Doctor of Philosophy
A mixed LP norm of a function on a product space is the
result of successive classical Lp 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 Lp spaces.
The two-variable problem (with four Lp exponents, two for
each mixed norm) is studied extensively. The problem's ``unpermuted"
case simply reduces to a question of Lp embeddings. The other,
``permuted" case further divides, depending on the values of the
Lp 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 Lp 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
The multivariable problem with
unweighted lp 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.
Grey, Wayne R., "Inclusions Among Mixed-Norm Lebesgue Spaces" (2015). Electronic Thesis and Dissertation Repository. 2803.