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Register a new userOne-dimensional interpolation inequalities, Carlson--Landau inequalities and magnetic Schrodinger operators
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In this paper we prove refined first-order interpolation inequalities for periodic functions and give applications to various refinements of the Carlson--Landau-type inequalities and to magnetic Schrodinger operators. We also obtain Lieb-Thirring inequalities for magnetic Schrodinger operators on multi-dimensional cylinders.
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We consider interpolation inequalities for imbeddings of the $l^2$-sequence spaces over $d$-dimensional lattices into the $l^infty_0$ spaces written as interpolation inequality between the $l^2$-norm of a sequence and its difference. A general method is developed for finding sharp constants, extremal elements and correction terms in this type of inequalities. Applications to Carlsons inequalities and spectral theory of discrete operators are given.
In this paper, we derive Carleman estimates for the fractional relativistic operator. Firstly, we consider changing-sign solutions to the heat equation for such operators. We prove monotonicity inequalities and convexity of certain energy functionals to deduce Carleman estimates with linear exponential weight. Our approach is based on spectral methods and functional calculus. Secondly, we use pseudo-differential calculus in order to prove Carleman estimates with quadratic exponential weight, both in parabolic and elliptic contexts. The latter also holds in the case of the fractional Laplacian.
We establish that trace inequalities $$|D^{k-1}u|_{L^{frac{n-s}{n-1}}(mathbb{R}^{n},dmu)} leq c |mu|_{L^{1,n-s}(mathbb{R}^{n})}^{frac{n-1}{n-s}}|mathbb{A}[D]u|_{L^{1}(mathbb{R}^{n},dmathscr{L}^{n})}$$ hold for vector fields $uin C^{infty}(mathbb{R}^{n};mathbb{R}^{N})$ if and only if the $k$-th order homogeneous linear differential operator $mathbb{A}[D]$ on $mathbb{R}^{n}$ is elliptic and cancelling, provided that $s<1$, and give partial results for $s=1$, where stronger conditions on $mathbb{A}[D]$ are necessary. Here, $|mu|_{L^{1,lambda}}$ denotes the $(1,lambda)$-Morrey norm of the measure $mu$, so that such traces can be taken, for example, with respect to the Hausdorff measure $mathscr{H}^{n-s}$ restricted to fractals of codimension $0<s<1$. The above class of inequalities give a systematic generalisation of Adams trace inequalities to the limit case $p=1$ and can be used to prove trace embeddings for functions of bounded $mathbb{A}$-variation, thereby comprising Sobolev functions and functions of bounded variation or deformation. We moreover establish a multiplicative version of the above inequality, which implies ($mathbb{A}$-)strict continuity of the associated trace operators on $text{BV}^{mathbb{A}}$.
This paper is devoted to an extension of rigidity results for nonlinear differential equations, based on carr{e} du champ methods, in the one-dimensional periodic case. The main result is an interpolation inequality with non-trivial explicit estimates of the constants in W1,p(S1) with p $ge$ 2. Mostly for numerical reasons, we relate our estimates with issues concerning periodic dynamical systems. Our interpolation inequalities have a dual formulation in terms of generalized spectral estimates of Keller-Lieb-Thirring type, where the differential operator is now a p-Laplacian type operator. It is remarkable that the carr{e} du champ method adapts to such a nonlinear framework, but significant changes have to be done and, for instance, the underlying parabolic equation has a nonlocal term whenever p$ e$2.
This paper is devoted to the symmetry and symmetry breaking properties of a two-dimensional magnetic Schr{o}dinger operator involving an Aharonov-Bohm magnetic vector potential. We investigate the symmetry properties of the optimal potential for the corresponding magnetic Keller-Lieb-Thir-ring inequality. We prove that this potential is radially symmetric if the intensity of the magnetic field is below an explicit threshold, while symmetry is broken above a second threshold corresponding to a higher magnetic field. The method relies on the study of the magnetic kinetic energy of the wave function and amounts to study the symmetry properties of the optimal functions in a magnetic Hardy-Sobolev interpolation inequality. We give a quantified range of symmetry by a non-perturbative method. To establish the symmetry breaking range, we exploit the coupling of the phase and of the modulus and also obtain a quantitative result.
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