Lieb–Thirring Type Inequalities for Schrödinger Operators with a Complex-Valued Potential

We present a possible generalization of the Lieb–Thirring inequalities for eigenvalues of Schrödinger operators to the case of complex potentials. We ask for a proof or disproof of this generalization. Some weaker results which have been obtained are reviewed.     

New constants in discrete Lieb–Thirring inequalities for Jacobi matrices

As is known, some results used for improving constants in the Lieb–Thirring inequalities for Schr?dinger operators in L ²(−∞, ∞) can be translated to discrete Schr?dinger operators and, more generally, to Jacobi matrices. We improve constants obtained earlier for Lieb–Thirring inequalities for the moments of eigenvalues larger than or equal to one. Bibliography: 9 titles.     

Global Nonlinear Brascamp–Lieb Inequalities

We prove global versions of certain known nonlinear Brascamp–Lieb inequalities under a natural homogeneity assumption. We also establish a conditional theorem allowing one to generally pass from local to global nonlinear Brascamp–Lieb estimates under such a homogeneity assumption.     

Lieb–Thirring inequality with semiclassical constant and gradient error term

In 1975, Lieb and Thirring derived a semiclassical lower bound on the kinetic energy for fermions, which agrees with the Thomas–Fermi approximation up to a constant factor. Whenever the optimal constant in their bound coincides with the semiclassical one is a long-standing open question. We prove an improved bound with the semiclassical constant and a gradient error term which is of lower order.     

Remarks on the Cwikel–Lieb–Rozenblum and?Lieb–Thirring Estimates for Schr?dinger Operators on?Riemannian Manifolds

Let M be a general complete Riemannian manifold and consider a Schr?dinger operator −Δ+V on L ²(M). We prove Cwikel–Lieb–Rozenblum as well as Lieb–Thirring type estimates for −Δ+V. These estimates are given in terms of the potential and the heat kernel of the Laplacian on the manifold. Some of our results hold also for Schr?dinger operators with complex-valued potentials.     

Maximal Inequalities for Fractional Lévy and Related Processes

In this article we study processes that are constructed by a convolution of a deterministic kernel with a martingale. A special emphasis is put on the case where the driving martingale is a centred Lévy process, which covers the popular class of fractional Lévy processes. As a main result we show that, under appropriate assumptions on the kernel and the martingale, the maximum process of the corresponding “convoluted martingale” is p-integrable and we derive maximal inequalities in terms of the kernel and of the moments of the driving martingale.     

Local Isoperimetric Inequalities for Sectors on Surfaces and Cones

On surfaces we give conditions under which the solution of a restricted local isoperimetric problem for sectors with small solid angle is the circular sector and we characterize these surfaces. Also we study this problem for general spherical cones on hypersurfaces in higher dimensional Riemannian manifolds.     

Hardy–Littlewood Inequalities for Fractional Derivatives of Invariant Harmonic Functions

We establish Hardy–Littlewood inequalities for fractional derivatives of M?bius invariant harmonic functions over the unit ball of mathbb Rⁿ{mathbb R^n} in mixed-norm spaces. In doing so we introduce a new criteria for the boundedness of operators in mixed-norm L ^p -spaces in terms of hyperbolic geometry of the real unit ball.     

Lieb–Thirring type inequalities for non-self-adjoint perturbations of magnetic Schrödinger operators

Let H:=_H0+V$H:=H_0+V$ and _H⊥:=_H0,⊥+V$H_{\perp }:=H_{0,\perp }+V$ be respectively perturbations of the unperturbed Schrödinger operators _H0$H_0$ on L²(R³)$L^2({\mathbb{R}}^3)$ and _H0,⊥$H_{0,\perp }$ on L²(R²)$L^2({\mathbb{R}}^2)$ with constant magnetic field of strength b>0$b>0$, and V a complex relatively compact perturbation. We prove Lieb–Thirring type inequalities on the discrete spectrum of H   and _H⊥$H_{\perp }$. In particular, these estimates give a priori information on the distribution of eigenvalues around the Landau levels of the operator, and describe how fast sequences of eigenvalues converge.     

Weighted Lorentz Norm Inequalities for Riemann-Liouville Fractional Integra

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Subadditivity of The Entropy and its Relation to Brascamp–Lieb Type Inequalities

We prove a general duality result showing that a Brascamp–Lieb type inequality is equivalent to an inequality expressing subadditivity of the entropy, with a complete correspondence of best constants and cases of equality. This opens a new approach to the proof of Brascamp–Lieb type inequalities, via subadditivity of the entropy. We illustrate the utility of this approach by proving a general inequality expressing the subadditivity property of the entropy on ${\mathbb {R}^n}We prove a general duality result showing that a Brascamp–Lieb type inequality is equivalent to an inequality expressing subadditivity of the entropy, with a complete correspondence of best constants and cases of equality. This opens a new approach to the proof of Brascamp–Lieb type inequalities, via subadditivity of the entropy. We illustrate the utility of this approach by proving a general inequality expressing the subadditivity property of the entropy on \mathbb Rⁿ{\mathbb {R}^n}, and fully determining the cases of equality. As a consequence of the duality mentioned above, we obtain a simple new proof of the classical Brascamp–Lieb inequality, and also a fully explicit determination of all of the cases of equality. We also deduce several other consequences of the general subadditivity inequality, including a generalization of Hadamard’s inequality for determinants. Finally, we also prove a second duality theorem relating superadditivity of the Fisher information and a sharp convolution type inequality for the fundamental eigenvalues of Schr?dinger operators. Though we focus mainly on the case of random variables in \mathbb Rⁿ{\mathbb {R}^n} in this paper, we discuss extensions to other settings as well.     

Weighted Norm Inequalities for Multipliers and Littlewood-Paley Functions on Local Fields

Let K be a local field, w(x) be a A_p-weight on K (1≤p≤∞). We say that the measurable function m(x) is a multiplier on L~p(K,w), if (m)~v ∈L~p(K,w) for all f∈L~p(K,w) and there is a constant c>0,independent of f such that ‖(m     

Hardy–Littlewood–Sobolev Inequalities with the Fractional Poisson Kernel and Their Applications in PDEs

The purpose of this paper is five-fold. First, we employ the harmonic analysis techniques to establish the following Hardy–Littlewood–Sobolev inequality with the fractional Poisson kernel on the upper half space ■ where f ∈ L~p(?R_+~n), g ∈ Lq(R_+~n) and p, q'∈(1, +∞), 2 ≤α n satisfying (n-1)/np+1/q'+(2-α)/n= 1.Second, we utilize the technique combining the rearrangement inequality and Lorentz interpolation to show the attainability of best constant C_(n,α,p,q'). Third, we apply the regularity lifting method to obtain the smoothness of extremal functions of the above inequality under weaker assumptions. Furthermore,in light of the Pohozaev identity, we establish the sufficient and necessary condition for the existence of positive solutions to the integral system of the Euler–Lagrange equations associated with the extremals of the fractional Poisson kernel. Finally, by using the method of moving plane in integral forms, we prove that extremals of the Hardy–Littlewood–Sobolev inequality with the fractional Poisson kernel must be radially symmetric and decreasing about some point ξ_0 ∈ ?R_+~n. Our results proved in this paper play a crucial role in establishing the Stein–Weiss inequalities with the Poisson kernel in our subsequent paper.     

Local and Nonlocal Boundary Value Problems for Degenerating and Nondegenerating Pseudoparabolic Equations with a Riemann–Liouville Fractional Derivative

We study local and nonlocal boundary value problems for degenerating and nondegenerating third-order pseudoparabolic equations of the general form with variable coefficients and with a Riemann–Liouville fractional derivative. For their solutions, we obtain a priori estimates that imply the uniqueness of the solution and its stability with respect to the right-hand side and the initial data.     

An Itô Formula of Generalized Functionals and Local Time for Fractional Brownian Sheet

Abstract

By using the white noise theory for a fractional Brownian sheet, we derive an Itô formula for the generalized functionals for the fractional Brownian sheet with arbitrary Hurst parameters H ₁, H ₂ ∈ (0,1). As an application, we give the integral representations for two versions of local times of a fractional Brownian sheet, respectively.     

Harnack Inequalities and Applications for Ornstein–Uhlenbeck Semigroups with Jump

The Harnack inequality established in Röckner and Wang (J Funct Anal 203:237–261, 2003) for generalized Mehler semigroup is improved and generalized. As applications, the log-Harnack inequality, the strong Feller property, the hyper-bounded property, and some heat kernel inequalities are presented for a class of O-U type semigroups with jump. These inequalities and semigroup properties are indeed equivalent, and thus sharp, for the Gaussian case. As an application of the log-Harnack inequality, the HWI inequality is established for the Gaussian case. Perturbations with linear growth are also investigated.