Positive Commutator Estimates - Math Berkeley
POSITIVE COMMUTATORS PETER HINTZ
1. Introduction In this short and very elementary note, we will discuss the technique of proving estimates for solutions to certain partial differential equations using positive commutators, which is a widely used and powerful technique in particular in microlocal analysis. In order to keep the necessary prerequisites at the bare minimum, we won’t make any use of pseudodifferential operators and related microlocal machinery in the central parts of this note, but references will be given. Positive commutator estimates are particularly useful for proving energy and regularity estimates for hyperbolic equations like the wave equation, or, more generally, equations which involve some sort of “flow” (see remark 4.1 for references). To be more specific, consider the wave equation on R × M , M a closed (compact, boundaryless) manifold, given by u ≡ (∂t2 − ∆)u = f , and suppose u|t 0. We then have h∂t u, χui + hχu, ∂t ui = h[χ, ∂t ]u, ui = −hχ0 u, ui, where we integrate by parts for the first equality. Choosing χ non-negative and such that −χ0 ≥ 0 for 0 ≤ t ≤ Te , this together with Cauchy-Schwartz and AM-GM gives p √ √ k −χ0 uk2L2 ([0,Te ]) ≤ 2k χ∂t ukL2 ([0,Te ]) k χukL2 ([0,Te ]) √ √ ≤ k χ∂t uk2L2 ([0,Te ]) + k χuk2L2 ([0,Te ]) , (2.2) √ √ 2 where we split χ = χ · χ in order to be able to take L norms on the right hand side without worries. Rewriting this as Z Te √ (−χ0 − χ)|u|2 dt ≤ k χ∂t uk2L2 ([0,Te ]) , 0
we now see what we need, namely, say, −χ0 − χ ≥ −χ0 /2 on [0, Te ]. 2In the language of microlocal analysis, the principal symbol r of 1 [, V ] is given by the i derivative of the symbol of V along the Hamilton vector field Hp of the principal symbol p = σ2 () 2 of , and if we can make r positive, it can be written as q for some symbol q > 0, thus 1 [, V ]u = Q∗ Qu+ lower order terms. A priori control on the lower order terms (paired with a i solution u of u = f ) then gives, with some additional technical details, control on Qu, thus we get regularity information on u where q is positive. 3A related inequality is the Poincar´ e inequality kukL2 ([0,T ]) ≤ Ck∂t ukL2 ([0,T ]) for u ∈ 1 H0 ([0, T ]).
POSITIVE COMMUTATORS
The common way to find such a χ is to consider the C ∞ (R) function ( e−1/t , t > 0 φ(t) = 0, t ≤ 0,
3
(2.3)
satisfying φ0 (t) = t−2 φ(t). Throughout the rest of this note, φ is always going to denote this function. We then choose χ(t) = φ(z−1 (Te − t)) and compute χ0 (t) = −z−1 · (z−1 (Te − t))−2 χ(t) = −z(Te − t)−2 χ(t). Since for 0 ≤ t ≤ Te , z(Te − t)−2 ≥ zTe−2 , we can choose z sufficiently large to get −χ0 − χ ≥ −χ0 /2 on [0, Te ], as required. Plugging this into (2.2), we can absorb √ the k χukL2 ([0,Te ]) term into the left hand side and thus obtain p √ k −χ0 uk2L2 ([0,Te ]) ≤ Ck χ∂t uk2L2 ([0,Te ]) (2.4) for some C > 0. Now we see that we do not quite get (2.1), since −χ0 is not bounded from below on all of [0, Te ]; instead, using that −χ0 does have a positive lower bound on [0, T ] (T < Te ), we only get the weaker estimate kukL2 ([0,T ]) ≤ Ck∂t ukL2 ([0,Te ]) . To fix this insufficiency, note that if we cut off χ at time T , i.e. replaced χ(t) by χ(t)H(T − t), H being the Heaviside function, and still got the estimate (2.4), interpreted in a distributional sense (the left hand side being the distributional pairing −hχ0 u, ui), we would get an additional term involving the δ-distribution at t = T , i.e. p √ k −χ0 uk2L2 ([0,T ]) + χ(T )|u(T )|2 ≤ Ck χ∂t uk2L2 ([0,T ]) , and the boundary term χ(T )|u(T )|2 would be positive (put differently, it has the same sign as −χ0 ), thus could just be dropped, and we would get (2.1)! In fact, this idea works. To see this formally, let χ(t) ˇ = χ(t)H(T − t), then we can compute just as above (now with h , i denoting the distributional pairing, one side being a distribution, the other side a test function, made antilinear in the second slot) h∂t u, χui ˇ + hχu, ˇ ∂t ui = h[χ, ˇ ∂t ]u, ui = −hχ ˇ0 u, ui = χ(T )|u(T )|2 − hχ0 u, uiL2 ([0,T ]) , where, for the first equality, we just use the definition of the distributional derivative (which of course does the same as integration by parts), and in the last equality we use −χ ˇ0 = −χ0 H(T − ·) + χ(T )δT . We now proceed as before, dropping the boundary term χ(T )|u(T )|2 , and finally obtain4 (2.1). Remark 2.1. We note that (2.1) and (2.4) hold more generally for functions u ∈ C ∞ (Rt , L2 (M )) vanishing for t < 0, where M is any measure space, e.g. a Riemannian manifold. 4If one is suspicious about the use of distributions here, one can also write everything out in terms of integrals from 0 to T , and the boundary term χ(T )|u(T )|2 is precisely the boundary term one gets from integrating by parts. However, this procedure makes it seem like there was something special about the boundary term, although there is not, really.
4
PETER HINTZ
Next, we look at a propagation estimate. Proposition 2.2. Let −∞ < t0 < t1 < t2 < ∞. Then there is a constant C > 0 such that for all u ∈ C ∞ (R), kukL2 ([t1 ,t2 ]) ≤ C(kukL2 ([t0 ,t1 ]) + k∂t ukL2 ([t0 ,t2 ]) ).
(2.5)
Proof. We need to choose our commutant χ a little more carefully: Namely, while we still need to be able to dominate χ by −χ0 on the interval [t1 , t2 ], we also have to require χ|t t2 is fixed, and χ1 (t) = φ1 ((t − t0 )/(t − t1 )) with φ1 non-negative, identically 0 on (−∞, 0] and identically 1 on [1, ∞). We then compute hχu, ˇ ∂t ui + h∂t u, χui ˇ = −hχ ˇ0 u, ui = −hχ0 u, uiL2 ([t0 ,t2 ]) + χ(t2 )|u(t2 )|2 p = k −χ0 uk2L2 ([t1 ,t2 ]) − hχ0 u, uiL2 ([t0 ,t1 ]) + χ(t2 )|u(t2 )|2 , hence, again using Cauchy-Schwartz and AM-GM, p p √ √ k −χ0 uk2L2 ([t1 ,t2 ]) ≤ k χuk2L2 ([t0 ,t2 ]) + k χ∂t uk2L2 ([t0 ,t2 ]) + k |χ0 |uk2L2 ([t0 ,t1 ]) p √ √ = k χuk2L2 ([t1 ,t2 ]) + k χ + |χ0 |uk2L2 ([t0 ,t1 ]) + k χ∂t uk2L2 ([t0 ,t2 ]) . (2.6) Choosing z big enough, we can absorb the first term on the right into the left hand side; thus, since −χ0 is bounded from below on [t1 , t2 ] and χ, χ0 are bounded in absolute value on all of [t0 , t2 ], we obtain (2.5). Sure enough, using positive commutators to prove these simple inequalities is a little overkill (although the given proofs are conceptually clearer than hands-on proofs), but we will see in the next section that without doing anything new, we can obtain similar energy estimates for the wave equation. 3. Energy estimates for the wave equation on R × M To keep things simple and computations easily manageable, we focus on the wave equation on a product manifold, that is, we consider the operator = ∂t2 − ∆M , where ∆M is the Laplace-Beltrami operator on the closed Riemannian manifold M . Proposition 3.1. There exists C > 0 such that for all u ∈ C ∞ (R × M ), u|t T fixed. Then for u ∈ C ∞ (R × M ), u|t
1. Introduction In this short and very elementary note, we will discuss the technique of proving estimates for solutions to certain partial differential equations using positive commutators, which is a widely used and powerful technique in particular in microlocal analysis. In order to keep the necessary prerequisites at the bare minimum, we won’t make any use of pseudodifferential operators and related microlocal machinery in the central parts of this note, but references will be given. Positive commutator estimates are particularly useful for proving energy and regularity estimates for hyperbolic equations like the wave equation, or, more generally, equations which involve some sort of “flow” (see remark 4.1 for references). To be more specific, consider the wave equation on R × M , M a closed (compact, boundaryless) manifold, given by u ≡ (∂t2 − ∆)u = f , and suppose u|t 0. We then have h∂t u, χui + hχu, ∂t ui = h[χ, ∂t ]u, ui = −hχ0 u, ui, where we integrate by parts for the first equality. Choosing χ non-negative and such that −χ0 ≥ 0 for 0 ≤ t ≤ Te , this together with Cauchy-Schwartz and AM-GM gives p √ √ k −χ0 uk2L2 ([0,Te ]) ≤ 2k χ∂t ukL2 ([0,Te ]) k χukL2 ([0,Te ]) √ √ ≤ k χ∂t uk2L2 ([0,Te ]) + k χuk2L2 ([0,Te ]) , (2.2) √ √ 2 where we split χ = χ · χ in order to be able to take L norms on the right hand side without worries. Rewriting this as Z Te √ (−χ0 − χ)|u|2 dt ≤ k χ∂t uk2L2 ([0,Te ]) , 0
we now see what we need, namely, say, −χ0 − χ ≥ −χ0 /2 on [0, Te ]. 2In the language of microlocal analysis, the principal symbol r of 1 [, V ] is given by the i derivative of the symbol of V along the Hamilton vector field Hp of the principal symbol p = σ2 () 2 of , and if we can make r positive, it can be written as q for some symbol q > 0, thus 1 [, V ]u = Q∗ Qu+ lower order terms. A priori control on the lower order terms (paired with a i solution u of u = f ) then gives, with some additional technical details, control on Qu, thus we get regularity information on u where q is positive. 3A related inequality is the Poincar´ e inequality kukL2 ([0,T ]) ≤ Ck∂t ukL2 ([0,T ]) for u ∈ 1 H0 ([0, T ]).
POSITIVE COMMUTATORS
The common way to find such a χ is to consider the C ∞ (R) function ( e−1/t , t > 0 φ(t) = 0, t ≤ 0,
3
(2.3)
satisfying φ0 (t) = t−2 φ(t). Throughout the rest of this note, φ is always going to denote this function. We then choose χ(t) = φ(z−1 (Te − t)) and compute χ0 (t) = −z−1 · (z−1 (Te − t))−2 χ(t) = −z(Te − t)−2 χ(t). Since for 0 ≤ t ≤ Te , z(Te − t)−2 ≥ zTe−2 , we can choose z sufficiently large to get −χ0 − χ ≥ −χ0 /2 on [0, Te ], as required. Plugging this into (2.2), we can absorb √ the k χukL2 ([0,Te ]) term into the left hand side and thus obtain p √ k −χ0 uk2L2 ([0,Te ]) ≤ Ck χ∂t uk2L2 ([0,Te ]) (2.4) for some C > 0. Now we see that we do not quite get (2.1), since −χ0 is not bounded from below on all of [0, Te ]; instead, using that −χ0 does have a positive lower bound on [0, T ] (T < Te ), we only get the weaker estimate kukL2 ([0,T ]) ≤ Ck∂t ukL2 ([0,Te ]) . To fix this insufficiency, note that if we cut off χ at time T , i.e. replaced χ(t) by χ(t)H(T − t), H being the Heaviside function, and still got the estimate (2.4), interpreted in a distributional sense (the left hand side being the distributional pairing −hχ0 u, ui), we would get an additional term involving the δ-distribution at t = T , i.e. p √ k −χ0 uk2L2 ([0,T ]) + χ(T )|u(T )|2 ≤ Ck χ∂t uk2L2 ([0,T ]) , and the boundary term χ(T )|u(T )|2 would be positive (put differently, it has the same sign as −χ0 ), thus could just be dropped, and we would get (2.1)! In fact, this idea works. To see this formally, let χ(t) ˇ = χ(t)H(T − t), then we can compute just as above (now with h , i denoting the distributional pairing, one side being a distribution, the other side a test function, made antilinear in the second slot) h∂t u, χui ˇ + hχu, ˇ ∂t ui = h[χ, ˇ ∂t ]u, ui = −hχ ˇ0 u, ui = χ(T )|u(T )|2 − hχ0 u, uiL2 ([0,T ]) , where, for the first equality, we just use the definition of the distributional derivative (which of course does the same as integration by parts), and in the last equality we use −χ ˇ0 = −χ0 H(T − ·) + χ(T )δT . We now proceed as before, dropping the boundary term χ(T )|u(T )|2 , and finally obtain4 (2.1). Remark 2.1. We note that (2.1) and (2.4) hold more generally for functions u ∈ C ∞ (Rt , L2 (M )) vanishing for t < 0, where M is any measure space, e.g. a Riemannian manifold. 4If one is suspicious about the use of distributions here, one can also write everything out in terms of integrals from 0 to T , and the boundary term χ(T )|u(T )|2 is precisely the boundary term one gets from integrating by parts. However, this procedure makes it seem like there was something special about the boundary term, although there is not, really.
4
PETER HINTZ
Next, we look at a propagation estimate. Proposition 2.2. Let −∞ < t0 < t1 < t2 < ∞. Then there is a constant C > 0 such that for all u ∈ C ∞ (R), kukL2 ([t1 ,t2 ]) ≤ C(kukL2 ([t0 ,t1 ]) + k∂t ukL2 ([t0 ,t2 ]) ).
(2.5)
Proof. We need to choose our commutant χ a little more carefully: Namely, while we still need to be able to dominate χ by −χ0 on the interval [t1 , t2 ], we also have to require χ|t t2 is fixed, and χ1 (t) = φ1 ((t − t0 )/(t − t1 )) with φ1 non-negative, identically 0 on (−∞, 0] and identically 1 on [1, ∞). We then compute hχu, ˇ ∂t ui + h∂t u, χui ˇ = −hχ ˇ0 u, ui = −hχ0 u, uiL2 ([t0 ,t2 ]) + χ(t2 )|u(t2 )|2 p = k −χ0 uk2L2 ([t1 ,t2 ]) − hχ0 u, uiL2 ([t0 ,t1 ]) + χ(t2 )|u(t2 )|2 , hence, again using Cauchy-Schwartz and AM-GM, p p √ √ k −χ0 uk2L2 ([t1 ,t2 ]) ≤ k χuk2L2 ([t0 ,t2 ]) + k χ∂t uk2L2 ([t0 ,t2 ]) + k |χ0 |uk2L2 ([t0 ,t1 ]) p √ √ = k χuk2L2 ([t1 ,t2 ]) + k χ + |χ0 |uk2L2 ([t0 ,t1 ]) + k χ∂t uk2L2 ([t0 ,t2 ]) . (2.6) Choosing z big enough, we can absorb the first term on the right into the left hand side; thus, since −χ0 is bounded from below on [t1 , t2 ] and χ, χ0 are bounded in absolute value on all of [t0 , t2 ], we obtain (2.5). Sure enough, using positive commutators to prove these simple inequalities is a little overkill (although the given proofs are conceptually clearer than hands-on proofs), but we will see in the next section that without doing anything new, we can obtain similar energy estimates for the wave equation. 3. Energy estimates for the wave equation on R × M To keep things simple and computations easily manageable, we focus on the wave equation on a product manifold, that is, we consider the operator = ∂t2 − ∆M , where ∆M is the Laplace-Beltrami operator on the closed Riemannian manifold M . Proposition 3.1. There exists C > 0 such that for all u ∈ C ∞ (R × M ), u|t T fixed. Then for u ∈ C ∞ (R × M ), u|t
