Let p, q, n be natural numbers such that p + q = n.

Abstract. We find optimal integrability conditions on the initial data f for the existence of solutions e −t∆ λ f (x) and e −t √ ∆ λ f (x) of the heat and Poisson initial data problems for the Bessel operator ∆ λ in R + . We also characterize the most general class of weights v for which the solutions converge a.e. to f for every f ∈ L p (v), with 1 ≤ p < ∞. Finally, we show that for such weights and 1 < p < ∞ the local maximal operators are bounded from L p (v) to L p (u), for some weight u.

Abstract. We study the problem of extending a complex structure to a given Lie algebra g, which is firstly defined on an ideal h ⊂ g. We consider the next situations: h is either complex or it is totally real. The next question is to equip g with an additional structure, such as a (non)-definite metric or a symplectic structure and to ask either h is non-degenerate, isotropic, etc. with respect to this structure, by imposing a compatibility assumption. We show that this implies certain constraints on the algebraic structure of g. Constructive examples illustrating this situation are shown, in particular computations in dimension six are given.

We consider a family of initial boundary value problems governed by a fractional diffusion equation with Caputo derivative in time, where the parameter is the Newton heat transfer coefficient linked to the Robin condition on the boundary. For each problem we prove existence and uniqueness of solution by a Fourier approach. This will enable us to also prove the convergence of the family of solutions to the solution of the limit problem, which is obtained by replacing the Robin boundary condition with a Dirichlet boundary condition.

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