Skip to main content
Log in

Bilinear embedding for divergence-form operators with complex coefficients on irregular domains

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

Let \(\Omega \subseteq {\mathbb {R}}^{d}\) be open and A a complex uniformly strictly accretive \(d\times d\) matrix-valued function on \(\Omega \) with \(L^{\infty }\) coefficients. Consider the divergence-form operator \({{\mathscr {L}}}^{A}=-\mathrm{div}\,(A\nabla )\) with mixed boundary conditions on \(\Omega \). We extend the bilinear inequality that we proved in Carbonaro and Dragičević (J Eur Math Soc, to appear) in the special case when \(\Omega ={\mathbb {R}}^{d}\). As a consequence, we obtain that the solution to the parabolic problem \(u^{\prime }(t)+{{\mathscr {L}}}^{A}u(t)=f(t)\), \(u(0)=0\), has maximal regularity in \(L^{p}(\Omega )\), for all \(p>1\) such that A satisfies the p-ellipticity condition that we introduced in Carbonaro and Dragičević (to appear). This range of exponents is optimal for the class of operators we consider. We do not impose any conditions on \(\Omega \), in particular, we do not assume any regularity of \(\partial \Omega \), nor the existence of a Sobolev embedding. The methods of Carbonaro and Dragičević (to appear) do not apply directly to the present case and a new argument is needed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In the definition of \({{\mathcal {E}}}\) we implicitly identify \(T^{A,{{\mathscr {V}}}}_{t}f\) and \(T^{B,{{\mathscr {V}}}^{\prime }}_{t}g\) with their real counterparts; see Sect. 2.4 for a more accurate statement.

  2. In [38] the author also considered systems. The results are stated under geometric assumptions on \(\Omega \) which are stronger than (\(\mathrm{SE}_{q}\)) with \(q=2^{*}\) . These stronger assumptions are used, for example, for proving results on Riesz transforms. However, for the specific result stated here (\(\mathrm{SE}_{q}\)) with \(q=2^{*}\) suffices; see [39].

References

  1. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, Second, Pure and Applied Mathematics (Amsterdam), p. 140. Elsevier, Amsterdam (2003)

    Google Scholar 

  2. Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Commun. Pure Appl. Math. 12, 623–727 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ambrosio, L., Dal Maso, G.: A general chain rule for distributional derivatives. Proc. Am. Math. Soc. 108(3), 691–702 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. Auscher, P.: Regularity theorems and heat kernel for elliptic operators. J. Lond. Math. Soc. (2) 54(2), 284–296 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Auscher, P.: On necessary and sufficient conditions for \(L^p\)-estimates of Riesz transforms associated to elliptic operators on \({\mathbb{R}}^n\) and related estimates. Mem. Am. Math. Soc. 186, 871 (2007)

    MathSciNet  MATH  Google Scholar 

  6. Auscher, P., McIntosh, A., Tchamitchian, P.: Heat kernels of second order complex elliptic operators and applications. J. Funct. Anal. 152(1), 22–73 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Auscher, P., Tchamitchian, P.: Square root problem for divergence operators and related topics. Astérisque 249 (1998)

  8. Bakry, D.: Sur l’interpolation complexe des semigroupes de diffusion. In: Séminaire de Probabilités, XXIII, pp. 1–20 (1989)

  9. Betancor, J.J., Dalmasso, E., Fariña, J.C., Scotto, R.: Bellman functions and dimension free \(L^p\)-estimates for the Riesz transforms in Bessel settings (2018). Nonlinear Anal. 197 (2020) (In progress, August 2020)

  10. Blunck, S., Kunstmann, P.C.: Calderón–Zygmund theory for non-integral operators and the \({H}^{\infty }\) functional calculus. Rev. Mat. Iberoam. 19, 919–942 (2003)

    Article  MATH  Google Scholar 

  11. Burenkov, V.I., Davies, E.B.: Spectral stability of the Neumann Laplacian. J. Differ. Equ. 186(2), 485–508 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Burkholder, D .L.: Boundary value problems and sharp inequalities for martingale transforms. Ann. Prob. 12(3), 647–702 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  13. Burkholder, D.L.: A proof of Pełczyński’s conjecture for the Haar system. Studia Math. 1, 79–83 (1988)

    Article  MATH  Google Scholar 

  14. Burkholder, D.L.: Explorations in martingale theory and its applications. In: École d’Été de Probabilités de Saint-Flour XIX—1989, pp. 1–66 (1991)

  15. Carbonaro, A., Dragičević, O.: Bellman function and dimension-free estimates in a theorem of Bakry. J. Funct. Anal. 265, 1085–1104 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Carbonaro, A., Dragičević, O.: Bounded holomorphic functional calculus for nonsymmetric Ornstein–Uhlenbeck operators. Ann. Sc. Norm. Super. Pisa Cl Sci. 4, 1497–1533 (2019)

  17. Carbonaro, A., Dragičević, O.: Convexity of power functions and bilinear embedding for divergence-form operators with complex coefficients. J. Eur. Math. Soc. (to appear)

  18. Carbonaro, A., Dragičević, O.: Functional calculus for generators of symmetric contraction semigroups. Duke Math. J. 166(5), 937–974 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cialdea, A., Maz’ya, V.: Criterion for the \(L^p\)-dissipativity of second order differential operators with complex coefficients. J. Math. Pures Appl. (9) 84(8), 1067–1100 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Costabel, M.: On the limit Sobolev regularity for Dirichlet and Neumann problems on Lipschitz domains. Math. Nachr. 10, 2165–2173 (2019)

  21. Courant, R., Hilbert, D.: Methoden der Mathematischen Physik, vol. 2. Springer, Berlin (1937)

    Book  MATH  Google Scholar 

  22. Cowling, M.: Harmonic analysis on semigroups. Ann. Math. 2(117), 267–283 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  23. Cowling, M., Doust, I., McIntosh, A., Yagi, A.: Banach space operators with a bounded \({H}^\infty \) functional calculus. Aust. Math. Soc. 60, 51–89 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. Crouzeix, M., Delyon, B.: Some estimates for analytic functions of strip or sectorial operators. Arch. Math. (Basel) 81(5), 559–566 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Dahmani, K.: Sharp dimension free bound for the Bakry–Riesz vector (2016). arXiv e-prints arXiv:1611.07696

  26. Davies, E.B., Simon, B.: Spectral properties of Neumann Laplacian of horns. Geom. Funct. Anal. 2(1), 105–117 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  27. Denk, R., Hieber, M., Prüss, J.: \({\mathbb{R}}\)-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 166(788), viii+114 (2003)

  28. Dindoš, M., Pipher, J.: Boundary value problems for second order elliptic operators with complex coefficients (2018). arXiv e-prints arXiv:1810.10366

  29. Dindoš, M., Pipher, J.: Perturbation theory for solutions to second order elliptic operators with complex coefficients and the \(L^p\) Dirichlet problem (2018). Acta Math. Sin.(Engl. Ser) 6, 749–770 (2019)

  30. Dindoš, M., Pipher, J.: Regularity theory for solutions to second order elliptic operators with complex coefficients and the \(L^p\) Dirichlet problem. Adv. Math. 341, 255–298 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  31. Domelevo, K., Petermichl, S.: Sharp \(L^p\) estimates for discrete second order Riesz transforms. Adv. Math. 262, 932–952 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. Dore, G.: \(L^p\) regularity for abstract differential equations. In: Functional Analysis and Related Topics, 1991 (Kyoto), pp. 25–38 (1993)

  33. Dore, G., Venni, A.: On the closedness of the sum of two closed operators. Math. Z. 196(2), 189–201 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  34. Dragičević, O., Volberg, A.: Bellman function, Littlewood–Paley estimates and asymptotics for the Ahlfors–Beurling operator in \(L^p({\mathbb{C}})\). Indiana Univ. Math. J. 54(4), 971–995 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  35. Dragičević, O., Volberg, A.: Bellman functions and dimensionless estimates of Littlewood–Paley type. J. Oper. Theory 56(1), 167–198 (2006)

    MathSciNet  MATH  Google Scholar 

  36. Dragičević, O., Volberg, A.: Bilinear embedding for real elliptic differential operators in divergence form with potentials. J. Funct. Anal. 261(10), 2816–2828 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Dragičević, O., Volberg, A.: Linear dimension-free estimates in the embedding theorem for Schrödinger operators. J. Lond. Math. Soc. (2) 85(1), 191–222 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. Egert, M.: \(L^p\)-estimates for the square root of elliptic systems with mixed boundary conditions. J. Differ. Equ. 265(4), 1279–1323 (2018)

    Article  MathSciNet  Google Scholar 

  39. Egert, M.: On \(p\)-elliptic divergence form operators and holomorphic semigroups (2018). arXiv e-prints arXiv:1812.09154

  40. Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, vol. 194. Springer, New York (2000)

    MATH  Google Scholar 

  41. Evans, W.D., Harris, D.J.: On the approximation numbers of Sobolev embeddings for irregular domains. Q. J. Math. Oxf. Ser. (2) 40(157), 13–42 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  42. Giga, Y., Sohr, H.: Abstract \(L^p\) estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains. J. Funct. Anal. 102(1), 72–94 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  43. Grisvard, P.: Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, vol. 24. Pitman (Advanced Publishing Program), Boston (1985)

    MATH  Google Scholar 

  44. Haase, Markus: The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, p. 169. Birkhäuser Verlag, Basel (2006)

    Book  Google Scholar 

  45. Hajłasz, P., Koskela, P.: Isoperimetric inequalities and imbedding theorems in irregular domains. J. Lond. Math. Soc. (2) 58(2), 425–450 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  46. Haller-Dintelmann, R., Jonsson, A., Knees, D., Rehberg, J.: Elliptic and parabolic regularity for second-order divergence operators with mixed boundary conditions. Math. Methods Appl. Sci. 39(17), 5007–5026 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  47. Hempel, R., Seco, L.A., Simon, B.: The essential spectrum of Neumann Laplacians on some bounded singular domains. J. Funct. Anal. 102(2), 448–483 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  48. Hofmann, S., Mayboroda, S., McIntosh, A.: Second order elliptic operators with complex bounded measurable coefficients in \(L^p\), Sobolev and Hardy spaces. Ann. Sci. Éc. Norm. Supér. 44(4), 5 (2011)

    MATH  MathSciNet  Google Scholar 

  49. Jerison, D., Kenig, C.E.: The functional calculus for the Laplacian on Lipschitz domains. Journées “équations aux Dérivées Partielles” (Saint Jean de Monts, 1989). École Polytech., Palaiseau, Exp. No. IV, 10 (1989)

  50. Kalton, N.J.: A remark on sectorial operators with an \(H^\infty \)-calculus. In: Trends in Banach Spaces and Operator Theory (Memphis, TN, 2001), pp. 91–99 (2003)

  51. Kalton, N.J., Weis, L.: The \(H^\infty \)-calculus and sums of closed operators. Math. Ann. 321(2), 319–345 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  52. Kato, T.: Perturbation Theory for Linear Operators. Grundlehren der Mathematischen Wissenschaften, vol. 132, 2nd edn. Springer, Berlin (1976)

    Google Scholar 

  53. Kriegler, Ch.: Analyticity angle for non-commutative diffusion semigroups. J. Lond. Math. Soc. (2) 83(1), 168–186 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  54. Kunstmann, P.C., Weis, L.: Maximal \(L_p\)-regularity for parabolic equations, Fourier multiplier theorems and \(H^\infty \)-functional calculus. In: Functional Analytic Methods for Evolution Equations, pp. 65–311 (2004)

  55. Kunstmann, P.C.: Uniformly elliptic operators with maximal \(L_p\)-spectrum in planar domains. Arch. Math. (Basel) 76(5), 377–384 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  56. Kunstmann, P.C.: \(L_p\)-spectral properties of the Neumann Laplacian on horns, comets and stars. Math. Z. 242(1), 183–201 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  57. Kunstmann, P.C.: Štrkalj, Željko, \(H^\infty \)-calculus for submarkovian generators. Proc. Am. Math. Soc. 131(7), 2081–2088 (2003)

    Article  MATH  Google Scholar 

  58. Leoni, G.: A First Course in Sobolev Spaces. Graduate Studies in Mathematics, vol. 181, 2nd edn. American Mathematical Society, Providence (2017)

    Book  MATH  Google Scholar 

  59. Leoni, G., Morini, M.: Necessary and sufficient conditions for the chain rule in \(W^{1,1}_{{\rm loc}}({\mathbb{R}}^N;{\mathbb{R}}^d)\) and \({{\rm BV}}_{{\rm loc}}({\mathbb{R}}^N;{\mathbb{R}}^d)\). J. Eur. Math. Soc. 9(2), 219–252 (2007)

    Article  MathSciNet  Google Scholar 

  60. Liskevich, V.A., Perel’muter, M.A.: Analyticity of sub-Markovian semigroups. Proc. Am. Math. Soc. 123(4), 1097–1104 (1995)

    MathSciNet  Google Scholar 

  61. Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, vol. 16. Birkhäuser Verlag, Basel (1995)

    MATH  Google Scholar 

  62. Mauceri, G., Spinelli, M.: Riesz transforms and spectral multipliers of the Hodge–Laguerre operator. J. Funct. Anal. 269(11), 3402–3457 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  63. McIntosh, A.: Operators which have an \(H_\infty \) functional calculus. In: Miniconference on Operator Theory and Partial Differential Equations (North Ryde, 1986), pp. 210–231 (1986)

  64. McIntosh, A., Yagi, A.: Operators of type \(\omega \) without a bounded \(H_\infty \) functional calculus. In: Miniconference on Operators in Analysis (Sydney, 1989), pp. 159–172 (1990)

  65. Nazarov, F., Treil, S., Volberg, A.: The Bellman functions and two-weight inequalities for Haar multipliers. J. Am. Math. Soc. 12(4), 909–928 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  66. Nazarov, F., Treil, S., Volberg, A., Bellman function in stochastic control and harmonic analysis. In: Systems, Approximation, Singular Integral Operators, and Related Topics (Bordeaux, 2000), pp. 393–423 (2001)

  67. Nazarov, F.L., Treĭl’, S.R.: The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis. Algebra i Analiz 8(5), 32–162 (1996)

    MathSciNet  MATH  Google Scholar 

  68. Nittka, R.: Projections onto convex sets and \(L^p\)-quasi-contractivity of semigroups. Arch. Math. (Basel) 98(4), 341–353 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  69. Osekowski, A.: Sharp martingale and semimartingale inequalities. In: Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series), vol. 72. Birkhäuser, Basel (2012)

  70. Ouhabaz, E.-M.: \(L^\infty \)-contractivity of semigroups generated by sectorial forms. J. Lond. Math. Soc. (2) 46(3), 529–542 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  71. Ouhabaz, E.-M.: Invariance of closed convex sets and domination criteria for semigroups. Potential Anal. 5(6), 611–625 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  72. Ouhabaz, E.M.: Analysis of Heat Equations on Domains, London Mathematical Society Monographs Series, vol. 31. Princeton University Press, Princeton (2005)

    Google Scholar 

  73. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44. Springer, New York (1983)

    MATH  Google Scholar 

  74. Petermichl, S., Slavin, L., Wick, B.D.: New estimates for the Beurling–Ahlfors operator on differential forms. J. Oper. Theory 65(2), 307–324 (2011)

    MathSciNet  MATH  Google Scholar 

  75. Petermichl, S., Volberg, A.: Heating of the Ahlfors–Beurling operator: weakly quasiregular maps on the plane are quasiregular. Duke Math. J. 112(2), 281–305 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  76. Prüss, J., Sohr, H.: On operators with bounded imaginary powers in Banach spaces. Math. Z. 203(3), 429–452 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  77. Schwartz, L.: Théorie des distributions. Publications de l’Institut de Mathématique de l’Université de Strasbourg. No. IX-X, Hermann, Paris (1966)

  78. Stein, E.M.: Topics in harmonic analysis related to the Littlewood–Paley theory. Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton (1970)

  79. ter Elst, A.F.M., Haller-Dintelmann, R., Rehberg, J., Tolksdorf, P.: On the \({L}^{p}\)-theory for second-order elliptic operators in divergence form with complex coefficients (2019). arXiv e-prints arXiv:1903.06692

  80. Tolksdorf, P.: \({\mathbb{R}}\)-sectoriality of higher-order elliptic systems on general bounded domains. J. Evol. Equ. 18(2), 323–349 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  81. Volberg, A., Nazarov, F.: Heat extension of the Beurling operator and estimates for its norm. Algebra i Analiz 15(4), 142–158 (2003)

    MathSciNet  Google Scholar 

  82. Volberg, A.: Bellman approach to some problems in harmonic analysis, Exposé no XIX, Ecole Polytéchnique, vol. 14 (2002)

  83. Weis, L.: Fourier, operator-valued, multiplier theorems and maximal \(L_p\)-regularity. Math. Ann. 319(4), 735–758 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  84. Wittwer, J.: Survey article: a user’s guide to Bellman functions. Rocky Mt. J. Math. 41(3), 631–661 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  85. Wróbel, B.: Dimension-free \(L^p\) estimates for vectors of Riesz transforms associated with orthogonal expansions. Anal. PDE 11(3), 745–773 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  86. Yan, M.: Extension of convex function. J. Convex Anal. 21(4), 965–987 (2014)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first author was partially supported by the “National Group for Mathematical Analysis, Probability and their Applications” (GNAMPA-INdAM). The second author was partially supported by the Ministry of Higher Education, Science and Technology of Slovenia (research program Analysis and Geometry, Contract No. P1-0291) and Slovenian Research Agency (Grant no. J1-1690). The authors would like to thank the referee for his/her valuable comments which improved the presentation of this paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Andrea Carbonaro.

Additional information

Communicated by A. Malchiodi.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A: Comparison with known results

Under stronger assumptions on A and/or \(\Omega \) than those of Theorem 3 it is known that \((T^{A,{{\mathscr {V}}}}_{t})_{t>0}\), where \({{\mathscr {V}}}\) is one of the subspaces of Sect. 1.1, extrapolates to a bounded strongly continuous semigroup on \(L^{p}(\Omega )\) in a range of p’s larger than the range given by p-ellipticity, and the negative generator \({{\mathscr {L}}}^{A}_{p}\) has parabolic maximal regularity in this larger range of exponents.

1.1 A.1 Semigroup extrapolation

Let \({{\mathscr {V}}}\) denote one of the subspaces of Sect. 1.1.

(i) For every \(\Omega \subseteq {\mathbb {R}}^{d}\) and every real-valued\(A\in {{\mathcal {A}}}(\Omega )\) the semigroup \((T^{A,{{\mathscr {V}}}}_{t})_{t>0}\) is sub-Markovian (see [70, 71] and [72, Corollary 4.3 and Corollary 4.10]), so \(\omega _{H^{\infty }}({{\mathscr {L}}}^{A}_{p})<\pi /2\) for all \(p>1\); see [18, 22, 57] for symmetric real-valued A and [51, Corollary 5.2] for nonsymmetric real-valued A. It follows from the Dore-Venni theorem [33, 76] that, in this case, \({{\mathscr {L}}}^{A}_{p}\) has parabolic maximal regularity for all \(p>1\).

(ii) As for complex-valued\(A\in {{\mathcal {A}}}(\Omega )\), define the upper and lower Sobolev exponent by the rule \(2^{*}=2d/(d-2)\) if \(d\geqslant 3\) and \(2^{*}=+\infty \) if \(d=1,2\); \(2_{*}=(2^{*})^{\prime }\). For \(A\in {{\mathcal {A}}}(\Omega )\), \(\delta \geqslant 0\) and \(\vartheta \in [0,\pi /2)\) denote by \(J(A,{{\mathscr {V}}},\delta ,\vartheta )\) the maximal open interval in \((1,+\infty )\) such that \((e^{-\delta z}T^{A,{{\mathscr {V}}}}_{z})_{z\in \mathbf{S}_{\vartheta }}\) is uniformly bounded in \(L^{p}(\Omega )\), for all \(p\in J(A,{{\mathscr {V}}},\delta ,\vartheta )\). Denote by \(\omega ({{\mathscr {L}}}^{A}_{2})\) the sectoriality angle of \({{\mathscr {L}}}^{A}_{2}\).

  1. (a)

    When \(\Omega ={\mathbb {R}}^{d}\), \(d=1,2\) and \(A\in {{\mathcal {A}}}({\mathbb {R}}^{d})\), we have \(J(A,W^{1,2}({\mathbb {R}}^{d}),0,\vartheta )=(1,+\infty )\), for all \(\vartheta >\pi /2-\omega ({{\mathscr {L}}}^{A}_{2})\) [6]. When \(\Omega ={\mathbb {R}}^{d}\), \(d\geqslant 3\) and \(A\in {{\mathcal {A}}}({\mathbb {R}}^{d})\), Auscher proved [5] that there exists \(\varepsilon >0\) depending only on dimension and the ellipticity constants of A such that \((2_{*}-\varepsilon ,2^{*}+\varepsilon )\subset J(A,W^{1,2}({\mathbb {R}}^{d}),0,\vartheta )\), for all \(\vartheta >\pi /2-\omega ({{\mathscr {L}}}^{A}_{2})\).

  2. (b)

    The results in (a) are sharp [48]: for all \(d\geqslant 3\) and all \(p\in [1,\infty ]{\setminus }[2_{*},2^{*}]\) there exists \(A\in {{\mathcal {A}}}({\mathbb {R}}^{d})\) such that \((T^{A}_{t})_{t>0}\) is not bounded on \(L^{p}({\mathbb {R}}^{d})\).

  3. (c)

    In the case when \({{\mathscr {V}}}\) has the embedding property (\(\mathrm{SE}_{q}\)) with \(q=2^{*}\), Egert implicitlyFootnote 2 proved [38, Theorem 1.6] that \((2_{*},2^{*})\subset J(A,{{\mathscr {V}}},\delta ,\vartheta )\), for every \(\delta >0\) and \(\vartheta >\pi /2-\omega ({{\mathscr {L}}}^{A}_{2})\). Also, Egert extended the result in (a) (\(d\geqslant 3\)) by proving that if \(\Omega \) is bounded and connected, the boundary is Lipschitz regular around the Neumann part \(\overline{\partial \Omega {\setminus } D}\) and D satisfies further geometric assumptions (see [38, 46, 79]) one always has that \((2_{*}-\varepsilon ,2^{*}+\varepsilon )\subset J(A,{{\mathscr {V}}},\delta ,\vartheta )\), for all \(\delta >0\) and \(\vartheta >\pi /2-\omega ({{\mathscr {L}}}^{A}_{2})\) and some \(\varepsilon >0\) depending only on dimension, the ellipticity constants of A and the geometry of \(\Omega \). Note that under the above mentioned geometric assumptions there exists a Sobolev extension operator \(E: {{\mathscr {V}}}\rightarrow W^{1,2}({\mathbb {R}}^{d})\) and so (\(\mathrm{SE}_{q}\)) holds true with \(q=2^{*}\). A result similar to (c), but without any further geometric assumption on D, has been previously obtained by Tolksdorf [80], who also proved maximal regularity in the range \((2_{*}-\varepsilon ,2^{*}+\varepsilon )\).

  1. (d)

    Recently, Egert [39] improved the result in (c) by combining our notion of p-ellipticity and its properties with the technology developed in [5, 38].

A similar result has been proved by ter Elst et al. [79, Theorem 3.1] by means of a different technique, but still using p-ellipticity.

For \(p\geqslant 2\) set

$$\begin{aligned} p^{\circ } := \frac{p}{2}2^{*}=\frac{pd}{d-2}. \end{aligned}$$

Proposition A.1

([39, Theorem 1]) Let \(d\geqslant 3\). Let \(\Omega \subset {\mathbb {R}}^{d}\) be open. Let \({{\mathscr {V}}}\) denote one of the subspaces of Sect. 1.1. Assume the Sobolev embedding (\(\mathrm{SE}_{q}\)) with \(q=2^{*}\). Let \(p^{}_{0}>2\) be such that \(\Delta _{p_{0}}(A)>0\). Then

$$\begin{aligned} \left[ (p^{\circ }_{0})^{\prime },p^{\circ }_{0}\right] \subset J(A,{{\mathscr {V}}},\delta ,\vartheta ), \end{aligned}$$

for all \(\delta >0\) and \(\vartheta >\pi /2-\omega ({{\mathscr {L}}}^{A}_{2})\).

1.2 A.2. Functional calculus and maximal regularity

The next result follows by combining Proposition A.1 with a result of Blunck and Kunstmann [10] that was simplified by Auscher in [5, Theorem 1.1] and extended to domains \(\Omega \subset {\mathbb {R}}^{d}\) by Egert in [38, Proposition 5.2].

Corollary A.2

([38, 39]) Under the assumptions of Proposition A.1 we have

$$\begin{aligned} \omega _{H^{\infty }}({{\mathscr {L}}}^{A}_{p}+\delta I)=\omega ({{\mathscr {L}}}^{A}_{2})<\pi /2, \end{aligned}$$

for every \(p\in ((p^{\circ }_{0})^{\prime },p^{\circ }_{0})\) and \(\delta >0\).

As a consequence [33, 76], \({{\mathscr {L}}}^{A}_{p}\) has parabolic maximal regularity, for every \(p\in [(p^{\circ }_{0})^{\prime },p^{\circ }_{0}]\).

1.3 A.3. Absence of Sobolev embeddings

It is well-known that (\(\mathrm{SE}_{q}\)) for \(q>2\) requires geometric assumptions on \(\Omega \) and does not hold in general, see [1, Theorem 4.46, Theorem 4.48 and Example 4.55] and [11, Proposition 3 and Example 6]. For simplicity, we only discuss the case when \({{\mathscr {V}}}=W^{1,2}(\Omega )\).

When \(\Omega \subset {\mathbb {R}}^{d}\) has finite measure, by the Rellich–Kondrachov theorem [45, Theorem 5 and Corollary 1], the Sobolev embedding (\(\mathrm{SE}_{q}\)) for some \(q>2\) implies the compactness of the resolvent operator \((I+{{\mathscr {L}}}^{A}_{2})^{-1}\) for all \(A\in {{\mathcal {A}}}(\Omega )\) (see also [11, Theorem 7]). So, in this case, the spectrum of \({{\mathscr {L}}}^{I}_{2}\) is discrete.

  1. (a)

    A classical example of Courant and Hilbert [21, p. 531] shows that there exists a “rooms and passages” connected bounded domain \(\Omega \subset {\mathbb {R}}^{2}\) for which \(0\in \sigma _{\mathrm{ess}}({{\mathscr {L}}}^{I}_{2})\). Actually, for \(d\geqslant 2\), given any closed subset S of \([0,+\infty )\) there exists an open connected subset \(\Omega \) of the unit ball in \({\mathbb {R}}^{d}\) such that \(\sigma _{\mathrm{ess}}({{\mathscr {L}}}^{I}_{2})=S\), see [47].

  2. (b)

    Let \(d\geqslant 2\). By using a criterion of Evans and Harris [41] (see also [26, p. 106]), one can easily construct unbounded “horn-shaped” \(\Omega \subset {\mathbb {R}}^{d}\) of finite measure for which the Neumann Laplacian \({{\mathscr {L}}}^{I}_{2}\) does not have compact resolvent operators. A simple example of this phenomenon is given by the regions

    $$\begin{aligned} \Omega _{\alpha }=\{(x,x^{\prime })\in {\mathbb {R}}\times {\mathbb {R}}^{d-1}: x>0,\ |x^{\prime }|<ce^{-\alpha x}\},\quad \alpha>0,\ c>0. \end{aligned}$$
    (A.1)

1.4 A.4. Sharpness of Theorem 3

For open sets like those described above in (b), the conclusions of Proposition A.1 and Corollary A.2 are false, because the analyticity angle of the semigroup and the functional calculus angle of the generator may depend on p, even for smooth A and pure Neumann boundary conditions.

Kunstmann [56] further developed a result of Davies and Simon [26] for \(p=2\) and proved that for \(d=2\) and \(p>1\) the \(L^{p}\) spectrum of the Neumann Laplacian in the region \(\Omega _{\alpha }\) defined in (A.1) satisfies the inclusions

$$\begin{aligned} \{0\}\cup \mathbf{P}_{p,\alpha }\subseteq \sigma ({{\mathscr {L}}}^{I}_{p})\subseteq \left[ 0,\alpha ^{2}/4\right) \cup \mathbf{P}_{p,\alpha }, \end{aligned}$$
(A.2)

where

$$\begin{aligned} \mathbf{P}_{p}=\left\{ x+iy\in {{\mathbb {C}}}: x=\frac{p^{2}}{(p-2)^{2}}y^{2}+\frac{1}{p}-\frac{1}{p^{2}}\right\} ,\quad \mathbf{P}_{p,\alpha }=\alpha ^{2}{} \mathbf{P}_{p}. \end{aligned}$$

Moreover, \(\sigma _{\mathrm{ess}}({{\mathscr {L}}}^{I}_{p})=\mathbf{P}_{p,\alpha }\).

For \(p\in (1,+\infty )\) and \(\alpha >0\) the parabolic region \(\mathbf{P}_{p,\alpha }\) is tangent to the critical sector \(\mathbf{S}_{\phi ^{*}_{p}}\), where \(\phi ^{*}_{p}=\arcsin |2/p-1|\). Recall that \(\phi _{p}=\pi /2-\phi ^{*}_{p}\) and \(\phi ^{*}_{p}\) are, respectively, the optimal analyticity angle in \(L^{p}\) [53, 60] and the optimal functional calculus angle in \(L^{p}\) [18], for all generators of symmetric contraction semigroups.

By attaching to, say, a ball in \({\mathbb {R}}^{2}\) countably many disjoint horns \({\widetilde{\Omega }}_{\alpha _{n}}\), each one congruent to some \(\Omega _{\alpha _{n}}\), \(\alpha _{n}>0\), Kunstmann [56] was able to construct a domain \(\Omega _{\max }\) of finite measure such that the associated Neumann Laplacian has maximal \(L^{p}\) spectrum:

$$\begin{aligned} \sigma ({{\mathscr {L}}}^{I}_{p})={\overline{\mathbf{S }}}_{\phi ^{*}_{p}},\quad 1<p<+\infty . \end{aligned}$$
(A.3)

Consider the region \(\Omega _{\mathrm{max}}\) described above. Recall from [17, Lemma 5.22] that for \(\phi \in (-\pi /2,\pi /2)\) and \(p>1\) we have

$$\begin{aligned} \Delta _p(e^{i\phi }I)=\cos \phi -\left| {1-2/p}\right| =\cos \phi -\cos \phi _{p}. \end{aligned}$$
(A.4)

Fix \(\phi \in (0,\pi /2)\). Since \((T^{I}_{t})_{t>0}\) is symmetric and sub-Markovian, we obtain from [18], (A.4) and (A.3) that

$$\begin{aligned} \left\{ p\in (1,+\infty ): \omega _{H^{\infty }}({{\mathscr {L}}}^{e^{i\phi }I}_{p})<\pi /2\right\} =\left\{ p\in (1,+\infty ):\Delta _{p}(e^{i\phi }I)>0\right\} . \end{aligned}$$

Let \(p>1\) be such that \(\Delta _{p}(e^{i\phi }I)\leqslant 0\), that is, such that \(\phi \geqslant \phi _{p}\). Then \({{\mathscr {L}}}^{e^{i\phi } I}_{p}\) does not have parabolic maximal regularity in \(L^{p}(\Omega _{\mathrm{max}})\), since otherwise, by [32, Theorem 2.2], there would exist \(\varepsilon ,\delta >0\) such that \( \sigma ({{\mathscr {L}}}^{I}_{p})\subseteq {\overline{\mathbf{S }}}_{\pi /2-\phi -\varepsilon }-\delta \), contradicting (A.3).

More generally, by combining [18] with Proposition 20 and [55, Example 2.4 and Remark 3.2], we deduce that for all \(d\geqslant 2\) there exist \(\Omega \subset {\mathbb {R}}^{d}\) and \(A\in {{\mathcal {A}}}(\Omega )\) such that the Neumann operator \({{\mathscr {L}}}^{A}_{p}\) has parabolic maximal regularity if and only if \(\Delta _{p}(A)>0\).

In this sense Theorem 3 is sharp.

1.5 A.5. Extrapolation in \(L^{\infty }\) for smooth coefficients: counterexamples

Consider the open set \(\Omega _{\mathrm{max}}\) of [56] described above. The equality (A.3) implies that if \(\phi >0\) then \((T^{e^{i\phi }I}_{t})_{t>0}\)=\((T^{I}_{e^{i\phi }t})_{t>0}\) is not exponentially bounded in \(L^{\infty }(\Omega _{\mathrm{max}})\). Indeed, assuming the contrary, by interpolation with \(L^{2}\) and the relation \(\overline{T^{I}_{z}f}=T^{I}_{{\bar{z}}}{\bar{f}}\), \(\mathrm{Re}\,z>0\), (because \((T^{I}_{t})_{t>0}\) is positive and analytic in \(L^{2}\) in \(\{\mathrm{Re}\,z>0\}\)), there would exist \(r_{0}>0\) and \(\phi _{0}\in (0,\pi /2)\) such that

$$\begin{aligned} \sup _{t>0}{\left\| {e^{-r_{0}t}T^{I}_{e^{{\pm } i\phi _{0}}t}}\right\| _{p}}<+\infty ,\quad \forall p>2. \end{aligned}$$

This implies that \(\sigma ({{\mathscr {L}}}^{I}_{p})\subseteq \overline{\mathbf{S}}_{\pi /2-\phi _{0}}-r_{0}\) for all \(p>2\), contradicting (A.3) when p is such that \(\phi ^{*}_{p}>\pi /2-\phi _{0}\).

Example A.3

We would expect that if \(\phi \ne 0\) then one has \(T^{I}_{e^{i\phi }t}(L^{\infty }(\Omega _{\mathrm{max}}))\not \subset L^{\infty }(\Omega _{\mathrm{max}})\) for some \(t>0\), but we could not extract this result from the existing literature. Therefore, we now construct a (disconnected) open set \(\Omega \subset {\mathbb {R}}^2\) such that there exists \(\phi >0\) for which \(T^{e^{i\phi }I}_{1}(L^{\infty }(\Omega ))\not \subset L^{\infty }(\Omega )\).

We first consider the Neumann Laplacian \(\Delta ^{\Omega _{\alpha }} := {{\mathscr {L}}}^{I}\) in the region \(\Omega _{\alpha }\) given by (A.1) with \(d=2\), \(\alpha >0\) and \(c=\alpha ^{-1}\). Note that \(\Omega _{\alpha }=\alpha ^{-1}\cdot \Omega _{1}\). Set \(T^{\Omega _{\alpha }}_{t}=\exp (-t\Delta ^{\Omega _{\alpha }})\).

By arguing much as in the case of \(\Omega _{\mathrm{max}}\) discussed above and using (A.2), we see that

$$\begin{aligned} \sup _{t>0}{\left\| {T^{\Omega _{1}}_{e^{i\phi }t}}\right\| _{\infty }}=+\infty \end{aligned}$$

for every \(\phi \in (0,\pi /2)\). Fix \(\phi \in (0,\pi /2)\). By the uniform boundeness principle, there exists a nonzero \(f_{1}\in L^{\infty }(\Omega _{1})\) and a sequence \(t_{n}>0\) such that

$$\begin{aligned} {\left\| {T^{\Omega _{1}}_{t^{2}_{n}e^{i\phi }}f_{1}}\right\| _{L^{\infty }(\Omega _{1})}}\geqslant n\, {\left\| {f_{1}}\right\| _{L^{\infty }(\Omega _{1})}},\quad \forall n\in {{\mathbb {N}}}_{+}. \end{aligned}$$

We now use a rescaling argument. For \(\alpha >0\), consider the operator

$$\begin{aligned} (J_{\alpha }f)(x,y) := \alpha f(\alpha x,\alpha y). \end{aligned}$$

It is not hard to see that

  1. (i)

    \(J_{\alpha }:L^{2}(\Omega _{1})\rightarrow L^{2}(\Omega _{\alpha })\) is a surjective isometry with \(J^{-1}_{\alpha }=J_{1/\alpha }\);

  2. (ii)

    \(J_{\alpha }\mathrm{D}(\Delta ^{\Omega _{1}}_{2})=\mathrm{D}(\Delta ^{\Omega _{\alpha }}_{2})\);

  3. (iii)

    \(J^{-1}_{\alpha }\Delta ^{\Omega _{\alpha }}_{2}J_{\alpha }=\alpha ^{2}\Delta ^{\Omega _{1}}_{2}\).

It follows that

$$\begin{aligned} J_{1/\alpha }T^{\Omega _{\alpha }}_{z}J_{\alpha }=T^{\Omega _{1}}_{\alpha ^{2}z},\quad \forall z\in {{\mathbb {C}}}_{+},\quad \forall \alpha >0. \end{aligned}$$

Hence, for \(\alpha =t_{n}\) and \(z=e^{i\phi }\),

$$\begin{aligned} \left( T^{\Omega _{t_{n}}}_{e^{i\phi }}\left( f_{1}(t_{n}\cdot )\right) \right) (\tfrac{\cdot }{t_{n}})=T^{\Omega _{1}}_{t^{2}_{n}e^{i\phi }}\left( f_{1}\right) (\cdot ). \end{aligned}$$

Set \(f_{n}=f_{1}(t_{n}\cdot )\). Then \({\left\| {f_{n}}\right\| _{L^{\infty }(\Omega _{t_{n}})}}={\left\| {f_{1}}\right\| _{L^{\infty }(\Omega _{1})}}\) and

$$\begin{aligned} \begin{aligned} n{\left\| {f_{1}}\right\| _{L^{\infty }(\Omega _{1})}}\leqslant \Vert T^{\Omega _{1}}_{t^{2}_{n}e^{i\phi }}f_{1}\Vert _{L^{\infty }(\Omega _{1})}=\Vert T^{\Omega _{t_{n}}}_{e^{i\phi }}f_{n}\Vert _{L^{\infty }(\Omega _{t_{n}})}. \end{aligned} \end{aligned}$$
(A.5)

For each \(n\in {{\mathbb {N}}}_{+}\) select a rigid motion \(R_{n}\) of the plane such that the congruent copies

$$\begin{aligned} {\widetilde{\Omega }}_{t_{n}} := R_{n}\left( \Omega _{t_{n}}\right) ,\quad n\in {{\mathbb {N}}}_{+} \end{aligned}$$

are pairwise disjoint; for example \(R_{1}=\mathrm{Id}\) and, for \(n\geqslant 2\), \(R_{n}(x,y)=(x,y+v_{n})\), where \(v_{n}=t^{-1}_{n}+2\sum ^{n-1}_{k=1}t^{-1}_{k}\). Define

Then, \(\Vert f\Vert _{L^{\infty }(\Omega )}={\left\| {f_{1}}\right\| _{L^{\infty }(\Omega _{1})}}\) and

It follows from (A.5) that \(T^{\Omega }_{e^{i\phi }}f\notin L^{\infty }(\Omega )\).

Appendix B: Rigidity of generalized convexity

Proposition B.1

Let \(A\in {{\mathbb {C}}}^{d\times d}\) be an elliptic matrix, \(t_{0}>0\) and let \(\gamma \in C^{1}([0,+\infty );{\mathbb {R}})\cap C^{2}((0,+\infty ){\setminus }\{t_{0}\};{\mathbb {R}})\). Set \(\Gamma (\zeta )=\gamma (|\zeta |)\), \(\zeta \in {\mathbb {R}}^{2}\). Suppose that \(\Gamma \) is A-convex in \({\mathbb {R}}^{2}{\setminus }\{|\zeta |=t_{0}\}\). Then,

  1. (i)

    The profile function \(\gamma \) is nondecreasing and convex, and \(\Gamma \) is convex.

  2. (ii)

    If \(\mathrm{Im}\,A\ne 0\), then either \(\gamma =\mathrm{const}\) or \(\gamma ^{\prime }(t)>0\) for all \(t>0\).

Proof

A rapid calculation shows that

$$\begin{aligned} (D^{2}\Gamma )(\zeta )=\gamma ^{\prime \prime }(|\zeta |) \frac{\zeta }{|\zeta |}\otimes \frac{\zeta }{|\zeta |}+\frac{\gamma ^{\prime }(|\zeta |)}{|\zeta |}\left[ I_{{\mathbb {R}}^{2}}-\frac{\zeta }{|\zeta |}\otimes \frac{\zeta }{|\zeta |}\right] , \end{aligned}$$
(B.1)

for all \(\zeta \in {\mathbb {R}}^{2}{\setminus }(\{|\zeta |=t_{0}\}\cup \{0\})\).

We first prove that \(\gamma ^{\prime }(t)\geqslant 0\) for all \(t>0\). By continuity it suffices to consider \(t\ne t_{0}\). For \(Z=(Z^{1},Z^{2})\in {\mathbb {R}}^{d}\times {\mathbb {R}}^{d}\) write

$$\begin{aligned} Z^{j}=(z^{j}_{1},\dots ,z^{j}_{d}),\quad j=1,2. \end{aligned}$$

Fix \(\zeta \in {\mathbb {R}}^{2}{\setminus }(\{|\zeta |=t_{0}\}\cup \{0\})\). Take \(X^{1}=(x^{1}_{1},0,\dots ,0)\), \(X^{2}=(x^{2}_{1},0,\dots ,0)\). Define \(X=(X^{1},X^{2})\), \(Y=(Y^{1},Y^{2})={{\mathcal {M}}}(A)X\), \(x=(x^{1}_{1},x^{2}_{1})\) and \(y=(y^{1}_{1},y^{2}_{1})\). Then, by using the notation of Sect. 2.3, we obtain

$$\begin{aligned} \begin{aligned} H^{A}_{\Gamma }[\zeta ;X]&=\left\langle (D^{2}\Gamma )(\zeta )x , y\right\rangle _{{\mathbb {R}}^{2}}\\&=\frac{\gamma ^{\prime }(|\zeta |)}{|\zeta |}\left\langle x , y\right\rangle _{{\mathbb {R}}^{2}}+\left( \gamma ^{\prime \prime }(|\zeta |)-\frac{\gamma ^{\prime }(\zeta )}{|\zeta |}\right) |\zeta |^{-2}\left\langle \zeta , x\right\rangle _{{\mathbb {R}}^{2}}\left\langle \zeta , y\right\rangle _{{\mathbb {R}}^{2}}. \end{aligned} \end{aligned}$$

Now take \(X\ne 0\) of the form above and such that x is orthogonal to \(\zeta \). Then, by the assumption of A-convexity of \(\Gamma \),

$$\begin{aligned} 0\leqslant H^{A}_{\Gamma }[\zeta ;X]=\frac{\gamma ^{\prime }(|\zeta |)}{|\zeta |}\left\langle x , y\right\rangle _{{\mathbb {R}}^{2}}. \end{aligned}$$

It follows that \(\gamma ^{\prime }(|\zeta |)\geqslant 0\), because \(\left\langle x , y\right\rangle = \left\langle X , {{\mathcal {M}}}(A)X\right\rangle _{{\mathbb {R}}^{2d}}>0 \) by (12) and the ellipticity of A.

We now prove that\(\gamma \)is convex. It is well-known (and easy to see by means of a convolution argument for regularising \(\gamma \)) that this is equivalent to proving that \(\gamma ^{\prime \prime }(t)\geqslant 0\) for all \(t\in (0,+\infty ){\setminus }\{t_{0}\}\). Fix \(t\notin \{t_{0},0\}\) and \(\zeta \in {\mathbb {R}}^{2}\) such that \(|\zeta |=t\). We rewrite (B.1) as

$$\begin{aligned} \begin{aligned} (D^{2}\Gamma )(\zeta )&=\gamma ^{\prime \prime }(t)I_{{\mathbb {R}}^{2}}+\left( \gamma ^{\prime }(t)-t\gamma ^{\prime \prime }(t)\right) |\zeta |^{-1}\left[ I_{{\mathbb {R}}^{2}}-\frac{\zeta }{|\zeta |}\otimes \frac{\zeta }{|\zeta |}\right] ,\\&=\gamma ^{\prime \prime }(t)I_{{\mathbb {R}}^{2}}+\left( \gamma ^{\prime }(t)-t\gamma ^{\prime \prime }(t)\right) (D^{2}F_{1})(\zeta ), \end{aligned} \end{aligned}$$

where \(F_{1}(\zeta )=|\zeta |\).

Therefore, for all \(X\in {\mathbb {R}}^{2d}\) we have

$$\begin{aligned} 0\leqslant H^{A}_{\Gamma }[\zeta ,X]=\gamma ^{\prime \prime }(t)\left\langle X , {{\mathcal {M}}}(A)X\right\rangle _{{\mathbb {R}}^{2d}}+\left( \gamma ^{\prime }(t)-t\gamma ^{\prime \prime }(t)\right) H^{A}_{F_{1}}[\zeta ;X]. \end{aligned}$$
(B.2)

It follows from (4) that \(\Delta _{1}(A)\leqslant 0\). Hence, by Lemma 4 (v) we have

$$\begin{aligned} \min _{|X|=1}H^{A}_{F_{1}}[\zeta ;X]\leqslant 0. \end{aligned}$$

From this, (B.2), the fact that \(\gamma ^{\prime }\geqslant 0\), and the inequality \(\left\langle X , {{\mathcal {M}}}(A)X\right\rangle _{{\mathbb {R}}^{2d}}\geqslant \lambda |X|^{2}\) we deduce that \(\gamma ^{\prime \prime }(t)\geqslant 0\).

Convexity of\(\Gamma \) is now clear and easily follows from the already proved properties of \(\gamma \). Indeed, since \(\gamma \) is nondecreasing and convex, we have

$$\begin{aligned} \Gamma (t\zeta _{1}+(1-t)\zeta _{2})\leqslant \gamma (t|\zeta _{1}|+(1-t)|\zeta _{2}|)\leqslant t\Gamma (\zeta _{1})+(1-t)\Gamma (\zeta _{2}), \end{aligned}$$

for every \(\zeta _{1},\zeta _{2}\in {\mathbb {R}}^{2}\) and every \(t\in [0,1]\).

We now prove (ii). Suppose that there exists \(s>0\) such that \(\gamma ^{\prime }(s)=0\), then by item (i) \(\gamma ^{\prime }(t)=0\) for all \(t\in [0,s]\). Assume that \(\gamma \) is nonconstant. Then,

$$\begin{aligned} t_{1} := \sup \{t>0: \gamma ^{\prime }(t)=0\}\in (0,+\infty ), \end{aligned}$$

\(\gamma ^{\prime }(t)>0\), for all \(t>t_{1}\) and \(\gamma ^{\prime }(t_{1})=0\). For simplifying the proof, we first assume that \(t_{1}=t_{0}\). For \(t>t_{0}\), define the function \(v(t)=\log (\gamma ^{\prime }(t))\). Then \(\lim _{t\rightarrow t^{+}_{0} }v(t)=-\infty \) and \(v^{\prime }(t)=\frac{\gamma ^{\prime \prime }(t)}{\gamma ^{\prime }(t)}\geqslant 0\), for all \(t>t_{0}\). It follows that

$$\begin{aligned} \limsup _{t\rightarrow t^{+}_{0}}\frac{\gamma ^{\prime \prime }(t)}{\gamma ^{\prime }(t)}=+\infty . \end{aligned}$$
(B.3)

We now show that (B.3) implies \(\mathrm{Im}\,A=0\). Define the function

$$\begin{aligned} p(t) := t\frac{\gamma ^{\prime \prime }(t)}{\gamma ^{\prime }(t)}+1\geqslant 1, \quad t>t_{0}. \end{aligned}$$

Then, by (B.1),

$$\begin{aligned} (D^{2}\Gamma )(\zeta )=\frac{\gamma ^{\prime }(|\zeta |)}{|\zeta |}\left[ I_{{\mathbb {R}}^{2}}+(p(|\zeta |)-2)\,\frac{\zeta }{|\zeta |}\otimes \frac{\zeta }{|\zeta |}\right] ,\quad \forall |\zeta |>t_{0}. \end{aligned}$$

The formula above expresses the Hessian of the radial function \(\Gamma \) in terms of Hessians of power functions. More specifically, by (17) we have

$$\begin{aligned} (D^{2}\Gamma )(\zeta )=g(|\zeta |)\cdot (D^{2}F_{p(|\zeta |)})(\zeta ),\quad \forall |\zeta |>t_{0}, \end{aligned}$$

where

$$\begin{aligned} g(t) := \frac{\gamma ^{\prime }(t)}{p(t)t^{p(t)-1}}\geqslant 0, \quad t>t_{0}. \end{aligned}$$

Since \(\Gamma \) is A-convex, it follows from the identity above that

$$\begin{aligned} H^{A}_{F_{p(|\zeta |)}}[\zeta ;X]\geqslant 0,\quad \forall X\in {\mathbb {R}}^{2d},\quad \forall |\zeta |>t_{0}, \end{aligned}$$

or, equivalently [see Lemma 4 (v)], \(\Delta _{p(|\zeta |)}(A)\geqslant 0\), for all \(|\zeta |>t_{0}\). Now by (B.3) we have \(\sup _{|\zeta |>t_{0}}p(|\zeta |)=+\infty \), so by Lemma 4 (i) and (iii) we have \(\Delta _{p}(A)\geqslant 0\), for all \(p>1\). Hence, Lemma 4 (vi) implies that \(\mathrm{Im}\,A=0\).

When \(t_{1}\ne t_{0}\) the proof is similar. Just consider the function \(v(t)=\log (\gamma ^{\prime }(t))\) for \(t\in (t_{1},t_{0})\) if \(t_{1}<t_{0}\), or for \(t>t_{1}\) if \(t_{1}>t_{0}\) (and so \(\gamma \) is in \(C^{2}({\mathbb {R}})\)). \(\square \)

Appendix C: Flow regularity

Let \(({{\mathcal {X}}},\mu )\) be a \(\sigma \)-finite measure space. Fix \(p\geqslant 2\) and set \(q=p/(p-1)\). Suppose that \((\exp (-t{{\mathcal {A}}}))_{t>0}\) and \((\exp (-t{{\mathcal {B}}}))_{t>0}\) are analytic and uniformly bounded both in \(L^{p}({{\mathcal {X}}},\mu )\) and \(L^{q}({{\mathcal {X}}},\mu )\). Let \({{\mathcal {Q}}}={{\mathcal {Q}}}_{p,\delta }\) be the Nazarov–Treil Bellman function defined in (10). Fix \(f,g\in (L^{p}\cap L^{q})({{\mathcal {X}}},\mu )\). Consider the flow

$$\begin{aligned} {{\mathcal {E}}}(t)=\int _{{{\mathcal {X}}}}{{\mathcal {Q}}}(e^{-t{{\mathcal {A}}}}f,e^{-t{{\mathcal {B}}}}g),\quad t>0, \end{aligned}$$

where we omit the subscript \({{\mathcal {W}}}\) (see Sect. 2.4).

Proposition C.1

Under the above assumptions, we have:

  1. (a)

    \({{\mathcal {E}}}\in C[0,+\infty )\);

  2. (b)

    \({{\mathcal {E}}}\in C^{1}(0,+\infty )\) and

    $$\begin{aligned} -{{\mathcal {E}}}^{\prime }(t)=2\mathrm{Re}\,\int _{{{\mathcal {X}}}}\left( (\partial _{\zeta }{{\mathcal {Q}}})(e^{-t{{\mathcal {A}}}}f,e^{-t{{\mathcal {B}}}}g){{\mathcal {A}}}e^{-t{{\mathcal {A}}}}+(\partial _{\eta }{{\mathcal {Q}}})(e^{-t{{\mathcal {A}}}}f,e^{-t{{\mathcal {B}}}}g){{\mathcal {B}}}e^{-t{{\mathcal {B}}}}g\right) . \end{aligned}$$

Proof

We start with (a) We prove only the continuity at 0 since the continuity at other points can be proved exactly in the same way, or it follows from item (b). Set

$$\begin{aligned} F(t,x)={{\mathcal {Q}}}(e^{-t{{\mathcal {A}}}}f(x),e^{-t{{\mathcal {B}}}}g(x)). \end{aligned}$$

By the mean value theorem applied to \({{\mathcal {Q}}}\),

$$\begin{aligned} \left| {F(t,x)-F(0,x)}\right|\leqslant & {} \max \left\{ \left| {(D{{\mathcal {Q}}})(\zeta ,\eta )}\right| : |\zeta |\leqslant \left| {e^{-t{{\mathcal {A}}}}f(x)}\right| +|f(x)|,\right. \\&\left. \times |\eta |\leqslant \left| {e^{-t{{\mathcal {B}}}}g(x)}\right| +|g(x)|\right\} \\&\times \sqrt{\left| {e^{-t{{\mathcal {A}}}}f(x)-f(x)}\right| ^{2}+\left| {e^{-t{{\mathcal {B}}}}g(x)-g(x)}\right| ^{2}}. \end{aligned}$$

Estimates (11) immediately give

$$\begin{aligned} \begin{aligned}&\max \left\{ \left| {(D{{\mathcal {Q}}})(\zeta ,\eta )}\right| : |\zeta |\leqslant \left| {e^{-t{{\mathcal {A}}}}f(x)}\right| +|f(x)|,\ |\eta |\leqslant \left| {e^{-t{{\mathcal {B}}}}g(x)}\right| +|g(x)|\right\} \\&\qquad \leqslant C\max \left\{ \left( |e^{-t{{\mathcal {A}}}}f(x)|+|f(x)|\right) ^{p-1},\left( |e^{-t{{\mathcal {A}}}}g(x)|+|g(x)|\right) ^{q-1}, |e^{-t{{\mathcal {A}}}}g(x)|+|g(x)|\right\} , \end{aligned} \end{aligned}$$

where C does not depend on x and t. Now item (a) follows from Hölder’s inequality and the strong continuity of the two semigroups in \(L^{p}({{\mathcal {X}}},\mu )\) and \(L^{q}({{\mathcal {X}}},\mu )\).

We now prove item (b) Analyticity implies that there exists \(C\geqslant 1\) such that, for \(r\in \{p,q\}\),

$$\begin{aligned} {\left\| {\left( \frac{d}{d t}\right) ^{k}e^{-t{{\mathcal {A}}}}}\right\| _{r}}={\left\| {{{\mathcal {A}}}^{k}e^{-t{{\mathcal {A}}}}}\right\| _{r}}\leqslant t^{-k}C^{k}k!,\quad \forall t>0, \end{aligned}$$

see, for example, [40, Chapter II, p.104]. Fix \(t_{0}>0\). Then for \(\delta <t_{0}/C\) we have

$$\begin{aligned} e^{-t{{\mathcal {A}}}}f=\sum ^{+\infty }_{k=0}\frac{(-1)^{k}}{k!}(t-t_{0})^{k}{{\mathcal {A}}}^{k}e^{-t_{0}{{\mathcal {A}}}}f,\quad \forall t\in \left[ t_{0}-\delta ,t_{0}+\delta \right] \end{aligned}$$

where the series converges in \((L^{p}\cap L^{q})({{\mathcal {X}}},\mu )\). Moreover,

$$\begin{aligned} \sup _{|t-t_{0}|\leqslant \delta }\left| {e^{-t{{\mathcal {A}}}}f(x)}\right| \leqslant \sum ^{+\infty }_{k=0}\frac{\delta ^{k}}{k!}\left| {{{\mathcal {A}}}^{k}e^{-t_{0}{{\mathcal {A}}}}f(x)}\right| \in (L^{p}\cap L^{q})({{\mathcal {X}}},\mu ) \end{aligned}$$

and similarly,

$$\begin{aligned} \sup _{|t-t_{0}|\leqslant \delta }\left| {{{\mathcal {A}}}e^{-t{{\mathcal {A}}}}f(x)}\right| \in (L^{p}\cap L^{q})({{\mathcal {X}}},\mu ). \end{aligned}$$

Possibly taking a smaller \(\delta \), we also get

$$\begin{aligned} \sup _{|t-t_{0}|\leqslant \delta }\left| {e^{-t{{\mathcal {B}}}}g(x)}\right| +\sup _{|t-t_{0}|\leqslant \delta }\left| {{{\mathcal {B}}}e^{-t{{\mathcal {B}}}}g(x)}\right| \in (L^{p}\cap L^{q})({{\mathcal {X}}},\mu ). \end{aligned}$$

By using the powers series expansion of \(e^{-t{{\mathcal {A}}}}f\) and \(e^{-t{{\mathcal {B}}}}g\) one can also prove that each \(e^{-t{{\mathcal {A}}}}f\) and each \(e^{-t{{\mathcal {B}}}}g\) can be redefined in a set of measure zero, in such a manner that for almost every \(x\in \Omega \) the functions \(t\mapsto e^{-t{{\mathcal {A}}}}f(x)\) is real-analytic on \((0,\infty )\), for a.e. \(x\in \Omega \); see [78, p. 72]. Now item (b) follows from estimates (11) and the usual theorems of derivation and passage of the limit under the integral sign that are consequence of the Lebesgue’s dominated convergence theorem. \(\square \)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carbonaro, A., Dragičević, O. Bilinear embedding for divergence-form operators with complex coefficients on irregular domains. Calc. Var. 59, 104 (2020). https://doi.org/10.1007/s00526-020-01751-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-020-01751-3

Mathematics Subject Classification

Navigation