Manuscript VII-D@1 · VII-D.tex · section · ef67c89c

Shell and within-shell bounds

Strong finite part and dyadic certificates › Shell and within-shell bounds · lines 1888–1903

VII-D@1#eq:dyadicE, VII-D@1#eq:dyadicJ

Let $ε_{n} = 2^{- n}$ , $n \geq 1$ , and define

E_{n}^{ρ} J_{n, η}^{ρ} = T_{ε_{n + 1}}^{ρ} - T_{ε_{n}}^{ρ} - (lo g 2) r, = T_{2^{- (n + η)}}^{ρ} - T_{2^{- n}}^{ρ} - η (lo g 2) r, 0 \leq η \leq 1.

The positive-cutoff integrals are continuous in the cutoff, locally uniformly on bounded sets of tests, so $J_{n, η}^{ρ}$ is continuous in $η$ .

Displayed formulas in this passage

VII-D@1#eq:dyadicE · line 1891
$E_{n}^{ρ} J_{n, η}^{ρ} = T_{ε_{n + 1}}^{ρ} - T_{ε_{n}}^{ρ} - (lo g 2) r, = T_{2^{- (n + η)}}^{ρ} - T_{2^{- n}}^{ρ} - η (lo g 2) r, 0 \leq η \leq 1.$
VII-D@1#eq:dyadicJ · line 1891
$E_{n}^{ρ} J_{n, η}^{ρ} = T_{ε_{n + 1}}^{ρ} - T_{ε_{n}}^{ρ} - (lo g 2) r, = T_{2^{- (n + η)}}^{ρ} - T_{2^{- n}}^{ρ} - η (lo g 2) r, 0 \leq η \leq 1.$