We extend several well-known tools from the theory of secondorder divergence-form elliptic equations to the case of higher-order equations. These tools are the Caccioppoli inequality, Meyers's reverse Hölder inequality for gradients, and the fundamental solution. Our construction of the fundamental solution may also be of interest in the theory of second-order operators, as we impose no regularity assumptions on our elliptic operator beyond ellipticity and boundedness of coefficients.2010 Mathematics Subject Classification. Primary 35J48, Secondary 31B10, 35C15.

Chapter 1. Introduction 1.1. History of the problem: L p setting 1.2. The nature of the problem and our main results 1.3. Outline of the monograph Acknowledgements Chapter 2. Definitions 2.1. Function spaces 2.2. Elliptic equations 2.3. Layer potentials 2.4. Boundary-value problems Chapter 3. The Main Theorems 3.1. Sharpness of these results Chapter 4. Interpolation, Function Spaces and Elliptic Equations 4.1. Interpolation functors 4.2. Function spaces 4.3. Solutions to elliptic equations Chapter 5. Boundedness of Integral Operators 5.1. Boundedness of the Newton potential 5.2. Boundedness of the double and single layer potentials Chapter 6. Trace Theorems Chapter 7. Results for Lebesgue and Sobolev Spaces: Historic Account and some Extensions Chapter 8. The Green's Formula Representation for a Solution

In this paper we establish square‐function estimates on the double and single layer potentials for divergence form elliptic operators, of arbitrary even order 2m, with variable t‐independent coefficients in the upper half‐space. This generalizes known results for variable‐coefficient second‐order operators, and also for constant‐coefficient higher‐order operators.

In this paper I investigate elliptic partial differential equations on Lipschitz domains in R 2 whose coefficient matrices have small (but possibly nonzero) imaginary parts and depend only on one of the two coordinates.I show that for Dirichlet boundary data in L p for p large enough, solutions exist and are controlled by the L p -norm of the boundary data.Similarly, for Neumann boundary data in L q , or for Dirichlet boundary data whose tangential derivative is in L q ("regularity" boundary data), for q small enough, I show that solutions exist and are controlled by the L q -norm of the boundary data. I prove similar results for Neumann or regularity boundary data in H 1 , and for Dirichlet boundary data in L ∞ or BMO. Finally, I show some converses: if the solutions are controlled in some sense, then Dirichlet, Neumann, or regularity boundary data must exist.iv1. They refer to it as K; I use Ǩ to differentiate it from the K in (2.9).

We show that if u is a solution to a linear elliptic differential equation of order 2m ≥ 2 in the half-space with t-independent coefficients, and if u satisfies certain area integral estimates, then the Dirichlet and Neumann boundary values of u exist and lie in a Lebesgue space L p (R n ) or Sobolev spacė W p ±1 (R n ). Even in the case where u is a solution to a second order equation, our results are new for certain values of p.