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Register a new userDivergent operator with degeneracy and related sharp inequalities
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In this paper we classify all positive extremal functions to a sharp weighted Sobolev inequality on the upper half space, which involves divergent operators with degeneracy on the boundary. As an application of the results, we can derive a sharp Sobolev type inequality involving Baouendi-Grushin operator, and classify certain extremal functions for all $tau>0$ and $m e2 $ or $ n e1$.
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We prove the sharp quantitative stability for a wide class of weighted isoperimetric inequalities. More precisely, we consider isoperimetric inequalities in convex cones with homogeneous weights. Inspired by the proof of such isoperimetric inequalities through the ABP method, we construct a new convex coupling (i.e., a map that is the gradient of a convex function) between a generic set $E$ and the minimizer of the inequality (as in Gromovs proof of the isoperimetric inequality). Even if this map does not come from optimal transport, and even if there is a weight in the inequality, we adapt the methods of Figalli-Maggi-Pratelli and prove that if $E$ is almost optimal for the inequality then it is quantitatively close to a minimizer up to translations. Then, a delicate analysis is necessary to rule out the possibility of translations. As a step of our proof, we establish a sharp regularity result for restricted convex envelopes of a function that might be of independent interest.
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.
The sharp trace inequality of Jose Escobar is extended to traces for the fractional Laplacian on R^n and a complete characterization of cases of equality is discussed. The proof proceeds via Fourier transform and uses Liebs sharp form of the Hardy-Littlewood-Sobolev inequality.
Let $M$ be a complete, simply connected Riemannian manifold with negative curvature. We obtain some Moser-Trudinger inequalities with sharp constants on $M$.
This paper presents a highly non-trivial two-fold study of the fractional differential couples - derivatives ($ abla^{0<s<1}_+=(-Delta)^frac{s}{2}$) and gradients ($ abla^{0<s<1}_-= abla (-Delta)^frac{s-1}{2}$) of basic importance in the theory of fractional advection-dispersion equations: one is to discover the sharp Hardy-Rellich ($sp<p<n$) $|$ Adams-Moser ($sp=n$) $|$ Morrey-Sobolev ($sp>n$) inequalities for $ abla^{0<s<1}_pm$; the other is to handle the distributional solutions $u$ of the duality equations $[ abla^{0<s<1}_pm]^ast u=mu$ (a nonnegative Radon measure) and $[ abla^{0<s<1}_pm]^ast u=f$ (a Morrey function).
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