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

The joint spatial transform

Joint Fourier estimates and convergent sectors › The joint spatial transform · lines 609–641

VII-D@1#eq:jointFT, VII-D@1#eq:FTbound

For fixed $(t, s, t^{'}, y)$ , define

H_{Φ} (t, s, t^{'}, y; α, β) = \int_{R^{6}} Φ (t, x; s, y; t^{'}, x^{'}) \times e^{i α \cdot (x - y) - i β \cdot (x^{'} - y)} d^{3} x d^{3} x^{'} .

This preserves all correlations of the joint test.

We shall use the fixed momentum-decay exponent $N = 12$ . Applying $(1 - Δ_{x})^{6} (1 - Δ_{x^{'}})^{6}$ under the integral gives

∣ \partial_{t}^{j} \partial_{t^{'}}^{j^{'}} H_{Φ} ∣ \leq C_{K} p_{K, 64} (Φ) (1 + ∣ α ∣)^{- 12} (1 + ∣ β ∣)^{- 12}, j + j^{'} \leq 8.

Only $24 + j + j^{'} \leq 32$ derivatives are needed for this estimate; order $64$ also accommodates the fixed localizations used below. The variables $(t, s, t^{'}, y)$ range over a fixed compact projection of $K$ .

Displayed formulas in this passage

VII-D@1#eq:jointFT · line 612
$H_{Φ} (t, s, t^{'}, y; α, β) = \int_{R^{6}} Φ (t, x; s, y; t^{'}, x^{'}) \times e^{i α \cdot (x - y) - i β \cdot (x^{'} - y)} d^{3} x d^{3} x^{'} .$
VII-D@1#eq:FTbound · line 630
$∣ \partial_{t}^{j} \partial_{t^{'}}^{j^{'}} H_{Φ} ∣ \leq C_{K} p_{K, 64} (Φ) (1 + ∣ α ∣)^{- 12} (1 + ∣ β ∣)^{- 12}, j + j^{'} \leq 8.$

The joint spatial transform

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