×

Franke-Jawerth embeddings for Besov and Triebel-Lizorkin spaces with variable exponents. (English) Zbl 1407.42013

We denote the Besov and Triebel-Lizorkin spaces by \(B_{p,q}^{s}(\mathbb R^n) \) and \(F_{p,q}^{s}(\mathbb R^n) \), respectively. The Franke-Jawerth embeddings, go back to B. Jawerth [Math. Scand. 40, 94–104 (1977; Zbl 0358.46023)] and J. Franke [Math. Nachr. 125, 29–68 (1986; Zbl 0617.46036)]. Using interpolation techniques and duality, the authors prove that if \(-\infty <s_{1}<s_{0}<\infty ,0<p_{0}<p_{1}\leq \infty \) and \(0<q\leq \infty \) with \[ s_{0}-\frac{n}{p_{0}}=s_{1}-\frac{n}{p_{1}}, \] then
(i)
\[ F_{p_{0},q}^{s_{0}}\left(\mathbb{R}^{n}\right) \hookrightarrow B_{p_{1},p_{0}}^{s_{1}}\left(\mathbb{R}^{n}\right) . \]
(ii)
If \(p_{1}<\infty ,\) then \[ B_{p_{0},p_{1}}^{s_{0}}\left(\mathbb{R}^{n}\right) \hookrightarrow F_{p_{1},q}^{s_{1}}\left(\mathbb{R}^{n}\right) . \]
In the present paper in Section 2, the authors introduce the necessary notations and definitions which are needed in this paper. In Section 3, they present another proof for Franke embedding in the constant exponent case. With the help of the results in Section 3, the authors state and prove the following Franke-Jawerth embeddings in the scales of Besov \(B_{p\left( .\right) ,q\left( .\right) }^{s\left( .\right) }\left(\mathbb{R}^{n}\right) \) and Triebel-Lizorkin spaces \(F_{p\left( .\right) ,q\left( .\right) }^{s\left( .\right) }\left(\mathbb{R}^{n}\right) \) with variable exponents in Section 4: \[ F_{p_{0}\left( .\right) ,q\left( .\right) }^{s_{0}\left( .\right) }\left(\mathbb{R}^{n}\right) \hookrightarrow B_{p_{1}\left( .\right) ,p_{0}\left( .\right) }^{s_{1}\left( .\right) }\left(\mathbb{R}^{n}\right) \text{ } \] and \[ B_{p_{0}\left( .\right) ,p_{1}\left( .\right) }^{s_{0}\left( .\right) }\left(\mathbb{R}^{n}\right) \hookrightarrow F_{p_{1}\left( .\right) ,q\left( .\right) }^{s_{1}\left( .\right) }\left(\mathbb{R}^{n}\right) , \] respectively, if \(\inf_{x\in\mathbb{R}^{n}}\left( s_{0}\left( x\right) -s_{1}\left( x\right) \right) >0\) and \[ s_{0}\left( x\right) -\frac{n}{p_{0}\left( x\right) }=s_{1}\left( x\right) - \frac{n}{p_{1}\left( x\right) },x\in\mathbb{R}^{n}. \] In Section 5, the authors transfer their results to 2-mikrolocal function spaces with variable exponents. Finally in the Last Section they pose some open problems.

MSC:

42B35 Function spaces arising in harmonic analysis
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] Almeida, A., and A. Caetano: Atomic and molecular decompositions in variable exponent 2-microlocal spaces and applications. - J. Funct. Anal. 270:5, 2016, 1888-1921. · Zbl 1353.46024
[2] Almeida, A., and A. Caetano: On 2-microlocal spaces with all exponents variable. - Nonlinear Anal. 135, 2016, 97-119. · Zbl 1355.46034
[3] Almeida, A., and P. Hästö: Besov spaces with variable smoothness and integrability. - J. Funct. Anal. 258:5, 2010, 1628-1655. · Zbl 1194.46045
[4] Beauzamy, B.: Espaces de Sobolev et de Besov d’ordre variable définis sur Lp. - C. R. Acad. Sci. Paris (Ser. A) 274, 1972, 1935-1938. · Zbl 0238.46034
[5] Bennett, C., and R. Sharpley: Interpolation of operators. - Academic Press, San Diego, 1988. · Zbl 0647.46057
[6] Besov, O. V.: Equivalent normings of spaces of functions of variable smoothness. - Proc. Steklov Inst. Math. 243:4, 2003, 80-88. · Zbl 1084.46023
[7] Bony, J. M.: Second microlocalization and propagation of singularities for semilinear hyperbolic equations. - In: Hyperbolic equations and related topics (Katata/Kyoto, 1984), Academic Press, Boston, MA, 1986, 11-49. · Zbl 0669.35073
[8] Diening, L.: Maximal function on generalized Lebesgue spaces Lp(·). - Math. Inequal. Appl. 7:2, 2004, 245-253. · Zbl 1071.42014
[9] Diening, L., P. Harjulehto, P. Hästö, Y. Mizuta, and T. Shimomura: Maximal functions in variable exponent spaces: limiting cases of the exponent. - Ann. Acad. Sci. Fenn. Math. 34:2, 2009, 503-522. · Zbl 1180.42010
[10] Diening, L., P. Harjulehto, P. Hästö, and M. Růžička: Lebesgue and Sobolev spaces with variable exponents. - Springer-Verlag, 2011. · Zbl 1222.46002
[11] Diening, L., P. Hästö, and S. Roudenko: Function spaces of variable smoothness and integrability. - J. Funct. Anal. 256:6, 2009, 1731-1768. · Zbl 1179.46028
[12] Drihem, D.: Atomic decomposition of Besov spaces with variable smoothness and integrability. - J. Math. Anal. Appl. 389, 2012, 15-31. · Zbl 1239.46026
[13] Edmunds, D. E., and J. Rákosník: Sobolev embeddings with variable exponent. - Studia Math. 143:3, 2000, 267-293. · Zbl 0974.46040
[14] Franke, J.: On the spaces Fpqsof Triebel-Lizorkin type: pointwise multipliers and spaces on domains. - Math. Nachr. 125, 1986, 29-68. · Zbl 0617.46036
[15] Fu, J., and J. Xu: Characterizations of Morrey type Besov and Triebel-Lizorkin spaces with variable exponents. - J. Math. Anal. Appl. 381:1, 2011, 280-298. · Zbl 1221.46032
[16] Giga, Y., and T. Miyakawa: Navier-Stokes flow in R3with measures as initial vorticity and Morrey spaces. - Comm. Partial Differential Equations 14:5, 1989, 577-618. · Zbl 0681.35072
[17] Gonçalves, H. F., S. D. Moura, and J. S. Neves: On trace spaces of 2-microlocal spaces. - J. Funct. Anal. 267, 2014, 3444-3468. · Zbl 1311.46032
[18] Gonçalves, H. F., and H. Kempka: Non-smooth atomic decomposition of 2-microlocal spaces and application to pointwise multipliers. - J. Math. Anal. Appl. 434, 2016, 1875-1890. · Zbl 1347.42038
[19] Hansen, M., and J. Vybíral: The Jawerth-Franke embedding of spaces with dominating mixed smoothness. - Georgian Math. J. 16:4, 2009, 667-682. · Zbl 1187.42022
[20] Haroske, D. D.: Envelopes and sharp embeddings of function spaces. - Research Notes in Mathematics 437, Chapman & Hall/CRC, Boca Raton, FL, 2007. · Zbl 1111.46002
[21] Haroske, D. D., and L. Skrzypczak: On Sobolev and Franke-Jawerth embeddings of smoothness Morrey spaces. - Rev. Mat. Complut. 27:2, 2014, 541-573. · Zbl 1311.46033
[22] Jaffard, S.: Pointwise smoothness, two-microlocalization and wavelet coefficients. - Publ. Mat. 35:1, 1991, 155-168. · Zbl 0760.42016
[23] Jaffard, S., and Y. Meyer: Wavelet methods for pointwise regularity and local oscillations of functions. - Mem. Amer. Math. Soc. 123, 1996. · Zbl 0873.42019
[24] Jawerth, B.: Some observations on Besov and Triebel-Lizorkin spaces. - Math. Scand. 40:1, 1977, 94-104. · Zbl 0358.46023
[25] Kempka, H.: Generalized 2-microlocal Besov spaces. - PhD thesis, University of Jena, Germany, 2008. · Zbl 1213.42146
[26] Kempka, H.: 2-microlocal Besov and Triebel-Lizorkin spaces of variable integrability. - Rev. Mat. Complut. 22:1, 2009, 227-251. · Zbl 1166.42011
[27] Kempka, H.: Atomic, molecular and wavelet decomposition of 2-microlocal Besov and Triebel- Lizorkin spaces with variable integrability. - Funct. Approx. Comment. Math. 43:2, 2010, 171- 208. · Zbl 1214.46020
[28] Kempka, H., and J. Vybíral: Spaces of variable smoothness and integrability: Characterizations by local means and ball means of differences. - J. Fourier Anal. Appl. 18:4, 2012, 852-891. · Zbl 1270.46028
[29] Kempka, H., and J. Vybíral: A note on the spaces of variable integrability and summability of Almeida and Hästö. - Proc. Amer. Math. Soc. 141:9, 2013, 3207-3212. · Zbl 1283.46022
[30] Kováčik, O., and J. Rákosník: On spaces Lp(x)and Wk,p(x). - Czechoslovak Math. J. 41:4, 1991, 592-618. · Zbl 0784.46029
[31] Leopold, H. G.: On function spaces of variable order of differentiation. - Forum Math. 3, 1991, 1-21. · Zbl 0737.46020
[32] Lévy Véhel, J., and S. Seuret: A time domain characterization of 2-microlocal spaces. - J. Fourier Anal. Appl. 9:5, 2003, 473-495. · Zbl 1064.42029
[33] Lévy Véhel, J., and S. Seuret: The 2-microlocal formalism, fractal geometry and applications. - In: A Jubilee of Benoit Mandelbrot, Proc. Sympos. Pure Math. 72:2, 2004, 153-215. · Zbl 1093.28006
[34] Meyer, Y.: Wavelets, vibrations and scalings. - CRM Monogr. Ser. 9, Amer. Math. Soc., 1998. · Zbl 0893.42015
[35] Noi, T.: Trace and extension operators for Besov spaces and Triebel-Lizorkin spaces with variable exponents. - Rev. Mat. Complut. 29:2, 2016, 341-404. · Zbl 1350.46029
[36] Orlicz, W.: Über konjugierte Exponentenfolgen. - Studia Math. 3, 1931, 200-212. · JFM 57.0251.02
[37] Peetre, J.: On spaces of Triebel-Lizorkin type. - Ark. Math. 13, 1975, 123-130. · Zbl 0302.46021
[38] R ‌užička, M.: Electrorheological fluids: modeling and mathematical theory. - Lecture Notes in Math. 1748, Springer-Verlag, Berlin, 2000. · Zbl 0962.76001
[39] Triebel, H.: Theory of function spaces II. - Monographs in Mathematics 84, Birkhäuser Verlag, Basel, 1992. · Zbl 0763.46025
[40] Triebel, H.: Theory of function spaces III. - Monographs in Mathematics 100, Birkhäuser Verlag, Basel, 2006. · Zbl 1104.46001
[41] Unterberger, A.: Sobolev spaces of variable order and problems of convexity for partial differential operators with constant coefficients. - Astérisque 2 et 3, 1973, 325-341. · Zbl 0278.46034
[42] Unterberger, A., and J. Bokobza: Les opérateurs pseudodifférentiels d’ordre variable. C. R. Acad. Sci. Paris 261, 1965, 2271-2273. · Zbl 0143.37003
[43] Višik, M. I., and G. I. Eskin: Convolution equations of variable order. - Trudy Moskov Mat. Obsc. 16, 1967, 26-49 (in Russian).
[44] Vybíral, J.: A new proof of the Jawerth-Franke embedding. - Rev. Mat. Complut. 21, 2008, 75-82. · Zbl 1154.46021
[45] Vybíral, J.: Sobolev and Jawerth embeddings for spaces with variable smoothness and integrability. - Ann. Acad. Sci. Fenn. Math. 34:2, 2009, 529-544. · Zbl 1184.46037
[46] Xu, J.: Variable Besov and Triebel-Lizorkin spaces. - Ann. Acad. Sci. Fenn. Math. 33:2, 2008, 511-522. · Zbl 1160.46025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.