The inequalities of embedding type in lebesgue and sobolev spaces with variable exponent

M SUMMARY In this thesis, the inequalities of embedding type, Sobolev and weighted Hardy type inequalities, are obtained in Lebesgue and Sobolev space with variable exponent. In the first chapter, as an introduction chapter, the exit point and physical motivation of Lebesgue and Sobolev spaces with variable exponent, and the studies that have been carried out up to now are briefly dealt with. The basic information used in the thesis and the spaces forming the basis of Lebesgue and Sobolev spaces with variable exponent are presented in the second chapter. In Hie third chapter, theory of Lebesgue and Sobolev space with, variable exponent is given, and in these spaces, especially the results about our studies are taken up. We proved Sobolev type inequality for measurable functions in Lebesgue and Sobolev spaces with variable exponent and Sobolev type inequality for fractional maximal function by making use of the relation between fractional maximal function and Riesz potential operator in the fourth chapter. In the fifth chapter, we were interested in Hardy type inequality, which is an indispensable tool for getting embedding theorems Lebesgue space with variable exponent. In this chapter, we obtained Hardy type inequality under the global log-Hölder continuity condition and also power type Hardy inequality under regularity condition of variable exponents defined in the domain of test function and the global log-Hölder continuity condition which is expressed in terms of boundary value of these exponents in Lebesgue spaces with variable exponent. Finally, we proved generalized Hardy inequality by investigating behaviors of weight functions and under the global log-Hölder continuity condition which is expressed in terms of boundary value of variable exponents and weight functions in the last chapter.