The L p primitive integral

Abstract

For each 1 ≤ p < ∞, a space of integrable Schwartz distributions L′p, is defined by taking the distributional derivative of all functions in L p. Here, L p is with respect to Lebesgue measure on the real line. If f ∈ L′p such that f is the distributional derivative of F ∈ L p, then the integral is defined as $\int\limits_{ - \infty }^\infty {fG} = - \int\limits_{ - \infty }^\infty {F(x)g(x)dx} $, where g ∈ L q, $G(x) = \int\limits_0^x {g(t)dt} $ and 1/p + 1/q =1. A norm is ‖f‖p′ = ‖F‖p. The spaces L′p and L p are isometrically isomorphic. Distributions in L′p share many properties with functions in L p. Hence, L′p is reflexive, its dual space is identified with L q, there is a type of Hölder inequality, continuity in norm, convergence theorems, Gateaux derivative. It is a Banach lattice and abstract L-space. Convolutions and Fourier transforms are defined. Convolution with the Poisson kernel is well defined and provides a solution to the half plane Dirichlet problem, boundary values being taken on in the new norm. A product is defined that makes L′1 into a Banach algebra isometrically isomorphic to the convolution algebra on L 1. Spaces of higher order derivatives of L p functions are defined. These are also Banach spaces isometrically isomorphic to L p.

Semicontinuous Banach spaces for Schrödinger's Eq. with Dirac-$δ'$ potential

“…Here we will not need to employ such generalized formalisms, though there is certainly some overlap with the afore mentioned in all cases. The approach here is a construction from first principals, in terms of functional methods on topological vector spaces with particular measure properties, differentiable manifolds [15,42,37], and the spaces of integrable distributions [38,40,39].…”

“…Another benefit of our construction is that many operators are naturally self adjoint on the semicontinuous manifold spaces defined below. Of particular relevance to our discussion here are the regulated classes of gauge integrals with Lebesgue-Stieltjes measure [38,40,39], as well as Krein function spaces [22,23,36,14]. The gauge integrals define a gauge function within subintervals I Ă R, and a tagged partition of I.…”

The one-dimensional heat equation in the Alexiewicz norm

“…We do have continuity in the L p norms for 1 ≤ p < ∞. Distributions that are the distributional derivative of an L p function are considered in [26]. When these are convolved with the heat kernel they give solutions that take on initial conditions in spaces of distributions that are isometrically isomorphic to L p .…”

Distributions, their primitives and integrals with applications to differential equations

“…Similarly, the completion of A L (I) in the 1-norm is the space of distributional derivatives of Lebesgue integrable functions. This space was studied in [25].…”