218 research outputs found
Sublinear equations and Schur's test for integral operators
We study weighted norm inequalities of -type, for , where
is an integral
operator associated with a nonnegative kernel on , and
is a locally finite positive measure in . We show that this
embedding holds if and only if provided is a quasi-symmetric
kernel which satisfies the weak maximum principle.
In the case , where , we prove that this condition
characterizes the existence of a non-trivial solution (or supersolution) , for , to the the sublinear integral equation We also give some
counterexamples in the end-point case , which corresponds to solutions of this integral equation. These problems appear in
the investigation of weak solutions to the sublinear equation involving the
(fractional) Laplacian, for and in domains with a positive Green function
Radial fractional Laplace operators and Hessian inequalities
In this paper we deduce a formula for the fractional Laplace operator
on radially symmetric functions useful for some applications.
We give a criterion of subharmonicity associated with , and
apply it to a problem related to the Hessian inequality of Sobolev type:
where is the -Hessian
operator on , , under some restrictions on
a -convex function . In particular, we show that the class of for
which the above inequality was established in \cite{FFV} contains the extremal
functions for the Hessian Sobolev inequality of X.-J. Wang \cite{W1}. This is
proved using logarithmic convexity of the Gaussian ratio of hypergeometric
functions which might be of independent interest
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