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MATH 256B: B-ANALYSIS AND B-PSEUDODIFFERENTIAL OPERATORS

This course focuses on so-called totally characteristic, or b-(pseudo)differential operators. These are operators on manifolds M with boundaries or corners, though they can also arise on complete manifolds without boundary after one appropriately compactifies or bordifies them, as we discuss below. We study both elliptic and hyperbolic operators, with the Laplacian or the d’Alembertian of a b-metric (i.e. a metric corresponding to this structure) giving main examples. An important aspect of b-analysis is, however, that the principal symbol, on which ellipticity, etc., is based, does not completely capture the problem: there is also an analytic family of operators at the boundary ∂M , depending on the dual of the normal variable to the boundary, the poles of whose inverse determine both Fredholm properties and asymptotic expansions. To be more concrete, consider the vector space of vector fields Vb(M) tangent to ∂M , M an n-dimensional manifold with boundary. Recall that V ∈ Vb(M) is equivalent to the statement that for all f ∈ C∞(M) with f |∂M = 0 one has V f |∂M = 0; this in turn is equivalent to the same statement for a single f with a non-degenerate differential at ∂M . This space Vb(M) is a left C∞(M) module, and is a Lie algebra: if f is a function that vanishes at ∂M , V,W ∈ Vb(M), then VWf , WV f vanish at ∂M , thus so does