Transforming bilinear vector integrals

Bilinear Identity for q - Hypergeometric Integrals

Singular Bilinear Integrals in Quantum Physics

Bilinear integrals of operator-valued functions with respect to spectral measures and integrals of scalar functions with respect to the product of two spectral measures arise in many problems in scattering theory and spectral analysis. Unfortunately, the theory of bilinear integration with respect to a vector measure originating from the work of Bartle cannot be applied due to the singular vari...

Vector-valued integrals

Quasi-complete, locally convex topological vector spaces V have the useful property that continuous compactly-supported V -valued functions have integrals with respect to finite Borel measures. Rather than constructing integrals as limits following [Bochner 1935], [Birkhoff 1935], et alia, we use the [Gelfand 1936][Pettis 1938] characterization of integrals, which has good functorial properties...

Preview of vector-valued integrals

In contrast to construction of integrals as limits of Riemann sums, the Gelfand-Pettis characterization is a property no reasonable notion of integral would lack. Since this property is an irreducible minimum, this definition of integral is called a weak integral. Uniqueness of the integral is immediate when the dual V ∗ separates points, meaning that for v 6 v′ in V there is λ ∈ V ∗ with λv 6=...

Weighted estimates for rough bilinear singular integrals via sparse domination

We prove weighted estimates for rough bilinear singular integral operators with kernel K(y1, y2) = Ω((y1, y2)/|(y1, y2)|) |(y1, y2)| , where yi ∈ R and Ω ∈ L∞(S2d−1) with ∫ S2d−1 Ωdσ = 0. The argument is by sparse domination of rough bilinear operators, via an abstract theorem that is a multilinear generalization of recent work by CondeAlonso, Culiuc, Di Plinio and Ou, 2016. We also use recent ...