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Young's theorem integrates a Hölder function of unbounded variation against itself
Example
There is a -Hölder function of unbounded variation for which the Young integral nevertheless exists.
Facts & Assumptions
Given: Let , put , and tile by consecutive intervals of lengths accumulating at zero. On , let be the symmetric triangular tent of height , and set .
The -series converges for and diverges for (For rational , converges iff ).
Rational powers are monotone and obey their exponent laws (Monotonicity of and of , Laws of rational exponents).
Young's theorem applies when the two Hölder exponents have sum greater than one (Young's Riemann–Stieltjes existence theorem for rational Hölder exponents).
Verification
By [L1], and , so the intervals tile . On one tent, the linear slope estimate and [L2] give . If lie in different tents, let be the endpoint of 's tent toward and the endpoint of 's tent toward . Both have value zero and , so the two one-tent estimates and the triangle inequality give . Taking the endpoints of successively smaller tents gives the same estimate at zero. Thus is -Hölder.
A partition through the endpoints and peaks of the first tents has variation at least [given] This is unbounded by [L1], so is not BV.
Since , [L3] nonetheless gives existence of . This is genuinely outside the BV existence theorem.
Depends on
- Young's Riemann–Stieltjes existence theorem for rational Hölder exponents
- Lipschitz map, $\alpha$-Hölder map for rational $0 < \alpha \le 1$, and contraction
- Rational powers $a^r$ of a positive base
- Monotonicity of $r \mapsto a^{r}$ and of $a \mapsto a^{r}$
- Laws of rational exponents
- For rational $p > 0$, $\sum 1/k^p$ converges iff $p > 1$
- Series, partial sums, convergence and the sum, divergence, and the tail series
- A series converges iff each of its tail series converges, and the sum splits as $s_N$ plus the $N$-th tail
- Every convergent sequence is Cauchy
- Sequences of reals: bounded, eventually, frequently, tails, subsequences
- Bounded variation and total variation on an interval
- Partition of $[a,b]$ as a finite strictly increasing list $a = t_0 < t_1 < \dots < t_n = b$, its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions
- The triangle inequality
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
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Sources
- Nourdin, Nualart, and Peccati, The Breuer–Major theorem in total variation: improved rates under minimal regularity, Section 2.2 (standard reference, not scraped)