Integral decompositions of varifolds

Some Applications of the Isoperimetric Inequality for Integral Varifolds

In this work the Isoperimetric Inequality for integral varifolds is used to obtain sharp estimates for the size of the set where the density quotient is small and to generalise Calderón’s and Zygmund’s theory of first order differentiability for functions in Lebesgue spaces from Lebesgue measure to integral varifolds.

A Sobolev Poincaré Type Inequality for Integral Varifolds

In this work a local inequality is provided which bounds the distance of an integral varifold from a multivalued plane (height) by its tilt and mean curvature. The bounds obtained for the exponents of the Lebesgue spaces involved are shown to be sharp.

Second Order Rectifiability of Integral Varifolds of Locally Bounded First Variation

In this work it is shown that for every integral n varifold in an open subset U of Rn+m, n,m ∈ N, of locally bounded first variation there exists a countable collection C of n dimensional submanifolds of U of class C2 such that μ(U ∼ S C) = 0 and for each member M of C ~ Hμ(x) = ~ HM (x) for μ almost all x ∈M.

Bilinear decompositions and commutators of singular integral operators

Let b be aBMO-function. It is well-known that the linear commutator [b, T ] of a Calderón-Zygmund operator T does not, in general, map continuously H(R) into L(R). However, Pérez showed that if H(R) is replaced by a suitable atomic subspace H b(R ) then the commutator is continuous from H b(R ) into L(R). In this paper, we find the largest subspace H b (R ) such that all commutators of Calderón...

Generalized Gaussian Quadratures and Singular Value Decompositions of Integral Operators