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The Riesz-transform formula for second derivatives of the Laplacian

Example

Assume Countable Choice. Let $n \geq 1$ and $u \in S (R^{n})$ . Then, as tempered distributions (equivalently, as $L^{2}$ classes almost everywhere), $\partial_{i} \partial_{j} u = - R_{i} R_{j} (Δ u) = R_{i} R_{j} (- Δ u) (i, j \in {1, \dots, n}),$ where $R_{1}, \dots, R_{n}$ are the Riesz transforms of Riesz transforms on Euclidean space; the symbol of $R_{i} R_{j}$ is $- ξ_{i} ξ_{j} / ∣ ξ ∣^{2}$ off the origin, and combining the $L^{p}$ bounds of The Riesz transforms are bounded on Lp gives $∥ \partial_{i} \partial_{j} u ∥_{L^{p}} \leq C_{n, p}^{2} ∥ Δ u ∥_{L^{p}} (1 < p < \infty) .$ For $n = 1$ the identity reads $u^{''} = - R_{1} R_{1} u^{''}$ and is consistent because $R_{1}^{2} = - i d$ makes both sides equal $u^{''}$ .

Facts & Assumptions

Given: ${A C}_{ω}$ , $n \geq 1$ , $1 < p < \infty$ , a Schwartz function $u \in S (R^{n})$ , and the negative-sign $2 π$ -normalized Fourier convention.

[A1]

The only choice assumption is Countable Choice ${A C}_{ω}$ ; it enters through the Plancherel and Riesz-transform interfaces. No full Axiom of Choice is used. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

The Riesz transforms are $R_{j} = F_{2}^{- 1} M_{m_{j}} F_{2}$ with $m_{j} (ξ) = - i ξ_{j} / ∣ ξ ∣$ for $ξ \neq 0$ and $m_{j} (0) = 0$ ; $∥ R_{j} ∥_{L^{2} \to L^{2}} \leq 1$ and $\sum_{j} R_{j}^{2} = - i d$ on $L^{2}$ . For $1 < p < \infty$ each $R_{j}$ extends boundedly to $L^{p} (R^{n}; C)$ with norm at most $C_{n, p}$ . (Riesz transforms on Euclidean space, Riesz transforms are L2 contractions and square to minus the identity in sum, The Riesz transforms are bounded on Lp, Exact L2 Fourier multiplier norm)

[F2]

$F (\partial^{α} u) = (2 π i ξ)^{α} F u$ on $S^{'} (R^{n})$ , and $F_{2}$ is a linear isometry that is injective on $L^{2}$ ; two tempered distributions with the same Fourier transform are equal. (Fourier differentiation and multiplication identities on tempered distributions, Plancherel theorem)

Verification

technique · direct

1.1F1F2algebraA1

Fourier multipliers. For $u \in S (R^{n})$ and any $i, j$ , [F2] gives $F (\partial_{i} \partial_{j} u) = (2 π i ξ_{i}) (2 π i ξ_{j}) \hat{u} = - 4 π^{2} ξ_{i} ξ_{j} \hat{u}$ and $F (Δ u) = - 4 π^{2} ∣ ξ ∣^{2} \hat{u}$ in $S^{'}$ ; since $u$ and its derivatives are Schwartz functions, these are also the $L^{2}$ Fourier transforms $F_{2}$ of the corresponding classes. Off the origin $m_{i} (ξ) m_{j} (ξ) = (- i ξ_{i} / ∣ ξ ∣) (- i ξ_{j} / ∣ ξ ∣) = - ξ_{i} ξ_{j} / ∣ ξ ∣^{2}$ , so $m_{i} m_{j} \cdot (- 4 π^{2} ∣ ξ ∣^{2}) = 4 π^{2} ξ_{i} ξ_{j}$ and hence $F_{2} (R_{i} R_{j} (- Δ u)) = F_{2} (\partial_{i} \partial_{j} u)$ almost everywhere: indeed $R_{i} R_{j} (- Δ u) = F_{2}^{- 1} (m_{i} m_{j} F_{2} (- Δ u))$ and $F_{2} (- Δ u) = 4 π^{2} ∣ ξ ∣^{2} \hat{u}$ , while the value of $m_{i} m_{j}$ at the single point $ξ = 0$ is immaterial. Injectivity of $F_{2}$ [F2] gives the $L^{2}$ identity $\partial_{i} \partial_{j} u = R_{i} R_{j} (- Δ u)$ , hence also the distributional identity and the sign rearrangement $R_{i} R_{j} (- Δ u) = - R_{i} R_{j} (Δ u)$ .

2.1step 1.1F1algebra

Symbol and $L^{p}$ bound. The multiplier of $R_{i} R_{j}$ is $m_{i} m_{j} = - ξ_{i} ξ_{j} / ∣ ξ ∣^{2}$ off the origin, so its absolute value is at most $1$ ; applying [F1] twice and using the identity of step 1.1, $∥ \partial_{i} \partial_{j} u ∥_{L^{p}} = ∥ R_{i} R_{j} (- Δ u) ∥_{L^{p}} \leq C_{n, p}^{2} ∥ Δ u ∥_{L^{p}}$ for $1 < p < \infty$ , the norms being those of the $L^{p}$ classes of the Schwartz functions involved.

2.2step 1.1F1algebra

The one-dimensional case. For $n = 1$ one has $m_{1} (ξ) = - i ξ / ∣ ξ ∣ = - i sign (ξ)$ off the origin, so $m_{1}^{2} = - 1$ and therefore $R_{1}^{2} = - i d$ on $L^{2}$ by [F1]; the identity of step 1.1 then reads $u^{''} = R_{1} R_{1} (- u^{''}) = - R_{1} R_{1} u^{''}$ , whose right-hand side equals $u^{''}$ , so the two sides agree.

3.1step 1.1step 2.1step 2.2given∎

Conclusion. For every $n \geq 1$ and $u \in S (R^{n})$ the second derivatives are the composition of the second-order Riesz multiplier with $- Δ u$ ; the strict range $1 < p < \infty$ is inherited from the Riesz-transform $L^{p}$ theorem, and the sign convention is the negative-sign $2 π$ -normalized Fourier transform used throughout. No endpoint $p = 1$ or $p = \infty$ bound is asserted.

Remarks

The formula identifies the Hessian of $u$ with a bounded combination of Riesz transforms of the Laplacian, which is the multiplier version of the Calderón–Zygmund representation of second derivatives; it is the whole-space model estimate behind the interior $W^{2, p}$ regularity on this page.

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Sources

Armin Schikorra, Partial Differential Equations I & II (version October 1, 2025; complete 281-page graduate lecture notes) (standard reference, not scraped)
John Villavert, Elementary Theory and Methods for Elliptic Partial Differential Equations (2017; complete 220-page lecture notes) (standard reference, not scraped)
Xu-Jia Wang, Schauder Estimates for Elliptic and Parabolic Equations (Australian National University, 2006; complete 7-page note) (standard reference, not scraped)