The modulus of smoothness in metric spaces and related problems (Q645038)

The paper studies generalized Besov and Sobolev spaces of functions on a metric measure space. The definition involves a rather general modulus of smoothness with two Banach lattices \(E\) and \(F\), and an auxiliary nondecreasing function. Let \(f^*\) stand for the decreasing rearrangement of \(f\) on \((0,\mu(X))\), and \(f^{**}\) for its maximal function. The main results provide estimates between the mentioned modulus of continuity, the oscillation \(f^{**}(t)-f^*(t)\), the sharp maximal function \(f^{\#}\), and the \(K\)-functional of real interpolation for \(E\) and related Besov or Sobolev spaces. Unfortunately, the paper contains several non-negligible typos which call for attentive reading. For instance, the condition of ``dyadic doubling'', as written, defines a void class of metric spaces.