Boundedness and Compactness Criteria for a~Generalized Truncated Potential

V. M. Kokilashvili

Informatics and Automation, Function spaces, harmonic analysis, and differential equations, Tome 232 (2001), pp. 164-178.

Résumé

Boundedness and compactness criteria are established for a generalized truncated Riesz potential $K_{\alpha} f(x,t) =\int _{|y|\leq 2|x|} (|x-y| +t)^{\alpha -n} f(y)\,dy$, $t\in [0,\infty)$, $x\in \mathbb R^n$, that acts from $L^p (\mathbb R^n)$ to $L_{\nu }^q (\mathbb R_+^{n+1})$, where ${1$, ${0$, ${\alpha >n/p}$, and $\nu$ is a positive Borel measure on $\mathbb R_+^{n+1}$. Also, two-sided estimates for a measure of noncompactness of the operator $K_{\alpha }$ are obtained.

Bibliographie

[1] Sawyer E., “Multipliers of Besov and power-weighted $L^2$ spaces”, Indiana Univ. Math. J., 33:3 (1984), 353–366 | DOI | MR | Zbl

[2] Gabidzashvili M. A., Genebashvili I. Z., Kokilashvili V. M., “Dvukhvesovye neravenstva dlya obobschennykh potentsialov”, Tr. MIAN, 194, 1992., 89–96. | MR | Zbl

[3] Genebashvili I., Gogatishvili A., Kokilashvili V., Krbec M., “Weight theory for integral transforms on spaces of homogeneous type”, Pitman Monogr. and Surv. Pure and Appl. Math., 92, Longman, Harlow, 1998 | MR | Zbl

[4] Meskhi A., “Criteria of the boundedness and compactness for generalized one-sided potentials”, Real Analysis Exchange, 26:1 (2000/2001), 217–236 | MR

[5] Meskhi A., “Boundedness and compactness criteria for the generalized Riemann–Liouville operator”, Proc. A. Razmadze Math. Inst., 121 (1999), 161–162 | Zbl

[6] Adams D. R., “A trace inequality for generalized potentials”, Stud. Math., 48 (1973), 99–105 | MR | Zbl

[7] Genebashvili I., “Carleson measures and potentials defined on the spaces of homogeneous type”, Bull. Georgian Acad. Sci., 135:3 (1989), 505–508 | MR | Zbl

[8] Sawyer E. T., Wheeden R. L., Zhao S., “Weighted norm inequalities for operators of potential type and fractional maximal functions”, Pot. Anal., 5 (1996), 523–580 | DOI | MR | Zbl

[9] Wheeden R. L., Wilson J. M., “Weighted norm estimates for gradients of half-space extensions”, Indiana Univ. Math. J., 44:3 (1995), 917–969 | DOI | MR

[10] Kokilashvili V., Meskhi A., “On the boundedness and compactness of generalized truncated potentials”, Bull. Tbilisi Intern. Centre Math. and Inform., 4 (2000), 28–31 | Zbl

[11] Mazya V. G., Sobolev spaces, Springer, Berlin, 1985 | MR

[12] Sinnamon G., “Weighted Hardy and Opial-type inequalities”, J. Math. Anal. and Appl., 160 (1991), 434–445 | DOI | MR | Zbl

[13] Bradley J. S., “Hardy inequality with mixed norms”, Canad. Math. Bull., 21 (1978), 405–408 | MR | Zbl

[14] Kokilashvili V. M., “O neravenstve Khardi”, Soobsch. AN GSSR, 96 (1979), 37–40 | MR | Zbl

[15] Drabek P., Heinig H., Kufner A., “Higher dimentional Hardy inequality”, General inequalities 7, 7th Intern. Conf. Oberwolfach (Nov. 13–18, 1995), Intern. Ser. Numer. Math., 123, Birkhäuser, Basel etc., 1997, 3–16 | MR | Zbl

[16] Edmunds D. E., Kokilashvili V., Meskhi A., “Boundedness and compactness of Hardy-type operators in Banach function spaces”, Proc. A. Razmadze Math. Inst., 117 (1998), 7–30 | MR | Zbl

[17] Kantorovich L. V., Akilov G. P., Functional analysi, Pergamon Press, Oxford, 1982 | MR | Zbl

[18] König H., Eigenvalue distribution of compact operators, Birkhäuser, Boston etc., 1986 | MR

[19] Ando T., “On the compactness of integral operators”, Indag. Math., 24 (1962), 235–239 | MR

[20] Krasnoselskii M. A., Zabreiko P. P., Pustylnik E. I., Sobolevskii P. E., Integralnye operatory v prostranstvakh summiruemykh funktsii, Nauka, M., 1966 | MR

[21] Edmunds D. E., Evans W. D., Spectral theory and differential operators, Oxford Univ. Press, Oxford, 1987 | MR

[22] Meskhi A., “Criteria for the boundedness and compactness of integral transforms with positive kernels”, Proc. Edinburgh Math. Soc., 44 (2001) (to appear) | DOI | MR | Zbl

[23] 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 | DOI | MR | Zbl

[24] Opic B., “On the distance of the Riemann–Liouville operators from compact operators”, Proc. Amer. Math. Soc, 122:2 (1994), 495–501 | DOI | MR | Zbl

[25] Edmunds D. E., Evans W. D., Harris D. J., “Approximation numbers of certain Volterra integral operators”, J. London Math. Soc., 37:2 (1988), 471–489 | DOI | MR | Zbl