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Extra resources for Functional Analysis (Springer Classics in Mathematics)
Example text
14), M is locally bounded and tends to zero at the origin, thus 0 < β < α. 24) It is known that β is close to α for very narrow cones, and it will be close to 0 for obtuse cones (for Θ close to π), at least in dimension d 2. 17) for the half-line). 3 Approximate Factorization of Green Function In this section we will consider a bounded Lipschitz domain D ⊂ Rd , d 2, with Lipschitz constant λ. To simplify formulas, we recall the notation ≈: we write f (y) ≈ g(y) for y ∈ A if there exist constants C1 , C2 not depending on y such that C1 f (y) g(y) C2 f (y), y ∈ A.
The gauge function of D and q is defined as follows: u(x) = E x eq (τD ) . We can interpret u(x) as the expected mass of the particle when it leaves the domain. We note that since τD is an unbounded random variable, the mass may be infinite if q is (say, positive and) large enough. When the gauge 2 Boundary Potential Theory for Schr¨ odinger Operators 41 function satisfies u(x) < ∞ for (some, hence for all) x ∈ D, we call the pair (D, q) gaugeable. We consider ut (x, y) = E x [1t<τD eq (t)|Yt = y], the integral kernel of Tt .
First, it seems important to obtain an approximate factorization of the Green function for general (non-Lipschitz) domains, by using [38]. Second, it is of interest to study the asymptotics of the Martin kernel for narrow cones, and use the setup of [5] to complete the results of [111]. Third, it is of paramount importance to give sharp estimates for the transition density of the killed process. Fourth, it seems important to generalize the results discussed above to other stable L´evy processes ([40]), to more general jump type Markov processes, and to more general additive perturbations of their generators ([36, 52, 102, 82, 83]).