Monotonicity of certain functionals under rearrangement
Annales de l'Institut Fourier, Volume 24 (1974) no. 2, p. 67-116
We show here that a wide class of integral inequalities concerning functions on [0,1] can be obtained by purely combinatorial methods. More precisely, we obtain modulus of continuity or other high order norm estimates for functions satisfying conditions of the type 0 1 0 1 Ψf(x)-f(y) p(x-y)dxdy< where Ψ(u) and p(u) are monotone increasing functions of |u|.Several applications are also derived. In particular these methods are shown to yield a new condition for path continuity of general stochastic processes
Nous montrons qu’une large classe d’inégalités intégrales concernant des fonctions sur [0,1] peut être obtenue par des méthodes purement combinatoires. De façon plus précise, nous obtenons des modules de continuité ou d’autres estimations de normes d’ordre élevé pour des fonctions vérifiant des conditions du type0101Ψf(x)-f(y)p(x-y)dxdy<Ψ(u) et p(u) sont des fonctions croissantes de |u|. On en déduit différentes applications. On montre en particulier que ces méthodes donnent une nouvelle condition pour la continuité des chemins d’un processus stochastique général.
@article{AIF_1974__24_2_67_0,
     author = {Garsia, Adriano and Rodemich, Eug\`ene},
     title = {Monotonicity of certain functionals under rearrangement},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {24},
     number = {2},
     year = {1974},
     pages = {67-116},
     doi = {10.5802/aif.507},
     mrnumber = {54 \#2894},
     zbl = {0274.26006},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_1974__24_2_67_0}
}
Monotonicity of certain functionals under rearrangement. Annales de l'Institut Fourier, Volume 24 (1974) no. 2, pp. 67-116. doi : 10.5802/aif.507. https://aif.centre-mersenne.org/item/AIF_1974__24_2_67_0/

[1] P. Bernard, Quelques propriétés des trajectoires des fonctions aléatoires stables sur Rn, Ann. Inst. H. Poincaré, Sect. B 6, 131-151. | Numdam | MR 42 #1198 | Zbl 0196.18601

[2] J. Delporte, Fonctions aléatoires de deux variables à échantillons continus sur un domaine rectangulaire borné, Z. Wahrsch., 20, 249-258. | Zbl 0147.15801

[3] R. M. Dudley, Sample functions of the Gaussian process, Annals of Prob. V. 1, No. 1 (1973), 66-103. | MR 49 #11605 | Zbl 0261.60033

[4] A. Garsia, On the smoothness of functions satisfying certain integral inequalities, Functional Analysis, Proceedings of a symposium, N. Y. Acad. Press, 1970, 127-161. | MR 42 #8267 | Zbl 0286.26004

[5] A. Garsia, Continuity properties of Gaussian processes with multi-dimensional time parameter, Proceedings VI Berkeley Symposium V. II (1970), 369-374. | MR 53 #14623 | Zbl 0272.60034

[6] A. Garsia, Martingale inequalities, Seminar Notes W. A. Benjamin, Lecture Series (to appear).

[7] A. Garsia, E. Rodemich and H. Rumsey Jr., A real variable lemma and the continuity of paths of Gaussian processes, Indiana U. Math. J. V., 20 (1970), 565-578. | MR 42 #2534 | Zbl 0252.60020

[8] R. Getoor and H. Kesten, Continuity of local times for Markov Processes, Compositio Math., V. 24, Fasc. 3 (1972), 277-303. | Numdam | MR 46 #10075 | Zbl 0293.60069

[9] C. Greenhall, Growth and continuity of functions satisfying quadratic integral inequalities, Indiana U. Math. J. V., 21, No. 2 (1971), 157-175. | MR 44 #5420 | Zbl 0212.08604

[10] C. S. Herz, Lipschitz spaces and Bernstein's theorem on absolutely convergent Fourier transforms, Jour. Math. Mech. V., 18, No. 4 (1968), 283-324. | MR 55 #11028 | Zbl 0177.15701

[11] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure and Appl. Math., V. 14 (1961), 415-426. | MR 24 #A1348 | Zbl 0102.04302

[12] J. Lamperti, Probability, W. A. Benjamin Inc., Amsterdam (1966). | Zbl 0147.15502

[13] M. Marcus and L. Shepp, Sample Behaviour of Gaussian processes, Proc. VI, Berkeley Symp., V. II (1970), 423-439. | Zbl 0379.60040

[14] H. J. Reyser, Combinatorial Mathematics, Carus Math. Monograph, No. 14, (1963). | MR 27 #51 | Zbl 0112.24806

[14] H. Taylor, Rearrangements of incidence tables, Jour. of Comb. Theory (A), 14 (1973), 30-36. | MR 47 #8322 | Zbl 0257.05019