Some Hardy-type estimates in realized homogeneous Besov and Triebel–Lizorkin spaces

Moussai, Madani

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 1, pp. 39-55.

Nous monterons que les espaces homogènes réalisés de Besov $\dot{\tilde{B}}{}_{p_0,p}^{s_0}\left(ℝ\right)$ et de Triebel–Lizorkin $\dot{\tilde{F}}{}_{p_0,p}^{s_0}\left(ℝ\right)$ s’injectent continûment dans les espaces quasi-Banach de Lebesgue à poids $L_p{(ℝ;|x|}^{-p(s_0-n/p_0+n/p)}\mathrm{d}x)$ pour $p_0<p$ et ${(n/p_0-n)}_{+}<s_0<n/p_0$.

We prove that the realized homogeneous Besov $\dot{\tilde{B}}{}_{p_0,p}^{s_0}\left(ℝ\right)$ and Triebel–Lizorkin $\dot{\tilde{F}}{}_{p_0,p}^{s_0}\left(ℝ\right)$ spaces are continuously embedded in the quasi-Banach weighted Lebesgue spaces $L_p{(ℝ;|x|}^{-p(s_0-n/p_0+n/p)}\mathrm{d}x)$ for $p_0<p$ and ${(n/p_0-n)}_{+}<s_0<n/p_0$.

Classification : 46E35
Mots clés : Homogeneous Besov spaces, homogeneous Triebel–Lizorkin spaces, realizations

[1] Bourdaud, Gérard Localisation et multiplicateurs des espaces de Sobolev homogènes, Manuscr. Math., Volume 60 (1988) no. 1, pp. 93-130 | DOI | Zbl

[2] Bourdaud, Gérard Réalisations des espaces de Besov homogènes, Ark. Mat., Volume 26 (1988) no. 1, pp. 41-54 | DOI

[3] Bourdaud, Gérard La propriété de Fatou dans les espaces de Besov homogènes, C. R. Math. Acad. Sci. Paris, Volume 349 (2011) no. 15-16, pp. 837-840 | DOI | Zbl

[4] Bourdaud, Gérard Realizations of homogeneous Besov and Lizorkin-Triebel spaces, Math. Nachr., Volume 286 (2013) no. 5-6, pp. 476-491 | DOI | MR | Zbl

[5] Bourdaud, Gérard; Lanza de Cristoforis, Massimo; Sickel, Winfried Superposition operators and functions of bounded $p$-variation, Rev. Mat. Iberoam., Volume 22 (2006) no. 2, pp. 455-487 | DOI | MR | Zbl

[6] Bourdaud, Gérard; Moussai, Madani; Sickel, Winfried Composition operators in Lizorkin-Triebel spaces, J. Funct. Anal., Volume 259 (2010) no. 5, pp. 1098-1128 | DOI | Zbl

[7] Bourdaud, Gérard; Moussai, Madani; Sickel, Winfried Composition operators acting on Besov spaces on the real line, Ann. Mat. Pura Appl., Volume 193 (2014) no. 5, pp. 1519-1554 | DOI | MR | Zbl

[8] Franke, Jens On the spaces ${\mathbf{F}}_p{q}^s$ of Triebel-Lizorkin type: pointwise multipliers and spaces on domains, Math. Nachr., Volume 125 (1986), pp. 29-68 | DOI | MR | Zbl

[9] Frazier, Michael; Jawerth, Björn A discrete transform and decompositions of distribution spaces, J. Funct. Anal., Volume 93 (1990) no. 1, pp. 34-170 | DOI | MR | Zbl

[10] Grisvard, Pierre Espaces intermédiaires entre espaces de Sobolev avec poids, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 17 (1963) no. 3, pp. 255-296

[11] Jawerth, Björn Some observations on Besov and Lizorkin-Triebel spaces, Math. Scand., Volume 40 (1977) no. 1, pp. 94-104 | DOI | MR | Zbl

[12] Lions, Jacques-Louis; Magenes, Enrico Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, 17, Dunod, 1968 | Zbl

[13] Moussai, Madani Composition operators on Besov algebras, Rev. Mat. Iberoam., Volume 28 (2012) no. 1, pp. 239-272 | DOI | MR | Zbl

[14] Moussai, Madani Realizations of homogeneous Besov and Triebel-Lizorkin spaces and an application to pointwise multipliers, Anal. Appl. (Singap.), Volume 13 (2015) no. 2, pp. 149-183 | DOI | MR | Zbl

[15] Runst, Thomas; Sickel, Winfried Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, De Gruyter Series in Nonlinear Analysis and Applications, 3, Walter de Gruyter, 1996 | MR | Zbl

[16] Schmeisser, Hans-Jürgen; Triebel, Hans Topics in Fourier analysis and function spaces, John Wiley & Sons, 1987, 300 pages | Zbl

[17] Sickel, Winfried; Skrzypczak, Leszek On the interplay of regularity and decay in case of radial functions II. Homogeneous spaces, J. Fourier Anal. Appl., Volume 18 (2012) no. 3, pp. 548-582 | DOI | MR | Zbl

[18] Stein, Elias M.; Weiss, Guido Fractional integrals on $n$-dimensional Euclidean space, J. Math. Mech., Volume 7 (1958), pp. 503-514 | MR | Zbl

[19] Triebel, Hans Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, 18, North-Holland, 1978, 528 pages | MR | Zbl

[20] Triebel, Hans Theory of function spaces, Monographs in Mathematics, 78, Birkhäuser, 1983, 284 pages | MR | Zbl

[21] Triebel, Hans Theory of function spaces. II, Monographs in Mathematics, 84, Birkhäuser, 1992 | MR | Zbl

[22] Triebel, Hans The structure of functions, Monographs in Mathematics, 97, Birkhäuser, 2001 | MR | Zbl

[23] Triebel, Hans Tempered homogeneous function spaces, EMS Series of Lectures in Mathematics, European Mathematical Society, 2015 | Zbl

[24] Yamazaki, Masao A quasi-homogeneous version of paradifferential operators. I: Boundedness on spaces of Besov type, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 33 (1986), pp. 131-174 | Zbl

[25] Youssfi, Abdellah Localisation des espaces de Besov homogènes, Indiana Univ. Math. J., Volume 37 (1988) no. 3, pp. 565-587 | DOI | MR | Zbl

[26] Youssfi, Abdellah Localisation des espaces de Lizorkin-Triebel homogènes, Math. Nachr., Volume 147 (1990), pp. 107-121 | DOI | MR | Zbl