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Let L be a second order, uniformly elliptic operator, and consider theequation L u=f under the homogeneous boundary condition u=0. It is well knownthat f in C(Om) (Om= Omega) does not guarantee second order derivatives D^2 uin C(Om). This gap led to look for functional spaces C_*(Om), contained inC(Om), as large as possible, for which f in C_*(Om) merely guarantees thecontinuity of D^2 u (but nothing more, say). H\"older continuity is toorestrictive to fulfill this minimal requirement since in this case D^2 uinherits the whole regularity enjoyed by f (we say that "full regularity"occurs). This two opposite situations led us to look for significant cases inwhich intermediate regularity (i.e., between "mere continuity" and "fullregularity") occurs. This holds for data in Log spaces D^{0,a}(Om) (a= alpha)where 0< a < infty, simply obtained by replacing in the modulus of continuityof H\"older spaces the quantity 1/|x-y| by log(1/|x-y|). If f in D^{0,a} forsome fixed a > 1, then D^2 u in D^{0,a-1}. This regularity is optimal. Theabove picture opened the way to further investigation. Below we study the moregeneral problem of data f in subspaces of continuous functions D_{om},characterized by a given modulus of continuity om(r). H\"older and Log spacesare particular cases. A significant new, lets say curious, case is shown by thefamily of functional spaces C^{0,l}_a (l= lambda), where 0 <= l < 1, and a inR. In particular, C^{0,l}_0= C^{0,l}, is a H\"older space, and C^{0,0}_a =D^{0,a}, is a Log space. Main point is that full regularity occurs for l > 0,and arbitrary a in R. If f in C^{0,l}_a then D^2 u in C^{0,l}_a.

On classical solutions to elliptic boundary value problems. The full regularity spaces $C^{0,\,\la}_\al(\Ov)$