Multipliers in Spaces of Bessel Potentials:

A. A. Belyaev ; A. A. Shkalikov

Matematičeskie zametki, Tome 102 (2017) no. 5, pp. 684-699.

Résumé

The aim of the paper is to study spaces of multipliers acting from the Bessel potential space $H^s_p(\mathbb{R}^n)$ to the other Bessel potential space $H^t_q(\mathbb{R}^n)$. We obtain conditions ensuring the equivalence of uniform and standard multiplier norms on the space of multipliers $$ M[H^s_p(\mathbb{R}^n) \to H^t_q(\mathbb{R}^n)]\qquad \text{for}\quad s,t \in \mathbb{R},\quad p,q > 1. $$ In the case $$ p,q > 1,\qquad p \le q,\qquad s > \frac np,\qquad t \ge 0,\qquad s-\frac np \ge t-\frac nq, $$ the space $M[H^s_p(\mathbb{R}^n) \to H^t_q(\mathbb{R}^n)]$ can be described explicitly. Namely, we prove in this paper that the latter space coincides with the space $H^t_{q,\mathrm{unif}}(\mathbb{R}^n)$ of uniformly localized Bessel potentials introduced by Strichartz. It is also proved that if both smoothness indices $s$ and $t$ are nonnegative, then such a description is possible only for the given values of the indices.

Détail

Keywords: Bessel potential space, multiplier, Strichartz theorem, uniform localization principle.

Bibliographie

[1] R. S. Strichartz, “Multipliers on fractional Sobolev spaces”, J. Math. Mech., 16 (1967), 1031–1060 | MR | Zbl

[2] J. Franke, “On the spaces $F^s_{p,q}$ of Triebel–Lizorkin type: pointwise multipliers and spaces on domains”, Math. Nachr., 125 (1986), 29–68 | MR | Zbl

[3] W. Sickel, “On pointwise multipliers for $F^s_{p,q}(\mathbb{R}^n)$ in case $\sigma_{p,q} s {n}/{p}$”, Ann. Mat. Pura Appl. (4), 176 (1999), 209–250 | DOI | MR | Zbl

[4] V. K. Nguyen, W. Sickel, On a Problem of Jaak Peetre Concerning Pointwise Multipliers of Besov Spaces, 2017, arXiv: 1703.03246

[5] V. G. Maz'ya, T. O. Shaposhnikova, Theory of Sobolev Multipliers. With Applications to Differential and Integral Operators, Grundlehren Math. Wiss., 337, Springer-Verlag, Berlin, 2009 | MR | Zbl

[6] M. I. Neiman-Zade, A. A. Shkalikov, “Operatory Shredingera s singulyarnymi potentsialami iz prostranstv multiplikatorov”, Matem. zametki, 66:5 (1999), 723–733 | DOI | MR | Zbl

[7] D.-G. Bak, A. A. Shkalikov, “Multiplikatory v dualnykh sobolevskikh prostranstvakh i operatory Shredingera s potentsialami-raspredeleniyami”, Matem. zametki, 71:5 (2002), 643–651 | DOI | MR | Zbl

[8] M. I. Neiman-Zade, A. A. Shkalikov, “Strongly elliptic operators with singular coefficients”, Russ. J. Math. Phys., 13:1 (2006), 70–78 | DOI | MR | Zbl

[9] A. A. Belyaev, “Characterization of spaces of multipliers for Bessel potential spaces”, Math. Notes, 96:5-6 (2014), 634–646 | DOI | MR | Zbl

[10] Kh. Tribel, Teoriya interpolyatsii, funktsionalnye prostranstva, differentsialnye operatory, Mir, M., 1980 | MR | Zbl

[11] J. Marschall, “Pseudo-differential operators with coefficients in Sobolev spaces”, Trans. Amer. Math. Soc., 307 (1988), 335–361 | MR | Zbl

[12] H. Triebel, Theory of Function Spaces. II, Monogr. Math., 84, Birkhäuser Verlag, Basel, 1992 | MR | Zbl

[13] G. G. Khardi, Dzh. E. Littlvud, G. Polia, Neravenstva, IL, M., 1948 | MR | Zbl

[14] I. Berg, I. Lefstrem, Interpolyatsionnye prostranstva. Vvedenie, Mir, M., 1980 | MR | Zbl

[15] T. Runst, W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators and Nonlinear Partial Differential Equations, De Gruyter Ser. Nonlinear Anal. Appl., 3, Walter de Gruyter, Berlin, 1996 | MR | Zbl