Annular variation and dyadic absolute bounds for the triangular Hilbert transform. Proves maximal and annular r-variation bounds, for every r > 2, from complex $L^3(\mathbb R^2)\times L^3(\mathbb R^2)$ to $L^{3/2}(\mathbb R^2)$. The maximal estimate controls both hard truncation endpoints and gives almost-everywhere and norm convergence. Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric $L^3\times L^3\times L^3$ point.
released 2026-10-05 | 2 theorems · 15 lemmas · 27 proofs · 13,494 words |
PLAY LEVEL 1 »(pdf)
We prove the annular r-variation estimate for the triangular Hilbert transform from complex $L^3\times L^3$ to L3/2 for every r > 2. The partitions may depend on the output point and range over all positive scales. The estimate yields the two-endpoint maximal bound and joint almost-everywhere and norm principal values, and resolves the symmetric scalar triangular Hilbert transform problem.
released 2026-10-05 | 1 theorem · 4 lemmas · 9 proofs · 3,738 words |
PLAY LEVEL 2 »(pdf)
We prove a uniform $L^3\times L^3\times L^3$ estimate for the dyadic triangular Hilbert form with unrestricted real inputs. The sum of the absolute local contributions over any finite set of scales is bounded by forty times the product of the input norms. In particular, the bound allows coefficients of modulus at most one to vary independently among admissible interval triples.
For arbitrary complex inputs in $L^3(\mathbb R^2)$, we prove the pointwise maximal $L^3\times L^3\to L^{3/2}$ estimate for the triangular Hilbert transform, with the supremum over both hard truncation endpoints. The estimate yields joint almost-everywhere and L3/2 convergence as the lower endpoint tends to zero and the upper endpoint tends to infinity. This also proves the conjectured scalar estimate at the symmetric point.