Embedding theorems for composition of homotopy and projection operators
Niu et al. Journal of Inequalities and Applications
Embedding theorems for composition of homotopy and projection operators
Jinling Niu 1
Shusen Ding 0
Yuming Xing 1
0 Department of Mathematics, Seattle University , Seattle, WA 98122 , USA
1 Department of Mathematics, Harbin Institute of Technology , Harbin, 150001 , China
We prove both local and global embedding theorems with Lϕ -norms for the composition of the homotopy operator and projection operator applied to differential forms. We also establish some Lϕ -norm inequalities for certain compositions of the related operators. MSC: Primary 35J60; secondary 35B45; 30C65; 47J05; 46E35
embedding inequalities; differential forms; homotopy operators; projection operators
-
T H(u) – T H(u)
W,ϕ( ) ≤ C u Lϕ( ),
(.)
where is any bounded domain in Rn, n ≥ , ϕ : [, ∞) → [, ∞) with ϕ() = is a Young
function satisfying certain conditions described later, and C is a constant independent
of the differential form u. In order to establish the above main Lϕ -embedding inequality,
we also prove the Poincaré inequality and some inequalities with Lϕ -norm for the related
compositions of operators.
We keep using the traditional notations throughout this paper. Let B and σ B be the balls
with the same center and diam(σ B) = σ diam(B). Let |E| be the n-dimensional Lebesgue
measure of a set E ⊆ Rn. In this paper, we treat a ball same as a cube and use uB = |B| B u dx
to denote the average of a function u. Let ∧l = ∧l(Rn) be the set of all l-forms in Rn,
D ( , ∧l) be the space of all differential l-forms in , and Lp( , ∧l) be the l-forms u(x) =
I uI (x) dxI in satisfying |uI |p < ∞ for all ordered l-tuples I, l = , , . . . , n. We denote
the exterior derivative by d and the Hodge star operator by .
The definition of the operator Ky with the case y = and its generalized version can
be found in [, ]. To each y ∈ there corresponds a linear operator Ky : C∞( , ∧l) →
C∞( , ∧l–) defined by (Kyω)(x; ξ, . . . , ξl–) = tl–ω(tx + y – ty; x – y, ξ, . . . , ξl–) dt and the
decomposition ω = d(Kyω) + Ky(dω). A homotopy operator T : C∞( , ∧l) → C∞( , ∧l–)
is defined by averaging Ky over all points y ∈ : T ω = φ(y)Kyω dy, where φ ∈ C∞( ) is
normalized so that φ(y) dy = . For each differential form u, we have the decomposition
(.)
(.)
(.)
(.)
× ∧l(Rn) → ∧l(Rn) and
and
u = d(Tu) + T (du)
∇(Tu) p,B ≤ C|B| u p,B and
Tu p,B ≤ C|B| diam(B) u p,B.
From [], p., we know that any open subset in Rn is the union of a sequence of cubes
Qk , whose sides are parallel to the axes, whose interiors are mutually disjoint, and whose
diameters are approximately proportional to their distances from F, where F is the
complement of in Rn. Specifically, (i) = k∞= Qk , (ii) Qj ∩ Qk = φ if j = k, (iii) there exist two
constants c, c > (we can take c = , and c = ), so that c diam(Qk) ≤ distance(Qk, F) ≤
c diam(Qk). Thus, the definition of the homotopy operator T can be generalized to any
domain in Rn: For any x ∈ , x ∈ Qk for some k. Let TQk be the homotopy operator
defined on Qk (each cube is bounded and convex). Thus, we can define the homotopy
operator T on any domain by T = k∞= TQk χQk (x). The nonlinear partial differential
equation for differential forms
d A(x, du) = B(x, du)
is called a non-homogeneous A-harmonic equation, where A :
B : × ∧l(Rn) → ∧l–(Rn) satisfy the conditions:
A(x, ξ ) ≤ a|ξ |p–,
A(x, ξ ) · ξ ≥ |ξ |p
and
B(x, ξ ) ≤ b|ξ |p–
for x ∈ a.e. and all ξ ∈ ∧l(Rn). Here p > is a constant related to the equation (.), and
a, b > . See [–] for recent results on the A-harmonic equations and related topics.
Assume that ∧l is the lth exterior power of the cotangent bundle, C∞(∧l ) is the space of
smooth l-forms on and W(∧l ) = {u ∈ Lloc(∧l ) : u has generalized gradient}. The
harmonic l-fields are defined by H(∧l ) = {u ∈ W(∧l ) : du = d u = , u ∈ Lp for some <
p < ∞}. The orthogonal complement of H in L is defined by H⊥ = {u ∈ L : u, h =
for all h ∈ H}. Then the Green’s operator G is defined as G : C∞(∧l ) → H⊥ ∩ C∞(∧l )
pϕ(t) ≤ tϕ (t) ≤ qϕ(t), < p ≤ q < ∞.
ctp – c ≤ ϕ(t) ≤ c tq + .
The first inequality in (.) is equivalent to that ϕt(pt) is increasing, and the second inequality
in (.) is equivalent to -condition, i.e., for each t > , ϕ(t) ≤ K ϕ(t), where K > , and
ϕ(qt) is decreasing with t. Also, condition (.) implies that ϕ(t) satisfies
t
by assigning G(u) be the unique element of H⊥ ∩ C∞(∧l ) satisfying Poisson’s equation
G(u) = u – H(u), where H is the harmonic projection operator that maps C∞(∧l ) onto
H so that H(u) is the harmonic part of u. See [] for more properties of these operators.
2 Local embedding theorem
The purpose of this section is to prove the local Lϕ -embedding theorem and some related
Lϕ -norm inequalities that will be used to prove the global embedding theorem in the next
section. We first recall the following subclass of Young functions that can be found in
[–].
Definitio (...truncated)