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        "description": "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.",
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          "citation": "Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires (original1927)",
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      "statement": "# Marcel Riesz: log-convex bilinear bounds and restricted-exponent operator interpolation\n\nFor 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.\n\n## Scientific scope\n\nThe 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.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nMarcel Riesz\n\nSource-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.\n\n\n\n## Original sources\n\n- [Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires (original1927)](https://archive.ymsc.tsinghua.edu.cn/pacm_download/117/5399-11511_2007_Article_BF02564121.pdf)\n",
      "summary": "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.",
      "title": "Marcel Riesz: log-convex bilinear bounds and restricted-exponent operator interpolation"
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    "title": "Marcel Riesz: log-convex bilinear bounds and restricted-exponent operator interpolation",
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    "what_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.",
    "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_summary": "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.",
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