## Illinois Journal of Mathematics

### A new interpolation approach to spaces of Triebel–Lizorkin type

Peer Christian Kunstmann

#### Abstract

We introduce in this paper new interpolation methods for closed subspaces of Banach function spaces. For $q\in[1,\infty]$, the $l^{q}$-interpolation method allows to interpolate linear operators that have bounded $l^{q}$-valued extensions. For $q=2$ and if the Banach function spaces are $r$-concave for some $r<\infty$, the method coincides with the Rademacher interpolation method that has been used to characterize boundedness of the $H^{\infty}$-functional calculus. As a special case, we obtain Triebel–Lizorkin spaces $F^{2\theta}_{p,q}(\mathbb{R}^{d})$ by $l^{q}$-interpolation between $L^{p}(\mathbb{R}^{d})$ and $W^{2}_{p}(\mathbb{R}^{d})$ where $p\in(1,\infty)$. A similar result holds for the recently introduced generalized Triebel–Lizorkin spaces associated with $R_{q}$-sectorial operators in Banach function spaces. So, roughly speaking, for the scale of Triebel–Lizorkin spaces our method thus plays the role the real interpolation method plays in the theory of Besov spaces.

#### Article information

Source
Illinois J. Math., Volume 59, Number 1 (2015), 1-19.

Dates
Revised: 13 February 2015
First available in Project Euclid: 11 February 2016

https://projecteuclid.org/euclid.ijm/1455203156

Digital Object Identifier
doi:10.1215/ijm/1455203156

Mathematical Reviews number (MathSciNet)
MR3459625

Zentralblatt MATH identifier
1346.46018

#### Citation

Kunstmann, Peer Christian. A new interpolation approach to spaces of Triebel–Lizorkin type. Illinois J. Math. 59 (2015), no. 1, 1--19. doi:10.1215/ijm/1455203156. https://projecteuclid.org/euclid.ijm/1455203156

#### References

• M. G. Cowling, I. Doust, A. McIntosh and A. Yagi, Banach space operators with a bounded $H^\infty$ functional calculus, J. Aust. Math. Soc. A 60 (1996), no. 1, 51–89.
• G. Dore, $H^\infty$-functional calculus in real interpolation spaces, Studia Math. 137 (1999), no. 2, 161–167.
• X. T. Duong and J. Li, Hardy spaces associated with operators satisfying Davies–Gaffney estimates and bounded holomorphic functional calculus, J. Funct. Anal. 264 (2013), no. 6, 1409–1437.
• D. Frey and P. C. Kunstmann, A $T(1)$-theorem for non-integral operators, Math. Ann. 357 (2013), no. 1, 215–278.
• J. Garc\`\ia-Cuerva and J. L. Rubio di Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, Notas de Matemática [Mathematical Notes], vol. 104, North-Holland, Amsterdam, 1985.
• M. Haase, The functional calculus for sectorial operators, Operator Theory: Advances and Applications, vol. 169, Birkhäuser, Basel, 2006.
• S. Hofmann and S. Mayboroda, Hardy and BMO spaces associated with divergence form elliptic operators, Math. Ann. 344 (2009), no. 1, 37–116.
• N. J. Kalton, P. C. Kunstmann and L. Weis, Perturbation and interpolation theorems for the $H^\infty$-calculus with applications to differential operators, Math. Ann. 336 (2006), no. 4, 747–801.
• N. J. Kalton and L. Weis, The $H^\infty$-calculus and sums of closed operators, Math. Ann. 321 (2001), no. 2, 319–345.
• N. J. Kalton and L. Weis, The $H^\infty$-functional calculus and square function estimates, manuscript, 2004.
• N. J. Kalton and L. Weis, Euclidean structures, manuscript, 2004.
• P. C. Kunstmann and A. Ullmann, $R_s$-sectorial operators and generalized Triebel–Lizorkin spaces, J. Fourier Anal. Appl. 20 (2014), no. 1, 135–185.
• P. C. Kunstmann and L. Weis, Maximal $L^p$-regularity for parabolic equations, Fourier multiplier theorems and $H^\infty$-functional calculus, Functional analytic methods for evolution equations, Lecture Notes in Math., vol. 1855, Springer, Berlin, 2004, pp. 65–311.
• F. Lancien and C. Le Merdy, Square functions and $H^\infty$ calculus on subspaces of $L^p$ and on Hardy spaces, Math. Z. 251 (2005), 101–115.
• C. Le Merdy, On square functions associated to sectorial operators, Bull. Soc. Math. France 132 (2004), no. 1, 137–156.
• J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I and II, Springer, Berlin, 1996. Reprint of the 1st edn.
• J. Suárez and L. Weis, Interpolation of Banach spaces by the $\ga$-method, Methods in Banach space theory, London Math. Soc. Lecture Note Ser., vol. 337, Cambridge University Press, Cambridge, 2006, pp. 293–306.
• H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, vol. 18, North-Holland, Amsterdam, 1978.
• H. Triebel, Characterizations of Besov–Hardy–Sobolev spaces via harmonic functions, temperatures, and related means, J. Approx. Theory 35 (1982), no. 3, 275–297.
• H. Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser, Basel, 1983.
• L. Weis, A new approach to maximal $L^p$-regularity, Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), Lecture Notes in Pure and Appl. Math., vol. 215, Dekker, New York, 2001, pp. 195–214.