19 research outputs found
On the best constant of Hardy-Sobolev Inequalities
We obtain the sharp constant for the Hardy-Sobolev inequality involving the
distance to the origin. This inequality is equivalent to a limiting
Caffarelli-Kohn-Nirenberg inequality. In three dimensions, in certain cases the
sharp constant coincides with the best Sobolev constant
Trace Hardy--Sobolev--Mazy'a inequalities for the half fractional Laplacian
In this work we establish trace Hardy-Sobolev-Maz'ya inequalities with best
Hardy constants, for weakly mean convex domains. We accomplish this by
obtaining a new weighted Hardy type estimate which is of independent inerest.
We then produce Hardy-Sobolev-Maz'ya inequalities for the spectral half
Laplacian. This covers a critical case left open in \cite{FMT1}
Improving estimates to Harnack inequalities
We consider operators of the form , where is an
elliptic operator and is a singular potential, defined on a smooth bounded
domain with Dirichlet boundary conditions. We allow the
boundary of to be made of various pieces of different codimension. We
assume that has a generalized first eigenfunction of which we
know two sided estimates. Under these assumptions we prove optimal Sobolev
inequalities for the operator , we show that it generates an
intrinsic ultracontractive semigroup and finally we derive a parabolic Harnack
inequality up to the boundary as well as sharp heat kernel estimates
A logarithmic Hardy inequality
We prove a new inequality which improves on the classical Hardy inequality in
the sense that a nonlinear integral quantity with super-quadratic growth, which
is computed with respect to an inverse square weight, is controlled by the
energy. This inequality differs from standard logarithmic Sobolev inequalities
in the sense that the measure is neither Lebesgue's measure nor a probability
measure. All terms are scale invariant. After an Emden-Fowler transformation,
the inequality can be rewritten as an optimal inequality of logarithmic Sobolev
type on the cylinder. Explicit expressions of the sharp constant, as well as
minimizers, are established in the radial case. However, when no symmetry is
imposed, the sharp constants are not achieved among radial functions, in some
range of the parameters
Sharp two-sided heat kernel estimates for critical Schr\"odinger operators on bounded domains
On a smooth bounded domain \Omega \subset R^N we consider the Schr\"odinger
operators -\Delta -V, with V being either the critical borderline potential
V(x)=(N-2)^2/4 |x|^{-2} or V(x)=(1/4) dist (x,\partial\Omega)^{-2}, under
Dirichlet boundary conditions. In this work we obtain sharp two-sided estimates
on the corresponding heat kernels. To this end we transform the Scr\"odinger
operators into suitable degenerate operators, for which we prove a new
parabolic Harnack inequality up to the boundary. To derive the Harnack
inequality we have established a serier of new inequalities such as improved
Hardy, logarithmic Hardy Sobolev, Hardy-Moser and weighted Poincar\'e. As a
byproduct of our technique we are able to answer positively to a conjecture of
E.B.Davies.Comment: 40 page
Sharp Trace Hardy-Sobolev-Maz'ya Inequalities and the Fractional Laplacian
In this work we establish trace Hardy and trace Hardy-Sobolev-Maz'ya
inequalities with best Hardy constants, for domains satisfying suitable
geometric assumptions such as mean convexity or convexity. We then use them to
produce fractional Hardy-Sobolev-Maz'ya inequalities with best Hardy constants
for various fractional Laplacians. In the case where the domain is the half
space our results cover the full range of the exponent of the
fractional Laplacians. We answer in particular an open problem raised by Frank
and Seiringer \cite{FS}.Comment: 42 page