Mixed norms and rearrangements: Sobolev’s inequality and Littlewood’s inequality. (English) Zbl 0639.46034

Let \(N(f)=N_ 1(f)+...+N_ K(f)\) for a measurable function f over \(R^ K\), where \(N_ k(f)\) is the mixed norm of power (1,...,1,\(\infty,1,...,1)\), \(\infty\) being on the k-th place, \(k=1,2,...,K\), \(K\geq 2\). It is shown that if \(N(f)<\infty\), then f is rearrangeable. Moreover, if g is a measure-preserving rearrangement of f such that for every \(\lambda >0\) the set where \(| g| >\lambda\) is essentially a K-cube with edges parallel to the coordinate axes, then \(N(g)\leq N(f)\). Finally, supposing \(N(f)<\infty\), f belongs to the Lorentz space \(L(r,1)\) with \(r=K/(K-1)\) and \(\| f\|_{L(r,1)}\leq N(f)/K\). These results are applied to prove sharper forms of the Sobolev inequality
Reviewer: M.Abel


46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Full Text: DOI


[1] Adams, R. A., Sobolev Spaces (1975), New York: Academic Press, New York
[2] Adams, R. A.; Fournier, J., Cone conditions and properties of Sobolev spaces, J. Math. Anal. Appl., 61, 713-734 (1977) · Zbl 0385.46024
[3] G. R.Allen,An inequality involving product measures, Radical Banach Algebras and Automatic Continuity (Long Beach, Calif., 1981), pp. 277-279; Lecture Notes in Math.,975, Springer, Berlin-New York (1983).
[4] Aubin, T., Nonlinear Analysis on Manifolds. Monge-Ampère Equations, Grundlehren der mathematischen Wissenschaften,252 (1982), New York-Heidelberg-Berlin: Springer-Verlag, New York-Heidelberg-Berlin · Zbl 0512.53044
[5] Benedek, A.; Panzone, R., The spaces L^p with mixed norms, Duke Math. J., 28, 301-324 (1961) · Zbl 0107.08902
[6] Bergh, J.; Löfstrom, J., Interpolation Spaces. An Introduction, Grundlehren der mathematischen Wissenschaften,223 (1976), Berlin-Heidelberg-New York: Springer-Verlag, Berlin-Heidelberg-New York · Zbl 0344.46071
[7] Blei, R. C., Sidon partitions and p-Sidon sets, Pacific J. Math., 65, 307-313 (1976) · Zbl 0335.43008
[8] Blei, R. C., Fractional cartesian products of sets, Annales Institut Fourier (Grenoble), 29, 79-105 (1979) · Zbl 0381.43003
[9] Calderón, A. P., An inequality for integrals, Studia Math., 57, 275-277 (1976) · Zbl 0343.26017
[10] Duff, G. F. D., A general integral inequality for the derivative of an equimeasurable rearrangement, Canadian J. Math., 28, 793-804 (1976) · Zbl 0342.26015
[11] Edwards, R. E.; Ross, K. A., p-Sidon sets, J. Functional Anal., 15, 404-427 (1974) · Zbl 0273.43007
[12] Faris, W. J., Weak Lebesgue spaces and quantum mechanical binding, Duke Math. J., 43, 365-373 (1976) · Zbl 0329.46037
[13] Gagliardo, E., Proprietà di alcune classi di funzioni in più variabili, Ricerche Mat., 7, 102-137 (1958) · Zbl 0089.09401
[14] Hardy, G. H.; Littlewood, J. E.; Pólya, G., Inequalities (1934), Cambridge: Cambridge University Press, Cambridge · JFM 60.0169.01
[15] Hardy, G. H.; Littlewood, J. E.; Pólya, G., Some simple inequalities satisfied by convex functions, Messenger of Math., 58, 145-152 (1929) · JFM 55.0740.04
[16] Littlewood, J. E., On bounded bilinear forms in an infinite number of variables, Quarterly J. Math., 1, 164-174 (1930) · JFM 56.0335.01
[17] Loomis, L. H.; Whitney, H., An inequality related to the isoperimetric inequality, Bull. Amer. Math. Soc., 55, 961-962 (1949) · Zbl 0035.38302
[18] Moser, J., A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20, 473-484 (1971)
[19] Nirenberg, L., On elliptic partial differential operators, Annali della Scuola Normale Sup. Pisa, 13, 116-162 (1959)
[20] O’Neil, R., Convolution operators and L(p, q) spaces, Duke Math. J., 30, 129-142 (1963) · Zbl 0178.47701
[21] Poornima, S. A., An embedding theorem for the Sobolev space W^1,1, Bull. Sci. Math., 107, 253-259 (1983) · Zbl 0529.46025
[22] Praciano-Pereira, T., On bounded multilinear forms on a class of l^p spaces, J. Math. Anal. Appl., 81, 561-568 (1981) · Zbl 0497.46007
[23] Schwenk, A. J.; Munro, J. I., How small can the mean shadow of a set be?, American Math. Monthly, 90, 325-329 (1983) · Zbl 0563.05006
[24] Sobolev, S. L., On a theorem of functional analysis, Mat. Sb., 46, 471-496 (1938)
[25] Stein, E. M., Singular Integrals and Differentiability Properties of Functions (1970), Princeton: Princeton University Press, Princeton · Zbl 0207.13501
[26] Stein, E. M.; Weiss, G., Introduction to Fourier Analysis an Euclidean Spaces (1971), Princeton: Princeton University Press, Princeton · Zbl 0232.42007
[27] Strichartz, R. S., Multipliers on fractional Sobolev spaces, J. Math. Mech., 16, 1031-1060 (1967) · Zbl 0145.38301
[28] Szelmeczka, J., On a property of a function belonging to various spaces with mixed norms, Functiones et Approximatio, 9, 25-28 (1980) · Zbl 0456.46026
[29] Talenti, G., Best constant in Sobolev’s inequality, Ann. Mat. Pura Appl., 110, 353-372 (1976) · Zbl 0353.46018
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.