Recently, by Z. Shen, resolvent estimates for the Stokes operator were established in L^p(Ω) when Ω is a Lipschitz domain in R^d, with d≥3 and |1/p-1/2|<1/(2d)+ε. This result implies that the Stokes operator generates a bounded analytic semigroup in L^p(Ω) in the case that Ω is a three-dimensional Lipschitz domain and 3/2-ε<p<3+ε. To fully understand the work of Z. Shen, a lot of background information is needed. In this thesis the resolvent estimates are studied in detail in the case d=3. In the end the results of Shen are extended to resolvent estimates in L^p(w,Ω), where Ω is a three-dimensional Lipschitz domain, |1/p-1/2|<1/6, and w∈A_2p/3∩RH_3/(3-p) is a weight function that belongs to an intersection of a Muckenhoupt weight class and satisfies a reverse Hölder inequality.

Wavelets are a recent development in signal processing. These kind of functions
are both well-localized in time and in frequency, and so using these to transform the signal gives insight where certain frequencies are needed. The classical way of constructing wavelets, as described by Daubechies and Meyer [3,9] is only well-suited for the real numbers, so new methods are developed for a broader range of spaces. In this paper, we describe the algorithm developed by Coifman and Maggioni [1], and the algorithm developed by Coulhon et al. [11]. Lastly, we modify the last algorithm using the finite speed of propagation property, and so we obtain a new way of developing wavelets.

In this bachelor's thesis we will solve the Dirichlet problem with an L^p(T) boundary function. First, we will focus on the holomorphic version of the Dirichlet problem and introduce Hardy space theory, from which will follow a sufficient condition on the Fourier coefficients of the boundary function. Then we will prove the Marcinkiewicz interpolation theorem. After that we introduce the conjugate function "tilde f", which equals the Hilbert transform of f, and use functional analysis to prove an important duality argument of the Hilbert transform. Finally, we will give several different proofs for the boundedness of the map f ↦ tilde f using the Marcinkiewicz interpolation theorem and the duality argument: the last proof will be done rigorously from scratch, i.e. without relying on (unproved) arguments from other literature.