## Degree

Doctor of Philosophy

## Program

Mathematics

## Supervisor

Gord Sinnamon

## Abstract

A mixed L^{P} norm of a function on a product space is the

result of successive classical L^{p} norms in each variable,

potentially with a different exponent for each. Conditions to

determine when one mixed norm space is contained in another are

produced, generalizing the known conditions for embeddings

of L^{p} spaces.

The two-variable problem (with four L^{p} exponents, two for

each mixed norm) is studied extensively. The problem's ``unpermuted"

case simply reduces to a question of L^{p} embeddings. The other,

``permuted" case further divides, depending on the values of the

L^{p} exponents. Often, they fit the ``Minkowski case", when

Minkowski's integral inequality provides an easy, complete solution.

In the ``non-Minkowski case", the solution is determined

by the structure of the measures in the component L^{p} spaces.

When no measure is purely atomic, there can be no mixed-norm

embedding in the non-Minkowski case, so for such measures the

problem is solved.

With at least one purely atomic measure, the non-Minkowski case

divides further based on the structure of the measures and the

values of the exponents. Various necessary conditions and

sufficient conditions are found, solving a number of subcases.

Other subcases are shown to be genuinely complicated, with

their solutions expressed in terms of an optimization problem known

to be computationally difficult.

With some difficult cases already present in the two-variable

problem, it is impractical to cover every case of the

multivariable problem, but results are presented which

fully solve some cases.

When no measure is purely atomic, the multivariable problem

is solved by a reduction to the Minkowski case of certain

two-variable subproblems.

The multivariable problem with

unweighted *l*^{p} spaces has a similar reduction to

easy two-variable subproblems. It is conjectured that

this applies more generally; that, regardless of the structures

of the involved measures, when every permuted two-variable

subproblem fits the Minkowski case, the full multivariable

mixed norm inclusion must hold.

## Recommended Citation

Grey, Wayne R., "Inclusions Among Mixed-Norm Lebesgue Spaces" (2015). *Electronic Thesis and Dissertation Repository*. 2803.

https://ir.lib.uwo.ca/etd/2803