Exact Norms and Morrey Estimates for Logarithmic Hardy Operators and Their Commutators on the Half-Line

We study logarithmic Hardy operators and their commutators on the half-line Morrey spaces defined using all bounded intervals. For 1≤p<∞, 0<λ<1, and γ>−1, the positive dilation principle gives the exact norm Γ(γ+1)[p/(p+λ−1)]γ+1, attained by a power function. We obtain exact norms for Hardy iterates and a reverse inequality for nonnegative nonincreasing functions. For the fractional operator, we prove a mapping theorem between distinct Morrey spaces, establish the necessary scaling relation, and give explicit upper bounds throughout a logarithmic range that is optimal for finiteness at every point. We distinguish this pointwise threshold from the almost-everywhere mapping endpoint. For the one-sided point-value symbol class, we prove embeddings into ordinary and central mean-oscillation spaces and identify its relation to the symmetric point-value class through even extension. This symbol condition yields explicit logarithmic and fractional commutator estimates. The symbol b(x)=logx raises the logarithmic order by one and gives an exact commutator norm with which the general upper bound can be compared.