The one-dimensional heat equation in the Alexiewicz norm

A distribution on the real line has a continuous primitive integral if it is the distributional derivative of a function that is continuous on the extended real line. The space of distributions integrable in this sense is a Banach space that includes all functions integrable in the Lebesgue and Henstock–Kurzweil senses. The one-dimensional heat equation is considered with initial data that is integrable in the sense of the continuous primitive integral. Let Θ

The L p primitive integral

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.

Monotonically Controlled Mappings

We study classes of mappings between finite and infinite dimensional Banach spaces that are monotone and mappings which are differences of monotone mappings (DM). We prove a Radó–Reichelderfer estimate for monotone mappings in finite dimensional spaces that remains valid for DM mappings. This provides an alternative proof of the Fréchet differentiability a.e. of DM mappings. We establish a Morrey-type estimate for the distributional derivative of monotone mappings. We prove that a locally DM mapping between finite dimensional spaces is also globally DM. We introduce and study a new class of the so-called UDM mappings between Banach spaces, which generalizes the concept of curves of finite variation.

Convolutions with the Continuous Primitive Integral

IfFis a continuous function on the real line andf=F′is its distributional derivative, then the continuous primitive integral of distributionfis∫abf=F(b)−F(a). This integral contains the Lebesgue, Henstock-Kurzweil, and wide Denjoy integrals. Under the Alexiewicz norm, the space of integrable distributions is a Banach space. We define the convolutionf∗g(x)=∫−∞∞f(x−y)g(y)dyforfan integrable distribution andga function of bounded variation or anL1function. Usual properties of convolutions are shown to hold: commutativity, associativity, commutation with translation. Forgof bounded variation,f∗gis uniformly continuous and we have the estimate‖f∗g‖∞≤‖f‖‖g‖ℬ&#x1D4B1;, where‖f‖=supI|∫If|is the Alexiewicz norm. This supremum is taken over all intervalsI⊂ℝ. Wheng∈L1, the estimate is‖f∗g‖≤‖f‖‖g‖1. There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.

Poincaré and Friedrichs inequalities in abstract Sobolev spaces II

This paper is a continuation of [4]; its aim is to extend the results of that paper to include abstract Sobolev spaces of higher order and even anisotropic spaces. Let Ω be a domain in ℝN, let X = X(Ω) and Y = Y(Ω) be Banach function spaces in the sense of Luxemburg (see [4] for details), and let W(X, Y) be the abstract Sobolev space consisting of all f ∈ X such that for each i ∈ {l, …, N} the distributional derivative belongs to Y; equipped with the normW(X, Y) is a Banach space. Given any weight function w on Ω, the triple [w, X, Y] is said to support the Poincaré inequality on Ω. if there is a positive constant K such that for all u ∈ W(X, Y)the pair [X, Y] is said to support the Friedrichs inequality if there is a positive constant K such that for all u ∈ W0(X, Y) (the closure of in W(X, Y))

Poincaré and Friedrichs inequalities in abstract Sobolev spaces

Given any domain Ω in ℝN we consider spaces X = X(Ω),Y = Y(Ω) which are Banach function spaces in the sense of Luxemburg [12]. From these we form what we call the abstract Sobolev space W(X, Y), which is denned to be the linear space of all ƒ ∈ X such that for i = 1, …, N, the distributional derivative ∂ƒ/∂xi belongs to Y; equipped with the normW(X, Y) is a Banach space. The closure of (Ω) in W(X, Y) is denoted by W0(X, Y). Let w to bea weight function on Ω, that is, a measurable function which is positive and finite almost everywhere on Ω. We say that [w, X, Y] supports the (weighted) Poincaré inequality if there is a positive constant K such that for all u ∈ W(X, Y),analogously, [X, Y] is said to support the Friedrichs inequality if there is a positive constant K such that for all u ∈ W0(X, Y),

Regularization and distributional derivatives of ( x 1 2 + x 2 2 + . . . + x p 2 ) –½ n in R p

Our main aim is to present the value of the distributional derivative ∂͞ N /∂ x 1 k 1 ∂ x 2 k 2 . . . ∂ x p k p (1/ r n ), where r = ( x 1 2 + x 2 2 + . . . + x p 2 ) ½ in R p , N = k 1 + k 2 + . . . + K p , and p , n , k 1 , k 2 , . . ., k p are positive integers. For this purpose, we first define a regularization of 1/ x n in R 1 , which in turn helps us to define the regularization of 1/ r n in R p . These regularizations are achieved as asymptotic limits of the truncated functions H ( x – ∊)/ x n and H ( r –∊)/ r n as ϵ → 0, plus certain terms concentrated at the origin, where H is the Heaviside function. In the process of the derivation of the distributional derivative formula mentioned, we also derive many other interesting results and introduce some simplifying notation.