On variable exponent analysis on homogeneous spaces
We start from a short survey of known results in Harmonic Analysis in Lebesgue spaces on metric measure spaces with doubling condition (homogeneous spaces) in the case of constant , this case having a long history, and in the case of variable , for this case only a single result on non-weighted boundedness of the maximal operator being recently obtained. After that we give a new result on the weighted boundedness of maximal Hardy-Littlewood operator on homogeneous metric measure spaces, for variable . We prove this result under certain sufficient condition which we call an "ersatz" of the Muckenhoupt condition. This sufficient condition nevertheless coincides with the necessary Muckenhoupt condition when is constant. A special class of weights is also considered, which includes "radial type" weights oscillating between two power functions (Zygmund-Bary-Stechkin type functions). In this case a stronger statement on the weighted boundeness is obtained. In connection with the weighted boundedness, we also introduce a new notion of lower and upper local dimensions of metric measure spaces. The talk is based on the joint work with Prof. Vakhtang Kokilashvili.
