Lebesgue Spaces Lp Research Articles

Some ergodic theorems

SynopsisA general ergodic theorem is proved for semi-group operators on B-space X. In particular X may be a Lebesgue space Lp(S, Σ, μ) where (S, Σ, μ) is a positive measure space.The discussion is based on the theory of semi-groups as developed by Hille [6] and results in the theory of product measures [3]. The reader need only be familiar with the basic concepts of these theories, as all pertinent results used in this note are proved as they are needed.

A note on invariant subspaces for finite maximal subdiagonal algebras

Let M be a von Neumann algebra with a faithful, normal, tracial state τ \tau and H ∞ {H^\infty } be a finite, maximal, subdiagonal algebra of M. Every left- (or right-) invariant subspace with respect to H ∞ {H^\infty } in the noncommutative Lebesgue space L p ( M , τ ) , 1 ⩽ p > ∞ {L^p}(M,\tau ),1 \leqslant p > \infty , is the closure of the space of bounded elements it contains.

Spectral analysis of a time dependent particles transport problem by an integral approach

Abstract A time-dependent problem arising in the diffusion of charged particles is studied by a recently proposed integral method. The Laplace transform technique leads to a linear integral equation containing the transform variable as a nonlinear parameter. The spectral analysis of this equation is then performed in the framework of Lebesgue spaces LP.

Local and global factorizations of matrix-valued functions

Let C be a simple closed Liapounov contour in the complex plane and A an invertible n × n n \times n matrix-valued function on C with bounded measurable entries. There is a well-known concept of factorization of the matrix function A relative to the Lebesgue space L p ( C ) {L_p}(C) . The notion of local factorization of A relative to L p {L_p} at a point t 0 {t_0} in C is introduced. It is shown that A admits a factorization relative to L p ( C ) {L_p}(C) if and only if A admits a local factorization relative to L p {L_p} at each point t 0 {t_0} in C. Several problems connected with local factorizations relative to L p {L_p} are raised.

Transport theory of a totally reflected parallelepiped reactor

An application of the linear integral Boltzmann equation for determining the stationary total flux distribution for monoenergetic neutrons in a totally reflected parallelepiped reactor — where both core and reflector are made by isotropically scattering and multiplying materials — is here presented. The general properties of the solution for the total flux are discussed in the context of the theory of linear integral equations in a Le-besgue space Lp. A practical version of the solution, holding in the whole reactor, is proposed in terms of a polynomials expansion.

Practical solution for a three-dimensional problem of integral neutron transport theory

The concept of Banach-adjoint transformation in a Lebesgue space Lp(p⩾1) is extensively used for proving the existence, the uniqueness and other basic properties of the solution to the stationary linear integral transport equation for the neutron flux in a three-dimensional inhomogeneous body. A practical solution of the problem is also constructed via a suitable polynomial expansion which is shown to converge in the mean of index one on the domain D occupied by the body considered.

Linear integral transformations generated by the three-dimensional neutron transport kernel

Two theorems are established by which the theory of linear integral transformations in a Lebesgue space Lp(D), p ≥ 1, can be appropriately extended to solve linear integral equations with kernel of nonfinite double norm with respect to the considered Lp(D). An application of these theorems to a physical problem of three-dimensional neutron transport theory is illustrated.

Convergence in the mean of solutions to the neutron integral Boltzmann equation in three-dimensional systems

The Neumann series solution as well as practical solutions for the stationary integral Boltzmann equation, which governs the flux distribution of monoenergetic neutrons in a three-dimensional system made by an isotropically scattering and multiplying material, are built up by extensively using the concept of double norm and the theory of bounded linear integral transformations in a Lebesgue space Lp. The convergence in the mean as well as other basic properties of the proposed solutions are studied for the cases of both distributed and isotropic deltalike sources.

On Hardy’s inequality and Laplace transforms in weighted rearrangement invariant spaces

Hardy’s well-known inequality relating the norm of a function and the norm of its integral mean in the Lebesgue spaces L p ( μ ) , d μ ( t ) = t σ − 1 d t {L^p}(\mu ),d\mu (t) = {t^{\sigma - 1}}dt , is extended to the class of rearrangement invariant spaces X ( μ ) X(\mu ) . These spaces include, for example, the L p ( μ ) {L^p}(\mu ) , the Lorentz and the Orlicz spaces. As an application, necessary and sufficient conditions are obtained for an operator related to the Laplace transform to be bounded as a linear operator between rearrangement invariant spaces of μ \mu -measurable functions.

Convolution Operators on Lebesgue Spaces of the Half-Line

In this paper we determine the lattice of closed invariant subspaces for certain convolution operators on Lebesgue spaces LP(da) where a is a suitable weighted measure on the half-line. We exploit the rather close relationship between convolution operators and the collection of right translation operators {TA}Aao on Lv(da). We show that a convolution operator K and the collection {TA}A,o have the same lattice of closed invariant subspaces provided the kernel k of K is a cyclic vector. The converse also holds if we assume in addition that the closed span of {TAk}A2o is all of LP(da). We show that the lattice of closed right translation invariant subspaces of LP(da) is totally ordered by set inclusion whenever a has compact support. Thus in this case a convolution operator K is unicellular if and only if its kernel is a cyclic vector. Finally, we show for suitable weighted measures a on the half-line that the convolution operators on LP(da) are Volterra.

Convolution operators on Lebesgue spaces of the half-line

In this paper we determine the lattice of closed invariant subspaces for certain convolution operators on Lebesgue spaces L p ( d σ ) {L^p}(d\sigma ) where σ \sigma is a suitable weighted measure on the half-line. We exploit the rather close relationship between convolution operators and the collection of right translation operators { T λ } λ ≧ 0 {\{ {T_\lambda }\} _{\lambda \geqq 0}} on L p ( d σ ) {L^p}(d\sigma ) . We show that a convolution operator K and the collection { T λ } λ ≧ 0 {\{ {T_\lambda }\} _{\lambda \geqq 0}} have the same lattice of closed invariant subspaces provided the kernel k of K is a cyclic vector. The converse also holds if we assume in addition that the closed span of { T λ k } λ ≧ 0 {\{ {T_\lambda }k\} _{\lambda \geqq 0}} is all of L p ( d σ ) {L^p}(d\sigma ) . We show that the lattice of closed right translation invariant subspaces of L p ( d σ ) {L^p}(d\sigma ) is totally ordered by set inclusion whenever σ \sigma has compact support. Thus in this case a convolution operator K is unicellular if and only if its kernel is a cyclic vector. Finally, we show for suitable weighted measures σ \sigma on the half-line that the convolution operators on L p ( d σ ) {L^p}(d\sigma ) are Volterra.

The existence of a solution of a linear integral equation

A theorem is proved concerning the existence of a solution of a linear integral equation in generalized Lebesgue space Lp (Δn),p=(p1, ..., pn), where An is an n-dimensional parallelepiped.

A Study of Holders Inequality in New Spaces

The central attraction of this paper is to prove holders inequality in some new spaces such as Grand Lebesgue Space LP). Small Lebesgue space LP)' and Banach Lattice space XP.Key words: Holders InequalityDOI: http://dx.doi.org/10.3126/jie.v8i1-2.5119Journal of the Institute of Engineering Vol. 8, No. 1&2, 2010/2011Page: 259-262Uploaded Date: 20 July, 2011