Norms of the composition of the maximal and projection operators. (English) Zbl 1189.35036

Xing, Yuming; Ding, Shusen

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 12, 4614-4624 (2010).

Summary: We develop different norm estimates for the compositions of the maximal and projection operators, applied to the solutions of a nonlinear elliptic partial differential equation. As applications, we also obtain some estimates related to the Jacobian and exponents of mappings.

MSC:

35B45	A priori estimates in context of PDEs
26B10	Implicit function theorems, Jacobians, transformations with several variables
31B10	Integral representations, integral operators, integral equations methods in higher dimensions
31C45	Other generalizations (nonlinear potential theory, etc.)
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J60	Nonlinear elliptic equations

Keywords:

differential forms; harmonic equations

References:

[1]	Warner, F. W., Foundations of Differentiable Manifolds and Lie Groups (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0516.58001
[2]	Xing, Y.; Ding, S., Norm comparison inequalities for the composite operator, J. Inequal. Appl. (2009), Article ID 212915 · Zbl 1165.26013
[3]	Xing, Y.; Wu, C., Global weighted inequalities for operators and harmonic forms on manifolds, J. Math. Anal. Appl., 294, 294-309 (2004) · Zbl 1058.58010
[4]	Ding, S., Two-weight Caccioppoli inequalities for solutions of nonhomogeneous \(A\)-harmonic equations on Riemannian manifolds, Proc. Amer. Math. Soc., 132, 2367-2375 (2004) · Zbl 1127.35021
[5]	Ding, S.; Sylvester, D., Weak reverse Hölder inequalities and imbedding inequalities for solutions to the \(A\)-harmonic equation, Nonlinear Anal., 51, 783-800 (2002) · Zbl 1012.31006
[7]	Liu, B., \(A_r(\lambda)\)-weighted Caccioppoli-type and Poincaré-type inequalities for \(A\)-harmonic tensors, Int. J. Math. Math. Sci., 31, 115-122 (2002) · Zbl 1014.30014
[8]	Nolder, C. A., Global integrability theorems for \(A\)-harmonic tensors, J. Math. Anal. Appl., 247, 236-245 (2000) · Zbl 0973.35074
[9]	Nolder, C. A., Hardy-Littlewood theorems for \(A\)-harmonic tensors, Illinois J. Math., 43, 613-631 (1999) · Zbl 0957.35046
[10]	Wang, Y.; Wu, C., Sobolev imbedding theorems and Poincaré inequalities for Green’s operator on solutions of the nonhomogeneous \(A\)-harmonic equation, Comput. Math. Appl., 47, 1545-1554 (2004) · Zbl 1155.31303
[11]	Xing, Y., Two-weight imbedding inequalities for solutions to the \(A\)-harmonic equation, J. Math. Anal. Appl., 307, 555-564 (2005) · Zbl 1112.31003
[12]	Cartan, H., Differential Forms (1970), Houghton Mifflin Co.: Houghton Mifflin Co. Boston · Zbl 0213.37001
[13]	Iwaniec, T.; Lutoborski, A., Integral estimates for null Lagrangians, Arch. Ration. Mech. Anal., 125, 25-79 (1993) · Zbl 0793.58002
[14]	Stein, E. M., Singular Integrals and Differentiability Properties of Functions (1970), Princeton University Press: Princeton University Press Princeton · Zbl 0207.13501
[15]	Stein, E. M., Harmonic Analysis (1993), Princeton University Press: Princeton University Press Princeton
[16]	Scott, C., \(L^p\)-theory of differntial forms on manifolds, Trans. Amer. Math. Soc., 347, 2075-2096 (1995) · Zbl 0849.58002