Weakly Canceling Operators and Singular Integrals

Singular Integrals and Littlewood–Paley Operators

We prove mixed Ap-Ar inequalities for several basic singular integrals, Littlewood–Paley operators, and the vector-valued maximal function. Our key point is that r can be taken arbitrarily big. Hence, such inequalities are close in spirit to those obtained recently in the works by T. Hytönen and C. Pérez, and M. Lacey. On one hand, the “Ap-A∞” constant in these works involves two independent su...

Multiresolution separated representations of singular and weakly singular operators ✩

For a finite but arbitrary precision, we construct efficient low separation rank representations for the Poisson kernel and for the projector on the divergence free functions in the dimension d = 3. Our construction requires computing only one-dimensional integrals. We use scaling functions of multiwavelet bases, thus making these representations available for a variety of multiresolution algor...

Bounds for Singular Fractional Integrals and Related Fourier Integral Operators

Monte Carlo approximation of weakly singular integral operators

We study the randomized approximation of weakly singular integral operators. For a suitable class of kernels having a standard type of singularity and being otherwise of finite smoothness, we develop a Monte Carlo multilevel method, give convergence estimates and prove lower bounds which show the optimality of this method and establish the complexity. As an application we obtain optimal methods...

A Nyström method for weakly singular integral operators on surfaces

We describe a modified Nyström method for the discretization of the weakly singular boundary integral operators which arise from the formulation of linear elliptic boundary value problems as integral equations. Standard Nyström and collocation schemes proceed by representing functions via their values at a collection of quadrature nodes. Our method uses appropriately scaled function values in l...