the Calderon-Zygmund estimates
/ kal-deh-RON ZIG-moond /
The Calderon-Zygmund estimates are the third great pillar of elliptic regularity, written in the L^p scale — the language of integrals raised to a power rather than pointwise size. They answer: if the right-hand side f of - Laplacian u = f is merely L^p-integrable, how integrable are the second derivatives of u? The clean and surprising answer is that all the second derivatives are just as integrable as f: if f is in L^p then every entry of the Hessian of u is in L^p too, for every p strictly between 1 and infinity.
Concretely the estimate reads: the L^p norm of D^2 u is not greater than a constant C(n,p) times the L^p norm of f, the same gain of two derivatives, now in the L^p sense. The engine underneath is one of the jewels of harmonic analysis. The Hessian of u is recovered from f by a singular integral operator — convolution with a kernel that decays like 1 over distance-to-the-n, sitting right at the borderline of integrability. Calderon and Zygmund proved that such singular integral operators are bounded on L^p for every 1 < p < infinity, by decomposing an arbitrary function into a good bounded part and a bad oscillating part supported on a controlled family of cubes (the Calderon-Zygmund decomposition). That single theorem about singular integrals delivers the elliptic L^p estimate as a corollary.
Why does L^p matter when we already have Sobolev and Schauder? Because the endpoints are different and the L^p scale interpolates between them flexibly. As p tends to infinity the L^p estimate connects to the Holder/Schauder world; as p tends to two it recovers the H^2 energy estimate. The borderline cases p = 1 and p = infinity are genuinely FALSE — at those endpoints the second derivatives can fail to be integrable, replaced by weaker substitutes (bounded mean oscillation at the top, weak-L^1 at the bottom). For nonlinear problems and for sharp control of how rough data propagate to the solution, the L^p flexibility is indispensable.
Solve - Laplacian u = f on a domain where f is in L^3 but not bounded — say f has a mild local spike. The Calderon-Zygmund estimate guarantees every second derivative u_(x_i x_j) is also in L^3, with norm controlled by the L^3 norm of f. By the Sobolev embedding this then gives Holder continuity of grad u, so even an unbounded f yields a solution with a continuous gradient.
L^p in gives L^p out for the full Hessian — for every p strictly between 1 and infinity, but not at the endpoints.
The estimate genuinely fails at p = 1 and p = infinity. A bounded (L^infinity) right-hand side does NOT guarantee bounded second derivatives — the classic example is a continuous f whose solution has a logarithmic blow-up in its Hessian. The good range is the OPEN interval 1 < p < infinity, and the constant C(n,p) degenerates as p approaches either endpoint.