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;∞.

This thesis consists of four papers, three papers about modelling animal populations and one paper about an area integral estimate for solutions of partial differential equations on non-smooth domains. The papers are: I. Å. Brännström, Single species population models from first principles. II. Å. Brännström and D. J. T. Sumpter, Stochastic analogues of deterministic single species population models. III. Å. Brännström and D. J. T. Sumpter, Coupled map lattice approximations for spatially explicit individual-based models of ecology. IV. Å. Brännström, An area integral estimate for higher order parabolic equations. In the first paper we derive deterministic discrete single species population models with first order feedback, such as the Hassell and Beverton-Holt model, from first principles. The derivations build on the site based method of Sumpter & Broomhead (2001) and Johansson & Sumpter (2003). A three parameter generalisation of the Beverton-Holtmodel is also derived, and one of the parameters is shown to correspond directly to the underlying distribution of individuals. The second paper is about constructing stochastic population models that incorporate a given deterministic skeleton. Using the Ricker model as an example, we construct several stochastic analogues and fit them to data using the method of maximum likelihood. The results show that an accurate stochastic population model is most important when the dynamics are periodic or chaotic, and that the two most common ways of constructing stochastic analogues, using additive normally distributed noise or multiplicative lognormally distributed noise, give models that fit the data well. The latter is also motivated on theoretical grounds. In the third paper we approximate a spatially explicit individual-based model with a stochastic coupledmap lattice. The approximation effectively disentangles the deterministic and stochastic components of the model. Based on this approximation we argue that the stable population dynamics seen for short dispersal ranges is a consequence of increased stochasticity from local interactions and dispersal. Finally, the fourth paper contains a proof that for solutions of higher order real homogeneous constant coefficient parabolic operators on Lipschitz cylinders, the area integral dominates the maximal function in the L2-norm.