Some new Besov and Triebel-Lizorkin spaces associated with para-accretive functions on spaces of homogeneous type. (English) Zbl 1093.42015
Summary: Let \((X,\rho,\mu)_{d,\theta}\) be a space of homogeneous type with \(d>0\) and \(\theta\in(0,1]\), \(b\) be a para-accretive function, \(\varepsilon\in(0, \theta]\), \(|s|<\varepsilon\), and \(a_0\in (0,1)\) be some constant depending on \(d\), \(\varepsilon\) and \(s\). The authors introduce the Besov space \(b\dot B^s_{pq}(X)\) with \(a_0 <p\leq\infty\) and \(0<q\leq\infty\), and the Triebel-Lizorkin space \(b \dot F^s_{pq}(X)\) with \(a_0<p<\infty\) and \(a_0<q\leq\infty\) by first establishing a Plancherel-Pólya-type inequality. Moreover, the authors establish the frame and the Littlewood-Paley function characterizations of these spaces. Furthermore, the authors introduce the new Besov space \(b^{-1}\dot B^s_{pq}(X)\) and the Triebel-Lizorkin space \(b^{-1}\dot F^s_{pq}(X)\). The relations among these spaces and the known Hardy-type spaces are presented. As applications, the authors also establish some real interpolation theorems, embedding theorems, \(Tb\) theorems, and the lifting property by introducing some new Riesz operators of these spaces.
MSC:
42B35 | Function spaces arising in harmonic analysis |
46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
42B25 | Maximal functions, Littlewood-Paley theory |
43A85 | Harmonic analysis on homogeneous spaces |
47B06 | Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators |
42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
47A30 | Norms (inequalities, more than one norm, etc.) of linear operators |
47B38 | Linear operators on function spaces (general) |
Keywords:
Plancherel-Pólya inequality; Calderón reproducing formula; interpolation; Riesz potential; lifting property; space of homogeneous type; para-accretive function; Besov space; Triebel-Lizorkin space; embedding theorems; \(Tb\) theoremReferences:
[1] | DOI: 10.2307/2160955 · Zbl 0830.42012 · doi:10.2307/2160955 |
[2] | DOI: 10.2307/2373450 · Zbl 0222.26019 · doi:10.2307/2373450 |
[3] | Han, Rev. Mat. Iberoam. 10 pp 51– (1994) · Zbl 0797.42009 · doi:10.4171/RMI/145 |
[4] | DOI: 10.1023/A:1023217617339 · Zbl 1016.43008 · doi:10.1023/A:1023217617339 |
[5] | Hajlasz, Mem. Amer. Math. Soc. 145 pp 1– (2000) |
[6] | DOI: 10.1360/03ys9020 · Zbl 1217.46021 · doi:10.1360/03ys9020 |
[7] | DOI: 10.1090/S0002-9947-03-03211-2 · Zbl 1018.60075 · doi:10.1090/S0002-9947-03-03211-2 |
[8] | Yang, Georgian Math. J. 9 pp 567– (2002) |
[9] | Gatto, Studia Math. 133 pp 19– (1999) |
[10] | DOI: 10.1007/s002080100301 · Zbl 1007.46034 · doi:10.1007/s002080100301 |
[11] | Kigami, Analysis on fractals (2001) · Zbl 0998.28004 · doi:10.1017/CBO9780511470943 |
[12] | Heinonen, Lectures on analysis on metric spaces (2001) · Zbl 0985.46008 · doi:10.1007/978-1-4613-0131-8 |
[13] | DOI: 10.4064/sm156-1-5 · Zbl 1032.42025 · doi:10.4064/sm156-1-5 |
[14] | DOI: 10.4064/dm403-0-1 · Zbl 1019.43006 · doi:10.4064/dm403-0-1 |
[15] | Han, Mem. Amer. Math. Soc. 110 pp 1– (1994) |
[16] | DOI: 10.1007/BF02513077 · Zbl 0991.43004 · doi:10.1007/BF02513077 |
[17] | Han, Math. Sci. Res. Hot-Line 3 pp 1– (1999) |
[18] | Han, Approx. Theory Appl. 15 pp 37– (1999) |
[19] | DOI: 10.1007/s00041-002-0014-5 · Zbl 1032.42024 · doi:10.1007/s00041-002-0014-5 |
[20] | Han, J. Geom. Anal. 14 pp 291– (2004) · Zbl 1059.42015 · doi:10.1007/BF02922074 |
[21] | DOI: 10.1007/BF02942219 · doi:10.1007/BF02942219 |
[22] | DOI: 10.1090/S0002-9939-98-04445-1 · Zbl 0920.42011 · doi:10.1090/S0002-9939-98-04445-1 |
[23] | DOI: 10.1002/mana.200310198 · Zbl 1069.46014 · doi:10.1002/mana.200310198 |
[24] | Triebel, Interpolation theory, function spaces, differential operators (1995) · Zbl 0830.46028 |
[25] | David, Rev. Mat. Iberoam. 1 pp 1– (1985) · Zbl 0604.42014 · doi:10.4171/RMI/17 |
[26] | Triebel, Theory of function spaces (1983) · Zbl 1235.46002 · doi:10.1007/978-3-0346-0416-1 |
[27] | DOI: 10.2307/2006946 · Zbl 0567.47025 · doi:10.2307/2006946 |
[28] | DOI: 10.1016/S0022-1236(02)00035-6 · Zbl 1023.46034 · doi:10.1016/S0022-1236(02)00035-6 |
[29] | DOI: 10.1090/S0002-9904-1977-14325-5 · Zbl 0358.30023 · doi:10.1090/S0002-9904-1977-14325-5 |
[30] | Coifman, Analyse harmonique non-commutative sur certains espaces homogènes (1971) · Zbl 0224.43006 · doi:10.1007/BFb0058946 |
[31] | Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals (1993) · Zbl 0821.42001 |
[32] | Christ, Colloq. Math. pp 601– (1990) |
[33] | Semmes, Notice Amer. Math. Soc. 50 pp 438– (2003) |
[34] | Bergh, Interpolation spaces (1976) · doi:10.1007/978-3-642-66451-9 |
[35] | DOI: 10.1006/jfan.1994.1007 · Zbl 0834.43008 · doi:10.1006/jfan.1994.1007 |
[36] | Nahmod, C. R. Acad. Sci. Paris Sér. l Math. 313 pp 721– (1991) |
[37] | Meyer, Wavelets. Calderón-Zygmund and multilinear operators (1997) |
[38] | McIntosh, C. R. Acad. Sci. Paris Sèr. l Math. 301 pp 395– (1985) |
[39] | DOI: 10.1016/0001-8708(79)90012-4 · Zbl 0431.46018 · doi:10.1016/0001-8708(79)90012-4 |
[40] | DOI: 10.1016/0001-8708(79)90013-6 · Zbl 0431.46019 · doi:10.1016/0001-8708(79)90013-6 |
[41] | DOI: 10.1002/1522-2616(200205)238:13.0.CO;2-5 · doi:10.1002/1522-2616(200205)238:13.0.CO;2-5 |
[42] | Gatto, Rev. Mat. Iberoam. 12 pp 111– (1996) · Zbl 0921.43005 · doi:10.4171/RMI/196 |
[43] | Triebel, Rev. Mat. Complut. 15 pp 1– (2002) · Zbl 1034.46033 · doi:10.5209/rev_REMA.2002.v15.n2.16910 |
[44] | Triebel, The structure of functions (2001) |
[45] | DOI: 10.1006/jfan.2001.3836 · Zbl 1031.43005 · doi:10.1006/jfan.2001.3836 |
[46] | DOI: 10.1016/0022-1236(90)90137-A · Zbl 0716.46031 · doi:10.1016/0022-1236(90)90137-A |
[47] | Triebel, Fractals and spectra (1997) · Zbl 1208.46036 · doi:10.1007/978-3-0348-0034-1 |
[48] | Yang, Z. Anal. Anwendungen 22 pp 53– (2003) · Zbl 1041.43006 · doi:10.4171/ZAA/1132 |
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