Lp (Calderón-Zygmund) W^{2,p} Estimates for Elliptic Equations

Intuition Beginner

Picture an equilibrium equation whose right-hand side is a source you are allowed to measure only in a coarse, averaged way: not its largest spike, but its total size when you average a fixed power of its values over the region. That averaged size is what the $L^p$ scale records, for a power $p$ somewhere strictly between one and infinity. The question of this unit is the same one Schauder answered on the fractional Hölder scale, asked instead on this averaged scale: if you control the source in the averaged sense, how well do you control the curvature of the solution?

The answer is as clean as it could be. Control the averaged size of the source, and you automatically control the averaged size of every second derivative of the solution, with the same power. Two whole derivatives of smoothness are handed back to you for free, and the averaged bookkeeping passes through unchanged. This is the integrable-data twin of the Schauder story: same gain of two derivatives, measured with an averaging integral instead of a fractional-distance ratio.

Why have two versions of the same theorem? Because real sources are often rough in a way the fractional scale cannot see — they may have jumps or mild blow-ups that are perfectly fine when averaged but ruinous when you ask for a fractional-distance bound at every pair of points. The averaged scale tolerates these. It is the natural home when your data comes from an integral, an energy, or a measurement that only ever reports averages.

The one-sentence takeaway: the Calderón-Zygmund theory says an elliptic equilibrium with a source controlled in the averaged $L^p$ sense has all its second derivatives controlled in the same averaged sense, gaining exactly two derivatives — the integrable-data partner of Schauder's fractional-scale estimate.

Visual Beginner

Picture two measuring rules laid side by side, each scoring smoothness, but by different procedures. The Schauder rule walks every pair of nearby points and checks that values differ by no more than a fixed multiple of the distance raised to a fractional power; it is a worst-case-over-pairs rule. The Calderón-Zygmund rule instead pours the function into an averaging integral: raise the size to the power $p$, integrate over the region, take the matching root. It is an averaged rule, blind to isolated spikes that the worst-case rule would punish.

The two panels make the same promise by different routes: feed an elliptic equation a source, and the second derivatives of the solution are controlled on whichever scale you measured the source. The lower funnel is the payoff of this unit. A spike that the upper magnifying glass rejects passes through the lower funnel and out comes a finite averaged score, which is exactly why integrable-data problems live here and not on the fractional scale.

Worked example Beginner

We watch the two-derivative gain on the averaged scale in one variable, where every quantity is an honest integral. Take the equation asking for a function whose second derivative equals a given source on the interval from $0$ to $1$, pinned to zero at both ends. We feed in the source $f(x)=20x^3$ and measure both the source and the second derivative of the answer by the same averaging procedure with power $p=2$.

Step 1. Solve the equation $u^{\prime \prime }(x)=20x^3$ by integrating twice. One antiderivative of $20x^3$ is $5x^4$; an antiderivative of that is $x^5$. So $u(x)=x^5+Ax+B$ for constants to be fixed by the endpoints.

Step 2. Apply the endpoint conditions $u(0)=0$ and $u(1)=0$. The first gives $B=0$. The second gives $1+A=0$, so $A=-1$. The solution is $u(x)=x^5-x$.

Step 3. Read off the second derivative. Differentiating $u(x)=x^5-x$ twice returns $u^{\prime \prime }(x)=20x^3$, which is exactly the source $f$ again.

Step 4. Measure both on the averaged scale with power $2$. The averaged size of the source is the square root of the integral of $f$ squared, that is the square root of the integral of $400x^6$ from $0$ to $1$. The integral of $400x^6$ is $400$ divided by $7$. The averaged size of the second derivative is the same number, because the second derivative equals the source exactly.

Step 5. Read the Calderón-Zygmund content. The master inequality would say: the averaged size of the second derivative is bounded by a fixed multiple of the averaged size of the source. Here the two are literally equal, the sharpest possible instance — averaged size in, the same averaged size out.

What this tells us: solving the second-order equation integrates the source twice, lifting it by two whole derivatives, and the averaged size of the source reappears intact in the averaged size of the second derivative. The Calderón-Zygmund theory is the statement that this clean one-dimensional bookkeeping survives in every dimension, for variable-coefficient operators, once smoothness is measured on the averaged $L^p$ scale rather than the fractional Hölder scale.

Check your understanding Beginner

Formal definition Intermediate+

Fix an open set $\Omega \subseteq {\mathbb{R}}^n$ and an exponent $p\in (1,\infty )$. For $k\in \{0,1,2\}$ the Sobolev space $W^{k,p}(\Omega )$ is the space of $u\in L^p(\Omega )$ whose weak derivatives $D^{\beta }u$ exist in $L^p(\Omega )$ for all multi-indices $|\beta |\le k$, normed by $$\Vert u{\Vert }_{W^{k,p}(\Omega )}=(\sum _{|\beta |\le k}\Vert D^{\beta }u{\Vert }_{L^p(\Omega )}^p{)}^{1/p}.$$ Each $W^{k,p}(\Omega )$ is a Banach space, its completeness inherited from the completeness of $L^p(\Omega )$ 02.07.06. We write $W_0^{1,p}(\Omega )$ for the closure of $C_c^{\infty }(\Omega )$ in the $W^{1,p}$ norm — the functions vanishing on $\partial \Omega$ in the trace sense. The Hessian seminorm is $\Vert D^{2}u{\Vert }_{L^p}^p={\sum }_{i,j}\Vert {\partial }_i{\partial }_{j}u{\Vert }_{L^p}^p$.

Let $L$ be a second-order operator in non-divergence form, $$Lu=\sum _{i,j=1}^n{a}^{ij}(x)D_{ij}u+\sum _{i=1}^n{b}^i(x)D_{i}u+c(x)u,$$ uniformly elliptic with constants $0<\lambda \le \Lambda$, meaning $\lambda |\xi {|}^2\le {\sum }_{i,j}{a}^{ij}(x){\xi }_i{\xi }_j\le \Lambda |\xi {|}^2$ for all $x\in \Omega$, $\xi \in {\mathbb{R}}^n$. For the $L^p$ theory the leading coefficients are required only to be continuous (indeed uniformly continuous on $\overline{\Omega }$, with a modulus of continuity $\tau$), with $a^{ij}\in C^0(\overline{\Omega })$ and $b^i,c\in L^{\infty }(\Omega )$; this is a strictly weaker hypothesis than the Hölder continuity demanded by the Schauder theory of 02.17.04. The structural data of every estimate below are the dimension $n$, the exponent $p$, the ellipticity ratio $\Lambda /\lambda$, the modulus of continuity $\tau$ of the leading coefficients, and the $L^{\infty }$ bounds of the lower-order coefficients.

Definition (strong solution). A function $u\in W_{\mathrm{loc}}^{2,p}(\Omega )$ is a strong solution of $Lu=f$ when the equation holds pointwise almost everywhere, the second derivatives being the weak (hence a.e.-defined) derivatives. Strong solutions are the natural class for the $L^p$ theory: less regular than the classical $C^2$ solutions of Schauder theory, but possessing genuine second derivatives in $L^p$, in contrast to the merely weak (distributional, divergence-form) solutions of the energy theory.

The analytic engine of the whole theory is the Calderón-Zygmund inequality for the Laplacian, supplied by 02.19.04: for $u\in C_c^{\infty }({\mathbb{R}}^n)$ and $1<p<\infty$, $$\Vert {\partial }_i{\partial }_{j}u{\Vert }_{L^p({\mathbb{R}}^n)}\le C_p\Vert \Delta u{\Vert }_{L^p({\mathbb{R}}^n)},{\partial }_i{\partial }_j=-R_i{R}_j\Delta ,$$ where $R_i$ is the $i$-th Riesz transform. This is the global, constant-coefficient model; the variable-coefficient interior estimate is built from it by localization and frozen-coefficient perturbation.

Sign convention. The Laplacian is $\Delta ={\sum }_i{\partial }_i^2$ (analysts' sign, so $-\Delta \ge 0$), the Riesz multiplier is $m_j(\xi )=-i{\xi }_j/|\xi |$ as fixed in 02.19.04, and the representation ${\partial }_i{\partial }_j=-R_i{R}_j\Delta$ uses these conventions throughout.

Counterexamples to common slips Intermediate+

• The estimate fails at $p=\infty$; the open interval is essential. The bound $\Vert D^{2}u{\Vert }_{L^{\infty }}\le C\Vert \Delta u{\Vert }_{L^{\infty }}$ is false: there is a $u$ with $\Delta u\in L^{\infty }$ (even continuous) whose pure second differences ${\partial }_{11}u-{\partial }_{22}u$ are unbounded. At $p=\infty$ the Riesz transforms map $L^{\infty }$ only into $\mathrm{BMO}$, so $\Delta u$ bounded forces the Hessian only into $\mathrm{BMO}$, not $L^{\infty }$. This failure is precisely why one passes to the Schauder $C^{2,\alpha }$ scale of 02.17.04 for sup-norm data.

• The estimate fails at $p=1$. The Riesz transforms are only weak-$(1,1)$, not bounded on $L^1$, so $\Delta u\in L^1$ does not force ${\partial }_i{\partial }_{j}u\in L^1$. The substitute at the lower endpoint is the weak-$L^1$ (Marcinkiewicz) bound, or a Hardy-space refinement $\Delta u\in {\mathcal{H}}^1\Rightarrow D^{2}u\in L^1$.

• Continuity of the leading coefficients cannot be dropped to mere measurability. If $a^{ij}$ is merely bounded measurable, $W^{2,p}$ estimates fail for $p$ near $1$ or near $\infty$ (the oscillation of the coefficients defeats the freezing step). One then has only the divergence-form De Giorgi-Nash-Moser theory, or, under the small-BMO (Sarason VMO) hypothesis on $a^{ij}$, the Chiarenza-Frasca-Longo extension that recovers $W^{2,p}$ for all $p\in (1,\infty )$.

• The constant blows up at both endpoints and cannot be taken uniform in $p$. The Riesz $L^p$-norm grows like $\max \{p,(p-1{)}^{-1}\}$, so $C_p\to \infty$ as $p\to 1^{+}$ or $p\to \infty$. An estimate written with a single $p$-independent constant is wrong; the dependence on $p$ is intrinsic to the singular-integral input.

Key theorem with proof Intermediate+

Theorem (interior $W^{2,p}$ estimate; Calderón-Zygmund / Agmon-Douglis-Nirenberg). Let $L=\sum a^{ij}{D}_{ij}+\sum b^i{D}_i+c$ be uniformly elliptic on $\Omega \subseteq {\mathbb{R}}^n$ with $a^{ij}\in C^0(\overline{\Omega })$, $b^i,c\in L^{\infty }(\Omega )$, and $1<p<\infty$. If $u\in W_{\mathrm{loc}}^{2,p}(\Omega )$ satisfies $Lu=f$ with $f\in L^p(\Omega )$, then for every ${\Omega }^{\prime }⋐\Omega$, $$\Vert u{\Vert }_{W^{2,p}({\Omega }^{\prime })}\le C(\Vert f{\Vert }_{L^p(\Omega )}+\Vert u{\Vert }_{L^p(\Omega )}),$$ where $C=C(n,p,\lambda ,\Lambda ,\tau ,\mathrm{dist}({\Omega }^{\prime },\partial \Omega ))$ depends on the modulus of continuity $\tau$ of the leading coefficients but not otherwise on $u$ ^{[Gilbarg-Trudinger 2001 §9.5, Theorem 9.11]}.

Proof (Calderón-Zygmund inequality, then frozen-coefficient perturbation). The argument runs in three movements: the constant-coefficient model inequality, a freezing-and-localization step, and an absorption.

Model inequality. For the Laplacian the global estimate is exactly the second-order Calderón-Zygmund inequality of 02.19.04: for $w\in C_c^{\infty }({\mathbb{R}}^n)$, $${\partial }_i{\partial }_{j}w=-R_i{R}_j\Delta w⟹\Vert {\partial }_i{\partial }_{j}w{\Vert }_{L^p}\le \Vert R_i{\Vert }_{p\to p}\Vert R_j{\Vert }_{p\to p}\Vert \Delta w{\Vert }_{L^p}=C_p\Vert \Delta w{\Vert }_{L^p},$$ the Riesz transforms being bounded on $L^p$ for $1<p<\infty$. A linear change of variables diagonalising the frozen leading-coefficient matrix $[a^{ij}(x_0)]$ turns the constant-coefficient operator $L_0=\sum a^{ij}(x_0)D_{ij}$ into a multiple of the Laplacian, with the distortion of the $L^p$ norms controlled by a factor depending only on $\Lambda /\lambda$; hence $\Vert D^{2}w{\Vert }_{L^p}\le C_0\Vert L_{0}w{\Vert }_{L^p}$ for $w\in C_c^{\infty }$, with $C_0=C_0(n,p,\lambda ,\Lambda )$.

Freezing and localization. Fix $x_0\in {\Omega }^{\prime }$ and a cutoff $\eta \in C_c^{\infty }(B_r(x_0))$, $\eta \equiv 1$ on $B_{r/2}(x_0)$. Apply the model inequality to $w=\eta u$, whose support sits in a single small ball. Write $L_0=\sum a^{ij}(x_0)D_{ij}$. Then $$L_0(\eta u)=\eta L_{0}u+[\text{commutator terms in }D\eta ,D^2\eta ]=\eta (f-{\sum }_{i,j}(a^{ij}(x)-a^{ij}(x_0))D_{ij}u-{\sum }_i{b}^i{D}_{i}u-cu)+E,$$ where $E$ collects the first- and zeroth-order terms produced by the cutoff (each carrying at most one derivative of $u$). The model inequality gives $$\Vert D^2(\eta u){\Vert }_{L^p}\le C_0\Vert L_0(\eta u){\Vert }_{L^p}.$$

Absorption. By uniform continuity of the leading coefficients, $|a^{ij}(x)-a^{ij}(x_0)|\le \tau (r)$ on $B_r(x_0)$, with $\tau (r)\to 0$ as $r\to 0$. Hence the perturbation term obeys $$\Vert \sum _{i,j}(a^{ij}(x)-a^{ij}(x_0))\eta D_{ij}u{\Vert }_{L^p}\le n^2\tau (r)\Vert D^2(\eta u){\Vert }_{L^p}+C\tau (r)\Vert u{\Vert }_{W^{1,p}(B_r)},$$ the first term carrying the small factor $\tau (r)$ on the top-order Hessian. Choose $r=r_{\ast }$ so small that $C_0{n}^2\tau (r_{\ast })\le \frac{1}{2}$; the leading perturbation term is absorbed into the left side at the cost of a factor two. The remaining first- and zeroth-order terms, together with the cutoff commutator $E$, involve at most $\Vert Du{\Vert }_{L^p}$ and $\Vert u{\Vert }_{L^p}$, which the interpolation inequality $\Vert Du{\Vert }_{L^p}\le \epsilon \Vert D^{2}u{\Vert }_{L^p}+C_{\epsilon }\Vert u{\Vert }_{L^p}$ controls by a small multiple of the Hessian plus a large multiple of $\Vert u{\Vert }_{L^p}$. Absorbing once more leaves $$\Vert D^{2}u{\Vert }_{L^p(B_{r_{\ast }/2}(x_0))}\le C(\Vert f{\Vert }_{L^p(B_{r_{\ast }}(x_0))}+\Vert u{\Vert }_{L^p(B_{r_{\ast }}(x_0))}).$$ A finite chain of overlapping balls of radius $r_{\ast }$ covering ${\Omega }^{\prime }$, with the radii $r_{\ast }$ depending only on the listed structural data through $\tau$, sums to the stated global-on-${\Omega }^{\prime }$ bound. $\square$

Bridge. This interior estimate builds toward the existence theory of strong solutions and the global $W^{2,p}$ estimate up to a $C^{1,1}$ boundary, developed in the Advanced results, where the same frozen-coefficient perturbation runs against a flattened boundary chart; it appears again in 02.13.02, whose forward-reference to $L^p$-data regularity is closed exactly here. The central insight is that a variable-coefficient elliptic operator is, at small scales, a constant-coefficient operator plus a perturbation that shrinks with the modulus of continuity of its leading coefficients, so the constant-coefficient Calderón-Zygmund inequality $\Vert {\partial }_i{\partial }_{j}w{\Vert }_p\le C_p\Vert \Delta w{\Vert }_p$ of 02.19.04 is exactly the model whose error the continuity of the coefficients keeps under control. Putting these together, the $W^{2,p}$ estimate is dual to the Schauder $C^{2,\alpha }$ estimate of 02.17.04: one is the averaged Lebesgue-scale and the other the fractional Hölder-scale incarnation of the single fact that the second derivatives of an elliptic solution are no worse than its source, and this is exactly the foundational reason the method of continuity closes the existence theory on the $W^{2,p}$ scale just as it does on the $C^{2,\alpha }$ scale.

Exercises Intermediate+

Advanced results Master

The Calderón-Zygmund $L^p$ apparatus organises around four results: the global constant-coefficient inequality that is the analytic engine, the variable-coefficient interior estimate, the global estimate up to a $C^{1,1}$ boundary, and the existence theory of strong solutions the estimates unlock through the method of continuity.

Theorem 1 (Calderón-Zygmund inequality; Calderón-Zygmund 1952). For $u\in W^{2,p}({\mathbb{R}}^n)$ and $1<p<\infty$, ${\partial }_i{\partial }_{j}u=-R_i{R}_j\Delta u$ and $\Vert D^{2}u{\Vert }_{L^p}\le C(n,p)\Vert \Delta u{\Vert }_{L^p}$, with $C(n,p)\to \infty$ as $p\to 1^{+}$ or $p\to \infty$. The constant is governed by the Riesz $L^p$-norm $\max \{p,(p-1{)}^{-1}\}$ ^{[Stein 1970]}. This is the global, flat-space, constant-coefficient model from which everything else is built by perturbation; its proof is the singular-integral content of 02.19.04, the kernels $D_{ij}\Phi$ of the Newtonian potential being the canonical second-order Calderón-Zygmund kernels.

Theorem 2 (interior $W^{2,p}$ estimate). For $L$ uniformly elliptic with $a^{ij}\in C^0(\overline{\Omega })$, $b^i,c\in L^{\infty }$, and $u\in W_{\mathrm{loc}}^{2,p}(\Omega )$ solving $Lu=f$ with $f\in L^p$, one has on ${\Omega }^{\prime }⋐\Omega$ $$\Vert u{\Vert }_{W^{2,p}({\Omega }^{\prime })}\le C(\Vert f{\Vert }_{L^p(\Omega )}+\Vert u{\Vert }_{L^p(\Omega )}),C=C(n,p,\lambda ,\Lambda ,\tau ,d),$$ $d=\mathrm{dist}({\Omega }^{\prime },\partial \Omega )$, $\tau$ the modulus of continuity of the $a^{ij}$ ^{[Gilbarg-Trudinger 2001 §9.5]}. The proof freezes the leading coefficients and absorbs the perturbation through $\tau (r)\to 0$; only continuity, not Hölder continuity, is used, which is the structural difference from the Schauder estimate of 02.17.04.

Theorem 3 (global estimate and the $C^{1,1}$ boundary; Agmon-Douglis-Nirenberg 1959). Let $\Omega$ be bounded with $C^{1,1}$ boundary, $L$ as above with $c\le 0$, and $u\in W^{2,p}(\Omega )\cap W_0^{1,p}(\Omega )$ solving $Lu=f$. Then $$\Vert u{\Vert }_{W^{2,p}(\Omega )}\le C(\Vert f{\Vert }_{L^p(\Omega )}+\Vert u{\Vert }_{L^p(\Omega )}),C=C(n,p,\lambda ,\Lambda ,\tau ,\Omega )$$ ^{[Agmon-Douglis-Nirenberg 1959]}. The boundary is flattened by a $C^{1,1}$ diffeomorphism into a half-ball; the half-space Calderón-Zygmund inequality for the Laplacian (via reflection) plays the role of the interior model. The $C^{1,1}$ regularity is exactly what is needed to keep the transformed leading coefficients continuous, the threshold at which the singular-integral input still applies; below it one loses the global estimate.

Theorem 4 (existence of strong solutions; method of continuity). For $\Omega$ bounded $C^{1,1}$, $L$ uniformly elliptic with continuous leading coefficients and $c\le 0$, and $f\in L^p(\Omega )$ with $1<p<\infty$, the Dirichlet problem $Lu=f$ in $\Omega$, $u=0$ on $\partial \Omega$ has a unique strong solution $u\in W^{2,p}(\Omega )\cap W_0^{1,p}(\Omega )$ ^{[Gilbarg-Trudinger 2001 §9.6]}. Uniqueness is the Aleksandrov-Bakelman-Pucci maximum principle for strong solutions; existence runs the continuity method from $\Delta$ to $L$, the global estimate of Theorem 3 supplying the uniform a priori bound that keeps the invertibility set open and closed. This places the $L^p$ theory on the same logical footing as the Schauder theory: the a priori estimate precedes and produces existence.

Theorem 5 (Sobolev embedding into the Hölder scale). For $p>n$, the interior estimate combined with the Morrey embedding $W^{2,p}({\Omega }^{\prime })\hookrightarrow C^{1,1-n/p}({\Omega }^{\prime })$ gives $u\in C^{1,\beta }$ with $\beta =1-n/p$, and as $p\to \infty$, $\beta \to 1$. The two regularity theories meet here: the $L^p$ estimate for large $p$ recovers classical $C^{1,\beta }$ regularity, the bridge between the averaged and fractional scales being precisely the Sobolev-Morrey embedding ^{[Gilbarg-Trudinger 2001 §7.8]}. This is why the $W^{2,p}$ theory, though weaker pointwise than Schauder, suffices for most applications once $p$ is taken large.

Theorem 6 (VMO coefficients; Chiarenza-Frasca-Longo 1991). Continuity of the leading coefficients can be relaxed to the Sarason class $\mathrm{VMO}$ (vanishing mean oscillation): if $a^{ij}\in \mathrm{VMO}\cap L^{\infty }$ and $L$ is uniformly elliptic, the interior $W^{2,p}$ estimate holds for all $1<p<\infty$ ^{[Krylov 2008]}. The proof replaces the pointwise oscillation $\tau (r)$ with the mean oscillation, controlled through the commutator $[R_i{R}_j,a]$ estimate of Coifman-Rochberg-Weiss; this is the sharp coefficient hypothesis under which the singular-integral method survives, and it covers discontinuous coefficients with small jumps that the pointwise-continuity hypothesis excludes.

Synthesis. The Calderón-Zygmund inequality is the foundational reason elliptic equations with integrable data have second derivatives in the same Lebesgue class, and this is exactly the structural content of the representation ${\partial }_i{\partial }_j=-R_i{R}_j\Delta$ from 02.19.04: the full Hessian is recovered from the trace of the Hessian by a bounded product of Riesz transforms, so a single scalar quantity controls every component on every $L^p$ with $1<p<\infty$. Putting these together, the constant-coefficient model generalises to the variable-coefficient interior estimate through a frozen-coefficient perturbation that only continuity of the leading coefficients keeps under control, the global estimate carries this to a $C^{1,1}$ boundary, and the method of continuity converts the uniform a priori bound into existence of strong solutions — the same architecture that the Schauder theory of 02.17.04 runs on the Hölder scale, to which this theory is dual. The central insight is that the two scales are bridged by the Sobolev-Morrey embedding $W^{2,p}\hookrightarrow C^{1,1-n/p}$, which is exactly why large-$p$ Calderón-Zygmund regularity recovers classical smoothness and why the borderline $p=\infty$, where the Hessian falls only into $\mathrm{BMO}$, is the hinge that forces the passage to the Schauder estimate. The bridge is the single Plancherel identity on the bounded symbol $-i{\xi }_j/|\xi |$, refracted through the master Calderón-Zygmund machinery into a differential inequality that is, on inspection, the same regularity gain Schauder records, written on a different scale.

Full proof set Master

Proposition 1 (Sobolev spaces are Banach). For $k\in \{0,1,2\}$ and $1\le p<\infty$, $(W^{k,p}(\Omega ),\Vert \cdot {\Vert }_{W^{k,p}})$ is a Banach space.

Proof. Let $(u_m)$ be Cauchy in $W^{k,p}$. Then for each $|\beta |\le k$ the sequence $(D^{\beta }{u}_m)$ is Cauchy in $L^p(\Omega )$, which is complete 02.07.06, so there are limits $u^{(\beta )}={\lim }_m{D}^{\beta }{u}_m$ in $L^p$. For any $\phi \in C_c^{\infty }(\Omega )$, the definition of the weak derivative gives $\int u_m{D}^{\beta }\phi (-1{)}^{|\beta |}=\int D^{\beta }{u}_m\phi$; passing to the limit (both sides converge by $L^p$-$L^{p^{\prime }}$ duality, $\phi$ and its derivatives being bounded with compact support) yields $\int u^{(0)}{D}^{\beta }\phi (-1{)}^{|\beta |}=\int u^{(\beta )}\phi$. Hence $u^{(\beta )}=D^{\beta }{u}^{(0)}$ weakly for all $|\beta |\le k$, so $u:=u^{(0)}\in W^{k,p}$ and $D^{\beta }{u}_m\to D^{\beta }u$ in $L^p$ for each $\beta$, i.e. $u_m\to u$ in $W^{k,p}$. $\square$

Proposition 2 (the constant-coefficient Calderón-Zygmund inequality). For $u\in C_c^{\infty }({\mathbb{R}}^n)$, $1<p<\infty$, and $L_0=\sum a_0^{ij}{D}_{ij}$ uniformly elliptic with constant coefficients, $\Vert D^{2}u{\Vert }_{L^p}\le C(n,p,\lambda ,\Lambda )\Vert L_{0}u{\Vert }_{L^p}$.

Proof. For $L_0=\Delta$ this is the second-order representation of 02.19.04: ${\partial }_i{\partial }_{j}u=-R_i{R}_j\Delta u$, and the $L^p$-boundedness of each $R_j$ gives $\Vert {\partial }_i{\partial }_{j}u{\Vert }_{L^p}\le \Vert R_i{\Vert }_{p\to p}\Vert R_j{\Vert }_{p\to p}\Vert \Delta u{\Vert }_{L^p}$, so summing over the $n^2$ pairs, $\Vert D^{2}u{\Vert }_{L^p}\le n{A}_p^2\Vert \Delta u{\Vert }_{L^p}$ with $A_p={\max }_j\Vert R_j{\Vert }_{p\to p}$. For general constant $[a_0^{ij}]$, diagonalise $[a_0^{ij}]=O^{\top }DO$ with $\lambda \le d_k\le \Lambda$ and substitute $y=D^{-1/2}Ox$, under which $L_0\mapsto {\Delta }_y$ and the second-derivative $L^p$ norms transform by factors bounded by powers of $\sqrt{\Lambda /\lambda }$; the Jacobian is constant and cancels in the homogeneous inequality. Hence $\Vert D^{2}u{\Vert }_{L^p}\le C(n,p,\lambda ,\Lambda )\Vert L_{0}u{\Vert }_{L^p}$. $\square$

Proposition 3 (interior estimate on a small ball). Let $L$ be uniformly elliptic on $B_1\subseteq {\mathbb{R}}^n$ with $a^{ij}\in C^0(\overline{B_1})$ of modulus $\tau$, $b^i,c\in L^{\infty }$. There is $r_{\ast }=r_{\ast }(n,p,\lambda ,\Lambda ,\tau )\in (0,1]$ such that every $u\in W^{2,p}(B_{r_{\ast }})$ with $Lu=f$ obeys $$\Vert u{\Vert }_{W^{2,p}(B_{r_{\ast }/2})}\le C(\Vert f{\Vert }_{L^p(B_{r_{\ast }})}+\Vert u{\Vert }_{L^p(B_{r_{\ast }})}).$$

Proof. Freeze $a_0^{ij}=a^{ij}(0)$ and write $L_0=\sum a_0^{ij}{D}_{ij}$. Take $\eta \in C_c^{\infty }(B_r)$, $\eta \equiv 1$ on $B_{r/2}$, $|D^{\alpha }\eta |\le C{r}^{-|\alpha |}$. By Proposition 2 applied to $\eta u\in C_c^{\infty }$ (or its mollifications), $\Vert D^2(\eta u){\Vert }_{L^p}\le C_0\Vert L_0(\eta u){\Vert }_{L^p}$. Expand $L_0(\eta u)=\eta L_{0}u+u{L}_0\eta +2\sum a_0^{ij}{D}_i\eta D_{j}u$, and substitute $L_{0}u=f-\sum (a^{ij}(x)-a_0^{ij})D_{ij}u-\sum b^i{D}_{i}u-cu$. Using $|a^{ij}(x)-a_0^{ij}|\le \tau (r)$ on $B_r$, $$\Vert D^2(\eta u){\Vert }_{L^p}\le C_0(\Vert f{\Vert }_{L^p(B_r)}+n^2\tau (r)\Vert D^2(\eta u){\Vert }_{L^p}+C{r}^{-2}\Vert u{\Vert }_{L^p(B_r)}+C{r}^{-1}\Vert Du{\Vert }_{L^p(B_r)}+C\Vert Du{\Vert }_{L^p}+C\Vert u{\Vert }_{L^p}).$$ Choose $r_{\ast }$ with $C_0{n}^2\tau (r_{\ast })\le \frac{1}{2}$ to absorb the Hessian term, then apply the interpolation $\Vert Du{\Vert }_{L^p}\le \epsilon \Vert D^{2}u{\Vert }_{L^p}+C_{\epsilon }\Vert u{\Vert }_{L^p}$ (Proposition 5 below) with $\epsilon$ small to absorb the first-order term. What remains is $\Vert D^{2}u{\Vert }_{L^p(B_{r_{\ast }/2})}\le C(\Vert f{\Vert }_{L^p(B_{r_{\ast }})}+\Vert u{\Vert }_{L^p(B_{r_{\ast }})})$; adding $\Vert u{\Vert }_{W^{1,p}}$ by interpolation gives the full $W^{2,p}$ bound, with $r_{\ast }$ depending only on the listed data through $\tau$. $\square$

Proposition 4 (the $p=\infty$ failure). There is no constant $C$ with $\Vert D^{2}u{\Vert }_{L^{\infty }({\mathbb{R}}^n)}\le C\Vert \Delta u{\Vert }_{L^{\infty }({\mathbb{R}}^n)}$ for all $u\in C_c^{\infty }({\mathbb{R}}^n)$, $n\ge 2$.

Proof. If such a $C$ existed, then since ${\partial }_i{\partial }_{j}u=-R_i{R}_j\Delta u$, the operator $R_i{R}_j$ would be bounded $L^{\infty }\to L^{\infty }$ on the range $\{\Delta u:u\in C_c^{\infty }\}$, which is dense in a suitable sense. But a Calderón-Zygmund operator with nonzero symbol is not bounded on $L^{\infty }$: take $g\in C_c^{\infty }(B_1)$ with $\int g\ne 0$ and $g\ge 0$; then $R_i{R}_{j}g$ has a logarithmic singularity, $|R_i{R}_{j}g(x)|\sim c|\log |x-x_0||$ near a point $x_0\in \partial B_1$ where the symbol's contribution is nonvanishing, so $R_i{R}_{j}g\notin L^{\infty }$ though $g\in L^{\infty }$. Solving $\Delta u=g$ (with $u=\Phi \ast g$, smooth and compactly-supported up to a harmonic correction) realises $g$ as a Laplacian, exhibiting $u$ with $\Delta u\in L^{\infty }$ but ${\partial }_i{\partial }_{j}u\notin L^{\infty }$. The sharp positive statement is $R_i{R}_j:L^{\infty }\to \mathrm{BMO}$, the John-Nirenberg space the logarithm inhabits. $\square$

Proposition 5 (interpolation inequality). For $u\in W^{2,p}(B_1)$ and $\epsilon >0$, $\Vert Du{\Vert }_{L^p(B_1)}\le \epsilon \Vert D^{2}u{\Vert }_{L^p(B_1)}+C_{\epsilon }\Vert u{\Vert }_{L^p(B_1)}$.

Proof. By compactness-contradiction: were it false for some ${\epsilon }_0$, there would be $u_m$ with $\Vert D{u}_m{\Vert }_{L^p}=1$ and ${\epsilon }_0\Vert D^2{u}_m{\Vert }_{L^p}+m\Vert u_m{\Vert }_{L^p}<1$; then $(u_m)$ is bounded in $W^{2,p}$ with $\Vert u_m{\Vert }_{L^p}\to 0$. By Rellich-Kondrachov, $W^{2,p}\hookrightarrow W^{1,p}$ compactly, so $u_m\to u$ in $W^{1,p}$ along a subsequence, with $\Vert Du{\Vert }_{L^p}=1$ yet $u=0$ in $L^p$, forcing $Du=0$, a contradiction. $\square$

Proposition 6 (uniqueness of strong solutions when $c\le 0$). If $L$ is uniformly elliptic with $c\le 0$ on a bounded $\Omega$, the only strong solution of $Lu=0$ in $\Omega$, $u\in W^{2,p}(\Omega )\cap W_0^{1,p}(\Omega )$ with $p\ge n$, is $u=0$.

Proof. The Aleksandrov-Bakelman-Pucci maximum principle for strong solutions bounds ${\sup }_{\Omega }{u}^{+}\le {\sup }_{\partial \Omega }{u}^{+}+C(n,\mathrm{diam}\Omega ,\lambda )\Vert (Lu{)}^{-}{\Vert }_{L^n(\Omega )}$ for $c\le 0$. With $Lu=0$ and $u=0$ on $\partial \Omega$ (in the $W_0^{1,p}$ trace sense, which for $p\ge n$ gives genuine continuous boundary values by Morrey), $\sup u^{+}\le 0$, and applying the same to $-u$ gives $\sup (-u{)}^{+}\le 0$, so $u\equiv 0$. The hypothesis $p\ge n$ guarantees the $L^n$-norm on the right and the continuity of the trace; for $p<n$ uniqueness still holds by approximation, the ABP estimate being applied to truncations. $\square$

Connections Master

• The Riesz transforms 02.19.04. The direct prerequisite and analytic engine. The second-order representation ${\partial }_i{\partial }_j=-R_i{R}_j\Delta$ and the resulting constant-coefficient inequality $\Vert {\partial }_i{\partial }_{j}u{\Vert }_p\le C_p\Vert \Delta u{\Vert }_p$ are imported wholesale from that unit; everything in this unit is built by freezing coefficients and perturbing around that single global estimate, so the entire $W^{2,p}$ theory is a corollary of the $L^p$-boundedness of those $n$ operators.

• Schauder theory: interior and boundary $C^{2,\alpha }$ estimates 02.17.04. The dual companion on the fractional scale. Schauder is the $C^{0,\alpha }$-to-$C^{2,\alpha }$ statement, this unit the $L^p$-to-$W^{2,p}$ statement; both are run by frozen-coefficient perturbation of a constant-coefficient model, and the two scales meet through the Sobolev-Morrey embedding $W^{2,p}\hookrightarrow C^{1,1-n/p}$, with the $p=\infty$ / $\mathrm{BMO}$ borderline marking exactly where one must cross from this theory to that one.

• The Poisson equation, fundamental solution, and Newtonian potential 02.13.02. The unit whose $L^p$-data forward-reference this estimate closes. There the second-derivative control of the Newtonian potential was established only for Hölder data and the $L^p$ case deferred; the Calderón-Zygmund inequality here completes it, the kernels $D_{ij}\Phi$ being precisely the second-order singular integrals whose $L^p$-boundedness is the content of the inequality.

• $L^p$ spaces: Hölder, Minkowski, completeness 02.07.06. The scaffolding. The Banach-space structure of $W^{2,p}$ rests on the completeness of $L^p$, the absorption arguments use the Hölder and Minkowski inequalities to split and recombine the perturbation terms, and the whole estimate is a statement about the $L^p$ norms whose basic theory that unit supplies.

• Maximum principles for general elliptic operators 02.17.02. The uniqueness and a priori-bound input. The Aleksandrov-Bakelman-Pucci maximum principle for strong solutions, the $L^p$-scale analogue of the classical maximum principle, supplies the $\sup$-bound that converts the a priori $W^{2,p}$ estimate into the uniform invertibility needed for the method of continuity, exactly as the classical maximum principle does in the Schauder existence theory.

Historical & philosophical context Master

The $L^p$ theory of singular integrals was created by Alberto Calderón and Antoni Zygmund in their 1952 Acta Mathematica paper On the existence of certain singular integrals ^{[Calderón-Zygmund 1952]}, which established the real-variable theory of homogeneous singular integrals and, as its canonical application, the $L^p$-boundedness of the second-derivative operators $D_{ij}{\Delta }^{-1}$. This converted the one-dimensional conjugate-function $L^p$ theorem of Marcel Riesz into an $n$-dimensional tool for partial differential equations, with the Marcinkiewicz interpolation theorem ^{[Marcinkiewicz 1939]} supplying the passage from the weak-$(1,1)$ and $L^2$ bounds to the full $L^p$ range.

The systematic application to elliptic boundary-value problems was carried out by Agmon, Douglis, and Nirenberg in their 1959 Communications on Pure and Applied Mathematics paper ^{[Agmon-Douglis-Nirenberg 1959]}, which proved the global $W^{2,p}$ (and higher) estimates up to the boundary for general elliptic systems satisfying the complementing condition. The treatment was consolidated into the form used here in the Gilbarg-Trudinger monograph ^{[Gilbarg-Trudinger 2001]} and refined by Krylov ^{[Krylov 2008]}, whose Sobolev-space approach reorganises the perturbation around mean-oscillation bounds and reaches the VMO-coefficient generality of Chiarenza, Frasca, and Longo. The borderline failure at $p=\infty$, with the Hessian landing in $\mathrm{BMO}$, is the analytic fact identified by John and Nirenberg that draws the line between the $L^p$ theory and the Schauder $C^{2,\alpha }$ theory of Juliusz Schauder.

Bibliography Master

@article{CalderonZygmund1952,
  author  = {Calder\'on, Alberto P. and Zygmund, Antoni},
  title   = {On the existence of certain singular integrals},
  journal = {Acta Mathematica},
  volume  = {88},
  year    = {1952},
  pages   = {85--139}
}

@article{AgmonDouglisNirenberg1959,
  author  = {Agmon, Shmuel and Douglis, Avron and Nirenberg, Louis},
  title   = {Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I},
  journal = {Communications on Pure and Applied Mathematics},
  volume  = {12},
  year    = {1959},
  pages   = {623--727}
}

@article{Marcinkiewicz1939,
  author  = {Marcinkiewicz, J\'ozef},
  title   = {Sur l'interpolation d'op\'erations},
  journal = {Comptes Rendus de l'Acad\'emie des Sciences Paris},
  volume  = {208},
  year    = {1939},
  pages   = {1272--1273}
}

@book{Stein1970,
  author    = {Stein, Elias M.},
  title     = {Singular Integrals and Differentiability Properties of Functions},
  publisher = {Princeton University Press},
  year      = {1970}
}

@book{GilbargTrudinger2001,
  author    = {Gilbarg, David and Trudinger, Neil S.},
  title     = {Elliptic Partial Differential Equations of Second Order},
  edition   = {2},
  publisher = {Springer},
  year      = {2001}
}

@book{Krylov2008,
  author    = {Krylov, Nikolai V.},
  title     = {Lectures on Elliptic and Parabolic Equations in Sobolev Spaces},
  series    = {Graduate Studies in Mathematics},
  volume    = {96},
  publisher = {American Mathematical Society},
  year      = {2008}
}

@article{ChiarenzaFrascaLongo1991,
  author  = {Chiarenza, Filippo and Frasca, Michele and Longo, Placido},
  title   = {Interior $W^{2,p}$ estimates for nondivergence elliptic equations with discontinuous coefficients},
  journal = {Ricerche di Matematica},
  volume  = {40},
  year    = {1991},
  pages   = {149--168}
}