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Register a new userImproved Sobolev trace inequalities in the unit ball under constraints
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We establish an improved Sobolev trace inequality of order two in the Euclidean unit ball under the vanishing of higher order moments of the boundary volume element, and construct precise test functions to show that such inequalities are almost optimal. Our arguments can be adapted to the fourth order Sobolev trace inequalities in higher dimensional unit ball.
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We give a new proof of Aubins improvement of the Sobolev inequality on $mathbb{S}^{n}$ under the vanishing of first order moments of the area element and generalize it to higher order moments case. By careful study of an extremal problem on $mathbb{S}^{n}$, we determine the constant explicitly in the second order moments case.
We show that gradient shrinking, expanding or steady Ricci solitons have potentials leading to suitable reference probability measures on the manifold. For shrinking solitons, as well as expanding soltions with nonnegative Ricci curvature, these reference measures satisfy sharp logarithmic Sobolev inequalities with lower bounds characterized by the geometry of the manifold. The geometric invariant appearing in the sharp lower bound is shown to be nonnegative. We also characterize the expanders when such invariant is zero. In the proof various useful volume growth estimates are also established for gradient shrinking and expanding solitons. In particular, we prove that the {it asymptotic volume ratio} of any gradient shrinking soliton with nonnegative Ricci curvature must be zero.
For the weight function $W_mu(x) = (1-|x|^2)^mu$, $mu > -1$, $lambda > 0$ and $b_mu$ a normalizing constant, a family of mutually orthogonal polynomials on the unit ball with respect to the inner product $$ la f,g ra = {b_mu [int_{BB^d} f(x) g(x) W_mu(x) dx + lambda int_{BB^d} abla f(x) cdot abla g(x) W_mu(x) dx]} $$ are constructed in terms of spherical harmonics and a sequence of Sobolev orthog onal polynomials of one variable. The latter ones, hence, the orthogonal polynomials with respect to $la cdot,cdotra$, can be generated through a recursive formula.
A classical result of Aubin states that the constant in Moser-Trudinger-Onofri inequality on $mathbb{S}^{2}$ can be imporved for furnctions with zero first order moments of the area element. We generalize it to higher order moments case. These new inequalities bear similarity to a sequence of Lebedev-Milin type inequalities on $mathbb{S}^{1}$ coming from the work of Grenander-Szego on Toeplitz determinants (as pointed out by Widom). We also discuss the related sharp inequality by a perturbation method.
Motivated by a recent work of Ache and Chang concerning the sharp Sobolev trace inequality and Lebedev-Milin inequalities of order four on the Euclidean unit ball, we derive such inequalities on the Euclidean unit ball for higher order derivatives. By using, among other things, the scattering theory on hyperbolic spaces and the generalized Poisson kernel, we obtain the explicit formulas of extremal functions of such inequations. Moreover, we also derive the sharp trace Sobolev inequalities on half spaces for higher order derivatives. Finally, we compute the explicit formulas of adapted metric, introduced by Case and Chang, on the Euclidean unit ball, which is of independent interest.
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