On the Gauge for the Third Boundary Value Problem

Abstract

For fairly general g, c if

$$E_x \left[ {\int_0^\infty {e_q (s)\hat e_c (s)I_A \left( {X(s)} \right)d\xi (s)} } \right] < \infty$$

for some x ∈D̄, where D is a bounded domain and A ⊂ ∂D is a nonempty open subset, it is shown that the gauge function for the third boundary value problem is bounded continuous. In the case of the Neumann problem, with the further assumption that q is Holder continuous, it is shown that the gauge is in C ² (D) ∩ C ¹(0304D).