Geodesic
On the boundedness of the integral convolution operator in a pair of classical Lebesgue spaces $L_p$ and $L_r$
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 83 (2023), pp. 52-58 Cet article a éte moissonné depuis la source Math-Net.Ru

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In terms of the kernel of an integral convolution operator, a constructive criterion for its boundedness in a pair of classical Lebesgue spaces $L_p$ and $L_r$ is obtained. It is shown that in order for the integral convolution operator to act boundedly from $L_p$ to $L_{r,p}$, it is necessary and sufficient that the kernel $K(t)$ of the operator belonged to the Marcinkiewicz space $M_{t^{1-1/q}}$.
Keywords: integral convolution operator, boundedness, boundedness criterion
Mots-clés : Lebesgue spaces 
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E. A. Pavlov; A. I. Furmenko. On the boundedness of the integral convolution operator in a pair of classical Lebesgue spaces $L_p$ and $L_r$. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 83 (2023), pp. 52-58. http://geodesic.mathdoc.fr/item/VTGU_2023_83_a4/

[1] P. Khalmosh, V. Sander, Ogranichennye integralnye operatory v prostranstve $L^2$, Nauka, M., 1985, 158 pp.

[2] L. V. Kantorovich, G. P. Akilov, Funktsionalnyi analiz, Nauka, M., 1977, 744 pp.

[3] V. B. Korotkov, Integralnye operatory, Nauka, Sib. otd-nie, Novosibirsk, 1983, 224 pp.

[4] M. A. Krasnoselskii, P. P. Zabreiko, E. N. Pustylnik, P. E. Sobolevskii, Integralnye operatory v prostranstvakh summiruemykh funktsii, Nauka, M., 1966, 499 pp.

[5] Pavlov E. A. Ob operatorakh, “invariantnykh otnositelno sdviga v simmetrichnykh prostranstvakh”, Sibirskii matematicheskii zhurnal, 18:1 (1977), 189–194 | MR | Zbl

[6] R. Edvards, Ryady Fure v sovremennom izlozhenii, v. 2, Mir, M., 1985, 399 pp.

[7] S. G. Krein, Yu. I. Petunin, E. M. Semenov, Interpolyatsiya lineinykh operatorov, Nauka, M., 1978, 400 pp.

[8] A. P. Blozinski, “On a convolution theorem for $L(p, q)$ spaces”, Trans. Amer. Math. Soc, 164 (1972), 255–265 | MR

[9] V. W. Daniel, “Convolution operators on Lebesgue spaces of the half-line”, Trans. Amer. Math. Soc, 164 (1972), 479–488 | DOI | MR | Zbl

[10] R. O'Neil, “Convolution operators and $L(p, q)$ spaces”, Duke Math. J, 30 (1963), 129–142 | MR | Zbl

[11] E. M. Semenov, “Teoremy vlozheniya dlya banakhovykh prostranstv izmerimykh funktsii”, Doklady AN SSSR, 156:6 (1964), 1292–1295 | Zbl

[12] E. A. Pavlov, “Ob ogranichennosti operatorov svertki v simmetrichnykh prostranstvakh”, Izvestiya vuzov. Matematika, 1982, no. 2, 36–40 | Zbl

[13] G. G. Lorentz, “Some new functional spaces”, Ann. of Math, 51:1 (1950), 37–55 | DOI | MR | Zbl

[14] J. Marcinkiewicz, “Sur l'interpolation d'operations”, C.R. Acad. Sci. Paris, 208 (1939), 1272–1273 | Zbl