A generalisation of the Malgrange-Ehrenpreis theorem to find fundamental solutions to fractional PDEs

We present and prove a new generalisation of the Malgrange–Ehrenpreis theorem to fractional partial differential equations, which can be used to construct fundamental solutions to all partial differential operators of rational order and many of arbitrary real order. We demonstrate with some examples and mention a few possible applications

Invertibility for a class of Fourier multipliers

In this paper, we establish invertibility for a class of multipliers in the setting of Hormander quantization of pseudo-differential operators on R-n More precisely, the existence of inverses and fundamental solutions of these operators are investigated

The Malgrange-Ehrenpreis Theorem

This paper collect of some proofs of the Malgrange-Ehrenpreis Theorem, which asserts the existence of fundamental solutions of linear partial differential operator with constant coefficients.Ribera Puchades, JM. (2011). The Malgrange-Ehrenpreis Theorem. http://hdl.handle.net/10251/15596Archivo delegad

Syzygies of modules and applications to propagation of regularity phenomena

Propagation of regularity is considered for solutions of rectangular systems of infinite order partial differential equations (resp. convolution equations) in spaces of hyperfunctions (resp. C∞ functions and distributions). Known results of this kind are recovered as particular cases, when finite order partial differential equations are considered

Lacunary non-continuable boundary-regular holomorphic functions with universal properties

A holomorphic function in a Jordan domain G in the complex plane is constructed with all its derivatives extending continuously up to the boundary G that happens to be a natural boundary of In addition the action of a certain class of operators on presents some universal properties related to the overconvergence phenomenon.Plan Andaluz de Investigación (Junta de Andalucía)Ministerio de Ciencia y Tecnologí

Fundamental solutions to elliptic multipliers with real-analytic symbol in Rn\mathbb{R}^n

Using a version of Hironaka's resolution of singularities for real-analytic functions, we show that any elliptic multiplier with real-analytic symbol has a tempered fundamental solution, and this can be weak$*$-approximated by entire functions belonging to a certain Paley-Wiener space. In some special cases of global symmetry, the construction can be specialized to become fully explicit. We use this to compute tempered fundamental solutions for sums of powers of the Laplacian on $\mathbb{R}^n$