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Showing 1 to 8 of 8 for “"Sobolev inequalities"”.
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Nodal geometry of eigenfunctions on smooth manifolds and hardy-littlewood-sobolev inequalities on the heisenberg group
… dimension Q=2n+2, we study the Hardy-Littlewood-Sobolev (HLS) inequality,</p> <p>and particularly its sharp version. Weighted Hardy-Littlewood-Sobolev inequalities with different weights shall also be investigated, and we solve the following problems.</p> <p>(1) Establish the existence results of …
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Noncommutative Sobolev type inequalities
In this thesis, we study Sobolev type inequalities in the noncommutative (quantum) settings. We establish the abstract Bakry-\'Emery criterion for the operator-valued $f$-Sobolev inequalities on the derivation triple. We recapture the celebrated Bakry-\'Emery theorem for operator-valued functions …
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Error estimates for finite difference solutions of second-order elliptic equations in discrete Sobolev spaces
… use the Fefferman-Stein theorem and discrete Sobolev inequalities to establish our purpose. Based on these lp-estimates, we obtain the convergence rate of the approximate solutions and their difference quotients in the sup norm.
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Smoothing estimates for non commutative spaces
… perspective to establish the noncommutaive Sobolev inequaties (namely, Hardy-Littlewood-Sobolev inequalites), and extend the Sobolev embedding from noncommutative $L_p$ spaces to general Orlicz function spaces related with Cowling and Meda's work. Also we will show some examples to …
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Convex Relaxations: Beyond Polynomials, Splitting Methods and Average Case Analysis
… with the problem of determining the logarithmic Sobolev constant of a finite Markov chain, which can be framed as a nonconvex optimisation problem involving an entropy-like objective function. We demonstrate that it is possible to apply techniques from sum-of-squares programming to these …
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New tools for Bayesian optimal experimental design and kernel-based generative modeling
… information bounds, obtained via log-Sobolev inequalities, to identify optimal projections of the parameters and observations, and then apply our transport-based EIG estimation scheme. We next study the problem of cardinality-constrained observation selection to maximize MI in …
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On the nature and decay of quantum relative entropy
… inequality and uncertainty-like entropy inequalities to subalgebras of operators on quantum systems for which usual independence assumptions fail. We construct new measures of non-classicality that simultaneously quantify entanglement and uncertainty, leading to a new resource theory of …
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Functional inequalities in quantum information theory
Functional inequalities constitute a very powerful toolkit in studying various problems arising in classical information theory, statistics and many-body systems. Extensions of these tools to the noncommutative setting have been introduced in the beginning of the 90's in order to study the …