Monotonicity of certain functionals under rearrangement
Annales de l'Institut Fourier, Volume 24 (1974) no. 2, pp. 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 type

0101Ψf(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},
     pages = {67--116},
     publisher = {Institut Fourier},
     volume = {24},
     number = {2},
     year = {1974},
     doi = {10.5802/aif.507},
     zbl = {0274.26006},
     mrnumber = {54 #2894},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.507/}
}
TY  - JOUR
AU  - Garsia, Adriano
AU  - Rodemich, Eugène
TI  - Monotonicity of certain functionals under rearrangement
JO  - Annales de l'Institut Fourier
PY  - 1974
DA  - 1974///
SP  - 67
EP  - 116
VL  - 24
IS  - 2
PB  - Institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.507/
UR  - https://zbmath.org/?q=an%3A0274.26006
UR  - https://www.ams.org/mathscinet-getitem?mr=54 #2894
UR  - https://doi.org/10.5802/aif.507
DO  - 10.5802/aif.507
LA  - en
ID  - AIF_1974__24_2_67_0
ER  - 
%0 Journal Article
%A Garsia, Adriano
%A Rodemich, Eugène
%T Monotonicity of certain functionals under rearrangement
%J Annales de l'Institut Fourier
%D 1974
%P 67-116
%V 24
%N 2
%I Institut Fourier
%U https://doi.org/10.5802/aif.507
%R 10.5802/aif.507
%G en
%F AIF_1974__24_2_67_0
Garsia, Adriano; Rodemich, Eugène. Monotonicity of certain functionals under rearrangement. Annales de l'Institut Fourier, Volume 24 (1974) no. 2, pp. 67-116. doi : 10.5802/aif.507. http://www.numdam.org/articles/10.5802/aif.507/

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

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

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

[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 | Zbl

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

[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 | Zbl

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

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

[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 | Zbl

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

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

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

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

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

Cited by Sources: