Matrix weighted Poincar\'e inequalities and applications to degenerate elliptic systems

0.4 MB·

by  Joshua Isralowitz

MATRIX WEIGHTED POINCARE´ INEQUALITIES
AND APPLICATIONS TO DEGENERATE ELLIPTIC
6 SYSTEMS
1
0
2 JOSHUA ISRALOWITZ AND KABE MOEN

r
p
A

3 Abstract. WeprovePoincar´eandSobolevinequalitiesinmatrix
1
A weighted spaces. We then use these Poincar´e inequalities to
p
] prove existence and regularity results for degenerate systems of
P
elliptic equations whose degeneracy is governed by a matrix A
p
A
weight. Such results parallel earlier results by Fabes, Kenig, and
.
h Serapioni for a single degenerate equation governed by a scalar
t A weight. In addition, we prove Cacciopoli and reverse Meyers
a p
m Ho¨lder inequalities for weak solutions of the degenerate systems.
[ As a means to prove the Poincar´e inequalities we show that the

Riesz potential and fractional maximal operators are bounded on
3
v matrix weighted Lp spaces and go on to develop an entire matrix
1 A theory.
p,q
1
1
0
0 1. Introduction
.
1
The classic Poincar´e inequality
0
6
1 1 1/q 1 1/p
v: Q |u(x)−uQ|qdx . |Q|d1 Q |∇u(x)|pdx ,
i (cid:18) | | ZQ (cid:19) (cid:18) | | ZQ (cid:19)
X
holds for all cubes Q in Rd when u is sufficiently smooth, 1 p < d,
r
≤
a and q = dp . Such inequalities are vital to the theory of regularity of
d−p
weak solutions to PDE. Fabes, Kenig, and Serapioni [10] studied the
degenerate elliptic equation
d
div(A(x) u(x)) = ∂ (Aβ(x)∂ u(x)) = (divf~)(x) (1.1)
∇ α α β −
α,β=1
X
2010 Mathematics Subject Classification. Primary 42B20.
Key words and phrases. Matrix A weights, Poincar´einequalities, elliptic PDE,
p
fractional operators.
The second is partially supported by the NSF under grant DMS 1201504.
1
2 JOSHUAISRALOWITZANDKABE MOEN
where A is a positive definite matrix that satisfies
w(x) ξ 2 A(x)ξ,ξ , ξ Rd
| | ≃ h i ∈
for some w A and f~ L2(Ω,w−1) for some domain Ω Rd. They
2
∈ | | ∈ ⊆
proved that weighted Poincar´e inequalities of the form
1/q
1
u(x) u qw(x)dx
Q
w(Q) | − |
(cid:18) ZQ (cid:19)
1/p
1
. Q d1 u(x) pw(x)dx
| | w(Q) |∇ |
(cid:18) ZQ (cid:19)
hold for some q > p when w A and used these inequalities to
p
∈
~
prove that weak solutions to (1.1) (under further assumptions on f)
are H¨older continuous.
In this paper we will more generally consider systems of degenerate
elliptic equations of the form
n d
∂ (Aαβ(x)∂ u (x)) = (divF) (x), i = 1,...,n (1.2)
α ij β j − i
j=1 α,β=1
X X
for n N not necessarily equal to d, where Aαβ C and
∈ ij ∈
n d
Aαijβ(x)ηβjηαi & kW(x)21ηk2, η ∈ Mn×d(C) (1.3)
i,j=1α,β=1
X X
and
n d
| Aαijβ(x)νβjηαi| . kW21(x)ηkkW21(x)νk, η,ν ∈ Mn×d(C)
i,j=1α,β=1
X X
(1.4)
for a matrix weight W (i.e. an a.e. positive definite (C) valued
n×n
M
function with locally integrable entries) and F L2(Ω,W−1) (which
∈
will be defined momentarily). To the best of our knowledge, it seems
that systems of elliptic equations whose degeneracies are governed by
matrix weights have never been considered before.
Given a matrix weight W and an exponent p > 0 we define Lp(Ω,W)
to be the collection of all Cn valued functions f~ such that
f~ p = Wp1(x)f~(x) pdx < .
k kLp(Ω,W) | | ∞
ZΩ
3
We will also sometimes let Lp(Ω,W) denote the space of all (C)
n×d
M
valued functions F whose norm above is finite. When Ω = Rd we will
write Lp(W).
A natural solution space for weak solutions of (1.2) is the matrix
weighted Sobolev space H1,p(Ω,W). Define the norm by
1 1
kf~kH1,p(Ω,W) = |Wp1(x)f~(x)|pdx p + kWp1(x)Df~(x)kpdx p
(cid:18)ZΩ (cid:19) (cid:18)ZΩ (cid:19)
where is any matrix norm. The space H1,p(Ω,W) is defined as the
k·k
completion of C∞(Ω) with respect to the norm .
k·kH1,p(Ω,W)
A matrix weight W belongs to A if
p
p
[W]Ap = sup Q1 Q1 kWp1(x)W−p1(y)kp′dy p′ dx < ∞.
Q | | ZQ(cid:18)| | ZQ (cid:19)
Note that when p = 2 we have that
1 1
[W] sup tr W(x)dx W−1(x)dx
A2 ≃ Q Q
Q (cid:18)| | ZQ (cid:19)(cid:18)| | ZQ (cid:19)
which says that the matrix A condition is especially easy to verify.
2
Treil-Volberg[26]showedthattheHilberttransform,definedcomponent-
wise, is bounded on L2(W) if and only if the matrix weight W belongs
to A . Nazarov-Treil and Volberg [20,27] when d = 1 and the first au-
2
thor [13] when d > 1 proved dyadic upper and lower matrix weighted
Littlewood-Paley Lp bounds when W is a matrix A weight. Further-
p
more, Goldberg [8] characterized the boundedness of singular integral
operators and the Hardy-Littlewood maximal operator by the matrix
A condition.
p
We are now ready to state our main results. We begin with Sobolev
and Poincar´e inequalities in the matrix weighted case.
Theorem 1.1. If 1 < p < and W is a matrix A weight, then there
p
∞
exists ǫ > 0 such that
1
1 Wp1(x)f~(x) p+ǫdx p+ǫ
Q | |
(cid:18)| | ZQ (cid:19)
1
. Q d1 1 Wp1(x)Df~(x) p−ǫdx p−ǫ
| | Q k k
(cid:18)| | ZQ (cid:19)
for each cube Q and f~ C1(Q).
∈ 0
4 JOSHUAISRALOWITZANDKABE MOEN
Theorem 1.2. If 1 < p < and W is a matrix A weight, then there
p
∞
exists ǫ > 0 such that
1
1 Wp1(x)(f~(x) f~Q) p+ǫdx p+ǫ
Q | − |
(cid:18)| | ZQ (cid:19)
1
. Q d1 1 Wp1(x)Df~(x) p−ǫdx p−ǫ
| | Q k k
(cid:18)| | ZQ (cid:19)
for each cube Q and f~ C1(Q).
∈
As will be apparent from the proof, note that we can in fact replace
the cube Q in Theorems 1.1 and 1.2 with an open ball B so long as we
have that f~ C1(B).
∈
It is well known that Poincar´e inequalities follow from bounds on the
fractional integral operators
f(y)
I f(x) = dy, 0 < α < d
α
x y d−α
ZRd | − |
and their corresponding fractional maximal operators
1
M f(x) = sup f(y) dy, 0 α < d.
α Q 1−α | | ≤
Q∋x | | d ZQ
Such operators play a crucial role in the theory of the smoothness of
functions. The fractional integral operator acts as an anti-derivative
and hence its boundedness implies the Sobolev embedding theorems.
While our Sobolev and Poincar´e inequalities will not, strictly speaking,
follow from matrix weighted bounds for fractional integral operators,
we will nevertheless be interested in proving such bounds for their own
sake.
First let us recall the results in the scalar case. Muckenhoupt and
Wheeden [19] characterized the weights w for which M and I are
α α
bounded on weighted Lebesgue spaces. In particular, they showed that
if 1 < p < d/α and q is defined by 1 = 1 α, then I and M are
q p − d α α
p
bounded from Lp(wq) to Lq(w) if and only if w Ap,q:
∈
q
[w]Ap,q = sup Q1 w(x)dx Q1 w(x)−pq′ dx p′ < ∞.
Q (cid:18)| | ZQ (cid:19)(cid:18)| | ZQ (cid:19)
5
Lacey et. al. [17], found the sharp uppers bounds on the operator
norms in terms of the constant [w] showing that
Ap,q
(1−α)p′
kMαkLp(wpq)→Lq(w) . [w]Ap,qd q (1.5)
and
(1−α)max(1,p′)
kIαkLp(wpq)→Lq(w) . [w]Ap,qd q . (1.6)
We will study the matrix weighted case of these results. Given a
matrix weight W and a pair of exponents p and q we define the matrix
A constant as follows
p,q
q
[W]Ap,q = sup Q1 Q1 kWq1(x)W−1q(y)kp′dy p′ dx,
Q | | ZQ(cid:18)| | ZQ (cid:19)
where the supremum is over all cubes contained in Rd. Amatrix weight
W belongs to A if [W] < . Moreover, we define the weighted
p,q Ap,q ∞
fractional maximal function as follows
1
MW,αf~(x) = sup Q 1−α |W1q(x)W−1q(y)f~(y)|dy
Q∋x | | d ZQ
where the supremum is over all cubes that contain x. We will be
concerned with Lp Lq bounds for M . Our first result is the
α,W
→
following.
Theorem 1.3. Suppose 0 < α < d, 1 < p < d and q is defined by
α
1 = 1 α. If W A then
q p − d ∈ p,q
p′(1−α)
M . [W]q d (1.7)
α,W Lp→Lq A
k k p,q
and this bound is sharp.
Inequality (1.7) is the matrix valued version of (1.5). In fact, the
sharpness of (1.7) follows from the scalar case because a better bound
for the matrix case would imply a better bound for the scalar case. We
remark that the proof is a modification of the arguments found in [1,8].
For the fractional integral operator we have the following result.
Theorem 1.4. Suppose 0 α < d, 1 < p < d/α and q is defined by
≤ p
1q = p1 − αd. If W ∈ Ap,q then Iα : Lp(Wq) → Lq(W) and
(1−α)p′+1
Iα p . [W]A d q (1.8)
k kLp(Wq)→Lq(W) p,q
6 JOSHUAISRALOWITZANDKABE MOEN
We do not believe the bound (1.8) is sharp and other methods will be
needed to find the sharp bound.
Using ourPoincar´einequalities weareabletoproveregularityresults
for weak solutions to (1.2). We begin with the following reverse H¨older
inequality. In the uniformly elliptic case, this result is due to Meyers.
Theorem 1.5. Let W be a matrix A weight, let Ω be a domain in
2
Rd, and let W−21F ∈ Lr(Ω) for some r > 2. If A = Aαijβ satisfies (1.3)
and (1.4), and if ~u H1,2(Ω,W) is a weak solution to (1.2), then there
∈
exists q > 2 such that given B Ω we have
2r
⊂
1
1 W12(x)D~u(x) qdx q . 1 W21(x)D~u(x) 2dx 21
B k k B k k
| r/2| ZBr/2 ! (cid:18)| r| ZBr (cid:19)
1
+ 1 W−21(x)F(x) qdx q .
B k k
(cid:18)| r| ZBr (cid:19)
In Section 6 we will also use some of the simple ideas in the very
recent paper [9] to extend this Meyers reverse H¨older inequality to
solutions of nonhomogenous degenerate p Laplacian systems with a
−
matrix A degeneracy (see the beginning of Section 5 for precise defi-
p
nitions.)
Finally, we end with the last of our main result: a regularity theorem
for weak solutions in dimension two.
Theorem 1.6. Let d = 2 and ~u be a weak solution to (1.2) when
F = 0. Then there exists ǫ > 0 such that for x,y Ω with x y <
∈ | − |
1dist( x,y ,Ωc), we have
2 { }
~u(x) ~u(y) . C x y ǫ
x,y
| − | | − |
where
1
Cx,y = sup Q11−ǫ kW−21(ξ)k2dξ 2
(cid:18) | | ZQ (cid:19)
where the supremum is over cubes Q Ω centered either at x or y,
⊂
and having side length smaller than x y .
| − |
As with Theorem 1.5, we will also extend 1.6 to solutions of homoge-
nous degenerate p-Laplacian systems setting in the last section. Note
that it would bevery interesting to know whether one can use Theorem
1.6 to prove continuity a.e. of weak solutions to (1.2) when F = 0.
7
In the special case when Aαβ(x) = B (x)δ for some (C)
ij ij αβ Mn×n
valued function B, the system (1.2) becomes
div(B(x)D~u(x)) = (divF)(x).
−
Such systems were considered by Iwaniec/Martin [14], Huang [12], and
Stroffolini [25]. Of particular interest is when B itself is a matrix A
2
weight, and Theorems 1.5 and 1.6 are of independent interest them-
selves in this case.
The plan of the paper will be as follows. In Section 2 we will state
some notation that will be used throughout the paper. In Section 3 we
will prove Theorem 1.3 and Theorem 1.4. We will prove the Poincar´e
and Sobolev inequalities in Section 4 and prove the existence results in
Section 5. Finally we finish the manuscript with the proof of the local
regularity of weak solutions including the proofs of the Meyers reverse
H¨older estimates (Theorem 1.5) and the local regularity in dimension
two (Theorem 1.6) in Section 6.
We will end this section by noting that K. Bickel, K. Lunceford,
and N. Mukhtar recently proved [4] that a matrix weight W : R
→
(C) of the form W (x) = a x γij is a matrix A weight if and
n×n ij ij 2
M | |
only if A = (a ) is positive definite, 1 < γ < 1, and each γ =
ij ii ij
−
(γ + γ )/2. Furthermore, it is possible (and will be investigated by
ii jj
the authors) that a similar result holds for matrix weights W : Rd
→
(C) of the same form, which would furnish a very concrete and
n×n
M
very interesting family of elliptic systems for which the results of this
paper hold.
2. Preliminaries
We will first need the notion of dyadic grid. Cubes will always be
assumedtohavesidesparalleltothecoordinateaxesandwewill denote
the side-length of a cube Q as ℓ(Q). A dyadic grid, usually denoted D
will be a collection of cubes that satisfy the following three properties:
(1) If Q D then ℓ(Q) = 2k for some k Z.
∈ ∈
(2) If Dk = Q D : ℓ(Q) = 2k , then Rd = Q.
{ ∈ } Q∈Dk
(3) If Q,P D then Q P is either ∅,Q, or P.
∈ ∩ S
We will use the following well known fact about dyadic grids whose
proof can be found in a recent manuscript by Lerner and Nazarov [18].
8 JOSHUAISRALOWITZANDKABE MOEN
Proposition 2.1. Let Dt = 2−k([0,1)d + m + ( 1)kt) : k Z,m
{ − ∈ ∈
Zd , then given any cube Q, there exists 1 t 2d and Q Dt such
t
} ≤ ≤ ∈
that Q Q and ℓ(Q ) 6ℓ(Q).
t t
⊂ ≤
We now establish the machinery of the matrix weights needed for
the paper. Given a cube Q, let V , V′ be a reducing operator (i.e. a
Q Q
positive definite n n matrix) where
×
e e
1 1
|VQ~e| ≈ Q1 |W−q1(x)~e|p′dx p′ , |VQ′~e| ≈ Q1 |Wq1(x)~e|qdx q .
(cid:18)| | ZQ (cid:19) (cid:18)| | ZQ (cid:19)
Ine fact we can pick the reducing operateors in such a way that
1 1
1 W−q1(x)~e p′dx p′ VQ~e √n 1 W−q1(x)~e p′dx p′
Q | | ≤ | | ≤ Q | |
(cid:18)| | ZQ (cid:19) (cid:18)| | ZQ (cid:19)
and a similar statement holds efor V′~e (see Proposition 1.2 in [8]).
| Q |
Using the reducing operators we see that
e
sup V V′ q [W]
k Q Qk ≈ Ap,q
Q
q
e e= sup 1 1 W1q(x)W−1q(y) p′dy p′ dx. (2.1)
Q Q k k
Q | | ZQ(cid:18)| | ZQ (cid:19)
Let ρ be a norm on Cn and let ρ∗ be the dual norm defined by
~
~e,f
ρ∗(~e) = sup Cn .
(cid:12)D ~E (cid:12)
f~∈Cn (cid:12) ρ(f) (cid:12)
(cid:12) (cid:12)
By elementary arguments we have that (ρ∗)∗ = ρ for any norm ρ. Also
let
1
ρq,Q(~e) = 1 Wq1(x)~e qdx q
Q | |
(cid:18)| | ZQ (cid:19)
so that ρ (~e) V′~e and by trivial arguments ρ∗ (~e) (V′)−1~e .
q,Q ≈ | Q | q,Q ≈ | Q |
3. Boeunds for fractional operators e
In this section we will prove Theorems 1.3 and 1.4. We begin with
some facts about the matrix A condition. Throughout this section
p,q
we will assume that 0 α < d and p and q and satisfy the Sobolev
≤
relationship
1 1 α
= .
q p − d
9
Proposition 3.1. W is an A weight if and only if the averaging
p,q
operators
1
~ Q ~
f f(x)dx
7→ Q 1−α
| | d ZQ
p
are uniformly bounded from Lp(Wq) to Lq(W).
Proof. The proofissimilar to Proposition2.1 in[8]. Inparticular, since
Lp(Wpq) is the dual space of Lp′(W−pq′) under the usual unweighted
pairing
~ ~
L (f) = f,~g
~g
L2(Rd;Cn)
D E
p
for ~g Lp(Wq), we have that
∈
1
Q ~ −1 ~
sup Q 1−α f(x)dx = sup |Q| p′ρq,Q f dx
kf~k p =1(cid:13)| | d ZQ (cid:13)Lq(W) kf~k p =1 (cid:18)ZQ (cid:19)
Lp(Wq) (cid:13) (cid:13) Lp(Wq)
(cid:13) (cid:13)
(cid:13) (cid:13) ~
f(x),~e dx
= sup sup Q −p1′ Q Cn
kf~k p =1~e∈Cn| | (cid:12)(cid:12)R D(ρq,Q)∗(E~e) (cid:12)(cid:12)
Lp(Wq) (cid:12) (cid:12)
1 ~e
Q p′
−1 k kLp′(W−q )
= sup Q p′
| | (ρ )∗(~e)
~e∈Cn q,Q
and the last term here being uniformly finite (with respect to all cubes
Q) is easily seen to be equivalent to W being an A weight. (cid:3)
p,q
3.1. The fractional maximal operator. Recallthatthenaturaldef-
inition of the maximal operator on matrix weighted spaces is given by
1
MW,αf~(x) = sup Q 1−α |W1q(x)W−q1(y)f~(y)|dy.
Q∋x | | d ZQ
We will also need the following auxiliary fractional maximal operator:
1
M′ f~(x) = sup V−1W−q1(y)f~(y) dy.
W,α Q 1−α | Q |
Q∋x | | d ZQ
Corollary 3.2. If M : Lp Lq bounededly then W is a matrix A
W,α p,q
→
weight.
Proof. For each cube Q containing x we have
1 (x) 1 (x)
Q 1 ~ Q 1 ~
Wq(x)f(y)dy Wq(x)f(y) dy
Q 1−α ≤ Q 1−α | |
(cid:12)| | d ZQ (cid:12) | | d ZQ
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
10 JOSHUAISRALOWITZANDKABE MOEN
1 ~
MW,α(Wqf)
≤
so that
1
Q ~ 1 ~
suQp(cid:13)|Q|1−αd ZQf(y)dy(cid:13)Lq(W) ≤ (cid:13)MW,α(Wqf)(cid:13)Lq
(cid:13) (cid:13)
(cid:13)(cid:13) (cid:13)(cid:13) . (cid:13)(cid:13)f~ p . (cid:13)(cid:13)
k kLp(Wq)
(cid:3)
Corollary 3.3. If W is a matrix A weight then for any unit vector
p,q
~e we have that Wq1~e q is a scalar Ap,q weight with Ap,q characteristic
| |
. [W]A .
p,q
~
Proof. Let φ be any scalar function and let f = φ~e. By Proposition
3.1, we have that
1
Q
φ φ(x)dx
7→ Q 1−α
| | d ZQ
are uniformly bounded from the scalar weighted space Lp( W1q~e p) to
| |
the scalar weighted space Lq( Wq1~e q). But Proposition 3.1 again in
| |
scalar setting then gives us that Wq1~e q is a scalar Ap,q weight with
| |
A characteristic . [W] . (cid:3)
p,q Ap,q
Corollary 3.4. W is an Ap,q weight if and only if W−pq′ is an Aq′,p′
weight.
Proof. ByProposition3.1andduality, wehavethatW isanA weight
p,q
if and only if the averaging operators
1
~ Q ~
f f(x)dx
7→ Q 1−α
| | d ZQ
are uniformly bounded from Lq′(W−qq′) to Lp′(W−pq′). However, we
also have that
1 1 α
= .
p′ q′ − d
Another application of Proposition 3.1 tells us that W−pq′ is an Aq′,p′
weight if and only if W is an A weight. (cid:3)
p,q
Remark. Let r = 1+ q. These two corollaries also imply that the A
p′ r
characteristic of each W−q1~e p′ is bounded by [W]r′−1.
| | Ap,q