O совпадении пространств J(//(Omega)и J(// для плоских областей Omega, имеющих выходы на бесконечность
We prove Obata’s rigidity theorem for metric measure spaces that satisfy a Riemannian curvaturedimension condition. Additionally,we show that a lower bound K for the generalizedHessian of a sufficiently regular function u holds if and only if u is K-convex. A corollary is also a rigidity result for higher order eigenvalues.
We consider elliptic nonlinear equations in a separable Hilbert space and their solutions in spaces of Sobolev type.
A definition of seminorm in the Sobolev space Ws,p (Γ) on a smooth compact manifold Gamma without boundary, using a localization procedure without partition of unity.
A modification of the Nikolskij extension theorem for functions from Sobolev spaces is presented. This modification requires the boundary to be only Lipschitz continuous for an arbitrary ; however, it is restricted to the case of two-dimensional bounded domains.
The Hardy inequality with holds for if is an open set with a sufficiently smooth boundary and if . P. Hajlasz proved the pointwise counterpart to this inequality involving a maximal function of Hardy-Littlewood type on the right hand side and, as a consequence, obtained the integral Hardy inequality. We extend these results for gradients of higher order and also for .
We prove an approximation theorem in generalized Sobolev spaces with variable exponent and we give an application of this approximation result to a necessary condition in the calculus of variations.
We obtain new variants of weighted Gagliardo-Nirenberg interpolation inequalities in Orlicz spaces, as a consequence of weighted Hardy-type inequalities. The weights we consider need not be doubling.
Let M be an N-function satisfying the Δ₂-condition, and let ω, φ be two other functions, with ω ≥ 0. We study Hardy-type inequalities , where u belongs to some set of locally absolutely continuous functions containing . We give sufficient conditions on the triple (ω,φ,M) for such inequalities to be valid for all u from a given set . The set may be smaller than the set of Hardy transforms. Bounds for constants are also given, yielding classical Hardy inequalities with best constants.
Define as the subspace of consisting of all harmonic functions in B, where B is the ball in the n-dimensional Euclidean space and E is any Banach space. Consider also the space consisting of all harmonic E*-valued functions g such that is bounded for some m>0. Then the dual is represented by through , . This extends the results of S. Bell in the scalar case.