Boundary values of solutions of elliptic equations satisfying $H^{p}$ conditions

Abstract:

Let A be an elliptic linear partial differential operator with ${C^\infty }$ coefficients on a manifold ${\mathbf {\Omega }}$ with boundary ${\mathbf {\Gamma }}$. We study solutions of $Au = \sigma$ which satisfy the ${H^p}$ condition that ${\sup _{0 < t < 1}}{\left \| {u( \cdot ,t)} \right \|_p} < \infty$, where we have chosen coordinates in a neighborhood of ${\mathbf {\Gamma }}$ of the form ${\mathbf {\Gamma }} \times [0,1]$ with ${\mathbf {\Gamma }}$ identified with $t = 0$. If A has a well-posed Dirichlet problem such solutions may be characterized in terms of the Dirichlet data $u( \cdot ,0) = {f_0},{(\partial /\partial t)^j}u( \cdot ,0) = {f_j},j = 1, \cdots ,m - 1$ as follows: ${f_0} \in {L^p}$ (or $\mathfrak {M}$ if $p = 1$) and ${f_j} \in {\mathbf {\Lambda }}( - j;p,\infty ),j = 1, \cdots ,m$ . Here ${\mathbf {\Lambda }}$ denotes the Besov spaces in Taibleson’s notation. If $m = 1$ then u has nontangential limits almost everywhere.