Generalized Besov-Morrey spaces and generalized Triebel-Lizorkin-Morrey spaces on domains. (English) Zbl 1428.42040
Summary: In this paper, we consider a non-smooth atomic decomposition by using a smooth atomic decomposition. Applying the non-smooth atomic decomposition, a local means characterization and a quarkonical decomposition, we obtain a pointwise multiplier and a trace operator for generalized Besov-Morrey spaces and generalized Triebel-Lizorkin-Morrey spaces on the whole space. We also develop the theory of those spaces on domains. We consider an extension operator and a trace operator on the upper half space and on compact oriented Riemannian manifolds.
MSC:
| 42B35 | Function spaces arising in harmonic analysis |
| 41A17 | Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) |
| 58D17 | Manifolds of metrics (especially Riemannian) |
Keywords:
Besov space; Morrey space; non-smooth atomic decomposition; trace operator; Triebel-Lizorkin spaceReferences:
| [1] | J.Franke and T.Runst, Regular elliptic boundary value problems in Besov-Triebel-Lizorkin spaces, Math. Nachr.174 (1995), 113-149. · Zbl 0843.35026 |
| [2] | M.Frazier and B.Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal.93 (1990), 34-170. · Zbl 0716.46031 |
| [3] | J.Fu and J.Xu, Characterizations of Morrey type Besov and Triebel-Lizorkin spaces with variable exponents, J. Math. Anal. Appl.381 (2011), 280-298. · Zbl 1221.46032 |
| [4] | H. F.Gonçalves and H.Kempka, Non‐smooth atomic decomposition of 2‐microlocal spaces and application to pointwise multiplies, J. Math. Anal. Appl.434 (2016), 1875-1890. · Zbl 1347.42038 |
| [5] | D. D.Haroske and L.Skrzypczak, Embeddings of Besov-Morrey spaces on bounded domains, Studia Math.218 (2013), no. 2, 119-144. · Zbl 1293.46021 |
| [6] | M.Izuki, Y.Sawano, and H.Tanaka, Weighted Besov-Morrey spaces and Triebel-Lizorkin spaces, RIMS Kôkyûroku Bessatsu Ser.B22 (2010), 21-60. · Zbl 1223.42015 |
| [7] | H.Kempka and J.Vybíral, Spaces of variable smoothness and integrability: characterizations by local means and ball means of differences, J. Fourier Anal. Appl.18 (2012), no. 4, 852-891. · Zbl 1270.46028 |
| [8] | H.Kozono and M.Yamazaki, Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differential Equations19 (1994), 959-1014. · Zbl 0803.35068 |
| [9] | Y.Liang, D.Yang, W.Yuan, Y.Sawano, and T.Ullrich, A new framework for generalized Besov‐type and Triebel-Lizorkin‐type spaces, Dissertationes Math.489 (2013), 1-114. · Zbl 1283.46027 |
| [10] | A. M.Najafov, Some properties of functions from the intersection of Besov-Morrey type spaces with dominant mixed derivatives, Proc. A. Razmadze Math. Inst.139 (2005), 71-82. · Zbl 1107.46028 |
| [11] | Y. V.Netrusov, Some imbedding theorems for spaces of Besov-Morrey type, Numerical Methods and Questions in the Organization of Calculations, vol. 7, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)139 (1984), 139-147 (Russian). · Zbl 0575.46033 |
| [12] | C. B.Morrey, On the solutions of quasi‐linear elliptic partial differential equations, Trans. Amer. Math. Soc.43 (1938), 126-166. · JFM 64.0460.02 |
| [13] | E.Nakai, Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces, Math. Nachr.166 (1994), 95-103. · Zbl 0837.42008 |
| [14] | E.Nakai, A characterization of pointwise multipliers on the Morrey spaces, Sci. Math Jpn.3 (2000), 445-454. · Zbl 0980.42005 |
| [15] | S.Nakamura, T.Noi, and Y.Sawano, Gereralized Morrey spaces and trace operator, Sci. China Math.59 (2016), 281-336. · Zbl 1344.42020 |
| [16] | T.Noi, Trace and extension operators for Besov spaces and Triebel-Lizorkin spaces with variable exponents, Rev. Mat. Complut.29 (2016), 341-404. · Zbl 1350.46029 |
| [17] | Y.Sawano, Wavelet characterization of Besov-Morrey and Triebel-Lizorkin-Morey spaces, Funct. Approx. Comment. Math.38 (2008), 93-107. · Zbl 1176.42020 |
| [18] | Y.Sawano, Generalized Morrey spaces for non‐doubling measures, NoDEA Nonlinear Differential Equations Appl.15 (2008), no. 4-5, 413-425. · Zbl 1173.42317 |
| [19] | Y.Sawano, Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces on domains, Math. Nachr.283 (2010), no. 10, 1456-1487. · Zbl 1211.46031 |
| [20] | Y.Sawano and H.Tanaka, Decompositions of Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces, Math. Z.257 (2007), no. 4, 871-905. · Zbl 1133.42041 |
| [21] | B.Scharf, Atomic representations in function spaces and applications to pointwise multipliers and diffemorphisms, a new approach, Math. Nachr.286 (2013), no. 2-3, 283-305. · Zbl 1267.46052 |
| [22] | L.Tang and J.Xu, Some properties of Morrey type Besov-Triebel spaces, Math. Nachr.278 (2005), no. 7-8, 904-917. · Zbl 1074.42011 |
| [23] | H.Triebel, Theory of function spaces, Birkhäuser, Basel, Boston, 1983. · Zbl 0546.46028 |
| [24] | H.Triebel, Theory of function spaces II, Birkhäuser, Basel, Boston, 1992. · Zbl 0763.46025 |
| [25] | H.Triebel, Fractals and spectra, Birkhäuser, Basel, Boston, 1997. · Zbl 0898.46030 |
| [26] | H.Triebel, The structure of functions, Birkhäuser, Basel, Boston, 2001. · Zbl 0984.46021 |
| [27] | T.Ullrich, Continuous characterizations of Besov-Lizorkin-Triebel spaces and new interpretations as coorbits, J. Funct. Spaces Appl. (2012), Article ID 163213, 47 pp. · Zbl 1246.46036 |
| [28] | H.Wang, Decomposition for Morrey type Besov-Triebel spaces, Math. Nachr.282, no. 5, 774-787. · Zbl 1196.42020 |
| [29] | W.Yuan, W.Sickel, and D.Yang, Morrey and Campanato meet Besov, Lizorkin and Triebel, Lecture Notes in Math., vol. 2005, Springer‐Verlag, Berlin, 2010. · Zbl 1207.46002 |
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