Any bounded linear operator between real (quasi-)Banach spaces T : X ® Y has a natural bounded complex linear extension TC : XC ® YC defined by the formula TC(x+iy)=Tx+iTy for x,yÎX, where XC and YC are so called reasonable complexifications of X and Y, respectively. We are interested in the exact relation between the norms of the operators TC and T. This relation can be expressed in terms of the constant gX,Y appearing in the inequality||TC|| £ gX,Y ||T|| considered for all bounded linear operators T : X®Y between (quasi-)Banach spaces. The work on the constant gLp,Lq for 0 &lt; p,q £ ¥, or shortly gp,q, is traced back to M. Riesz, Thorin, Marcinkiewicz, Zygmund, Verbitckii, Krivine, Gasch, Maligranda, Defant and others. In this thesis we try to summarize the results of these authors. We also present some new estimates for gp,q in the case when at least one of the spaces is quasi-Banach as well as in the case when the spaces are supplied with discrete measures. For example, we get that gp,q £ 2 for all 0 &lt; p,q £ ¥. Furthermore we obtain some new results concerning the relation between complex and real norms of the operators between spaces of functions of bounded p-variation and between mixed norm Lebesgue spaces. Looking for the criteria of the equality of real and complex norms of operators from a Banach lattice into the same Banach lattice we find a number of examples of two dimensional Orlicz spaces different from Lebesgue spaces and a simple operator between them with non-equal real and complex norms. We also consider in detail the Clarkson inequality which can be interpreted in terms of a certain operator norm inequality appearing as an example in many parts of the thesis. It turns out that complex norm of this operator can be easily obtained but to find the real one is not so trivial. With the help of the Clarkson inequality we construct an operator between Lebesgue spaces with non-atomic measures which has different real and complex norms. Finally, we consider both complex and real versions of the Riesz-Thorin interpolation theorem in the first quadrant and by using numbers gp,q find, for example, that the real Riesz constant is bounded by 2 for all 0 &lt; p,q £ ¥. / <p>Godkänd; 2007; 20070220 (ysko)</p>

This thesis consists of an introduction and three papers, which deal with some new discrete Hardy type inequalities. In the introduction we give an overview of the area that serves as a frame for the rest of the thesis. In particular, a short description of the development of Hardy type inequalities is given.In Paper 1 we prove a new discrete Hardy-type inequality $$ \|Af\|_{q,u}\leq C\|f\|_{p,v},~~~~1$$where the matrix operator $A$ is defined by $\left(Af\right)_i:=\sum\limits_{j=1}^ia_{i,j}f_j,$ ~$a_{i, j}\geq 0$, where the entries $a_{i, j}$ satisfies less restrictive additional conditions than studied before. Moreover, we study the problem of compactness for the operator $A$, and also the dual result is stated, proved and discussed.In Paper 2 we derive the necessary and sufficient conditions for inequality (1) to hold for the case $1 In Paper 3 we consider an operator of multiple summation with weights in weighted sequence spaces, which cover a wide class of matrix operators and we state, prove and discuss both boundedness and compactness forthis operator, for the case $1

The thesis consists of two closely related papers (A and B). Paper A is concerned with the study of the behaviour at infinity of solutions to second order elliptic equation with first order terms stated in a half-cylinder. The coefficients of the equation are assumed to be measurable and bounded; Neumann boundary condition is imposed on the lateral boundary of the cylinder, while on the base we assign the Dirichlet boundary condition. Under the assumption that the coefficients of the equation stabilize to a periodic regime exponentially, and the functions on the right-hand side decay sufficiently fast at infinity, we prove the existence and the uniqueness of a bounded solution and its stabilization to a constant at the exponential rate at infinity. Also we provide a necessary and sufficient condition for the uniqueness of a bounded solution. Our approach relies on the results from local qualitative elliptic theory, such as Harnack's inequality, Nash and De Giorgi estimates, the maximum principle, positive operator theory and a number of nontrivial a priori estimates. The problems of this type have many interesting applications in physics and mechanics and also appear while constructing the asymptotic expansions of solutions to equations describing different phenomena in highly inhomogeneous media. In particular, these results allow one to construct boundary layer correctors. Paper B is devoted to the homogenization of a stationary convectiondiffusion equation in a thin cylinder being a union of two nonintersecting rods with a junction at the origin. It is assumed that each of these cylinders has a periodic microstructure, and that the microstructure period is of the same order as the cylinder diameter. Under some natural assumptions on the data we construct and justify the asymptotic expansion of a solution which consists of the interior expansion and the boundary layer correctors, arising both in the vicinity of the rod ends and the vicinity of the junction. In contrast to the divergence form operators, in the case of convectiondiffusion equation the asymptotic behaviour of solutions depends crucially on the direction of the so-called effective convection (effective axial drift). In the present work we only consider the case when in each of the two cylinders (being the constituents of the rod) the effective convection is directed from the end of the cylinder towards the junction.

This Licentiate thesis consists of an introduction and three papers, which deal with some spaces of infinite matrices. In the introduction we give an overview of the area that serves as a frame for the rest of the thesis.In Paper 1 we introduce the space $B_w(\ell^2)$ of linear (unbounded) operators on $\ell^2$ which map decreasing sequences from $\ell^2$ into sequences from $\ell^2$ and we find some classes of operators belonging either to $B_w(\ell^2)$ or to the space of all Schur multipliers on $B_w(\ell^2)$.In Paper 2 we further continue the study of the space $B_w(\ell^p)$ in the range $1 <\infty$. In particular, we characterize the upper triangular positive matrices from $B_w(\ell^p)$.In Paper 3 we prove a new characterization of the Bergman-Schatten spaces $L_a^p(D,\ell^2)$, the space of all upper triangular matrices such that $\|A(\cdot)\|_{L^p(D,\ell^2)}<\infty$, where \[\|A(r)\|_{L^p(D,\ell^2)}=\left(2\int_0^1\|A(r)\|^p_{C_p}rdr\right)^\frac{1}{p}. \]This characterization is similar to that for the classical Bergman spaces. We also prove a duality between the little Bloch space and the Bergman-Schatten classes in the case of infinite matrices.

This Licentiate thesis is devoted to derive and discuss some new Friedrichs-type inequalities for functions, which belong to the Sobolev space $H^1$ in domains with microinhomogeneous structure and which vanish on a part of the boundary. The classical Friedrics inequality holds for functions from the space $\mathop{H^{\smash 1}}\limits^{\circ}$ with a constant depending only on the measure of the domain. It is well known that if the function has not zero trace on the whole boundary, but only on a subset of the boundary of a positive measure, then the Friedrichs inequality is still valid. Moreover, in such a case the constant in the inequality increases when the measure of the set where the function vanishes tends to zero. In particular, in this thesis we derive and discuss the corresponding behavior of the constant in our new Friedrichs-type inequalities. In paper A we prove a Friedrichs-type inequality for functions, having zero trace on small pieces of the boundary of a two-dimensional domain, which are periodically alternating. The total measure of the set, where the function vanishes, tends to zero. It turns out that for this case the constant in the Friedrichs inequality is bounded. Moreover, we describe the precise asymptotics of the constant in the derived Friedrichs inequality as the small parameter, describing the microinhomogeneous structure of the boundary, tends to zero. Paper B is devoted to the asymptotic analysis of functions depending on a small parameter, which characterizes the microinhomogeneous structure of the domain, where the functions are defined. We consider a boundary-value problem in a two-dimensional domain perforated nonperiodically along the boundary in the case when the diameter of circles and the distance between them have the same order. In particular, we prove that the limit of the original problems is a Dirichlet problem. Moreover, we use numerical simulations to illustrate the results. We also derive a new version of the Friedrichs inequality for functions, vanishing on the boundary of the cavities, and prove that the constant in the obtained inequality is close to the constant in the corresponding inequality for functions from $\mathop{H^{\smash 1}}\limits^{\circ}$. In paper C we consider a boundary-value problem in a three-dimensional domain, which is periodically perforated along the boundary in the case when the diameter of the holes and the distance between them have the same order. We suppose that the Dirichlet boundary condition holds on the boundary of the cavities. In particular, we derive the limit (homogenized) problem for the original problem. Moreover, we establish strong convergence in $H^1$ for the solutions of the considered problems to the corresponding solution of the limit problem. Moreover, we prove that the eigenelements of the original spectral problems converge to the corresponding eigenelement of the limit spectral problem. We apply these results to obtain that the constant in the derived Friedrichs inequality tends to the constant of the classical Friedrichs inequality for functions from $\mathop{H^{\smash 1}}\limits^{\circ}$, when the small parameter describing the size of perforation tends to zero.

This Licentiate thesis consists of an introduction and three papers, which deal with some new spaces of infinite matrices and Lorentz sequence spaces.In the introduction we give an overview of the area that serves as a frame for the rest of the thesis. In particular, a short description of Schur multipliers is given.In Paper 1 we prove that the space of all bounded operators on $\ell^2$ is contained in the space of all Schur multipliers on $B_w(\ell^2)$, where $B_w(\ell^2)$ is the space of linear (unbounded) operators on $\ell^2$ which map decreasing sequences from $\ell^2$ into sequences from $\ell^2$.In Paper 2 using a special kind of Schur multipliers and G. Bennett's factorization technique we characterize the upper triangular positive matrices from $B_w(\ell^p)$, $1In Paper 3 we consider the Lorentz spaces $\ell^{p,q}$ in the range $1\[\|x\|_{p,q}=\left(\sum_{n=1}^\infty (x^*)^q n^{\frac{q}{p}-1}\right)^\frac{1}{q}\]is only a quasi-norm. In particular, we derive the optimal constant in the triangle inequality for this quasi-norm, which leads us to consider the following decomposition norm:\[\|x\|_{(p,q)}=\inf\{\sum_k \|x^{(k)}\|_{p,q}\},\]where the infimum is taken over all finite representations $x=\sum_k x^{(k)}$.

This Licentiate Thesis is devoted to the study of Fourier series and Fourier transform multipliers and contains three papers (papers A - C) together with an introduction, which put these papers into a general frame. In paper A a generalization of the Lizorkin theorem on Fourier multipliers is proved. The proof is based on using the so-called net spaces and interpolation theorems. An example is given of a Fourier multiplier which satisfies the assumptions of the generalized theorem but does not satisfy the assumptions of the Lizorkin theorem.In paper B we prove and discuss a generalization and sharpening of the Lizorkin theorem concerning Fourier multipliers between $L_p$ and $L_q$. Some multidimensional Lorentz spaces and an interpolation technique (of Sparr type) are used as crucial tools in the proofs. The obtained results are discussed in the light of other generalizations of the Lizorkin theorem and some open questions are raised.Paper C deals with the Fourier series multipliers in the case with a regular system. This system is rather general. For example, trigonometrical systems, the Walsh system and all multiplicative system with bounded elements are regular. A generalization and sharpening of the Lizorkin type theorem concerning Fourier series multipliers between $L_p$ and $L_q$ in this general case is proved and discussed.

The aim of this Licentiate Thesis is to discuss special transformations and so-called adjoint symmetries of nonlinear partial differential equations. Nonlinear partial differential equations play an important role in the description of many physical phenomena. In order to understand the phenomena, modelled by the equations mentioned above, it is therefore necessary to obtain and analyze the solutions and the conservation laws of these equations. In this Thesis we investigate some methods to obtain conservation laws and transformations between nonlinear partial differential equations and moreover to classify nonlinear partial differential equations with respect to those methods.The main emphasis is on adjoint symmetries and transformations of evolution equations. In particular we study the adjoint symmetries and the construction of reciprocal Bäcklund transformations for evolution equations.

In this Licentiate thesis, we study some regular non-definite Sturm-Liouville problems. In this case, the weight function takes on both positive and negative signs on a given interval [a, b]. One feature of the non-definite Sturm-Liouville problem is the possible existence of non-real eigenvalues, unlike in the definite case (i.e. left or right definite) in which only real eigenvalues exist.This thesis consists of three papers (papers A-C) and an introduction to this area, which puts these papers into a more general frame.In paper A we give some precise estimates on the Richardson number for the two turning point case, thereby complementing the work of Jabon and Atkinson in an essential way. We also give a corrected version of their result since there seems to be a typographical error in their paper.In paper B we show that the interlacing property which holds in the one turning point case does not hold in the two turning point case. The paper consists of a detailed presentation of numerical results of the case in which the weight function is allowed to change its sign twice in the interval (−1, 2). We also present some theoretical results which support the numerical results.In paper C we extend results found in the paper by Jussi Behrndt et.al, in an essential way, to a case in which the weight function vanishes identically in a subinterval of [a, b]. In particular, we present some surprising numerical results on a specific problem in which the weight function is allowed to vanish identically on a subinterval of [−1, 2]. We also give some theoretical results which support these numerical examples.

In this licentiate thesis we study some mathematical problems in hydrodynamic lubrication theory. It is composed of two papers (A and B) and a complementary appendix. Lubrication theory is devoted to fluid flow in thin domains. The main purpose of lubrication is to reduce friction and wear between two solid surfaces in relative motion. The mathematical foundations of lubrication theory is given by the Navier--Stokes equation which describes the motion of viscous fluids. In thin domains several approximations are possible which leads to the so called Reynolds equation. This equation is crucial to describe the pressure in the lubricant film. When the pressure is found it is possible to predict different important physical quantities such as friction (stresses on the bounding surfaces), load carrying capacity and velocity field.In many practical situations the surface roughness amplitude and the film thickness are of the same order. Therefore, any realistic model should account for the effect of surface roughness. This implies that the mathematical modelling leads to partial differential equations with coefficients that will oscillate rapidly in space and time due to the relative motion of the surfaces. A direct numerical analysis is very difficult since an extremely fine mesh is required to describe the different scales. One method which has proved successful to handle such problems is to do some averaging (asymptotic analysis). The branch in mathematics which has been developed for this purpose is called homogenization.In Paper A the connection between the Stokes equation and the Reynolds equation is investigated. More precisely, the asymptotic behavior as both the film thickness ε and wavelength μ of the roughness tend to zero is analyzed and described. The results are obtained using the formal method of multiple scale expansion. The limit equation depends on how fast the two small parameters ε and μ go to zero relative to each other. Three different limit equations are derived. Time-dependent equations of Reynolds type are obtained in all three cases (Stokes roughness, Reynolds roughness and high frequency roughness regime).In paper B we present a mathematical model in hydrodynamic lubrication that takes into account cavitation (formation of air bubbles), surface roughness and compressibility of the fluid. We compute the homogenized coefficients in the case of unidirectional roughness. A one-dimensional problem describing a step bearing is also solved explicitly and by numerical methods.