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Additional info for Advanced Courses of Mathematical Analysis II: Proceedings of the Second International School, Granada, Spain, 20 - 24 September 2004
Ip (2) llP, is the norm in L P ( R ) . We observe the following. (i) For p = 00 the result is trivial: llMflloo 5 I l f l l m ; (ii) but for p = 1 the result is false: indeed, M f ( x ) 1z1-l when IC is large if f is not identically zero (integrability can fail even locally as was shown in ["I). : M f ( x )> A}l L C ,Ilflll forA > 0. ) It is good enough because 2 can be obtained for p > 1 from this estimate together with the trivial L" result (see the interpolation theorem below), and 3 is also good because Lebesgue's differentiation theorem comes as a corollary.
Amer. Math. ,No. 16, 1955. 9. A. Grothendieck, Rdsume' des resultats essentiels duns la the'orie des produits tensoriels topologiques et des espaces nucle'aires. Ann. Inst. Fourier Grenoble 4 (1952), 73-112 (appeared in 1954). 10. A. Grothendiek, Sur les applications line'aires faiblement compactes d'espaces du type C(K). Canadian J. , 5 (1953), 129-173. 11. A. Grothendieck, Re'sume' de la the'orie me'trique des produits tensoriels topologiques. Bol. Mat. Sao Paulo 8 (1953), 1-79 (appeared in 1956).
J. of Math. 74 (1952), 168-186. (7) Sur les applications line'aires faiblement compactes d'espaces du type C(K). Canadian J. Math. 5 (1953), 129-173. (8) Sur les espaces de solutions d'une classe ge'ne'rale d'e'quations aux de'rive'es partielles. J. Analyse Math. 2 (1953), 243-280. (9) Sur certains espaces de fonctions holomorphes, I. J. Reine Angew. Math. 192 (1953), 35-64. (10) Sur certains espaces de fonctions holomorphes, II. J. Reine Angew. Math. 192 (1953), 77-95. (11) Quelques points de la the'orie des produits tensoriels topologiques.