}, \end{aligned}$$

where \(\Delta ^{2}:=\Delta (\Delta )\) is the biharmonic operator, \(a,b>0\) are constants, \(N\le 7,\) \(p\in (4,2_{*})\) for \(2_{*}\) defined below, and \(V(x)\in C(\mathbb {R}^{N},\mathbb {R})\). Under appropriate assumptions on V(x), the existence of least energy sign-changing solution is obtained by combining the variational methods and the Nehari method.