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We continue our investigation of $E$-operators, in particular theirconnection with $G$-operators; these differential operators are fundamental inunderstanding the diophantine properties of Siegel's $E$ and $G$-functions. Westudy in detail microsolutions (in Kashiwara's sense) of Fuchsian differentialoperators, and apply this to the construction of basis of solutions at $0$ and$\infty$ of any $E$-operator from microsolutions of a $G$-operator; thisprovides a constructive proof of a theorem of Andr\'e. We also focus on thearithmetic nature of connection constants and Stokes constants betweendifferent bases of solutions of $E$-operators. For this, we introduce and studyin details an arithmetic (inverse) Laplace transform that enables one to getrid of transcendental numbers inherent to Andr\'e's original approach. As anapplication, we define a set of special values of arithmetic Gevrey series, anddiscuss its conjectural relation with the ring of exponential periods.

Microsolutions of differential operators and values of arithmetic Gevrey series