Parabolic wavelet transforms associated with the singular heat operators $-{\Delta }_{\gamma }+\partial /\partial t$ and $I-{\Delta }_{\gamma }+\partial /\partial t$, where ${\Delta }_{\gamma }={\sum }_{k=1}^n\partial ²/\partial x{²}_k+(2\gamma /xₙ)\partial /\partial xₙ$, are introduced. These transforms are defined in terms of the relevant generalized translation operator. An analogue of the Calderón reproducing formula is established. New inversion formulas are obtained for generalized parabolic potentials representing negative powers of the singular heat operators.

For a smooth curve $\Gamma$ and a set $\Lambda$ in the plane ${ℝ}^2$, let $AC(\Gamma ;\Lambda )$ be the space of finite Borel measures in the plane supported on $\Gamma$, absolutely continuous with respect to the arc length and whose Fourier transform vanishes on $\Lambda$. Following [12], we say that $(\Gamma ,\Lambda )$ is a Heisenberg uniqueness pair if $AC(\Gamma ;\Lambda )=\left\{0\right\}$. In the context of a hyperbola $\Gamma$, the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets $\Lambda$ of a collection of solutions to the Klein-Gordon equation. In this work, we mainly address the...

We prove that a function belonging to a fractional Sobolev space $L^{\alpha ,p}\left(ℝⁿ\right)$ may be approximated in capacity and norm by smooth functions belonging to $C^{m,\lambda }\left(ℝⁿ\right)$, 0 < m + λ < α. Our results generalize and extend those of [12], [4], [14], and [11].

We get a class of pointwise inequalities for Sobolev functions. As a corollary we obtain a short proof of Michael-Ziemer’s theorem which states that Sobolev functions can be approximated by $C^m$ functions both in norm and capacity.