journal article
Mar 26, 2026
L p estimates for wave equations with specific C 0 , 1 coefficients
Abstract
Peral–Miyachi’s celebrated theorem states that the operator
(
I
-
Δ
)
-
α
2
exp
(
i
-
Δ
)
is bounded on
L
p
(
ℝ
d
)
if and only if
α
≥
s
p
:
=
(
d
-
1
)
1
p
-
1
2
.
We extend this result to operators of the form
ℒ
=
-
∑
j
=
1
d
a
j
+
d
∂
j
a
j
∂
j
, such that, for
j
=
1
,
⋯
,
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
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.
(
I
-
Δ
)
-
α
2
exp
(
i
-
Δ
)
is bounded on
L
p
(
ℝ
d
)
if and only if
α
≥
s
p
:
=
(
d
-
1
)
1
p
-
1
2
.
We extend this result to operators of the form
ℒ
=
-
∑
j
=
1
d
a
j
+
d
∂
j
a
j
∂
j
, such that, for
j
=
1
,
⋯
,
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
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.
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References
35
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References
Details
- Published
- Mar 26, 2026
- Pages
- 1-46
Authors
Cite This Article
Dorothee Frey, Pierre Portal (2026).
L
p
estimates for wave equations with specific
C
0
,
1
coefficients. Annales de l'Institut Fourier, 1-46. https://doi.org/10.5802/aif.3763
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