×

zbMATH — the first resource for mathematics

Norms of the composition of the maximal and projection operators. (English) Zbl 1189.35036
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Warner, F.W., Foundations of differentiable manifolds and Lie groups, (1983), 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
[6] S. Ding, Norm estimates for the maximal operator and Green’s operator, in: Proceedings of the 6th International Conference on Differential Equations and Dynamical Systems, DCDIS, 2009, pp. 37-43 (a supplement).
[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. 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 · Zbl 0207.13501
[15] Stein, E.M., Harmonic analysis, (1993), 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.