High order representation formulas and embedding theorems on stratified groups and generalizations

• Autorzy: Guozhen Lu , Richard L. Wheeden
• Języki publikacji: EN

Abstrakty

EN

We derive various integral representation formulas for a function minus a polynomial in terms of vector field gradients of the function of appropriately high order. Our results hold in the general setting of metric spaces, including those associated with Carnot-Carathéodory vector fields, under the assumption that a suitable $L^1$ to $L^1$ Poincaré inequality holds. Of particular interest are the representation formulas in Euclidean space and stratified groups, where polynomials exist and $L^1$ to $L^1$ Poincaré inequalities involving high order derivatives are known to hold. We apply the formulas to derive embedding theorems and potential type inequalities involving high order derivatives.

Słowa kluczowe

EN

Poincaré inequalities; doubling measures; stratified groups; polynomials; representation formulas; vector fields; embedding theorems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2000
• Tom: 142
• Numer: 2
• Strony: 101-133

Daty

wydano

2000

otrzymano

1998-11-24

poprawiono

2000-04-25

Twórcy

autor

Guozhen Lu

• Department of Mathematics, Wayne State University, Detroit, MI 48202, U.S.A.

autor

Richard L. Wheeden

• Department of Mathematics, Rutgers University, Hill Center for Mathematical Studies, New Brunswick, NJ 08903, U.S.A.

Bibliografia

• [AH] D. Adams and L. Hedberg, Function Spaces and Potential Theory, Grund- lehren Math. Wiss. 314, Springer, New York, 1996.
• [B] B. Bojarski, Remarks on Sobolev imbedding theorems, in: Lecture Notes in Math. 1351, Springer, New York, 1988, 52-68.
• [BH] B. Bojarski and P. Hajłasz, Pointwise inequalities for Sobolev functions and some applications, Studia Math. 106 (1993), 77-92.
• [C1] S. Campanato, Proprietà di inclusione per spazi di Morrey, Ricerche Mat. 12 (1963), 67-86.
• [C2] S. Campanato, Proprietà di hölderianità di alcune classi di funzioni, Ann. Scuola Norm. Sup. Pisa 17 (1963), 175-188.
• [C3] S. Campanato, Proprietà di una famiglia di spazi funzionali, ibid. 18 (1964), 137-160.
• [CDG] L. Capogna, D. Danielli and N. Garofalo, Subelliptic mollifiers and a basic pointwise estimate of Poincaré type, Math. Z. 226 (1997), 147-154.
• [CW] S. Chanillo and R. L. Wheeden, Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions, Amer. J. Math. 107 (1985), 1191-1226.
• [FP] C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems, in: Proc. Conf. on Harmonic Analysis (Chicago 1980), W. Beckner et al. (eds.), Wads- worth, 1981, Vol. 2, 590-606.
• [FS] G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Princeton Univ. Press, Princeton, 1980.
• [F] B. Franchi, Inégalités de Sobolev pour des champs de vecteurs lipschitziens, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), 329-332.
• [FGW] B. Franchi, C. Gutiérrez and R. L. Wheeden, Weighted Sobolev-Poincaré inequalities for Grushin type operators, Comm. Partial Differential Equations 19 (1994), 523-604.
• [FHK] B. Franchi, P. Hajłasz and P. Koskela, Definitions of Sobolev classes on metric spaces, preprint, 1998.
• [FL] B. Franchi and E. Lanconelli, Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa 10 (1983), 523-541.
• [FLW1] B. Franchi, G. Z. Lu and R. L. Wheeden, Representation formulas and weighted Poincaré inequalities for Hörmander vector fields, Ann. Inst. Fourier (Grenoble) 45 (1995), 577-604.
• [FLW2] B. Franchi, G. Z. Lu and R. L. Wheeden, A relationship between Poincaré-type inequalities and representation formulas in spaces of homogeneous type, Internat. Math. Res. Notices 1996, no. 1, 1-14.
• [FW] B. Franchi and R. L. Wheeden, Some remarks about Poincaré type inequalities and representation formulas in metric spaces of homogeneous type, J. Inequal. Appl. 3 (1999), 65-89.
• [GT] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, Berlin, 1983.
• [H] P. Hajłasz, Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), 403-415.
• [HK] P. Hajłasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc., to appear.
• [He] L. Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505-510.
• [LLW] Y. Liu, G. Z. Lu and R. L. Wheeden, Several equivalent definitions of high order Sobolev spaces on stratified groups and generalizations to metric spaces, preprint, 1998.
• [L1] G. Z. Lu, Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications, Rev. Mat. Iberoamericana 8 (1992), 367-439.
• [L2] G. Z. Lu, The sharp Poincaré inequality for free vector fields: an endpoint result, ibid. 10 (1994), 453-466.
• [L3] G. Z. Lu, Local and global interpolation inequalities for the Folland-Stein Sobolev spaces and polynomials on the stratified groups, Math. Res. Lett. 4 (1997), 777-790.
• [L4] G. Z. Lu, Polynomials, higher order Sobolev extension theorems and interpolation inequalities on weighted Folland-Stein spaces on stratified groups, Acta Math. Sinica Ser. B, to appear.
• [L5] G. Z. Lu, Embedding theorems on Campanato-Morrey spaces for vector fields and applications, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 429-434.
• [L6] G. Z. Lu, Embedding theorems on Campanato-Morrey spaces for vector fields of Hörmander type, Approx. Theory Appl. 14 (1998), 69-80.
• [LP] G. Z. Lu and C. Pérez, $L^p$ to $L^1$ Poincaré inequalities for 0<p<1 imply representation formulas, preprint, 1999.
• [LW1] G. Z. Lu and R. L. Wheeden, Poincaré inequalities, isoperimetric estimates and representation formulas on product spaces, Indiana Univ. Math. J. 47 (1998), 123-151.
• [LW2] G. Z. Lu and R. L. Wheeden, An optimal representation formula for Carnot-Carathéodory vector fields, Bull. London Math. Soc. 30 (1998), 578-584.
• [N] D. Nhieu, Extension of Sobolev spaces on the Heisenberg group, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), 1559-1564.
• [PW] C. Pérez and R. L. Wheeden, Uncertainty principle estimates for vector fields, to appear.
• [SW] E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), 813-874.
• [SWZ] E. Sawyer, R. L. Wheeden and S. V. Zhao, Weighted norm inequalities for operators of potential type and fractional maximal functions, Potential Anal. 5 (1996), 523-580.
• [So1] S. L. Sobolev, On a boundary value problem for polyharmonic equations, Amer. Math. Soc. Transl. 33 (1963), 1-40.
• [So2] S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, Transl. Math. Monogr. 90, Amer. Math. Soc., Providence, RI, 1991.
• [S1] G. Stampacchia, The spaces $L^{p,λ}$, $N^{p,λ}$ and interpolation, Ann. Scuola Norm. Sup. Pisa 19 (1965), 443-462.
• [S2] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), no. 1, 189-258.
• [Z] W. Ziemer, Weakly Differentiable Functions, Grad. Texts in Math. 120, Springer, New York, 1989.