On higher order extensions for the fractional Laplacian

By Ray A. Yang. 2013

88 citations recorded by the source corpus.

The technique of Caffarelli and Silvestre, characterizing the fractional Laplacian as the Dirichlet-to-Neumann map for a function U satisfying an elliptic equation in the upper half space with one extra spatial dimension, is shown to hold for general positive, non-integer orders of the fractional Laplace operator, by showing an equivalence between the H^s norm on the boundary and a suitable higher-order seminorm of U.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7eea85242d3cfba5954663f253418c9308b11c16

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

On Fractional GJMS Operators (2014, background): “Yang [33] observed that both sides of (4.”
Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems (2020, background): “ng UCP, one replaces the condition uj V = 0 by the requirement that uvanishes to innite order at some point x 0 2V. The higher order case s2R+ n(Z[(0;1)) has been studied recently by several authors [22, 25, 81]. These results however assume some special conditions on the function u, i.e. they requi”
Sharp commutator estimates via harmonic extensions (2016, methodology): “operator (1 − ∆)s/2, and thus for which our arguments might be useful. We treat only differential orders < 2, sometimes even < 1. Since there are higher order extensions to local operators, see [57, 41], it should, in principle, be also possible to obtain higher order commutator estimates. Fur”
Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities (2019, methodology): “g theblowup. Whenσisnot aninteger, thelowerbound ismuchsubtler. Itrelies on thelocalization formula for fractional Laplacian operators developed by Caffarelli-Silvestre [7], Chang-Gonza´lez [9], Yang [47], Case-Chang [8], as well as some technical analysis of Poisson integrals. Note that if uis a so”
The Nirenberg problem and its generalizations: a unified approach (2014, background): “5] and Case-Chang [22], which also provide geometric interpretation of the extension formulations for fractional Laplacian (−∆)σ established by Caffarelli-Silvestre [20] when σ ∈ (0,1) and by R. Yang [107] when σ > 1. Prescribing such fractional Q-curvature of order2σ on Sn can be considered as g”
The Generalized Caffarelli‐Kohn‐Nirenberg Theorem for the Hyperdissipative Navier‐Stokes System (2017, methodology): “The following theorem is essentially due to Yang [21] (see also the survey [5]): for our analysis we need however some adjustments of his statements and some further properties and for this reason we give a self-contained proof of the theorem and of other related facts in Appendix B.”