In a few months ago I post a note about a family of an integro-partial differential equations (2), there the main objective was to expose, without details, a first work to treat questions such as existence and symmetries of solutions in certain critical space, namely, homogeneous Morrey Spaces. On that work the Mittag-Leffler family below was central on the estimates,

which is an integral form of the integro-partial differential equation

where , , for and and , here stands for Gamma function. In this post we will prove (1). Before, we need some properties of the Laplace transform. Given a “well” real-valued function on , we define the Laplace transform as

Given , let be certain -valued functions for each fixed, then

and

where stands for convolution operator on variable , that is, . Hence, applying the Laplace transform in (2), we obtain

In view of

and

we have that the equation (6) can be written as

For , let (Schwartz space) and let be the Fourier transform of given by

If with compact support, easily seems that

As it follows from (7) and (9) that

which yields

This motivates to define Mittag-Leffler function as a complex integral on certain curve. Indeed, firstly notice that the complex valued function has singularities in

Let be the standard Hankel’s curve in , positive oriented, such that and , that is, let be a parametrized curve given by , for and for . As we want to chose such that and , we shall suppose that and one defines Mittag-Leffler function as

Using residue theorem, a characterization very important of this definition was obtained in [Fujita] (see also [Hirata-Miao], [Fujita2]).

**Proposition 1** *If and , we have*

*where*

Some corollaries can be obtained by Lemma above. Indeed, taking one has

more general (see de Almeida, Ferrreira, L.F.C).

**Proposition 2** *Let and There is such that*

*for all with and for all *

Using an exercise,

we have

Taking in (11) the inverse Laplace transform and using (16), we obtain

-10.933922
-37.065357

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