Abstract. Let Ω ⊂ R N be a compact imbedded Riemannian manifold of dimension d ≥ 1 and define the (d + 1)-dimensional Riemannian manifold M := {(x, r(x)ω) : x ∈ R, ω ∈ Ω} with r > 0 and smooth, and the natural metric ds 2 = (1 + r (x) 2 )dx 2 + r 2 (x)ds 2 Ω . We require that M has conical ends:
We consider two types of multilinear pseudodifferential operators. First, we prove the boundedness of multilinear pseudodifferential operators with symbols which are only measurable in the spatial variables in weighted Lebesgue spaces. These results generalise earlier work of the present authors concerning linear pseudo-pseudodifferential operators. Secondly, we investigate the boundedness of bilinear pseudodifferential operators with symbols in the Hörmander S m ρ,δ classes. These results are new in the case ρ < 1, that is, outwith the scope of multilinear Calderón-Zygmund theory.
This paper studies the scattering matrix S(E;h) of the problemfor positive potentials V ∈ C ∞ (R) with inverse square behavior as x → ±∞. It is shown that each entry takes the formx + V has a zero energy resonance, then S(E;h) exhibits different asymptotic behavior as E → 0. The resonant case is excluded here due to V > 0.
Let R be a compact Riemann surface and Γ be a Jordan curve separating R into connected components Σ 1 and Σ 2 . We consider Calderón-Zygmund type operators T (Σ 1 , Σ k ) taking the space of L 2 anti-holomorphic one-forms on Σ 1 to the space of L 2 holomorphic one-forms on Σ k , which we call the Schiffer operators. We extend results of Menahem M. Schiffer and others, which where confined to analytic Jordan curves Γ, to general quasicircles in a characterizing manner, and prove new identities for adjoints of the Schiffer operators. Furthermore, we show that if V is the space of anti-holomorphic oneforms orthogonal to L 2 forms on R with respect to the inner product on Σ 1 , then the Schiffer operator T (Σ 1 , Σ 2 ) is an isomorphism onto the set of exact one-forms on Σ 2 .Using the relation between the Schiffer operator and a Cauchy-type integral involving Green's function, we also derive a jump decomposition (on arbitrary Riemann surfaces) for quasicircles and initial data which are boundary values of Dirichlet-bounded harmonic functions and satisfy the classical algebraic constraints. In particular we show that the jump operator is an isomorphism on the subspace determined by these constraints.2010 Mathematics Subject Classification. 58J05, 30F15.
We study the boundedness of rough Fourier integral and pseudodifferential operators, defined by general rough Hörmander class amplitudes, on Banach and quasi-Banach L p spaces. Thereafter we apply the aforementioned boundedness in order to improve on some of the existing boundedness results for Hörmander class bilinear pseudodifferential operators and certain classes of bilinear (as well as multilinear) Fourier integral operators. For these classes of amplitudes, the boundedness of the aforementioned Fourier integral operators turn out to be sharp. Furthermore we also obtain results for rough multilinear operators.
We consider bordered Riemann surfaces which are biholomorphic to compact Riemann surfaces of genus g with n regions biholomorphic to the disc removed. We define a refined Teichmüller space of such Riemann surfaces and demonstrate that in the case that 2g + 2 − n > 0, this refined Teichmüller space is a Hilbert manifold. The inclusion map from the refined Teichmüller space into the usual Teichmüller space (which is a Banach manifold) is holomorphic.We also show that the rigged moduli space of Riemann surfaces with non-overlapping holomorphic maps, appearing in conformal field theory, is a complex Hilbert manifold. This result requires an analytic reformulation of the moduli space, by enlarging the set of nonoverlapping mappings to a class of maps intermediate between analytically extendible maps and quasiconformally extendible maps. Finally we show that the rigged moduli space is the quotient of the refined Teichmüller space by a properly discontinuous group of biholomorphisms.2010 Mathematics Subject Classification. Primary 30F60 ; Secondary 30C55, 30C62, 32G15, 46E20, 81T40. Key words and phrases. Refined Teichmüller space, Hilbert manifold, quasiconformal maps, moduli space of rigged Riemann surfaces, conformal field theory.Eric Schippers is partially supported by the National Sciences and Engineering Research Council. He would like to thank Nina Zorboska for several helpful conversations.
Abstract. We define a class of pseudodifferential operators with symbols a(x, ξ) without any regularity assumptions in the x variable and explore their L p boundedness properties. The results are applied to obtain estimates for certain maximal operators associated with oscillatory singular integrals.
A complex harmonic function of finite Dirichlet energy on a Jordan domain has boundary values in a certain conformally invariant sense, by a construction of H. Osborn. We call the set of such boundary values the Douglas-Osborn space. One may then attempt to solve the Dirichlet problem on the complement for these boundary values. This defines a reflection of harmonic functions. We show that quasicircles are precisely those Jordan curves for which this reflection is defined and bounded.We then use a limiting Cauchy integral along level curves of Green's function to show that the Plemelj-Sokhotski jump formula holds on quasicircles with boundary data in the Douglas-Osborn space. This enables us to prove the well-posedness of a Riemann-Hilbert problem with boundary data in the Douglas-Osborn space on quasicircles.2010 Mathematics Subject Classification. Primary 35Q15, 30C62, 30E25, 31C25 ; Secondary 31A20.
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