# American Institute of Mathematical Sciences

November  2012, 11(6): 2213-2219. doi: 10.3934/cpaa.2012.11.2213

## The maximal regularity operator on tent spaces

 1 Univ. Paris-Sud, laboratoire de Mathématiques, UMR 8628, F-91405, Orsay; CNRS, F-91405, Orsay, France 2 LATP-UMR 6632, FST Saint-Jérôme - Case Cour A, Univ. Paul Cézanne, F-13397 Marseille Cédex 20, France 3 Université Lille 1, Laboratoire Paul Painlevé, F-59655, Villeneuve d'Ascq, France

Received  November 2010 Revised  December 2010 Published  April 2012

Recently, Auscher and Axelsson gave a new approach to non-smooth boundary value problems with $L^2$ data, that relies on some appropriate weighted maximal regularity estimates. As part of the development of the corresponding $L^p$ theory, we prove here the relevant weighted maximal estimates in tent spaces $T^{p, 2}$ for $p$ in a certain open range. We also study the case $p=\infty$.
Citation: Pascal Auscher, Sylvie Monniaux, Pierre Portal. The maximal regularity operator on tent spaces. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2213-2219. doi: 10.3934/cpaa.2012.11.2213
##### References:
 [1] P. Auscher, On necessary and sufficient conditions for $L^p$ estimates of Riesz transforms associated to elliptic operators on $R^n$ and related estimates, Mem. Amer. Math. Soc., 871 (2007).  Google Scholar [2] P. Auscher and A. Axelsson, Weighted maximal regularity estimates and solvability of elliptic systems I, Inventiones Math., 184 (2011), 47-115. doi: 10.1007/s00222-010-0285-4.  Google Scholar [3] P. Auscher and A. Axelsson, Remarks on maximal regularity estimates,, Parabolic Problems: Herbert Amann Festschrift, ().   Google Scholar [4] P. Auscher, S. Hofmann, M. Lacey, A. McIntosh and P. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on $\mathbbR^n$, Ann. of Math., 156 (2002), 633-654.  Google Scholar [5] P. Auscher, A. McIntosh and E. Russ, Hardy spaces of differential forms and Riesz transforms on Riemannian manifolds, J. Geom. Anal., 18 (2008), 192-248. doi: 10.1007/s12220-007-9003-x.  Google Scholar [6] R. Coifman, Y. Meyer and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal., 62 (1985), 304-335. doi: 10.1016/0022-1236(85)90007-2.  Google Scholar [7] C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-193. doi: 10.1007/BF02392215.  Google Scholar [8] T. Hytönen, A. McIntosh and P. Portal, Kato's square root problem in Banach spaces, J. Funct. Anal., 254 (2008), 675-726. doi: 10.1016/j.jfa.2007.10.006.  Google Scholar [9] T. Hytönen, J. van Neerven and P. Portal, Conical square function estimates in UMD Banach spaces and applications to $H^{\infty}$-functional calculi, J. Analyse Math., 106 (2008), 317-351. doi: 10.1007/s11854-008-0051-3.  Google Scholar [10] N. Kalton and G. Lancien, A solution to the problem of $L_p$ maximal-regularity, Math. Z., 235 (2000), 559-568. doi: 10.1007/PL00004816.  Google Scholar [11] H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937.  Google Scholar [12] H. Koch and T. Lamm, Geometric flows with rough initial data,, preprint, ().   Google Scholar [13] N. V. Krylov, A parabolic Littlewood-Paley inequality with applications to parabolic equations, Topol. Methods Nonlinear Anal., 4 (1994), 355-364.  Google Scholar [14] P. C. Kunstmann and L. Weis, Maximal $L^p$ regularity for parabolic problems, Fourier multiplier theorems and $H^{\infty}$-functional calculus, in "Functional Analytic Methods for Evolution Equations" (M. Iannelli, R. Nagel and S.Piazzera eds.), Lect. Notes in Math., 1855, Springer-Verlag (2004).  Google Scholar [15] J. van Neerven, M. Veraar and L. Weis, Stochastic maximal $L^p$ regularity,, submitted, ().   Google Scholar [16] L. de Simon, Un'applicazione della theoria degli integrali singolari allo studio delle equazioni differenziali lineare astratte del primo ordine, Rend. Sem. Mat., Univ. Padova, (1964), 205-223.  Google Scholar [17] L.Weis, Operator-valued Fourier multiplier theorems and maximal $L_p$-regularity, Math.Ann., 319 (2001), 735-758. doi: 10.1007/PL00004457.  Google Scholar

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##### References:
 [1] P. Auscher, On necessary and sufficient conditions for $L^p$ estimates of Riesz transforms associated to elliptic operators on $R^n$ and related estimates, Mem. Amer. Math. Soc., 871 (2007).  Google Scholar [2] P. Auscher and A. Axelsson, Weighted maximal regularity estimates and solvability of elliptic systems I, Inventiones Math., 184 (2011), 47-115. doi: 10.1007/s00222-010-0285-4.  Google Scholar [3] P. Auscher and A. Axelsson, Remarks on maximal regularity estimates,, Parabolic Problems: Herbert Amann Festschrift, ().   Google Scholar [4] P. Auscher, S. Hofmann, M. Lacey, A. McIntosh and P. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on $\mathbbR^n$, Ann. of Math., 156 (2002), 633-654.  Google Scholar [5] P. Auscher, A. McIntosh and E. Russ, Hardy spaces of differential forms and Riesz transforms on Riemannian manifolds, J. Geom. Anal., 18 (2008), 192-248. doi: 10.1007/s12220-007-9003-x.  Google Scholar [6] R. Coifman, Y. Meyer and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal., 62 (1985), 304-335. doi: 10.1016/0022-1236(85)90007-2.  Google Scholar [7] C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-193. doi: 10.1007/BF02392215.  Google Scholar [8] T. Hytönen, A. McIntosh and P. Portal, Kato's square root problem in Banach spaces, J. Funct. Anal., 254 (2008), 675-726. doi: 10.1016/j.jfa.2007.10.006.  Google Scholar [9] T. Hytönen, J. van Neerven and P. Portal, Conical square function estimates in UMD Banach spaces and applications to $H^{\infty}$-functional calculi, J. Analyse Math., 106 (2008), 317-351. doi: 10.1007/s11854-008-0051-3.  Google Scholar [10] N. Kalton and G. Lancien, A solution to the problem of $L_p$ maximal-regularity, Math. Z., 235 (2000), 559-568. doi: 10.1007/PL00004816.  Google Scholar [11] H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937.  Google Scholar [12] H. Koch and T. Lamm, Geometric flows with rough initial data,, preprint, ().   Google Scholar [13] N. V. Krylov, A parabolic Littlewood-Paley inequality with applications to parabolic equations, Topol. Methods Nonlinear Anal., 4 (1994), 355-364.  Google Scholar [14] P. C. Kunstmann and L. Weis, Maximal $L^p$ regularity for parabolic problems, Fourier multiplier theorems and $H^{\infty}$-functional calculus, in "Functional Analytic Methods for Evolution Equations" (M. Iannelli, R. Nagel and S.Piazzera eds.), Lect. Notes in Math., 1855, Springer-Verlag (2004).  Google Scholar [15] J. van Neerven, M. Veraar and L. Weis, Stochastic maximal $L^p$ regularity,, submitted, ().   Google Scholar [16] L. de Simon, Un'applicazione della theoria degli integrali singolari allo studio delle equazioni differenziali lineare astratte del primo ordine, Rend. Sem. Mat., Univ. Padova, (1964), 205-223.  Google Scholar [17] L.Weis, Operator-valued Fourier multiplier theorems and maximal $L_p$-regularity, Math.Ann., 319 (2001), 735-758. doi: 10.1007/PL00004457.  Google Scholar
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