6
∇(RH)∗[(1−ϕ)(RHm
Lf1)mL]
6 Z
L\B(z,2A)
1
|x−y|d+1|RmHLf1(y)|dmL(y) 6
Z
L\B(z,2A)
dmL(y)
|x−y|2d+2 1/2Z
L\B(z,2A)
|RHmLf1(y)|2dmL(y) 1/2
6[CA−(d+2)]1/2[CAdkfk2L∞(mL)]1/2
=CA−1kfkL∞(mL).
8. Weak limits
This section has two main goals. The first one is to define the Riesz transform operators Rµ (and theirH-restricted versionsRHµ)in L2(µ)for arbitrary good measures µas weak limits of the regularized operators Rµ,δ as δ!0+. The second one is to show that when a sequence of uniformly good measures µk tends weakly (over the space of compactly supported continuous functions in Rd+1) to some other measure µ in Rd+1, then the limiting measure µ is also good and for all compactly supported Lipschitz functions f (scalar-valued)and g (vector-valued)in Rd+1, we have R
hRµkf, gidµk!R
hRµf, gidµ.
Our starting point is to fix two compactly supported Lipschitz functionsf andgin Rd+1, where f is scalar-valued andg is vector-valued, and to use the antisymmetry of the kernelsKδ to write the scalar producthRµ,δf, giµ as
Iδ(f, g) = Z Z
hKδ(x−y)f(y), g(x)idµ(x)dµ(y) = Z Z
hKδ(x−y), H(x, y)idµ(x)dµ(y),
where
H(x, y) =12[f(y)g(x)−f(x)g(y)].
The vector-valued functionH(x, y) is compactly supported and Lipschitz onRd+1×Rd+1, so the integralIδ(f, g) converges absolutely as an integral of a bounded function over a set of finite measure for everyδ >0 and every locally finite measure µ. Moreover, since H vanishes on the diagonalx=y, we have
|H(x, y)|6C(f, g)|x−y|
for allx, y∈Rd+1. Ifµis nice, then
Z
B(x,r)
dµ(y)
|x−y|d−16Cr
for allx∈Rd+1 andr>0. Therefore, denoting suppf∪suppg byS, we get Z Z
x,y:|x−y|<r
|H(x, y)|
|x−y|d dµ(x)dµ(y)6C(f, g) Z
S
Z
y:|x−y|<r
dµ(y)
|x−y|d−1
dµ(x) 6C(f, g)µ(S)r.
In particular, takingr=diamS here, we conclude that the full integral Z Z |H(x, y)|
|x−y|d dµ(x)dµ(y) = Z Z
S×S
<∞.
Since|K(x)|=|x|−d and|Kδ(x)−K(x)|6|x|−dχB(0,δ)(x), we infer that the integral I(f, g) =
Z Z
hK(x−y), H(x, y)idµ(x)dµ(y)
converges absolutely and, moreover, there exists a constant C depending onf, g, and the growth constant ofµonly such that|Iδ(f, g)−I(f, g)|6Cδfor allδ >0.
This already allows one to define the bilinear form hRµf, giµ=I(f, g)
and to establish the existence of the limit operatorRµ=limδ!0+Rµ,δ as an operator from the space of Lipschitz functions to its dual for every nice measureµ.
However, ifµis good, we can say much more. Indeed, in this case the bilinear forms hRµ,δf, giµ=
Z
hRµ,δf, gidµ
make sense and satisfy the inequality
|hRµ,δf, giµ|6CkfkL2(µ)kgkL2(µ)
for allf, g∈L2(µ). Since the space of compactly supported Lipschitz functions is dense inL2(µ), we can write anyL2(µ) functionsf and gas f1+f2 andg1+g2, where f1 and g1 are compactly supported Lipschitz functions in Rd+1, and f2 and g2 have as small norms inL2(µ) as we want. Splitting
hRµ,δf, giµ=hRµ,δf1, g1iµ+[hRµ,δf1, g2iµ+hRµ,δf2, giµ],
we see thathRµ,δf, giµ can be written as a sum of the quantityhRµ,δf1, g1iµ=Iδ(f1, g1), which converges to a finite limit I(f1, g1) asδ!0+ and another quantity that stays as small as we want asδ!0+if theL2(µ) norms off2andg2are chosen small enough. From here we conclude that the limit ofhRµ,δf, giµ as δ!0+exists for all f, g∈L2(µ). More-over, this limit is a bilinear form inL2(µ) and it is still bounded by CkfkL2(µ)kgkL2(µ). By the Riesz–Fischer theorem, there exists a unique bounded linear operatorRµinL2(µ) such that this bilinear form is equal tohRµf, giµ. The convergence
hRµ,δf, giµ!hRµf, giµ as δ!0+
can be restated as the weak convergence of the operatorsRµ,δ toRµ.
Similarly, one can consider the duality coupling of Lp(µ) and Lq(µ), where p, q >1 andp−1+q−1=1, and use the uniform boundedness of the operatorsRµ,δ inLp(µ) to es-tablish the existence of the weak limit of the operatorsRµ,δinLp(µ) asδ!0+. Note that, iff∈Lp1(µ)∩Lp2(µ), then for everyg∈L∞(µ) withµ(suppg)<∞, the valuehRµ,δf, giµ
can be computed using the pointwise integral definition ofRµ,δf as Rδ(f µ), so it does not depend on whetherf is considered as an element ofLp1(µ) or an element ofLp2(µ).
Thus
hRµf, giµ= lim
δ!0+
hRµ,δf, giµ
also does not depend upon that (note thatg∈Lq1(µ)∩Lq2(µ), so the left-hand side makes sense in both cases). Sincegis arbitrary here, we conclude thatRµf(as a function defined µ-almost everywhere) is the same in both cases.
Another important observation is that if the pointwise limit limδ!0+Rµ,δf exists on a set E with µ(E)>0, then Rµf coincides with that limit µ-almost everywhere on E.
To prove it, just observe that, by Egorov’s theorem, we can exhaustEby sets of finiteµ measure on which the convergence is uniform.
At last, if Rµ,δ converges strongly in L2(µ), then the limit is still the same as the weak limit we constructed.
The analogous theory can be built for RH, R∗, and (RH)∗. We built it only for the full operatorRbecause projecting everything toH is trivial andR∗ does not really require a separate theory due to relation (3), which shows that, at least in principle, we can always view R∗ just as a fancy notation for the right-hand side of (3). From now on, we will always understand R(f µ) on suppµ as Rµf whenever µ is good and f∈Lp(µ) for somep∈(1,∞). As we have shown above, this convention is consistent with other reasonable definitions in the sense that when some other definition is applicable somewhere on suppµas well, the value it gives coincides withRµf except, maybe, on a set of zeroµmeasure.
The idea of defining Rµ as a weak limit of Rµ,δ goes back to Mattila and Verdera [MV]. They prove its existence in a slightly more general setting and their approach is somewhat different from ours. They also show that Rµf can be defined pointwise by some formula that is almost the expression for the principal value
lim
δ!0+
Z
y:|x−y|>δ
K(x−y)f(y)dµ(y)
but not quite. Note that Mattila, Preiss, and Tolsa showed that the existence of the principal valueµ-almost everywhere is strong enough to imply the rectifiability ofµ(see [MP] and [T1]), so for a while there was a hope that the Mattila–Verdera result would eventually lead to the proof of the rectifiability conjecture. However, as far as we can tell, nobody still knows how to get a proof in this way and we will use a different route below.
We have just attained the first goal of this section: the construction of the limiting operator Rµ for one fixed good measure µ. We now turn to the relations between the operatorsRµ corresponding to different measures µ.
We start with the case when a positive measureν has a bounded Borel measurable densitypwith respect to a good measureµ. Sinceν(B(x, r))6kpkL∞(µ)µ(B(x, r)), we see thatν is nice. To show thatν is good, note that for everyf∈L2(ν), we havepf∈L2(µ).
Moreover, we have the identity
Rδ(f ν) =Rδ(pf µ) pointwise inRd+1, whence
Z
|Rδ(f ν)|2dν= Z
|Rδ(pf µ)|2p dµ6CkpkL∞(µ) Z
|pf|2dµ6Ckpk2L∞(µ)
Z
|f|2dν due to the goodness ofµ. Thus, both operators Rν and Rµ exist. Now take anyf, g∈ L2(ν) and write
hRν,δf, giν=hRµ,δ(pf), pgiµ.
Passing to the limit on both sides asδ!0+, we conclude that hRνf, giν=hRµ(pf), pgiµ=hRµ(pf), giν
(note that the functionRµ(pf) is definedµ-almost everywhere, so it is also defined ν -almost everywhere). However, the mapping f7!Rµ(pf) is a bounded linear operator fromL2(ν) toL2(µ)⊂L2(ν), so we conclude that
Rνf=Rµ(pf) ν-almost everywhere.
This identity is, of course, by no means surprising. Still, since we will use it several times without mentioning, we decided it would be prudent to include a proof. The next property we need is a bit subtler.
Suppose thatµk (k>1) is a sequence of uniformly nice measures that converges to some locally finite measure µ weakly over the spaceC0(Rd+1) of compactly supported continuous functions inRd+1. We shall start with showing that µ is also nice. Indeed, take any ball B(x, r). Then µ(B(x, r)) can be found as the supremum of all integrals Rf dµwith continuous functionsf such that 06f61 and suppf⊂B(x, r). However, for every suchf, we have
Z
f dµ= lim
k!∞
Z
f dµk6sup
k
µk(B(x, r))6Crd,
whereCis the uniform growth constant ofµk, so we have the same bound forµ(B(x, r)).
Fix two compactly supported Lipschitz functionsf andg. The bilinear form hRµf, giµ
can be defined asI(f, g) for every nice measureµ. Once we know thatµis nice, we can say that
|hRµk,δf, giµk−hRµkf, giµk|6Cδ for allk>1 and also
|hRµ,δf, giµ−hRµf, giµ|6Cδ
with someC >0 depending only onf, g, and the uniform growth constant ofµk. Note, however, that for every fixedδ >0,
hRµk,δf, giµk= Z Z
hKδ(x−y)f(y), g(x)idµk(x)dµk(y)
and the integrand is a compactly supported Lipschitz function inRd+1×Rd+1, which is more than enough to ensure that
hRµk,δf, giµk!hRµ,δf, giµ
for every fixedδ >0 ask!∞. Since the convergencehRµk,δf, giµk!hRµkf, giµkasδ!0+ is uniform ink, we conclude that
hRµkf, giµk!hRµf, giµ as well.
It remains to show that if µk are uniformly good, then µ is also good, so all the bilinear forms in question can be also interpreted as L2(µ) couplings. Return to the regularized operatorsRµ,δ and note that the uniformC-goodness ofµk implies that
|hRµk,δf, giµk|6CkfkL2(µk)kgkL2(µk).
Since|f|2 and|g|2are compactly supported Lipschitz functions, we can pass to the limit on both sides and get
|hRµ,δf, giµ|6CkfkL2(µ)kgkL2(µ).
However, the operators Rµ,δf are well defined pointwise for every f∈L2(µ) and are bounded fromL2(µ) toL2loc(µ) as soon asµis merely nice. Using the fact that, for every bounded open setU, the space of Lipschitz functions compactly supported inside U is dense inL2(U, µ) and this a-priori boundedness, we conclude thatkRµ,δkL2(µ)!L2(U,µ)6C regardless of the choice of U. The monotone convergence lemma then shows that kRµ,δkL2(µ)!L2(µ)6C as well, finishing the story.