Some higher norm inequalities for composition of power operators

In this paper, we first prove the local Ls\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{s}$\end{document} norm estimate of composite operators △kGm(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\triangle^{k}G^{m}(u)$\end{document} by use of the Ls\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{s}$\end{document} norm of u. Then we establish the local and global higher norm inequalities of △kGm(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\triangle^{k}G^{m}(u)$\end{document}. Simultaneously, we also give a global higher norm estimate with Radon measure. Finally, as applications of these results, we give two examples to estimate the higher norm of △kGm(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\triangle^{k}G^{m}(u)$\end{document}.


Introduction
In this paper, our purpose is investigating some higher norm inequalities for composition of power operators k G m on a bounded domain M, where k, m are positive integers, is Laplace-Beltrami operator, and G is Green's operator. The norm estimate of operators applied to differential forms is an important and interesting research topic in some areas of mathematic analysis and has achieved fruitful results; see [1][2][3][4][5][6][7][8][9][10][11] for more detail. Some these results improved the development of some other branches of mathematics and mathematical physics; see [12][13][14][15][16][17][18] for details. In previous related research about norm estimates of operators the study mostly concentrated on estimates of the L t norm of operators and applying them to differential forms in terms of the L t norm of differential forms. Therefore, if s > t, then we could not estimate the L s norm of operators by the L t norm of differential forms from the literature. This motivated us to research the higher norm of operators than of differential forms. Since the norm estimate of a composite operator is more complicated than that of a single operator, in this paper, we choose the composition of power operators k G m to be the research object. In this paper, we first give the local L s norm estimate of k G m (u) by the L s norm of u in Theorem 2.5. Then based on Theorem 2.5, we prove the local and global higher norm inequalities of k G m (u) separately presented in Theorems 2.6-2.8 and 3.2. Simultaneously, we also establish the global higher norm estimate with Radon measure in Theorem 3.3. Finally, we give two examples as applications of Theorem 3.2.
We start this paper by introducing some notations and definitions in [19]. Let M ⊂ R n (n ≥ 2) be a bounded domain, B be a ball, and σ B be the ball with the same center as B satisfying diam(σ B) = σ diam(B). We do not distinguish balls from cubes in this paper. By Λ k = Λ k (R n ) (k = 1, 2, . . . , n) we denote the linear space of all k-forms u(x) = I u I (x) dx I = u i 1 i 2 ···i k dx i 1 Λ dx i 2 Λ · · · Λ dx i k with summation over all ordered k-tuples I = (i 1 , i 2 , . . . , i k ), 1 ≤ i 1 < i 2 < · · · < i k ≤ n, k = 1, 2, . . . , n. If the coefficient u I (x) of k-forms is differentiable on M, then we call u(x) a differential k-form on M. As usual, we use C ∞ (M, Λ k ) to denote the space of smooth k-forms in a domain M, D (M, Λ k ) to denote the space of all differential k-forms. Let L p (M, Λ k ) be the set of differential k-forms and then L p (M, Λ k ) is a Banach space. As usual, we use to denote the Hodge star operator and dist(x, M) to denote the distance of the point x from the set M. Also, we use d : D (M, Λ k ) → D (M, Λ k+1 ) to denote the differential operator and d : The n-dimensional Lebesgue measure of a set E ⊆ R n is denoted by |E|. For any differential form u, the average of u over B is defined as u B = 1 |B| B u dx. All integrals involved in this paper are the Lebesgue integrals. The Laplace-Beltrami operator is defined by = dd + d d. We define Green's operator G on the space of smooth k-forms in M by setting G(u) to be a solution of Poisson's equation where H is the harmonic projection; see [1,7,[19][20][21][22] for more detail about the Laplace-Beltrami operator , Green's operator G, and projection operator H. We call w a weight if w ∈ L 1 loc (R n ) and w > 0 a.e. For any Radon measure ν defined by dν = w(x) dx, we define the L p -norm of a measurable function f with Radon measure over M by (1.2) and the Radon measure of E by ν(E) = E dν = E w(x) dx. The nonlinear partial differential equation satisfies the following conditions: for almost every x ∈ M and all ξ ∈ Λ k (R n ), where a > 0 is a constant, and 1 < p < ∞ is a fixed exponent associated with (1.3). For more details about A-harmonic equation, see [3,4,8,19,23,24].

Local higher norm inequalities for k G m (u)
In this section, we first give the L s norm estimate of k G m (u) in terms of the L s norm of u.
Then based on this result, we establish local higher norm inequalities for k G m (u) in two cases.
Let ψ be a strictly increasing convex function on [0, +∞) with ψ(0) = 0, and let u be a differential form on a bounded domain M. Then ψ(κ|u| + |u M |) ∈ L 1 (M, ν) for any real number κ > 0 and ν({x ∈ M : |uu M | > 0}) > 0, where ν is the Radon measure defined by dν = w(x) dx for a weight w(x). For any positive constant a, we have for some constants C 1 > 0 and C 2 > 0. Let ψ(u) = u s , s > 1, w(x) = 1, and let M be a ball B in this inequality. Then there exist two positive constants C 3 and C 4 , independent of u, such that for a cube or a ball Q in R n , where l = 0, 1, 2, . . . , n -1 and 1 < p < n.

Lemma 2.4 Let u be a smooth differential form defined on M, G be Green's operator, and be the Laplace-Beltrami operator defined by
Then the Laplace-Beltrami operator and Green's operator G are commutable, that is, Proof From [19, p. 88] we find that Green's operator G commutes with d and d . Therefore, for any differential form u, we have

From (2.3) and the definition of the Laplace-Beltrami operator we obtain
Thus we complete the proof of Lemma 2.4. Now we will give the following local L s -norm estimate of composite operator k G m , which will be used in the proof of higher norm theorems.
. . , n) be a smooth differential form defined on M, G be Green's operator, and be the Laplace-Beltrami operator. Then there exists a constant C, independent of u, such that for all balls B with σ B ⊂ M and any positive integer k ≤ m, where σ > 1.
Next, based on Theorem 2.5, we will prove local higher norm inequalities for the composite operator k G m in two cases. Theorem 2.6 Let u ∈ L t loc (M, Λ l ) be a smooth differential form on M, G be Green's operator, and be the Laplace-Beltrami operator, l = 1, 2, . . . , n, 1 < t < n. Then k G m (u) ∈ L s loc (M, Λ l ) for any s such that 0 < s < nt/(nt). Moreover, there exists a constant C, independent of u, such that for all balls B with σ B ⊂ M and |B| > d 0 and any positive integer k < m, where σ > 1 and d 0 > 0 are constants.
Proof We prove this theorem in the following two cases: (1) First, assume that the mea- where σ 2 > σ 1 > 1 with σ 2 B ⊂ M.
(2) Second, if the the measure |{x ∈ B : | k G m (u)-( k G m (u)) B | > 0}| > 0, then (2.1) holds for k G m (u), and thus we have Note that G (u) = G(u) and k < m. Then d k G m (u) = dG( k G m-1 (u)) and k ≤ m -1. Thus, combining Lemma 2.1 and Theorem 2.5, we have where σ 4 > σ 3 > 1 are constants such that σ 4 B ⊂ M. Since 1 < t < n, from Lemma 2.3 and (2.18) we have By the monotonicity property of the L t -space, we obtain the inequality for any s such that 0 < s < nt/(nt In Theorem 2.6, since 1 < t < n, then nt n-t → +∞ as t → n -. Since 0 < s < nt/(nt), s can be greater than t, and thus the composite operator k G m (u) has higher norm than the differential form u. Next, we prove (2.14) for t ≥ n. (2) Next, let the measure |{x ∈ B : | k G m (u) -( k G m (u)) B | > 0}| > 0. Select p = max{1, s/t} and r = npt n+pt . Then 1 < r < n. Since t ≥ n, that is, nt ≤ 0, we have rt =

t(p(n-t)-n) n+pt
< 0, that is, r < t. Since 1 < r < n, Lemma 2.3 holds for k G m (u) and r. Thus replacing u and p by k G m (u) and r, respectively, in Lemma 2.3, we have  Since the measure |{x ∈ B : | k G m (u) -( k G m (u)) B | > 0}| > 0, estimates (2.1) hold for k G m (u). In the second half of (2.1), we replace u and s by k G m (u) and nr/(nr): Since p = max{1, s/t}, we have pt ≥ s. Since nr/(nr) = pt, we have nr/(nr) ≥ s, and thus applying the monotonicity property of the L t space, we have Inequality (2.30) implies (2.23). Therefore we complete the proof of this theorem.
In Theorem 2.7, from the condition t ≥ n we have 1 s + 1 n -1 t > 0, which is also presented in Theorem 2.6, and thus combining Theorems 2.6 and 2.7, we easily obtain the following theorem for any t > 1.

Global higher norm inequalities for composite operator k G m
In this section, based on Theorem 2.8, we will prove the global higher norm estimate for the composite operator k G m (u) in any bounded domain M ⊂ R n . Then we will establish the corresponding global higher norm estimate with Radon measure. In the following proof of related theorems, we need the following modified Whitney cover in [25]; see [23] for more detail about Whitney covers.

Lemma 3.1
Each Ω ⊂ R n has a modified Whitney cover of cubes W = {Q i } that satisfy for all x ∈ R n and some N > 1, and if Q i ∩ Q j = ∅, then there exists a cube R in Q i ∩ Q j such that Q i ∪ Q j ⊂ NR. Moreover, if Ω is a δ-John, then there is a distinguished cube Q 0 ∈ W that can be connected with every cube Q ∈ W by a chain of cubes Q 0 , Q 1 , . . . , Q k from W and such that Q ⊂ ρQ i , i = 1, 2, . . . , k, for some ρ = ρ(n, δ). Now we will give the global higher norm inequality for the composite operator k G m (u) based on Theorem 2.8.
where N > 1 is some constant. Hence, for u ∈ L t loc (M, Λ l ) , using Theorem 2.8, we have where C 2 = C 1 N is independent of u and all B i . Thus we complete the proof of Theorem 3.2.
In Theorem 3.2, if we assume that s > t, then Theorem 3.2 reduces to the global higher norm estimate for composite operator k G m . Next, we consider the following global norm comparison equipped with Radon measure based on Theorem 3.2. Therefore by the continuity and nonnegativity of 1 we have that there exists C 2 > 0 such that Therefore by the continuity and nonnegativity of 2 we have that there exists a constant C 5 > 0 such that for all x ∈ M 1 . Therefore we obtain where C 6 = max{ where C 7 is independent of u. Thus we complete the proof of Theorem 3.3.
In Theorem 3.3, choosing 1 (y) = y p and 2 (y) = y -q , 0 < p, q < ∞, we easily obtain the following corollary. for any positive integer k < m and any real numbers 0 < p, q < ∞.

Applications
In many cases, it is very difficult to give the norm estimate for a composite operator. In this section, we give two examples to obtain the upper bounds for the norm of the composite operator k G m (u) as applications of Theorem 3.2.
1 + x 2 2 ≤ a 2 } ⊂ R 2 , and let u be the differential 1-form defined on M. Then |M| = πa 2 , and u is a differential form satisfying the conditions of x 1 for any positive integer k < m.
Remark Examples 4.1 and 4.2 can be generalized to the n-dimensional space.

Conclusion
In this paper, we first give the local L s norm estimate for the composite operators k G m (u) in terms of the L s norm of u with commuting Laplace-Beltrami operator and Green's operator G. At the same time, we also obtain the local and global L s norms of k G m (u) in terms of the L t norm of differential forms u for any constant s > 0 such that 1 s + 1 n -1 t > 0. Then we establish a global higher norm estimate with Radon measure for composite operators k G m . At last, as applications of these results, we give two examples to estimate the higher norm of k G m (u).