## Rearrangement inequalities for functionals with monotone integrands.(English)Zbl 1102.26014

The authors consider integral functionals of the form $I_F(u_1,\dots,u_m) = \int_{X} F(u_1(x),\dots ,u_m(x))\,dx$ and $J_F(u_1,\dots ,u_m) = \int_{X^m} F(u_1(x_1), u(x_2),\dots, u(x_m))\prod_{i<j} K_{ij}(d(x_i,x_j))\,dx_1\dots dx_m,$ where $$X$$ is $${\mathbb R}^n,\;$$ the sphere $$S^n,$$ or the hyperbolic space $$H^n,\,dx$$ is the canonical measure on $$X,$$ and $$d(x,y)$$ the canonical distance on $$X.$$ The functions $$u_1,\dots ,u_m$$ are nonnegative Lebesgue measurable functions on $$X$$ which are small at infinity in an appropriate sense, the $$K_{ij}$$ are nonnegative decreasing functions on $$[0,\infty)$$, and $$F$$ is a real-valued Borel measurable function on $$[0,\infty)\times \dots \times [0,\infty).$$ Let $$u^*$$ denote the symmetric decreasing rearrangement of $$u.$$ When $$X = \mathbb R$$ and $$F(y_1,y_2) = y_1 y_2,$$ there are inequalities $$I_F(u_1,u_2) \leq I_F(u_1^*, u_2^*)$$ and $$J_F(u_1,u_2) \leq J_F(u_1^*, u_2^*).$$
The $$I$$-inequality is due to Hardy and Littlewood, the $$J$$-inequality to F. Riesz. The Hardy-Littlewood and Riesz inequalities have been extended and generalized in many directions. The immediate precursors of the article under review are papers by F. Brock (2000) and C. Draghici (2005) in which it is proved that $$I_F(u_1,\dots ,u_m),$$ respectively $$J_F(u_1,\dots ,u_m)$$, does not increase when the $$u_i$$ are replaced by their symmetric decreasing rearrangements, provided $$F$$ is continuous and nonnegative, $$F(0) = 0,$$ and $$F$$ satisfies the inequality $F(y + he_i + ke_j)\, + F(y) \geq F(y+he_i) + F(y+ke_j),$ for all $$y\in [0,\infty) \times \dots \times [0,\infty),$$ all positive numbers $$h$$ and $$k,$$ and all indices $$i \neq j,$$ where $$e_i$$ is the $$i$$th coordinate vector in $${\mathbb R}^m.$$ The authors call such functions $$F$$ supermodular. The term Schur-convex has also been used in the literature. The nonnegativity assumption on $$F$$ can be replaced by an integrability condition on the negative part of $$F.$$ The authors’ principal results assert that the theorems of Brock and Draghici still hold when $$F$$ is discontinuous. This does not readily follow from the proofs in the continuous case, so a new tool is required. The authors devise one via a factorization of $$F$$ into continuous and monotonic parts, inspired by a theorem in statistics of A. Sklar. The authors also prove some sufficient conditions for equality to hold in their inequalities.

### MSC:

 26D15 Inequalities for sums, series and integrals
Full Text:

### References:

 [1] L.V. Ahlfors, Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill Series in Higher Mathematics, 1973.; L.V. Ahlfors, Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill Series in Higher Mathematics, 1973. · Zbl 0272.30012 [2] Almgren, F.; Lieb, E. H., Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc., 2, 683-773 (1989) · Zbl 0688.46014 [3] A. Baernstein II, A unified approach to symmetrization, in: Partial differential equations of elliptic type (Cortona, 1992), Sympos. Math., XXXV, Cambridge University Press, Cambridge, 1994, pp. 47-91.; A. Baernstein II, A unified approach to symmetrization, in: Partial differential equations of elliptic type (Cortona, 1992), Sympos. Math., XXXV, Cambridge University Press, Cambridge, 1994, pp. 47-91. · Zbl 0830.35005 [4] Baernstein, A.; Taylor, B. A., Spherical rearrangements, subharmonic functions, and *-functions in $$n$$-space, Duke Math. J., 43, 245-268 (1976) · Zbl 0331.31002 [5] Beckner, W., Sobolev inequalities, the Poisson semigroup and analysis on the sphere $$S^n$$, Proc. Natl. Acad. Sci., 89, 4816-4819 (1992) · Zbl 0766.46012 [6] Beckner, W., Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. Math. (2), 138, 213-242 (1993) · Zbl 0826.58042 [7] Brascamp, H. J.; Lieb, E. H.; Luttinger, J. M., A general rearrangement inequality for multiple integrals, J. Funct. Anal., 17, 227-237 (1974) · Zbl 0286.26005 [8] Brock, F., A general rearrangement inequality à la Hardy-Littlewood, J. Inequality Appl., 5, 309-320 (2000) · Zbl 0982.26015 [9] Brock, F.; Solynin, A. Yu., An approach to symmetrization via polarization, Trans. Amer. Math. Soc., 352, 1759-1796 (2000) · Zbl 0965.49001 [10] Burchard, A., Cases of equality in the Riesz rearrangement inequality, Ann. Math. (2), 143, 499-527 (1996) · Zbl 0876.26016 [11] Burchard, A.; Schmuckenschläger, M., Comparison theorems for exit times, Geom. Funct. Anal., 11, 651-692 (2001) · Zbl 0995.60018 [12] Carlier, G., On a class of multidimensional optimal transportation problems, J. Convex Anal., 10, 517-529 (2003) · Zbl 1084.49037 [13] Crandall, M. G.; Tartar, L., Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc., 78, 385-390 (1980) · Zbl 0449.47059 [14] Crowe, J. A.; Zweibel, J. A.; Rosenbloom, P. C., Rearrangements of functions, J. Funct. Anal., 66, 432-438 (1986) · Zbl 0612.46027 [15] Draghici, C., Rearrangement inequalities with applications to ratios of heat kernels, Potential Anal., 22, 351-374 (2005) · Zbl 1128.26014 [16] Draghici, C., A general rearrangement inequality, Proc. Amer. Math. Soc., 133, 735-743 (2005) · Zbl 1128.26013 [17] Friedberg, R.; Luttinger, J. M., (a) Rearrangement inequalities for periodic functions, (b) A new rearrangement inequality for multiple integrals, Arch. Rat. Mech. Anal., 61 (1976), 35-44 and 45-64 · Zbl 0334.26008 [18] Hadwiger, H.; Ohmann, D., Brunn-Minkowskischer Satz und Isoperimetrie, Math. Z., 66, 1-8 (1956) · Zbl 0071.38001 [19] Hajaiej, H., Cases of equality and strict inequality in the extended Hardy Littlewood inequalities, Proc. Royal Soc. Edinburgh A, 135, 643-661 (2005) · Zbl 1102.26013 [20] H. Hajaiej, Extended Hardy-Littlewood inequalities and applications. Trans. Amer. Math. Soc., 357 (2005) 4885-4896.; H. Hajaiej, Extended Hardy-Littlewood inequalities and applications. Trans. Amer. Math. Soc., 357 (2005) 4885-4896. · Zbl 1071.26014 [21] Hajaiej, H.; Stuart, C. A., Symmetrization inequalities for composition operators of Carathéodory type, Proc. London Math. Soc., 87, 396-418 (2003) · Zbl 1052.26020 [22] Hajaiej, H.; Stuart, C. A., Extensions of the Hardy-Littlewood inequalities for Schwarz symmetrization, Int. J. Math. Math. Sci., 59, 3129-3150 (2004) · Zbl 1086.26013 [23] Hajaiej, H.; Stuart, C. A., Existence and non-existence of Schwarz symmetric ground states for eigenvalue problems, Mat. Pura Appl., 186 (2005) · Zbl 1099.49002 [24] Hardy, G. E.; Littlewood, J. E.; Pólya, G., Inequalities (1934/1952), Cambridge University Press: Cambridge University Press London and New York · Zbl 0010.10703 [25] B. Kawohl, Rearrangements and convexity of level sets in PDE, Springer Lecture Notes in Mathematics, vol. 1150, 1980.; B. Kawohl, Rearrangements and convexity of level sets in PDE, Springer Lecture Notes in Mathematics, vol. 1150, 1980. [26] Lieb, E. H., Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math., 57, 93-105 (1977) · Zbl 0369.35022 [27] E.H. Lieb, M. Loss, Analysis, first/second ed., AMS Graduate Studies in Mathematics, 1996/2001.; E.H. Lieb, M. Loss, Analysis, first/second ed., AMS Graduate Studies in Mathematics, 1996/2001. · Zbl 0873.26002 [28] Lorentz, G. G., An inequality for rearrangements, Amer. Math. Monthly, 60, 176-179 (1953) · Zbl 0050.28201 [29] Luttinger, J. M., Generalized isoperimetric inequalities, I, II, III, J. Math. Phys., 14 (1973), 586-593, 1444-1447 and 1448-1450 · Zbl 0261.52006 [30] Morpurgo, C., Sharp inequalities for functional integrals and traces of conformally invariant operators, Duke Math. J., 114, 477-553 (2002) · Zbl 1065.58022 [31] Riesz, F., Sur une inégalité intégrale, J. London Math. Soc., 5, 162-168 (1930) · JFM 56.0232.02 [32] Sklar, A., Fonctions de réparticion à $$n$$ dimensions et leurs marges, Inst. Statist. Univ. Paris, 8, 229-231 (1959) · Zbl 0100.14202 [33] Sklar, A., Random variables, joint distribution functions, and copulas, Kybernetika (Prague), 9, 449-460 (1973) · Zbl 0292.60036 [34] Amer. Math. Soc. (Transl.) (2) 34 (1963) 39-68.; Amer. Math. Soc. (Transl.) (2) 34 (1963) 39-68. · Zbl 0131.11501 [35] Stuart, C. A., Guidance properties of nonlinear planar wave guides, Arch. Rat. Mech. Anal., 125, 145-200 (1993) · Zbl 0801.35136 [36] R. Tahraoui, (a) Symmetrization inequalities, (b) Corrigendum, Nonlinear Anal. 27:933-955, 1996 and 39:535, 2000.; R. Tahraoui, (a) Symmetrization inequalities, (b) Corrigendum, Nonlinear Anal. 27:933-955, 1996 and 39:535, 2000. · Zbl 0879.35013 [37] Van Schaftingen, J.; Willem, M., Set transformations, symmetrizations and isoperimetric inequalities, (Nonlinear Analysis and Applications to Physical Sciences (2004), Springer Italia: Springer Italia Milan), 135-152 · Zbl 1453.26034 [38] Zygmund, A., On an integral inequality, J. London Math. Soc., 8, 175-178 (1933) · JFM 59.0992.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.