Calderón–Zygmund estimates

For a compactly supported smooth function on Euclidean space, Fourier transformation converts second derivatives into quadratic frequency factors. Dividing by the Laplacian factor gives the multiplier xi-i times xi-j divided by the squared frequency magnitude, up to the chosen sign convention. Its degree is zero: it does not increase derivative order relative to the source. In L-two, Plancherel directly bounds the multiplier, and summing all Hessian entries gives a particularly clean identity.

For p other than two, the inverse transform involves a singular kernel with directional cancellation. Taking absolute values inside its near-origin integral would destroy that cancellation and produce a divergent bound. The singular-integral theorem supplies boundedness for every exponent strictly between one and infinity. The constants depend on exponent and dimension. General strong L-one and L-infinity estimates of this same form fail; weak-type or mean-oscillation substitutes belong to distinct endpoint statements.

On a ball, multiply the solution by a cutoff and apply a full-space estimate. Commutators introduce first derivatives of the solution and derivatives of the cutoff. Interpolation and nested interior regions control these extra terms, producing a local W-two-p estimate involving the L-p source norm and an L-p norm of the solution on a larger ball. The second term cannot generally be omitted: a nonzero harmonic polynomial has zero source and nonzero Hessian.

For the constant-coefficient Laplacian, distributional solutions u and sources f in L-p locally gain two local L-p derivatives. Boundary versions require boundary geometry and data regularity; standard global zero-Dirichlet W-two-p theory uses a bounded C-one-one domain. One should not infer that a weak H-one solution on an arbitrary polygon has global W-two-p regularity for every p. Corners, boundary conditions, and coefficient changes can introduce additional restrictions even when the forcing is smooth.

Variable-coefficient estimates need a suitable coefficient class and operator interpretation. For nondivergence operators, continuity or more refined oscillation hypotheses can support W-two-p theory; arbitrary bounded measurable leading coefficients do not supply this full conclusion. For divergence-form operators, first derivatives of coefficients appear when expanding to nondivergence form. A flux solution with a coefficient jump can have a delta second derivative, exactly as the earlier interface example showed. The principal ellipticity inequality alone does not resolve those issues.

Finite-p estimates control integrals, not pointwise maxima. A Hessian can have a large value on a small set while keeping its L-p norm bounded. Combining W-two-p with a Sobolev embedding gives additional continuity only when the dimension and exponent satisfy the corresponding embedding hypotheses. This separates three useful forms of information: an averaged derivative bound, an oscillation bound, and a pointwise classical derivative bound. A numerical plot that looks smooth cannot replace the hypotheses linking them.