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Stepanov, Vladimir D.
Weighted norm inequalities for integral operators and related topics. (English). In: Krbec, Miroslav, Kufner, Alois, Opic, Bohumír and Rákosník, Jiří (eds.): Nonlinear Analysis, Function Spaces and Applications, Proceedings of the Spring School held in Prague, May 23-28, 1994. Vol. 5. Prometheus Publishing House, Praha, 1994. pp. 139-175
MSC: 26D10, 46E35, 47B38, 47G10 | MR 1322312 | Zbl 0866.47025
References:
[A] Andersen, K.: Weighted generalized Hardy inequalities for nonincreasing functions. Canad. J. Math. 43 (1991), 1121–1135. MR 1145581 | Zbl 0757.26018
[AM] Andersen, K., Muckenhoupt, B.: Weighted weak type Hardy inequalities with applications to Hilbert transform and maximal functions. Studia Math. 72 (1982), 9–26. MR 0665888
[An] Ando, T.: On compactness of integral operators. Indag. Math. (N.S.) 24 (1962), 235–239. MR 0139016 | Zbl 0100.11201
[AO] Apyshev, O.D., Otelbaev, M.: On the spectrum of a class of differential operators and embedding theorems. Math. USSR-Izv. 15 (1980), 1–24.
[ArM] Arino, M., Muckenhoupt, B.: Maximal function on classical Lorentz spaces and Hardy’s inequality with weights for nonincreasing functions. Trans. Amer. Math. Soc. 320 (1990), 727–735. MR 0989570
[AS] Aronszajn, N., Szeptycki, P.: On general integral transformations. Math. Ann. 163 (1966), 127–154. MR 0190799 | Zbl 0171.12402
[BL] Bergh, J., Löfström, J.: Interpolation spaces: an introduction. Springer Verlag, 1976. MR 0482275
[BS] Birman, M.Sh., Solomjak, M.Z.: Estimates for the singular numbers of integral operators. Russian Math. Surveys 32 (1977). MR 0438186
[BK$_1$] Bloom, S., Kerman, R.: Weighted norm inequalities for operators of Hardy type. Proc. Amer. Math. Soc. 113 (1991), 135–141. MR 1059623 | Zbl 0753.42010
[BK$_2$] Bloom, S., Kerman, R.: Weighted $L_\Phi $-integral inequalities for operators of Hardy type. Preprint. MR 1279373
[B] Bradley, J.S.: Hardy’s inequalities with mixed normes. Canad. Math. Bull. 21 (1978), 405–408. MR 0523580
[Br] Braverman, M.Sh.: On a class of operators. J. London Math. Soc. (2) 47 (1993), 119–128. MR 1200982 | Zbl 0732.47033
[BrS] Braverman, M.Sh., Stepanov, V.D.: On the discrete Hardy inequality. Bull. London Math. Soc. (1994), to appear. MR 1289048 | Zbl 0806.26011
[CS$_1$] Carro, M.J., Soria, J.: Weighted Lorentz spaces and the Hardy operator. J. Funct. Anal. 112 (1993), 480–494. MR 1213148 | Zbl 0784.46022
[CS$_2$] Carro, M.J., Soria, J.: Boundedness of some integral operators. Canad. J. Math. 45 (1993), 1155–1166. MR 1247539 | Zbl 0798.42010
[EE] Edmunds, D.E., Evans, W.D.: Spectral theory and differential operators. Univ. Press, Oxford, 1987. MR 0929030 | Zbl 0628.47017
[EEH] Edmunds, D.E., Evans, W.D., Harris, D.J.: Approximation numbers of certain Volterra integral operators. J. London Math. Soc. (2) 37 (1988), 471–489. MR 0939123 | Zbl 0658.47049
[EGP] Edmunds, D.E., Gurka, P., Pick, L.: Compactness of Hardy type integral operators in weighted Banach function spaces. Studia Math. 109 (1994), 73–90. MR 1267713 | Zbl 0821.46036
[ES$_1$] Edmunds, D.E., Stepanov, V.D.: The measure of non-compactness and approximation numbers of certain Volterra integral operators. Math. Ann. 298 (1994), 41–66. MR 1252816 | Zbl 0788.45013
[ES$_2$] Edmunds, D.E., Stepanov, V.D.: On singular numbers of certain Volterra integral operators. Preprint. MR 1359927
[GHS] Gol, M.L. ’dma, Heinig, H.P., Stepanov, V.D.: On the principle of duality in the Lorentz spaces. Russian Acad. Sci. Dokl. Math. (1994), to appear. (Russian) MR 1414066
[G] Glazman, I.M.: Direct methods of qualitative spectral analysis of singular differential operators. Nauka, Moscow, 1963, pp. . (Russian) MR 0185471
[GK] Gohberg, I.C., Krein, M.G.: Introduction to the theory of linear non-selfadjoint operators. AMS Transl. Math. Monographs, 18 vol. Providence, R.I., 1969. MR 0246142
[G$_1$] Gol, M.L. ’dma: Function spaces and their applications. Patrice Lumumba Univ., 1991, pp. 35–67.
[G$_2$] Gol, M.L. ’dma: On integral inequalities on a cone of functions with monotonicity properties. Soviet Math. Dokl. 42 (1992), 581–587. MR 1152406
[G$_3$] Gol, M.L. ’dma: On integral inequalities on the set of functions with some properties of monotonicity. Function Spaces, Differential Operators and Nonlinear Analysis, Teubner Texte zur Math. 133, Leipzig, 1993, pp. 274–279. MR 1242589
[HLP] Hardy, G.G., Littlewood, J.E., Polya, G.: Inequalities. Cambridge Univ. Press, 1934. Zbl 0010.10703
[H] Heinig, H.P.: Weighted inequalities in Fourier analysis. Teubner-Texte zur Math. 119, Leipzig, 1990, pp. 42–85. MR 1151431 | Zbl 0773.42008
[HM] Heinig, H.P., Maligranda, L.: Interpolation with weights in Orlicz spaces. Department of Applied Mathematics, Lulea University of Technology, 1992, Research Report 03.
[HSt] Heinig, H.P., Stepanov, V.D.: Weighted Hardy inequalities for increasing functions. Canad. J. Math. 45 (1993), 104–116. MR 1200323 | Zbl 0796.26008
[Hern] Hernandez, E.: Weighted inequalities through factorization. Publ. Mat. 35 (1991), 141–153. MR 1103612 | Zbl 0722.42012
[KA] Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon, Oxford, 1982. MR 0664597 | Zbl 0484.46003
[Ko] Kokilashvili, V.M.: On Hardy’s inequalities in weighted spaces. Soobshch. Akad. Nauk. Gruzin. SSR 96 (1979), 37–40. (Russian) MR 0564755 | Zbl 0434.26007
[K] König, H.: Eigenvalue distribution of compact operators. Birkhäuser, Boston, 1986. MR 0889455
[Kor] Korotkov, V.B.: Integral operators. Nauka, Novosibirsk, 1983. (Russian) MR 0721475 | Zbl 1032.45018
[KR] Krasnosel, M.A. ’sk, Ruticki, Ya.B.: Convex functions and Orlicz spaces. Noordhoff, Groningen, 1961. MR 0126722
[L] Lai, Qinsheng: Weighted integral inequalities for the Hardy type operator and the fractional maximal operator. J. London Math. Soc. (2) 49 (1994), 244–266. MR 1260111 | Zbl 0794.42015
[LS] Lomakina, E., Stepanov, V.D.: On the compactness and approximation numbers of Hardy type integral operators in Lorentz spaces. Preprint. MR 1373067 | Zbl 0853.42013
[L$_1$] Lorentz, G.G.: On the theory of spaces $\Lambda $. Pacific J. Math. 1 (1951), 411–429. MR 0044740 | Zbl 0043.11302
[L$_2$] Lorentz, G.G. : Bernstein polynomials. Univ. of Toronto Press, 1953.. MR 0057370 | Zbl 0051.05001
[MS] Martín, J.F. -Reye, Sawyer, E.: Weighted inequalities for Riemann-Liouville fractional integrals of order one and greater. Proc. Amer. Math. Soc. 106 (1989), 727–733. MR 0965246
[Ma] Maz, V.G. ’y: Sobolev spaces. Springer-Verlag, 1985. MR 0817985
[M] Muckenhoupt, B.: Hardy’s inequality with weights. Studia Math. 44 (1972), 31–38. MR 0311856 | Zbl 0236.26015
[MPS] Myasnikov, E.A., Persson, L.E., Stepanov, V.D.: On the best constants in certain integral inequalities for monotone functions. Lulea University of Technology, Department of Mathematics, 1993, Research Report 15. MR 1317179
[N] Neugebauer, C.J.: Weighted norm inequalities for averaging operators on monotone functions. Publ. Mat. 35 (1991), 429–447. MR 1201565
[NS] Newman, J., Solomyak, M.: Two-sided estimates on singular values for a class of integral operators on the semi-axes. Preprint. MR 1299892
[Nik] Nikol, N.K. ’ski: Lectures on the shift operator. Nauka, Moscow, 1980. (Russian) MR 0575166
[Now] Nowak, K.: Schatten ideal behaviour of a generalized Hardy operator. Proc. Amer. Math. Soc. 118 (1993), 479–483. MR 1152990
[O$_1$] Oinarov, R.: Weighted inequalities for a class of integral operators. Soviet Math. Dokl. 44 (1992), 291–293. MR 1152890
[O$_2$] Oinarov, R.: Two-sides estimates of a certain classes of integral operators. Proc. Steklov Inst. Math. 204 (1993), 240–250. (Russian) MR 1320028
[OK] Opic, B., Kufner, A.: Hardy-type Inequalities. Pitman Research Notes in Math. Series 219, Longman Scientific and Technical, Harlow, 1990. MR 1069756 | Zbl 0698.26007
[P$_1$] Pietsch, A.: Eigenvalues and $s$-numbers. Geest & Portig, Leipzig, 1987. MR 0917067 | Zbl 0615.47019
[Riem] Riemenschneider, S.D.: Compactness of a class of Volterra operators. Tôhoku Math. J. (2) 26 (1974), 385–387. MR 0361939 | Zbl 0289.47028
[Saw$_1$] Sawyer, E.T.: Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator. Trans. Amer. Math. Soc. 281 (1984), 329–337. MR 0719673 | Zbl 0538.47020
[Saw$_2$] Sawyer, E.T.: Boundedness of classical operators in classical Lorentz spaces. Studia Math. 96 (1990), 145–158. MR 1052631
[Sc] Schep, A.R.: Kernel operators. Thesis, Leiden, 1977. MR 0440404
[Sim] Simon, B.: Trace ideals and their applications. Univ. Press, Cambridge, 1979. MR 0541149 | Zbl 0423.47001
[S$_1$] Sinnamon, G.: Weighted Hardy and Opial type inequalities. J. Math. Anal. Appl. 160 (1991), 433–445. MR 1126128 | Zbl 0756.26011
[S$_2$] Sinnamon, G.: Spaces defined by level functions and their duals. Preprint, 1993, pp. 30. MR 1292851
[SS] Sinnamon, G., Stepanov, V.D.: The weighted Hardy inequality: new proofs and the case $p=1$. Preprint, 1994, pp. 15. MR 1395069
[SW] Stein, E.M., Weiss, G.: Introduction to Fourier analysis on euclidean spaces. Univ. Press, Princeton, 1971. MR 0304972 | Zbl 0232.42007
[St$_1$] Stepanov, V.D.: Two-weighted estimates for Riemann-Liouville integrals. Report No 39, Math. Inst., Czechoslovak Academy of Sciences, 1988, pp. 28.
[St$_2$] Stepanov, V.D.: Weighted inequalities of Hardy type for higher order derivatives, and their applications. Soviet Math. Dokl. 38 (1989), 389–393. MR 0981053 | Zbl 0674.26008
[St$_3$] Stepanov, V.D.: Weighted inequalities of Hardy type for higher order derivatives. Proc. Steklov Inst. Math. 187 (1990), no. 3, 205–220.
[St$_4$] Stepanov, V.D.: Weighted inequalities of Hardy type for Riemann-Liouville fractional integrals. Siberian Math. J. 31 (1990), 513–522. MR 1084772
[St$_5$] Stepanov, V.D.: On the boundedness and compactness of a class of integral operators. Soviet Math. Dokl. 41 (1990), 468–470. MR 1071870
[St$_6$] Stepanov, V.D.: Two-weighted estimates of Riemann-Liouville integrals. Math. USSR-Izv. 36 (1991), 669–681. MR 1072699 | Zbl 0724.26011
[St$_7$] Stepanov, V.D.: Integral operators on the cone of monotone functions and embeddings of Lorentz spaces. Soviet Math. Dokl. 43 (1991), 620–623. MR 1118029
[St$_8$] Stepanov, V.D.: Weighted inequalities for a class of Volterra convolution operators. J. London Math. Soc. (2) 45 (1992), 232–242. MR 1171551 | Zbl 0703.42011
[St$_9$] Stepanov, V.D.: The weighted Hardy’s inequality for nonincreasing functions. Trans. Amer. Math. Soc. 338 (1993), 173–186. MR 1097171 | Zbl 0786.26015
[St$_{10}$] Stepanov, V.D.: On weighted estimates for a class of integral operators. Siberian Math. J. 34 (1993), 755–766. MR 1248804 | Zbl 0815.47064
[St$_{11}$] Stepanov, V.D.: Integral operators on the cone of monotone functions. J. London Math. Soc. (2) 48 (1993), 465–487. MR 1241782 | Zbl 0837.26011
[St$_{12}$] Stepanov, V.D.: Weighted norm inequalities of Hardy type for a class of integral operators. J. London Math. Soc. (1994), to appear. MR 1277757 | Zbl 0837.26012
[St$_{13}$] Stepanov, V.D.: On singular numbers of certain class of integral operators. Russian Acad. Sci. Dokl. Math. 336 (1994), 457–458. (Russian) MR 1289518
[Sz] Szeptycki, P.: Notes on integral transformations. Dissertationes Math. (Rozprawy Mat.) 231 (1984), 1–52. MR 0763933 | Zbl 0558.44001
[Tal] Talenti, G.: Osservazioni sopra una classe di disuguaglianze. Rend. Sem. Mat. Fis. Milano 39 (1969), 171–185. MR 0280661 | Zbl 0218.26011
[Tom] Tomaselli, G.: A class of inequalities. Boll. Un. Mat. Ital. (4) 2 (1969), 622–631. MR 0255751 | Zbl 0188.12103
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