$L^p$ estimates for wave equations with specific $C^{0,1}$ coefficients
Frey, Dorothee1; Portal, Pierre2
1Karlsruhe Institute of Technology, Department of Mathematics, 76131 Karlsruhe (Germany)
2Australian National University, Mathematical Sciences Institute, Hanna Neumann Building, Ngunnawal and Ngambri Country, Canberra ACT 2601 (Australia)
Annales de l'Institut Fourier, Online first, 46 p.

Peral–Miyachi’s celebrated theorem states that the operator $(I-\Delta )^{- \frac{\alpha }{2}}\exp (i \sqrt{-\Delta })$ is bounded on $L^{p}(\mathbb{R}^{d})$ if and only if

\[ \alpha \ge s_{p} := (d-1)\biggl \vert \frac{1}{p}-\frac{1}{2}\biggr \vert . \]

We extend this result to operators of the form $\mathcal{L} = -\sum _{j=1} ^{d} a_{j+d}\partial _{j}a_{j}\partial _{j}$, such that, for $j=1,\dots ,d$, the functions $a_{j}$ and $a_{j+d}$ only depend on $x_{j}$, are bounded above and below, but are merely Lipschitz continuous. This is below the $C^{1,1}$ regularity that is required in general situations. We construct spaces on which $\exp (i\sqrt{\mathcal{L}})$ is bounded by lifting $L^{p}$ functions to tent spaces, using wave packets adapted to the coefficients. The result then follows from Sobolev embedding properties of these spaces.

Les célèbres théorèmes de Peral et Miyachi montrent que $(I-\Delta )^{- \frac{\alpha }{2}}\exp (i \sqrt{-\Delta })$ est borné sur $L^{p}(\mathbb{R}^{d})$ si et seulement si

\[ \alpha \ge s_{p} := (d-1)\biggl \vert \frac{1}{p}-\frac{1}{2}\biggr \vert . \]

Nous généralisons ce résultat à des opérateurs de la forme $\mathcal{L} = -\sum _{j=1} ^{d} a_{j+d}\partial _{j}a_{j}\partial _{j}$ à coefficients $a_{j}$ et $a_{j+d}$ ne dépendant que de $x_{j}$, bornés et bornés inférieurement, mais seulement lipschitziens. Ceci nous place donc en dessous de l’hypothèse de régularité $C^{1,1}$ nécessaire dans des situations plus générales. Pour ce faire, nous construisons des espaces invariants par l’action de $\exp (i\sqrt{\mathcal{L}})$ via une analyse des fonctions $L^{p}$ par plongement dans les espaces de tentes et montrons des plongements de Sobolev pour ces espaces.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3763
Classification: 42B35, 35L05, 42B30, 42B37, 35S30
Keywords: wave equation, rough coefficients, Hardy spaces, tent spaces, wave packets
Mots-clés : equations des ondes, coefficients irréguliers, espaces de Hardy, espaces de tentes, paquets d’ondes

Frey, Dorothee  1 ; Portal, Pierre  2

1 Karlsruhe Institute of Technology, Department of Mathematics, 76131 Karlsruhe (Germany)
2 Australian National University, Mathematical Sciences Institute, Hanna Neumann Building, Ngunnawal and Ngambri Country, Canberra ACT 2601 (Australia) 
Frey, Dorothee; Portal, Pierre. $L^p$ estimates for wave equations with specific $C^{0,1}$ coefficients. Annales de l'Institut Fourier, Online first, 46 p.
@unpublished{AIF_0__0_0_A52_0,
     author = {Frey, Dorothee and Portal, Pierre},
     title = {$L^p$ estimates for wave equations with specific $C^{0,1}$ coefficients},
     journal = {Annales de l'Institut Fourier},
     year = {2026},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     doi = {10.5802/aif.3763},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Frey, Dorothee
AU  - Portal, Pierre
TI  - $L^p$ estimates for wave equations with specific $C^{0,1}$ coefficients
JO  - Annales de l'Institut Fourier
PY  - 2026
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3763
LA  - en
ID  - AIF_0__0_0_A52_0
ER  - 
%0 Unpublished Work
%A Frey, Dorothee
%A Portal, Pierre
%T $L^p$ estimates for wave equations with specific $C^{0,1}$ coefficients
%J Annales de l'Institut Fourier
%D 2026
%V 0
%N 0
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3763
%G en
%F AIF_0__0_0_A52_0

[1] Amenta, Alex Interpolation and embeddings of weighted tent spaces, J. Fourier Anal. Appl., Volume 24 (2018) no. 1, pp. 108-140 | DOI | MR | Zbl

[2] Arendt, Wolfgang; Batty, Charles J. K.; Hieber, Matthias; Neubrander, Frank Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics, 96, Springer, 2011 | DOI | MR | Zbl

[3] Auscher, Pascal; Martell, José María Weighted norm inequalities, off-diagonal estimates and elliptic operators. II. Off-diagonal estimates on spaces of homogeneous type, J. Evol. Equ., Volume 7 (2007) no. 2, pp. 265-316 | DOI | MR | Zbl

[4] Auscher, Pascal; McIntosh, Alan; Morris, Andrew J. Calderón reproducing formulas and applications to Hardy spaces, Rev. Mat. Iberoam., Volume 31 (2015) no. 3, pp. 865-900 | DOI | MR | Zbl

[5] Auscher, Pascal; McIntosh, Alan; Nahmod, Andrea The square root problem of Kato in one dimension, and first order elliptic systems, Indiana Univ. Math. J., Volume 46 (1997) no. 3, pp. 659-695 | DOI | MR | Zbl

[6] Auscher, Pascal; McIntosh, Alan; Russ, Emmanuel Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal., Volume 18 (2008) no. 1, pp. 192-248 | DOI | MR | Zbl

[7] Auscher, Pascal; McIntosh, Alan; Tchamitchian, Philippe Heat kernels of second order complex elliptic operators and applications, J. Funct. Anal., Volume 152 (1998) no. 1, pp. 22-73 | DOI | MR | Zbl

[8] Auscher, Pascal; Tchamitchian, Philippe Calcul fontionnel précisé pour des opérateurs elliptiques complexes en dimension un (et applications à certaines équations elliptiques complexes en dimension deux), Ann. Inst. Fourier, Volume 45 (1995) no. 3, pp. 721-778 | DOI | MR | Zbl

[9] Bui, Huy-Qui; Duong, Xuan Thinh; Yan, Lixin Calderón reproducing formulas and new Besov spaces associated with operators, Adv. Math., Volume 229 (2012) no. 4, pp. 2449-2502 | DOI | MR | Zbl

[10] Cohn, W. S.; Verbitsky, I. E. Factorization of tent spaces and Hankel operators, J. Funct. Anal., Volume 175 (2000) no. 2, pp. 308-329 | DOI | MR | Zbl

[11] Coifman, R. R.; Meyer, Y.; Stein, Elias M. Some new function spaces and their applications to harmonic analysis, J. Funct. Anal., Volume 62 (1985) no. 2, pp. 304-335 | DOI | MR | Zbl

[12] Duong, Xuan Thinh; Li, Ji Hardy spaces associated to operators satisfying Davies–Gaffney estimates and bounded holomorphic functional calculus, J. Funct. Anal., Volume 264 (2013) no. 6, pp. 1409-1437 | DOI | MR | Zbl

[13] Duong, Xuan Thinh; Yan, Lixin Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Am. Math. Soc., Volume 18 (2005) no. 4, pp. 943-973 | DOI | MR | Zbl

[14] Frey, Dorothee; McIntosh, Alan; Portal, Pierre Conical square function estimates and functional calculi for perturbed Hodge–Dirac operators in L P , J. Anal. Math., Volume 134 (2018) no. 2, pp. 399-453 | DOI | MR | Zbl

[15] Harboure, Eleonor; Torrea, José L.; Viviani, Beatriz E. A vector-valued approach to tent spaces, J. Anal. Math., Volume 56 (1991), pp. 125-140 | DOI | MR | Zbl

[16] Hassell, Andrew; Portal, Pierre; Rozendaal, Jan Off-singularity bounds and Hardy spaces for Fourier integral operators, Trans. Am. Math. Soc., Volume 373 (2020) no. 8, pp. 5773-5832 | DOI | MR | Zbl

[17] Hassell, Andrew; Rozendaal, Jan L p and ℋ FIO p regularity for wave equations with rough coefficients (2010) | arXiv

[18] Hofmann, Steve; Lu, Guozhen; Mitrea, Dorina; Mitrea, Marius; Yan, Lixin Hardy spaces associated to non-negative self-adjoint operators satisfying Davies–Gaffney estimates, Memoirs of the American Mathematical Society, 1007, American Mathematical Society, 2011 | DOI | MR | Zbl

[19] Hofmann, Steve; Mayboroda, Svitlana Hardy and BMO spaces associated to divergence form elliptic operators, Math. Ann., Volume 344 (2009) no. 1, pp. 37-116 | DOI | MR | Zbl

[20] Hytönen, Tuomas; van Neerven, Jan; Portal, Pierre Conical square function estimates in UMD Banach spaces and applications to H ∞ -functional calculi, J. Anal. Math., Volume 106 (2008), pp. 317-351 | DOI | MR | Zbl

[21] Hytönen, Tuomas; van Neerven, Jan; Veraar, Mark; Weis, Lutz Analysis in Banach spaces. Vol. II. Probabilistic methods and operator theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 67, Springer, 2017 | DOI | MR | Zbl

[22] McIntosh, Alan; Morris, Andrew J. Finite propagation speed for first order systems and Huygens’ principle for hyperbolic equations, Proc. Am. Math. Soc., Volume 141 (2013) no. 10, pp. 3515-3527 | DOI | MR | Zbl

[23] Miyachi, Akihiko On some estimates for the wave equation in L p and H p , J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 27 (1980) no. 2, pp. 331-354 | MR | Zbl

[24] Müller, Detlef; Seeger, Andreas Sharp L p bounds for the wave equation on groups of Heisenberg type, Anal. PDE, Volume 8 (2015) no. 5, pp. 1051-1100 | DOI | MR | Zbl

[25] Peral, Juan C. L p estimates for the wave equation, J. Funct. Anal., Volume 36 (1980) no. 1, pp. 114-145 | DOI | MR | Zbl

[26] Rozendaal, Jan Characterizations of Hardy spaces for Fourier integral operators, Rev. Mat. Iberoam., Volume 37 (2021) no. 5, pp. 1717-1745 | DOI | MR | Zbl

[27] Rozendaal, Jan Rough pseudodifferential operators on Hardy spaces for Fourier integral operators II (2022) | arXiv | Zbl

[28] Seeger, Andreas; Sogge, Christopher D.; Stein, Elias M. Regularity properties of Fourier integral operators, Ann. Math. (2), Volume 134 (1991) no. 2, pp. 231-251 | DOI | MR | Zbl

[29] Smith, Hart F. A Hardy space for Fourier integral operators, J. Geom. Anal., Volume 8 (1998) no. 4, pp. 629-653 | DOI | MR | Zbl

[30] Smith, Hart F. A parametrix construction for wave equations with C 1,1 coefficients, Ann. Inst. Fourier, Volume 48 (1998) no. 3, pp. 797-835 | DOI | MR | Numdam | Zbl

[31] Smith, Hart F.; Sogge, Christopher D. On Strichartz and eigenfunction estimates for low regularity metrics, Math. Res. Lett., Volume 1 (1994) no. 6, pp. 729-737 | DOI | MR | Zbl

[32] Smith, Hart F.; Tataru, Daniel Sharp counterexamples for Strichartz estimates for low regularity metrics, Math. Res. Lett., Volume 9 (2002) no. 2-3, pp. 199-204 | DOI | MR | Zbl

[33] Stein, Elias M. Interpolation of linear operators, Trans. Am. Math. Soc., Volume 83 (1956), pp. 482-492 | DOI | MR | Zbl

[34] Štrkalj, Željko; Weis, Lutz On operator-valued Fourier multiplier theorems, Trans. Am. Math. Soc., Volume 359 (2007) no. 8, pp. 3529-3547 | MR | DOI | Zbl

[35] Tataru, Daniel Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. II, Am. J. Math., Volume 123 (2001) no. 3, pp. 385-423 | MR | DOI

Cited by Sources: