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 \(\kappa \).     

Regularity for Inhomogeneous Dirichlet Problems of Some Schrödinger Equations on Domains

Let \(n\ge 3, \Omega \) be a bounded, simply connected and semiconvex domain in \({\mathbb {R}}^n\) and \(L_{\Omega }:=-\Delta +V\) a Schrödinger operator on \(L^2 (\Omega )\) with the Dirichlet boundary condition, where \(\Delta \) denotes the Laplace operator and the potential \(0\le V\) belongs to the reverse Hölder class \(RH_{q_0}({\mathbb {R}}^n)\) for some \(q_0\in (\max \{n/2,2\},\infty ]\). Assume that the growth function \(\varphi :\,{\mathbb {R}}^n\times [0,\infty ) \rightarrow [0,\infty )\) satisfies that \(\varphi (x,\cdot )\) is an Orlicz function and \(\varphi (\cdot ,t)\in {\mathbb {A}}_{\infty }({\mathbb {R}}^n)\) (the class of uniformly Muckenhoupt weights). Let \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) be the Musielak–Orlicz–Hardy space whose elements are restrictions of elements of the Musielak–Orlicz–Hardy space, associated with \(L_{{\mathbb {R}}^n}:=-\Delta +V\) on \({\mathbb {R}}^n\), to \(\Omega \). In this article, the authors show that the operators \(VL^{-1}_\Omega \) and \(\nabla ^2L^{-1}_\Omega \) are bounded from \(L^1(\Omega )\) to weak-\(L^1(\Omega )\), from \(L^p(\Omega )\) to itself, with \(p\in (1,2]\), and also from \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) to the Musielak–Orlicz space \(L^\varphi (\Omega )\) or to \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) itself. As applications, the boundedness of \(\nabla ^2{\mathbb {G}}_D\) on \(L^p(\Omega )\), with \(p\in (1,2]\), and from \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) to \(L^\varphi (\Omega )\) or to \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) itself is obtained, where \({\mathbb {G}}_D\) denotes the Dirichlet Green operator associated with \(L_\Omega \). All these results are new even for the Hardy space \(H^1_{L_{{\mathbb {R}}^n},\,r}(\Omega )\), which is just \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) with \(\varphi (x,t):=t\) for all \(x\in {\mathbb {R}}^n\) and \(t\in [0,\infty )\).     

ANCOVA: a heteroscedastic global test when there is curvature and two covariates

For two independent groups, let \(M_j(\mathbf {X})\) be some conditional measure of location for the jth group associated with some random variable Y given \(\mathbf {X}=(X_1, X_2)\). Let \(\Omega =\{\mathbf {X}_1, \ldots , \mathbf {X}_K\}\) be a set of K points to be determined. An extant technique can be used to test \(H_0\): \(M_1(\mathbf {X})=M_2(\mathbf {X})\) for each \(\mathbf {X} \in \Omega \) without making any parametric assumption about \(M_j(\mathbf {X})\). But there are two general reasons to suspect that the method can have relatively low power. The paper reports simulation results on an alternative approach that is designed to test the global hypothesis \(H_0\): \(M_1(\mathbf {X})=M_2(\mathbf {X})\) for all \(\mathbf {X} \in \Omega \). The main result is that the new method offers a distinct power advantage. Using data from the Well Elderly 2 study, it is illustrated that the alternative method can make a practical difference in terms of detecting a difference between two groups.     

The fixed points of the multivariate smoothing transform

Let \((\mathbf {T}_1, \mathbf {T}_2, \ldots )\) be a sequence of random \(d\times d\) matrices with nonnegative entries, and let Q be a random vector with nonnegative entries. Consider random vectors \(X\) with nonnegative entries, satisfying

$$\begin{aligned} X\mathop {=}\limits ^{{\mathcal {L}}}\sum _{i \ge 1} \mathbf {T}_i X_i + Q, \end{aligned}$$

(*)

where \(\mathop {=}\limits ^{{\mathcal {L}}}\) denotes equality of the corresponding laws, \((X_i)_{i \ge 1}\) are i.i.d. copies of \(X\) and independent of \((Q, \mathbf {T}_1, \mathbf {T}_2, \ldots )\). For \(d=1\), this equation, known as fixed point equation of the smoothing transform, has been intensively studied. Under assumptions similar to the one-dimensional case, we obtain a complete characterization of all solutions \(X\) to (*) in the non-critical case, and existence results in the critical case.

     

Bochner representable operators on Banach function spaces

Let \((E,\Vert \cdot \Vert _E)\) be a Banach function space, \(E'\) the Köthe dual of E and \((X,\Vert \cdot \Vert _X)\) be a Banach space. It is shown that every Bochner representable operator \(T:E\rightarrow X\) maps relatively \(\sigma (E,E')\)-compact sets in E onto relatively norm compact sets in X. If, in particular, the associated norm \(\Vert \cdot \Vert _{E'}\) on \(E'\) is order continuous, then every Bochner representable operator \(T:E\rightarrow X\) is \((\gamma _E,\Vert \cdot \Vert _X)\)-compact, where \(\gamma _E\) stands for the natural mixed topology on E. Applications to Bochner representable operators on Orlicz spaces are given.     

Properties of singular integral operators $$\varvec{S}_{\varvec{\alpha ,\beta }}$$

For \(\alpha , \beta \in L^{\infty } (S^1),\) the singular integral operator \(S_{\alpha ,\beta }\) on \(L^2 (S^1)\) is defined by \(S_{\alpha ,\beta }f:= \alpha Pf+\beta Qf\), where P denotes the orthogonal projection of \(L^2(S^1)\) onto the Hardy space \(H^2(S^1),\) and Q denotes the orthogonal projection onto \(H^2(S^1)^{\perp }\). In a recent paper, Nakazi and Yamamoto have studied the normality and self-adjointness of \(S_{\alpha ,\beta }\). This work has shown that \(S_{\alpha ,\beta }\) may have analogous properties to that of the Toeplitz operator. In this paper, we study several other properties of \(S_{\alpha ,\beta }\).     

Weak type (1, 1) of some operators for the Laplacian with drift

Let \(v = (v_1, \ldots , v_n)\) be a vector in \(\mathbb {R}^n {\setminus } \{ 0 \}\). Consider the Laplacian on \(\mathbb {R}^n\) with drift \(\Delta _{v} = \sum _{i = 1}^n \Big ( \frac{\partial ^2}{\partial x_i^2} + 2 v_i \frac{\partial }{\partial x_i} \Big )\) and the measure \(d\mu (x) = e^{2 \langle v, x \rangle } dx\), with respect to which \(\Delta _{v}\) is self-adjoint. Let d and \(\nabla \) denote the Euclidean distance and the gradient operator on \(\mathbb {R}^n\). Consider the space \((\mathbb {R}^n, d, d\mu )\), which has the property of exponential volume growth. We obtain weak type (1, 1) for the Riesz transform \(\nabla (- \Delta _{v} )^{-\frac{1}{2}}\) and for the heat maximal operator, with respect to \(d\mu \). Further, we prove that the uncentered Hardy–Littlewood maximal operator is bounded on \(L^p\) for \(1 < p \le +\infty \) but not of weak type (1, 1) if \(n \ge 2\).     

Spectral multiplier theorems and averaged <Emphasis Type="Italic">R</Emphasis>-boundedness

Let A be a 0-sectorial operator with a bounded \(H^\infty (\Sigma _\sigma )\)-calculus for some \(\sigma \in (0,\pi ),\) e.g. a Laplace type operator on \(L^p(\Omega ),\, 1< p < \infty ,\) where \(\Omega \) is a manifold or a graph. We show that A has a \(\mathcal {H}^\alpha _2(\mathbb {R}_+)\) Hörmander functional calculus if and only if certain operator families derived from the resolvent \((\lambda - A)^{-1},\) the semigroup \(e^{-zA},\) the wave operators \(e^{itA}\) or the imaginary powers \(A^{it}\) of A are R-bounded in an \(L^2\)-averaged sense. If X is an \(L^p(\Omega )\) space with \(1 \le p < \infty \), R-boundedness reduces to well-known estimates of square sums.     

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.     

Ergodic Property of Stable-Like Markov Chains

A stable-like Markov chain is a time-homogeneous Markov chain on the real line with the transition kernel \(p(x,\hbox {d}y)=f_x(y-x)\hbox {d}y\), where the density functions \(f_x(y)\), for large \(|y|\), have a power-law decay with exponent \(\alpha (x)+1\), where \(\alpha (x)\in (0,2)\). In this paper, under a certain uniformity condition on the density functions \(f_x(y)\) and additional mild drift conditions, we give sufficient conditions for recurrence in the case when \(0<\liminf _{|x|\longrightarrow \infty }\alpha (x)\), sufficient conditions for transience in the case when \(\limsup _{|x|\longrightarrow \infty }\alpha (x)<2\) and sufficient conditions for ergodicity in the case when \(0<\inf \{\alpha (x):x\in \mathbb {R}\}\). As a special case of these results, we give a new proof for the recurrence and transience property of a symmetric \(\alpha \)-stable random walk on \(\mathbb {R}\) with the index of stability \(\alpha \ne 1\).     

Splines,lattice points,and arithmetic matroids

Let X be a \((d\times N)\)-matrix. We consider the variable polytope \(\varPi _X(u) = \{ w \ge 0 : X w = u \}\). It is known that the function \(T_X\) that assigns to a parameter \(u \in \mathbb {R}^d\) the volume of the polytope \(\varPi _X(u)\) is piecewise polynomial. The Brion–Vergne formula implies that the number of lattice points in \(\varPi _X(u)\) can be obtained by applying a certain differential operator to the function \(T_X\). In this article, we slightly improve the Brion–Vergne formula and we study two spaces of differential operators that arise in this context: the space of relevant differential operators (i.e. operators that do not annihilate \(T_X\)) and the space of nice differential operators (i.e. operators that leave \(T_X\) continuous). These two spaces are finite-dimensional homogeneous vector spaces, and their Hilbert series are evaluations of the Tutte polynomial of the arithmetic matroid defined by the matrix X. They are closely related to the \(\mathscr {P}\)-spaces studied by Ardila–Postnikov and Holtz–Ron in the context of zonotopal algebra and power ideals.     

Existence of positive solution for nonlocal singular fourth order Kirchhoff equation with Hardy potential

This paper is concerned with the existence of positive solution to a class of singular fourth order elliptic equation of Kirchhoff type

$$\begin{aligned} \triangle ^2 u-\lambda M(\Vert \nabla u\Vert ^2)\triangle u-\frac{\mu }{\vert x\vert ^4}u=\frac{h(x)}{u^\gamma }+k(x)u^\alpha , \end{aligned}$$

under Navier boundary conditions, \(u=\triangle u=0\). Here \(\varOmega \subset {\mathbf {R}}^N\), \(N\ge 1\) is a bounded \(C^4\)-domain, \(0\in \varOmega \), h(x) and k(x) are positive continuous functions, \(\gamma \in (0,1)\), \(\alpha \in (0,1)\) and \(M:{\mathbf {R}}^+\rightarrow {\mathbf {R}}^+\) is a continuous function. By using Galerkin method and sharp angle lemma, we will show that this problem has a positive solution for \(\lambda > \frac{\mu }{\mu ^*m_0}\) and \(0<\mu <\mu ^*\). Here \(\mu ^*=\Big (\frac{N(N-4)}{4}\Big )^2\) is the best constant in the Hardy inequality. Besides, if \(\mu =0\), \(\lambda >0\) and h, k are Lipschitz functions, we show that this problem has a positive smooth solution. If \(h,k\in C^{2,\,\theta _0}(\overline{\varOmega })\) for some \(\theta _0\in (0,1)\), then this problem has a positive classical solution.

     

Gradient Estimates for Commutators of Square Roots of Elliptic Operators with Complex Bounded Measurable Coefficients

Let \(L=-\mathrm{div}(A\nabla )\) be a second order divergence form elliptic operator and A an accretive \(n\times n\) matrix with bounded measurable complex coefficients in \({\mathbb R}^n\). Let \(\nabla b\in L^n({\mathbb R}^n)\,(n>2)\). In this paper, we prove that the commutator generated by b and the square root of L, which is defined by \([b,\sqrt{L}]f(x)=b(x)\sqrt{L}f(x)-\sqrt{L}(bf)(x)\), is bounded from the homogenous Sobolev space \({\dot{L}}_1^2({\mathbb R}^n)\) to \(L^2({\mathbb R}^n)\).     

Endpoint estimates of positive operators in a filtered measure space

In a filtered measure space, a necessary and sufficient Sawyer type condition is given under which a positive operator is bounded from \(L^1(d\mu )\) to \(L^{q,\infty }(d\mu )\) with \(1<q <\infty \). This result can be recognized as the endpoint case of Tanaka and Terasawa (Arch Math (Basel) 101:559–568, 2013, Theorem 1.1). We also establish a similar weak type inequality for generalized Doob’s maximal operators.     

Coalescece ad Meetig Times o $$$$-Block Markov Chais

We consider finite-state, discrete-time, mixing Markov chains \((V,P)\), where \(V\) is the state space and \(P\) is the transition matrix. To each such chain \((V,P)\), we associate a sequence of chains \((V_n,P_n)\) by coding trajectories of \((V,P)\) according to their overlapping \(n\)-blocks. The chain \((V_n,P_n)\), called the \(n\)-block Markov chain associated with \((V,P)\), may be considered an alternate version of \((V,P)\) having memory of length \(n\). Along such a sequence of chains, we characterize the asymptotic behavior of coalescence times and meeting times as \(n\) tends to infinity. In particular, we define an algebraic quantity \(L(V,P)\) depending only on \((V,P)\), and we show that if the coalescence time on \((V_n,P_n)\) is denoted by \(C_n\), then the quantity \(\frac{1}{n} \log C_n\) converges in probability to \(L(V,P)\) with exponential rate. Furthermore, we fully characterize the relationship between \(L(V,P)\) and the entropy of \((V,P)\).     

Averaged Wave Operators and Complex-symmetric Operators

We study the behaviour of sequences \(U_2^n X U_1^{-n}\), where \(U_1, U_2\) are unitary operators, whose spectral measures are singular with respect to the Lebesgue measure, and the commutator \(XU_1-U_2X\) is small in a sense. The conjecture about the weak averaged convergence of the difference \(U_2^n X U_1^{-n}-U_2^{-n} X U_1^n\) to the zero operator is discussed and its connection with complex-symmetric operators is established in a general situation. For a model case where \(U_1=U_2\) is the unitary operator of multiplication by z on \(L^2(\mu )\), sufficient conditions for the convergence as in the Conjecture are given in terms of kernels of integral operators.     

On the Dilation of Truncated Toeplitz Operators

An operator \(S_{\varphi ,\psi }^{u}\in \mathcal {L}(L^2)\) is called the dilation of a truncated Toeplitz operator if for two symbols \(\varphi ,\psi \in L^{\infty }\) and an inner function u,

$$\begin{aligned} S_{\varphi ,\psi }^{u}f=\varphi P_uf+\psi Q_uf \end{aligned}$$

holds for \(f\in {L}^{2}\) where \(P_{u}\) denotes the orthogonal projection of \(L^2\) onto the model space \(\mathcal { K}_{u}^2=H^2{\ominus }{{u}H^2}\) and \(Q_u=I-P_u.\) In this paper, we study properties of the dilation of truncated Toeplitz operators on \(L^{2}\). In particular, we provide conditions for the dilation of truncated Toeplitz operators to be normal. As some applications, we give several examples of such operators.

     

Optimal bounds for convergence of expected spectral distributions to the semi-circular law

Let \(\mathbf {X}=(X_{jk})_{j,k=1}^n\) denote a Hermitian random matrix with entries \(X_{jk}\), which are independent for \(1\le j\le k\le n\). We consider the rate of convergence of the empirical spectral distribution function of the matrix \(\mathbf {X}\) to the semi-circular law assuming that \(\mathbf{E}X_{jk}=0\), \(\mathbf{E}X_{jk}^2=1\) and that

$$\begin{aligned} \sup _{n\ge 1}\sup _{1\le j,k\le n}\mathbf{E}|X_{jk}|^4=:\mu _4<\infty , \end{aligned}$$

and

$$\begin{aligned} \sup _{1\le j,k\le n}|X_{jk}|\le D_0n^{\frac{1}{4}}. \end{aligned}$$

By means of a recursion argument it is shown that the Kolmogorov distance between the expected spectral distribution of the Wigner matrix \(\mathbf {W}=\frac{1}{\sqrt{n}}\mathbf {X}\) and the semicircular law is of order \(O(n^{-1})\).

     

Double ramification cycles on the moduli spaces of curves

Curves of genus \(g\) which admit a map to \(\mathbf {P}^{1}\) with specified ramification profile \(\mu\) over \(0\in \mathbf {P}^{1}\) and \(\nu\) over \(\infty\in \mathbf {P}^{1}\) define a double ramification cycle \(\mathsf{DR}_{g}(\mu,\nu)\) on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated the cycles for curves of compact type. We study here double ramification cycles on the moduli space of Deligne-Mumford stable curves.The cycle \(\mathsf{DR}_{g}(\mu,\nu)\) for stable curves is defined via the virtual fundamental class of the moduli of stable maps to rubber. Our main result is the proof of an explicit formula for \(\mathsf{DR}_{g}(\mu,\nu)\) in the tautological ring conjectured by Pixton in 2014. The formula expresses the double ramification cycle as a sum over stable graphs (corresponding to strata classes) with summand equal to a product over markings and edges. The result answers a question of Eliashberg from 2001 and specializes to Hain’s formula in the compact type case.When \(\mu=\nu=\emptyset\), the formula for double ramification cycles expresses the top Chern class \(\lambda_{g}\) of the Hodge bundle of \(\overline {\mathcal{M}}_{g}\) as a push-forward of tautological classes supported on the divisor of non-separating nodes. Applications to Hodge integral calculations are given.