Wavelet frame bijectivity on Lebesgue and Hardy spaces

We prove a sufficient condition for frame-type wavelet series in $L^p$, the Hardy space $H^1$, and BMO. For example, functions in these spaces are shown to have expansions in terms of the Mexican hat wavelet, thus giving a strong answer to an old question of Meyer. Bijectivity of the wavelet frame operator acting on Hardy space is established with the help of new frequency-domain estimates on the Calder\'on-Zygmund constants of the frame kernel.Comment: 23 pages, 7 figure

Magnetic spectral bounds on starlike plane domains

We develop sharp upper bounds for energy levels of the magnetic Laplacian on starlike plane domains, under either Dirichlet or Neumann boundary conditions and assuming a constant magnetic field in the transverse direction. Our main result says that $\sum_{j=1}^n \Phi \big( \lambda_j A/G \big)$ is maximal for a disk whenever $\Phi$ is concave increasing, $n \geq 1$, the domain has area $A$, and $\lambda_j$ is the $j$-th Dirichlet eigenvalue of the magnetic Laplacian $\big( i\nabla+ \frac{\beta}{2A}(-x_2,x_1) \big)^2$. Here the flux $\beta$ is constant, and the scale invariant factor $G$ penalizes deviations from roundness, meaning $G \geq 1$ for all domains and $G=1$ for disks