On s-sets and mutual absolute continuity of measures on homogeneous spaces.
C. Watari [12] obtained a simple characterization of Lipschitz classes on the dyadic group using the -modulus of continuity and the best approximation by Walsh polynomials. Onneweer and Weiyi [4] characterized homogeneous Besov spaces on locally compact Vilenkin groups, but there are still some gaps to be filled up. Our purpose is to give the characterization of Besov spaces by oscillations, atoms and others on the dyadic groups. As applications, we show a strong capacity inequality of the...
Let be a metric measure space satisfying the doubling condition and an -Poincaré inequality. Consider the nonnegative operator generalized by a Dirichlet form on . We will show that a solution to on satisfies an -Carleson condition if and only if can be represented as the Poisson integral of the operator with the trace in the generalized Morrey space , where is a nonnegative function defined on a class of balls in . This result extends the analogous characterization founded...