Abstract: Homogeneous operators are a class of operators in $\R^N$ that include the Laplacian and all its powers (including fractional ones) while homogeneous semigroups are the ones generated by homogeneous operators. Homogeneous spaces are a class of spaces that include, among others, Lebesgue, Lorentz and Morrey ones. When homogeneous operators or semigroups act on homogeneous spaces, homogeneity implies much more precise results than in the general case. For example, we will show in this talk that homogeneous semigroups must satisfy quite sharp estimates. Also, we sill show that the resolvent set of an homogeneous operator must be formed by semilines passing through zero in the complex plane and a sharp estimate is available on the resolvent operator on each semiline. These properties, in turn, give that for homogeneous operators, Hille-Yosida and Lumer-Phillips theorems for the generation of a semigroup are much easier to check. Also, conditions to check that an homogeneous operator is sectorial (and hence it generates an analytic semigroup) are much easier to met. These conditions become specially simple in the case of an homogeneous Hilbert space. We will also discuss the problem of perturbing an homogeneous operator by homogeneous operators of lower degree (non resonant case) or the same degree (resonant one). We will analyse the spectrum and resolvent of the perturbed operator and show in particular that any non resonant perturbation of a sectorial operator is still sectorial. We will also show smoothing estimates for the perturbed semigroup. Finally we apply these results to some linear diffusion problems, including fractional diffusion with Hardy type potentials. In this case we show that resonant and non resonant conditions correspond to some Hardy and Gagliardo-Nirenberg type inequalities respectively. The evolution problem in the non resonant and resonant cases is also analysed.

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