THE BOUNDEDNESS OF BESSEL-RIESZ OPERATORS ON MORREY SPACES - IDRIS (NIM: 30113011), MOCH.
The boundedness of fractional integral operators on Lebesgue spaces was first proved in 1927. Many researchers continue to prove its boundedness on larger spaces, i.e Morrey spaces and generalized Morrey spaces. This dissertation discusses about the boundedness of Bessel-Riesz operators, one of the extension of fractional integral operators, on Lebesgue spaces, Morrey spaces, and generalized Morrey spaces. Our proofs use dyadic decomposition, HÃƒÂ¶lder's inequality, Hedberg's inequality, and the fact that the kernel of operator is contained in Lebesgue spaces and Morrey spaces. In the results, we also show that the norm of the kernel dominates the norm of the operator. In Lebesgue spaces, an estimate of the norm of the operator is obtained by showing the equivalence between the norm of the operator and the norm of the kernel. Meanwhile in Morrey spaces and generalized Morrey spaces, an upper bound and a lower bound of the norm of the operator are obtained from the norms of kernel on different Morrey spaces. In particular, we also reprove the boundedness of fractional integral operators on these spaces using the fact that its kernel is a member of Morrey spaces. Next, the norm of fractional integral operators is estimated by the norm of its kernel.