Monotonicity of certain functionals under rearrangement

Adriano Garsia; Eugène Rodemich

Annales de l'institut Fourier (1974)

  • Volume: 24, Issue: 2, page 67-116
  • ISSN: 0373-0956

Abstract

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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 Ψ f ( x ) - f ( y ) p ( x - y ) d x d y < 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

How to cite

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Garsia, Adriano, and Rodemich, Eugène. "Monotonicity of certain functionals under rearrangement." Annales de l'institut Fourier 24.2 (1974): 67-116. <http://eudml.org/doc/74178>.

@article{Garsia1974,
abstract = {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 $\int ^1_0\int ^1_0\Psi \big (\{f(x)-f(y)\over p(x-y)\}\big )dxdy&lt; \infty $ where $\Psi (u)$ and $p(u)$ are monotone increasing functions of $\vert u\vert $.Several applications are also derived. In particular these methods are shown to yield a new condition for path continuity of general stochastic processes},
author = {Garsia, Adriano, Rodemich, Eugène},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {2},
pages = {67-116},
publisher = {Association des Annales de l'Institut Fourier},
title = {Monotonicity of certain functionals under rearrangement},
url = {http://eudml.org/doc/74178},
volume = {24},
year = {1974},
}

TY - JOUR
AU - Garsia, Adriano
AU - Rodemich, Eugène
TI - Monotonicity of certain functionals under rearrangement
JO - Annales de l'institut Fourier
PY - 1974
PB - Association des Annales de l'Institut Fourier
VL - 24
IS - 2
SP - 67
EP - 116
AB - 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 $\int ^1_0\int ^1_0\Psi \big ({f(x)-f(y)\over p(x-y)}\big )dxdy&lt; \infty $ where $\Psi (u)$ and $p(u)$ are monotone increasing functions of $\vert u\vert $.Several applications are also derived. In particular these methods are shown to yield a new condition for path continuity of general stochastic processes
LA - eng
UR - http://eudml.org/doc/74178
ER -

References

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  1. [1] P. BERNARD, Quelques propriétés des trajectoires des fonctions aléatoires stables sur Rn, Ann. Inst. H. Poincaré, Sect. B 6, 131-151. Zbl0196.18601MR42 #1198
  2. [2] J. DELPORTE, Fonctions aléatoires de deux variables à échantillons continus sur un domaine rectangulaire borné, Z. Wahrsch., 20, 249-258. Zbl0147.15801
  3. [3] R. M. DUDLEY, Sample functions of the Gaussian process, Annals of Prob. V. 1, No. 1 (1973), 66-103. Zbl0261.60033MR49 #11605
  4. [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. Zbl0286.26004MR42 #8267
  5. [5] A. GARSIA, Continuity properties of Gaussian processes with multi-dimensional time parameter, Proceedings VI Berkeley Symposium V. II (1970), 369-374. Zbl0272.60034MR53 #14623
  6. [6] A. GARSIA, Martingale inequalities, Seminar Notes W. A. Benjamin, Lecture Series (to appear). 
  7. [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. Zbl0252.60020MR42 #2534
  8. [8] R. GETOOR and H. KESTEN, Continuity of local times for Markov Processes, Compositio Math., V. 24, Fasc. 3 (1972), 277-303. Zbl0293.60069MR46 #10075
  9. [9] C. GREENHALL, Growth and continuity of functions satisfying quadratic integral inequalities, Indiana U. Math. J. V., 21, No. 2 (1971), 157-175. Zbl0212.08604MR44 #5420
  10. [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. Zbl0177.15701MR55 #11028
  11. [11] F. JOHN and L. NIRENBERG, On functions of bounded mean oscillation, Comm. Pure and Appl. Math., V. 14 (1961), 415-426. Zbl0102.04302MR24 #A1348
  12. [12] J. LAMPERTI, Probability, W. A. Benjamin Inc., Amsterdam (1966). Zbl0147.15502
  13. [13] M. MARCUS and L. SHEPP, Sample Behaviour of Gaussian processes, Proc. VI, Berkeley Symp., V. II (1970), 423-439. Zbl0379.60040
  14. [14] H. J. REYSER, Combinatorial Mathematics, Carus Math. Monograph, No. 14, (1963). Zbl0112.24806MR27 #51
  15. [14] H. TAYLOR, Rearrangements of incidence tables, Jour. of Comb. Theory (A), 14 (1973), 30-36. Zbl0257.05019MR47 #8322

Citations in EuDML Documents

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  1. Matthias Kurzke, A nonlocal singular perturbation problem with periodic well potential
  2. Matthias Kurzke, A nonlocal singular perturbation problem with periodic well potential
  3. G. Pisier, Conditions d'entropie assurant la continuité de certains processus et applications à l'analyse harmonique
  4. Constantin Nanopoulos, Photis Nobelis, Régularité et propriétés limites des fonctions aléatoires
  5. Talenti, Giorgio, Inequalities in rearrangement invariant function spaces
  6. Kolyada, Viktor I., On embedding theorems
  7. Guido Trombetti, Metodi di simmetrizzazione nelle equazioni alle derivate parziali

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