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We prove a number of \textit{a priori} estimates for weak solutions ofelliptic equations or systems with vertically independent coefficients in theupper-half space. These estimates are designed towards applications to boundaryvalue problems of Dirichlet and Neumann type in various topologies. We work inclasses of solutions which include the energy solutions. For those solutions,we use a description using the first order systems satisfied by their conormalgradients and the theory of Hardy spaces associated with such systems but themethod also allows us to design solutions which are not necessarily energysolutions. We obtain precise comparisons between square functions,non-tangential maximal functions and norms of boundary trace. The main thesisis that the range of exponents for such results is related to when those Hardyspaces (which could be abstract spaces) are identified to concrete spaces oftempered distributions. We consider some adapted non-tangential sharp functionsand prove comparisons with square functions. We obtain boundedness results forlayer potentials, boundary behavior, in particular strong limits, which is new,and jump relations. One application is an extrapolation for solvability ''\'ala {\v{S}}ne{\u\i}berg". Another one is stability of solvability in perturbingthe coefficients in $L^\infty$ without further assumptions. We stress that ourresults do not require De Giorgi-Nash assumptions, and we improve the availableones when we do so.

A priori estimates for boundary value elliptic problems via first order systems