Andreas Rosén

Layer potentials beyond singular integral operators

We prove that the double layer potential operator and the gradient of the single layer potential operator are L_2 bounded for general second order divergence form systems. As compared to earlier results, our proof shows that the bounds for the layer potentials are independent of well posedness for the Dirichlet problem and of De Giorgi-Nash local estimates. The layer potential operators are shown to depend holomorphically on the coefficient matrix A\in L_\infty, showing uniqueness of the extension of the operators beyond singular integrals. More precisely, we use functional calculus of differential operators with non-smooth coefficients to represent the layer potential operators as bounded Hilbert space operators. In the presence of Moser local bounds, in particular for real scalar equations and systems that are small perturbations of real scalar equations, these operators are shown to be the usual singular integrals. Our proof gives a new construction of fundamental solutions to divergence form systems, valid also in dimension 2.

On the Carleson duality

As a tool for solving the Neumann problem for divergence-form equations, Kenig and Pipher introduced the space

Riesz continuity of the Atiyah–Singer Dirac operator under perturbations of the metric

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Teaching and Learning Electromagnetics: An Analytical Problem-Solving Approach Education Corner

of functions on the half-space, such that the non-tangential maximal function of their L2 Whitney averages belongs to L2 on the boundary. In this paper, answering questions which arose from recent studies of boundary value problems by Auscher and the second author, we find the pre-dual of

Evolution of time-harmonic electromagnetic and acoustic waves along waveguides

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A spin integral equation for electromagnetic and acoustic scattering

, and characterize the pointwise multipliers from

Weighted maximal regularity estimates and solvability of nonsmooth elliptic systems, II

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Cauchy non-integral formulas

to L2 on the half-space as the well-known Carleson-type space of functions introduced by Dahlberg. We also extend these results to Lp generalizations of the space

Boundary value problems for degenerate elliptic equations and systems

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A local

. Our results elaborate on the well-known duality between Carleson measures and non-tangential maximal functions.