In this talk, we will revisit these classical theorems and discuss some recent results which identify a set of conditions on the measure that are both necessary and sufficient to ensure their veracity. This is joint work with Przemyslaw Gorka (Warsaw University of Technology), Piotr Hajlasz (University of Pittsburgh), and Marius Mitrea (University of Missouri, Columbia).

The result presented are obtained in collaboration with Sergio Polidoro.

Joint work with T. Ransford.

In this talk I will review some of these developments (starting from elementary considerations to more sophisticated versions) and I will discuss recents result regarding a sharp divergence theorem with non-tangential traces. This is joint work with Dorina Mitrea and Marius Mitrea from University of Missouri, Columbia.

The talk is based on the joint work with H. Dong (Brown University).

A set of functions $\{\psi_\gamma\}_{\gamma\in\Gamma}\subset L^2(\Rd)$ is called {\it almost-orthogonal\/} if there is a finite $R_1$ so that, for all finite subsets $\Eff\subset\Gamma$ and all linear combinations $\sum_{\gamma\in\Eff}\lambda_\gamma\psi_\gamma$, $$\bignrm{\sum_{\gamma\in\Eff}\lambda_\gamma\psi_\gamma}_{L^2(\Rd)}\leq R_1\left(\sum_{\gamma\in\Eff}\modd{\lambda_\gamma}^2\right)^{1/2}.$$ If $\{\phi_\gamma\}_{\gamma\in\Gamma}$ is another almost-orthogonal family, with constant $R_2$, then $$T(f)\equiv\sum_{\gamma\in\Gamma}\langle f,\psi_\gamma\rangle\,\phi_\gamma\eqno(1)$$ (where $\langle\cdot,\cdot\rangle$ is the inner product in $L^2(\Rd)$) converges unconditionally for all $f\in L^2(\Rd)$ to define a linear operator mapping $L^2(\Rd)\to L^2(\Rd)$, with bound $\leq R_1R_2$. We look at familiar (\dq wavelet-like'') almost-orthogonal families indexed over $\Gamma=$ the dyadic cubes. We show that they stay almost-orthogonal when they are replaced by their averages over rectangles defined by \dq irregular'' grids (an operation we call irregular discretization), even though the discretized functions can have many jump discontinuities in bad places. We show that if the grids are fine enough, then the resulting discretizations of (1) provide good approximations to the operator $T$.

We study a function space $JN_p$ based on a condition introduced by John and Nirenberg as a variant of BMO. It is known that $L^p\subset JN_{p}\subsetneq L^{p,\infty}$, but otherwise the structure of $JN_p$ is largely a mystery. Our first main result is the construction of a function that belongs to $JN_p$ but not $L^p$, showing that the two spaces are not the same. Nevertheless, we prove that for monotone functions, the classes $JN_{p}$ and $L^p$ do coincide. Our second main result describes $JN_p$ as the dual of a new Hardy kind of space $HK_{p'}$.