16:00
Lipschitz (or H\"older) spaces $C^\delta, \, k< \delta <k+1$, $k\in\mathbb{N}_0$, are the set of functions that are more regular than the $\mathcal{C}^k$ functions and less regular than the $\mathcal{C}^{k+1}$ functions. The classical definitions of H\"older classes involve pointwise conditions for the functions and their derivatives. This implies that to prove regularity results for an operator among these spaces we need its pointwise expression. In many cases this can be a rather involved formula, see for example the expression of $(-\Delta)^\sigma$ in (Stinga, Torrea, Regularity Theory for the fractional harmonic oscilator, J. Funct. Anal., 2011.)
In the 60's of last century, Stein and Taibleson, characterized bounded H\"older functions via some integral estimates of the Poisson semigroup, $e^{-y\sqrt{-\Delta}},$ and of the Gauss semigroup, $e^{\tau{\Delta}}$. These kind of semigroup descriptions allow to obtain regularity results for fractional operators in these spaces in a more direct way.
In this talk we shall see that we can characterize H\"older spaces adapted to other differential operators $\mathcal{L}$ by means of semigroups and that these characterizations will allow us to prove the boundedness of some fractional operators, such as $\mathcal{L}^{\pm \beta}$, Riesz transforms or Bessel potentials, avoiding the long, tedious and cumbersome computations that are needed when the pointwise expressions are handled.