Mapping properties of multi-parameter multipliers
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Date
30/11/2017Author
Bakas, Odysseas
Metadata
Abstract
This thesis is motivated by the problem of understanding the endpoint mapping
properties of higher-dimensional Marcinkiewicz multipliers. The one-dimensional
case was definitively characterised by Tao and Wright. In particular, they proved
that Marcinkiewicz multipliers acting on functions over the real line map the
Hardy space H¹(ℝ) to L¹;∞(ℝ) and they locally map L log¹/² L to L¹;∞ and that
these results are sharp.
The classical inequalities of Paley and Zygmund involving lacunary sequences
can be regarded as rudimentary prototypes of the aforementioned results of Tao
and Wright on the behaviour of Marcinkiewicz multipliers "near" L¹(ℝ). Motivated
by this fact, in Chapter 3 we obtain higher-dimensional variants of these
two inequalities and we establish sharp multiplier inclusion theorems on the torus
and on the real line. In Chapter 4 we extend the multiplier inclusion theorem on
T of Chapter 3 to higher dimensions.
In the last chapter of this thesis, we study endpoint mapping properties of
the classical Littlewood-Paley square function which can essentially be regarded
as a model Marcinkiewicz multiplier. More specifically, we give a new proof to a
theorem due to Bourgain on the growth of the operator norm of the Littlewood-
Paley square function as p → 1+ and then extend this result to higher dimensions.
We also obtain sharp weak-type inequalities for the multi-parameter Littlewood-
Paley square function and prove that the two-parameter Littlewood-Paley square
function does not map the product Hardy space H¹ to L¹;∞.