APPROXIMATIONS IN GENERALIZED MORREY SPACES - Hasanah, Dahliatul
Morrey spaces were first introduced by C.B. Morrey in 1938 while investigating elliptic partial differential equations. Morrey spaces can be considered as a generalization of Lebesgue spaces. While every function in Lebesgue spaces can be approximated by smooth functions, this property does not apply in Morrey spaces. C. T. Zorko in 1986 gave a function in Morrey spaces that smooth or continuous functions could not approximate. A subset of a Morrey space with a specific condition is defined so that each member can be approximated by compactly supported smooth functions. The subset, later, is called Zorko space and can be described as {f 2 Lp,!(Rn) : ||f(· + y) ? f(·)|| ! 0 as |y|!0}. The difference of function of first order utilized in defining Zorko spaces is a suitable tool to describe smoothness spaces. Many studies have been conducted to investigate the connection between smoothness spaces and spaces defined using the differences of function. For instance, Besov space as a smoothness space can be described using the difference of function of the first order in the form of f(.+h)?f(.) and higher order differences with sufficiently large smoothness exponents. The consequences of employing higher-order differences of function in Zorko spaces to the approximation property by smooth functions are now of interest to investigate further. This dissertation discusses a set of all functions f in Morrey spaces and generalized Morrey spaces with the property kf(· + h) + f(·?h) ? 2f(·)k ! 0 as |h| ! 0. The new set is a modification of Zorko space using the second-order difference of function. The dissertation presents research results concerning the relation between the Zorko space and its modification, as well as approximation properties by smooth functions in modified Zorko spaces. The inclusion property between Zorko spaces and modified Zorko spaces is studied and some examples of functions related to the spaces are provided. Each member in the modified Zorko spaces can be approximated by functions in the form of convolution with compactly supported smooth functions. By employing the inclusion relation between the Zorko space and the diamond space, it turns out that the modified Zorko space is identical to the classical one in Morrey spaces and generalized Morrey spaces.