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We extend in this paper the definition of Caputo derivatives of order in$(0,1)$ to a certain class of locally integrable functions using a convolutiongroup. Our strategy is to define a fractional calculus for a certain class ofdistributions using the convolution group. When acting on causal functions,this fractional calculus agrees with the traditional Riemann-Liouvilledefinition for $t>0$ but includes some singularities at $t=0$ so that the groupproperty holds. Then, making use of this fractional calculus, we introduce thegeneralized definition of Caputo derivatives. The new definition is consistentwith various definitions in literature while reveals the underlying groupstructure. Since the new definition is valid for a class of locally integrablefunctions that can blow up in finite time, it provides a framework forsolutions to fractional ODEs and fractional PDEs with very weak conditions. Theunderlying group property makes many properties of Caputo derivatives natural.In particular, it allows us to de-convolve the fractional differentialequations to integral equations with completely monotone kernels, which thenenables us to prove the general Gr\"onwall inequality (or comparison principle)with the most general conditions. This then provides the essential tools for{\it a priori} energy estimates of fractional PDEs. Some other fundamentalresults for fractional ODEs are also established within this frame under veryweak conditions.

A generalized definition of Caputo derivatives and its application to fractional ODEs