theoremroutine
Bochner–Riesz bounds from local smoothing
Bochner–Riesz means bounded on for , with maximal bounds
the headline application to Bochner–Riesz
Theorem·
thm:bochner-riesz·§10 Bochner--Riesz meansStatement (paper)
Let , , and put
.
If , then
In this range in as if
. If and , then
and almost everywhere as .
In plain English
For and , the Bochner–Riesz means are bounded on uniformly in , and in as when . For and , the maximal function is bounded on and almost everywhere.
Why it's here
This is the paper's application of local smoothing to Bochner–Riesz summation. The multiplier is decomposed into dyadic shells of width near the sphere, each treated using the shell maximal bound with loss . The gain from the multiplier's smoothness at the sphere makes the sum over shells converge once exceeds the relevant exponent, and this gives the uniform bounds, the maximal bound, and the convergence statements.
Classified routine:
This follows the known implication from local smoothing/shell estimates to Bochner–Riesz bounds (as in Sogge's and Carbery's work). It decomposes the multiplier into dyadic shells near the sphere, applies the shell maximal lemma, and sums the resulting geometric series in δ^{α−σ}.