On Hardy’s inequality in weighted rearrangement invariant spaces and applications. I

Maligranda, Lech

doi:10.1090/S0002-9939-1983-0691280-6

On Hardy’s inequality in weighted rearrangement invariant spaces and applications. I
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by Lech Maligranda

Proc. Amer. Math. Soc. 88 (1983), 67-74

DOI: https://doi.org/10.1090/S0002-9939-1983-0691280-6

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Abstract:

We give inequalities relating the norm of a function and the norm of its average operators ${P_\psi },{Q_\psi }$ and ${S_\psi },{T_\psi }$ in weighted rearrangement invariant spaces ${E_{\kappa ,\delta }}$ and $E(\mu ),d\mu (t) = \tau ’(t)dt$. These average operators include, for example, the integral mean, the ${P_p},{Q_p}$ operators of Boyd [4] and Butzer and Fehér [6], the average operators ${P_\varphi },{Q_\varphi }$ and ${S_E},{T_E}$ from [14,15,16]. In the particular case, for some $\psi ,\kappa ,\delta ,\tau$ and $E$ these inequalities were obtained by many authors and applied to a study of interpolation operators and imbedding theorems for Sobolev weight spaces.

References

Kenneth F. Andersen, On Hardy’s inequality and Laplace transforms in weighted rearrangement invariant spaces, Proc. Amer. Math. Soc. 39 (1973), 295–299. MR 315071, DOI 10.1090/S0002-9939-1973-0315071-4
Colin Bennett, Intermediate spaces and the class $\textrm {L} \log ^{+L}$, Ark. Mat. 11 (1973), 215–228. MR 352966, DOI 10.1007/BF02388518
D. W. Boyd, The Hilbert transform on rearrangement-invariant spaces, Canadian J. Math. 19 (1967), 599–616. MR 212512, DOI 10.4153/CJM-1967-053-7
David W. Boyd, Indices of function spaces and their relationship to interpolation, Canadian J. Math. 21 (1969), 1245–1254. MR 412788, DOI 10.4153/CJM-1969-137-x
David W. Boyd, Indices for the Orlicz spaces, Pacific J. Math. 38 (1971), 315–323. MR 306887, DOI 10.2140/pjm.1971.38.315
P. L. Butzer and F. Fehér, Generalized Hardy and Hardy-Littlewood inequalities in rearrangement-invariant spaces, Comment. Math. Spec. Issue 1 (1978), 41–64. Special issue dedicated to Władysław Orlicz on the occasion of his seventy-fifth birthday. MR 504152
F. Fehér, A note on weak-type interpolation and function spaces, Bull. London Math. Soc. 12 (1980), no. 6, 443–451. MR 593962, DOI 10.1112/blms/12.6.443

Inequalities

Hans P. Heinig, The Marcinkiewicz interpolation theorem extends to weighted spaces, Studia Math. 62 (1978), no. 2, 163–168. MR 499962, DOI 10.4064/sm-62-2-163-168
S. G. Kreĭn, Ju. I. Petunin, and E. M. Semënov, Interpolyatsiya lineĭ nykh operatorov, “Nauka”, Moscow, 1978 (Russian). MR 506343
Alois Kufner, Imbedding theorems for general Sobolev weight spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 23 (1969), 373–386. MR 253038
G. G. Lorentz, On the theory of spaces $\Lambda$, Pacific J. Math. 1 (1951), 411–429. MR 44740, DOI 10.2140/pjm.1951.1.411

An introduction to the theory of real variable functions

Lech Maligranda, A generalization of the Shimogaki theorem, Studia Math. 71 (1981/82), no. 1, 69–83. MR 651325, DOI 10.4064/sm-71-1-69-83
Lech Maligranda, Generalized Hardy inequalities in rearrangement invariant spaces, J. Math. Pures Appl. (9) 59 (1980), no. 4, 405–415. MR 607047

Indices and interpolation

Benjamin Muckenhoupt, Hardy’s inequality with weights, Studia Math. 44 (1972), 31–38. MR 311856, DOI 10.4064/sm-44-1-31-38
E. A. Pavlov, A generalization of the Hardy inequality, Collection of articles on applications of functional analysis (Russian), Voronež. Tehnolog. Inst., Voronezh, 1975, pp. 109–114 (Russian). MR 0453947
E. A. Pavlov, Some properties of the Hardy-Littlewood operator, Mat. Zametki 26 (1979), no. 6, 909–912, 973 (Russian). MR 567227

On Hardy’s operator in weighted

spaces