AWhat changed
For a fixed finite bilinear form, Riesz proves log-convexity of its norm in reciprocal-exponent coordinates α=1/p and β=1/r on0≤α,β≤1 with α+β≥1, including positive-weight versions. Via duality and Stieltjes–Lebesgue approximation this yields operator-norm log-convexity and endpoint interpolation for a compatible linear operator L^p→L^q in the region1≤p≤q≤∞: intermediate norms are bounded by the geometric interpolation of endpoint bounds.
Scientific scope
For a fixed finite bilinear form, Riesz proves log-convexity of its norm in reciprocal-exponent coordinates α=1/p and β=1/r on0≤α,β≤1 with α+β≥1, including positive-weight versions. Via duality and Stieltjes–Lebesgue approximation this yields operator-norm log-convexity and endpoint interpolation for a compatible linear operator L^p→L^q in the region1≤p≤q≤∞: intermediate norms are bounded by the geometric interpolation of endpoint bounds. The general 1927 result is restricted to that exponent triangle; it is not the later unrestricted complex Riesz–Thorin theorem. The original also treats complex coefficients/variables, so it should not be described as only a real-valued finite-matrix statement. For real coefficients, existence of real maximizing vectors in the triangular region does not extend generally to the complementary region: the original gives real/complex maximizing distinctions and a real-only convexity counterexample. A special nonnegative-coefficient extension is not a general all-operator extension. The functional formulation uses the paper's fixed Stieltjes–Lebesgue measures and compatible bounded endpoint actions; no nonlinear, arbitrary-vector-valued or endpoint-weak-type theorem is asserted. The specific Fourier applications developed later in the paper are outside this contribution description.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The general 1927 result is restricted to that exponent triangle; it is not the later unrestricted complex Riesz–Thorin theorem. The original also treats complex coefficients/variables, so it should not be described as only a real-valued finite-matrix statement. For real coefficients, existence of real maximizing vectors in the triangular region does not extend generally to the complementary region: the original gives real/complex maximizing distinctions and a real-only convexity counterexample. A special nonnegative-coefficient extension is not a general all-operator extension. The functional formulation uses the paper's fixed Stieltjes–Lebesgue measures and compatible bounded endpoint actions; no nonlinear, arbitrary-vector-valued or endpoint-weak-type theorem is asserted. The specific Fourier applications developed later in the paper are outside this contribution description.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
Provisional WRS assessment, edition 2026-10-23.3. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
Full scientific record · Browse contribution grades · Request a correction