Linear wave estimates

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Fixed-time estimates for free solutions f :
- (Energy estimate) If $f (0)$ is in $H s$ , then $f (t)$ is also.
- (Decay estimate) If $f (0)$ has more than $(d + 1) / 2$ derivatives in $L 1$ , then $\| f (t)\|_{L^\infty}$ decays like $< t > - (d - 1) / 2$ . One can obtain the endpoint of $(d + 1) / 2$ derivatives if one is willing to localize in frequency or use Hardy spaces and BMO.
- One can interpolate between these estimates to get $(L p, L p')$ estimates with the sharp loss of regularity Br1975. This is useful for Strichartz estimates and for scattering theory.
Strichartz estimates: A free solution is in if
- (Scaling) $d / 2 - s = 1 / q + d / r$
- (Parallel interactions) $(d-1)/4 \geq 1/q + (d-1)/2r$
- (Increase of integrability) $q, r \geq 2$
- (No double endpoints)
  - This estimate can be recovered for radial functions KlMa1993, or when a small amount of smoothing (either in the Sobolev sense, or in relaxing the integrability) in the radial variable [MacNkrNaOz-p].However in the general case one cannot recover the estimate even if one uses the BMO norm or attempts Littlewood-Paley frequency localization Mo1998
  - Actually even when $n > 3$ , the $(q,r) = (2,\infty)$ estimate is slightly subtle; one has BMO and Besov space estimates but not directly $L^\infty$ estimates. However, the endpoint $(q, r) = (2,2(d - 1) / (d - 3))$ is OK; see KeTa1998.
- In the case $s = 1 / 2, d = 3, q = r = 4$ , a maximizer exists (e.g. with initial position zero and initial velocity given by the Cauchy distribution $1 / 1 + | x | 2)$ , witih best constant $(3 p i / 4) 1 / 4$ [Fc-p4]
- These results extend globally outside of a convex obstacle [Bu-p], SmhSo1995, [SmhSo-p], [Met-p]; see [So-p] for a survey of this issue and applications to nonlinear wave equations outside of an obstacle.
- For data which is radial (or otherwise enjoys additional angular regularity) a much larger range of Strichartz estimates is possible (basically because the parallel interaction obstruction is substantially weakened); see [Stz-p4] for further discussion.

These estimates extend to some extent to the Klein-Gordon equation $\Box u = m^2 u$ .A useful heuristic to keep in mind is that this equation behaves like the Schrodinger equation $u + i u t + 1 / (2 m)Δ u = 0$ when the frequency $ξ$ has magnitude less than $m$ , but behaves like the wave equation for higher frequencies.Some basic Strichartz estimates here are in MsSrWa1980; see for instance [Na-p], [MacNaOz-p], [MacNkrNaOz-p] for more recent treatments.

For inhomogeneous estimates it is known that a solution with zero initial data and forcing term containing s-1 derivatives in a dual space ${L^{Q'}_t} {L^{R'}_x}$ will lie in $L^q_t L^r_x$ if both $(q, r)$ and $(Q, R)$ are admissible in the above sense, if $s \geq 0$ , and if one has the scaling condition

$1 / Q + d / R + 1 / q + d / r = d + s + 1$ .The \u201cs-1\u201d represents a smoothing effect of one derivative, though this full gain is only attainable if one uses the energy exponents $L^1_t L^2_x$ and ${L^{\infty}_t} {L^2_x}$ . It is possible to obtain inhomogeneous estimates in which only one of the exponents are admissible; this phenomenon was first observed in Har1990, Ob1989 (see also KeTa1998).More recently in [Fc-p2], inhomogeneous estimates are obtained with the above scaling condition assuming the weaker conditions $1 \leq q,r \leq \infty$ and $1 / q < (n - 1)(1 / 2 - 1 / r)$ or $(q,r) = (\infty,2)$ and similarly for $Q, R$ , and if the following additional conditions hold:

In $d = 1,2$ no further conditions are required;

When $d = 3$ , r,R are required to be finite;

When $d > 3$ , either $1 / q + 1 / Q < 1$ , $(n-3)/r \leq (n-1)/R$ , and $(n-3)/R \leq (n-1)/r$ , or $1 / q + 1 / Q = 1$ , $(n-3)/r < (n-1)/R, (n-3)/R < (n-1)/r, r \geq q$ , and $R \geq Q$ .

Strichartz estimates extend to situations in which there is a potential or when the metric is variable.For local-in-time estimates and smooth potentials or

metrics this is fairly straightforward (the potential can be treated by iterative methods, and the metric by parametrix methods).More interesting issues

arise for global-in-time estimates with smooth potentials/metrics or local-in-time estimates with rough potentials/metrics (the two types of results are linked by

scaling).For potentials of power-type decay, the global results are as follows:

For potentials of the form $V = a / | x | 2$ with $d \geq 3$ and $a > - (d - 2) 2 / 4$ , one has global Strichartz estimates [BuPlStaTv-p]; a simplified proof and more general result dealing with inverse square-like potentials which are not too negative is in [BuPlStaTv-p2]. The condition on a is necessary to avoid bound states.For potentials decaying slower than this, Strichartz estimates can fail. For potentials decaying an epsilon faster than $1 / | x | 2$ and assumed to be nonnegative, dispersive and Strichartz estimates were obtained when $d = 3$ in GeVis2003.

(More results to be added in future).

Linear wave estimates

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