Quantum symmetric $L^{p}$ derivatives

Abstract:

For $1\leq p\leq \infty$, a one-parameter family of symmetric quantum derivatives is defined for each order of differentiation as are two families of Riemann symmetric quantum derivatives. For $1\leq p\leq \infty$, symmetrization holds, that is, whenever the $L^{p}$ $k$th Peano derivative exists at a point, all of these derivatives of order $k$ also exist at that point. The main result, desymmetrization, is that conversely, for $1\leq p\leq \infty$, each $L^{p}$ symmetric quantum derivative is a.e. equivalent to the $L^{p}$ Peano derivative of the same order. For $k=1$ and $2$, each $k$th $L^{p}$ symmetric quantum derivative coincides with both corresponding $k$th $L^{p}$ Riemann symmetric quantum derivatives, so, in particular, for $k=1$ and $2$, both $k$th $L^{p}$ Riemann symmetric quantum derivatives are a.e. equivalent to the $L^{p}$ Peano derivative.