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At $p = 1$ bounded difference quotients need not give an $L^{1}$ weak derivative

Statement

The converse direction (2) of The difference-quotient characterisation of $W^{1, p}$ for $1 < p < \infty$ is false at $p = 1$ and is not asserted there. For the Heaviside step $u = 1_{(0, \infty)}$ on $Ω = (- 1, 1)$ one has $δ_{h}^{1} u = h^{- 1} 1_{(- h, 0)}$ on $Ω_{1, h} = (- 1, 1 - h)$ for $0 < h < 1$ , so $∥ δ_{h}^{1} u ∥_{L^{1} (Ω_{1, h})} = 1 (0 < h < 1),$ and likewise on any open $Ω^{'} \subseteq Ω_{1, h}$ with $(- h, 0) \subseteq Ω^{'}$ , while $u$ has no locally integrable weak derivative ([F2]); its distributional derivative is the Dirac mass at $0$ , and $u$ has bounded variation on every compact subinterval of $Ω$ (Bounded variation and total variation on an interval). For this witness the uniformly bounded local $L^{1}$ quotients correspond to a finite measure derivative and bounded variation, while $W^{1, 1}$ membership fails. No general multidimensional BV characterization is proved here. No consumer on this page may use part (2) at $p = 1$ .

Facts & Assumptions

Given: Countable Choice; the interval $Ω = (- 1, 1)$ ; the Heaviside class $u = 1_{(0, \infty)}$ on $Ω$ ; the difference-quotient operator of Difference quotients on a shrunken domain with $τ_{h} u (x) = u (x - h)$ ; and the companion theorem The difference-quotient characterisation of $W^{1, p}$ for $1 < p < \infty$ .

[F1]

Difference quotients on $Ω$ : for $0 < h < 1$ and $x \in (- 1, 1)$ with $x + h \in (- 1, 1)$ one has $δ_{h}^{1} u (x) = (u (x + h) - u (x)) / h$ . (Difference quotients on a shrunken domain)

[F2]

The Heaviside class on $(- 1, 1)$ has no locally integrable weak derivative, and its distributional derivative is the Dirac mass $δ_{0}$ at $0$ : if $v \in L_{l o c}^{1} (- 1, 1)$ satisfied $\int_{- 1}^{1} u φ^{'} d x = - \int_{- 1}^{1} v φ d x$ for every $φ \in C_{c}^{\infty} (- 1, 1)$ , then $\int_{0}^{1} φ^{'} d x = φ (1) - φ (0) = - φ (0)$ would force $\int_{- 1}^{1} v φ d x = φ (0)$ for every test $φ$ , whereas the shrinking bumps $φ_{ϵ} (x) = η (x / ϵ)$ , with $η$ the published smooth bump equal to $1$ on $[- 1 / 2, 1 / 2]$ and supported in $(- 1, 1)$ , satisfy $φ_{ϵ} (0) = 1$ and $∣ \int_{- 1}^{1} v φ_{ϵ} d x ∣ \leq \int_{[- ϵ, ϵ]} ∣ v ∣ d x \to 0$ as $ϵ ↓ 0$ by absolute continuity of the integral, a contradiction; hence $u \notin W^{1, p} ((- 1, 1))$ for every $1 \leq p \leq \infty$ . (Weak derivative of a locally integrable function, Absolute continuity of the integral, A smooth bump between concentric Euclidean balls)

[F3]

Bounded variation: for $a < b$ and $f : [a, b] \to R$ , the variation over a partition $P = (n, t)$ is $V (f, P) = \sum_{i < n} ∣ f (t_{i + 1}) - f (t_{i}) ∣$ and $f$ has bounded variation when these sums are bounded above, with total variation the supremum over partitions. (Bounded variation and total variation on an interval)

[F4]

The companion theorem asserts part (2) only for $1 < p < \infty$ , and its proof in Uniformly bounded difference quotients represent a weak derivative represents the limit in $L^{p^{'}} (Ω^{'})$ -duality, so it consumes the finiteness of $p^{'}$ — equivalently $p > 1$ — and does not extend to $p = 1$ , where $p^{'} = \infty$ . (The difference-quotient characterisation of $W^{1, p}$ for $1 < p < \infty$ )

Proof

technique · direct

1.1F1algebra

For $0 < h < 1$ and $x \in (- h, 0)$ one has $x + h \in (0, h) \subseteq (0, \infty)$ , so $u (x + h) = 1$ and $u (x) = 0$ , giving $δ_{h}^{1} u (x) = 1 / h$ ; for $x \in (0, 1 - h)$ both values are $1$ , and for $x \in (- 1, - h)$ both values are $0$ , giving $δ_{h}^{1} u (x) = 0$ there. Hence $δ_{h}^{1} u = h^{- 1} 1_{(- h, 0)}$ on $Ω_{1, h} = (- 1, 1 - h)$ , with the endpoint conventions immaterial for the almost-everywhere class.

1.2F3algebra

On each compact subinterval $[- a, b] \subseteq (- 1, 1)$ with $0 < a, b < 1$ the Heaviside is nondecreasing, so for every partition $P$ of $[- a, b]$ the identity $∣ u (t_{i + 1}) - u (t_{i}) ∣ = u (t_{i + 1}) - u (t_{i})$ holds and the variation telescopes: $V (u, P) = u (b) - u (- a) = 1$ ; the sums are therefore bounded by $1$ and $u$ has bounded variation there with total variation $1$ .

2.1step 1.1algebra

Integrating the identity of step 1.1 over $Ω_{1, h} = (- 1, 1 - h)$ : since $(- h, 0) \subseteq Ω$ for $0 < h < 1$ , $\int_{Ω_{1, h}} ∣ δ_{h}^{1} u ∣ d x = h^{- 1} λ ((- h, 0)) = h^{- 1} h = 1$ , and the same computation applies on any open $Ω^{'} \subseteq Ω_{1, h}$ containing $(- h, 0)$ . Thus the family $δ_{h}^{1} u$ , $0 < h < 1$ , is uniformly bounded on its shrunken domains. For $h < 0$ , the quotient has magnitude $1 / ∣ h ∣$ on $(0, - h)$ and is zero elsewhere on $Ω_{1, h}$ , so the same bound holds. In particular on $Ω^{'} = (- 1 / 2, 1 / 2)$ both signs satisfy a uniform bound for $0 < ∣ h ∣ < 1 / 4$ .

3.1F2step 2.1

By [F2], $u$ has no locally integrable weak derivative on $(- 1, 1)$ and in particular $u \notin W^{1, 1} (Ω)$ ; combined with step 2.1 this exhibits a class with a uniform local $L^{1}$ difference-quotient bound and no $W^{1, 1}$ membership, so part (2) of the companion theorem fails at $p = 1$ .

4.1F2F4step 2.1step 1.2∎

The witness of steps 1.1-2.1 has, by [F2], distributional derivative the Dirac mass at $0$ — a finite Borel measure, not an $L^{1}$ class — and bounded variation by step 1.2, so a finite measure derivative rather than an $L^{1}$ weak derivative occurs for this witness; by [F4] the companion theorem's duality proof consumes $p > 1$ , which is why part (2) is stated only there and no consumer on this page may apply it at $p = 1$ .

Source notes

Hunter's Theorem 4.53(2) (printed p. 125) and Laugesen's Proposition 5.7(ii) (printed p. 110) are both stated for $1 < p < \infty$ . The scaffold phrase $∥ δ_{h}^{1} u ∥_{L^{1} (Ω^{'})} = 1$ for all $0 < h < dist (Ω^{'}, \partial Ω)$ was made precise: the identity holds on the appropriate shrunken domain and on every open $Ω^{'}$ within it containing the interval $(- h, 0)$ , which is the interval actually carrying the quotient.

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Sources

John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page two-quarter graduate notes) (standard reference, not scraped)
Richard S. Laugesen, Linear Analysis and Partial Differential Equations (University of Illinois, 2020, complete 158-page graduate notes) (standard reference, not scraped)

At p=1p=1 bounded difference quotients need not give an L1L1 weak derivative