Quadratic variation functionals and dilation equations

Here we present some distribution function inequalities between certain functionals defined relative to a convolution approximation procedure. Such inequalities are best known when the approximation is made using dilations of the Gaussian or Cauchy kernels. In these cases, classical differential equations, the heat equation or Laplace's equation, provide the basis for comparisons; in the latter case, the quadratic functional is known as the Lusin area integral. The kernels we consider are compactly supported, and satisfy a dilation equation, rather than a differential equation. For these kernels, there is an intrinsic quadratic variation, defined from the dilation structure. We obtain good lambda distribution function inequalities between a maximal function and the quadratic variation functional.