Sharp pointwise gradient estimates for Riesz potentials with a bounded density
Tóm tắt
We establish sharp inequalities for the Riesz potential and its gradient in
$$\mathbb {R}^{n}$$
and indicate their usefulness for potential analysis, moment theory and other applications.
Tài liệu tham khảo
Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards Applied Mathematics Series, vol. 55. For Sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. (1964). http://people.math.sfu.ca/~cbm/aands/frameindex.htm
Adams, D.R.: A note on Riesz potentials. Duke Math. J. 42(4), 765–778 (1975). http://projecteuclid.org/euclid.dmj/1077311348
Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1996). https://doi.org/10.1007/978-3-662-03282-4
Aheizer, N.I., Krein, M.: Some questions in the theory of moments. Translated by W. Fleming and D. Prill. Translations of Mathematical Monographs, Vol. 2. American Mathematical Society, Providence (1962)
Curto, R.E., Fialkow, L.A.: Truncated \(K\)-moment problems in several variables. J. Oper. Theory 54(1), 189–226 (2005)
Fraenkel, L.E.: An Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge Tracts in Mathematics, vol. 128. Cambridge University Press, Cambridge (2000). https://doi.org/10.1017/CBO9780511569203
Garg, R., Spector, D.: On the role of Riesz potentials in Poisson’s equation and Sobolev embeddings. Indiana Univ. Math. J. 64(6), 1697–1719 (2015). https://doi.org/10.1512/iumj.2015.64.5706
Gustafsson, B., He, C., Milanfar, P., Putinar, M.: Reconstructing planar domains from their moments. Inverse Prob. 16(4), 1053–1070 (2000). https://doi.org/10.1088/0266-5611/16/4/312
Gustafsson, B., Putinar, M.: The exponential transform: a renormalized Riesz potential at critical exponent. Indiana Univ. Math. J. 52(3), 527–568 (2003). https://doi.org/10.1512/iumj.2003.52.2304
Gustafsson, B., Putinar, M.: Hyponormal quantization of planar domains: complex orthogonal polynomials and the exponential transform. Lecture Notes in Mathematics, vol. 2199. Springer (2017)
Gustafsson, B., Putinar, M., Saff, E.B., Stylianopoulos, N.: Bergman polynomials on an archipelago: estimates, zeros and shape reconstruction. Adv. Math. 222(4), 1405–1460 (2009). https://doi.org/10.1016/j.aim.2009.06.010
Gustafsson, B., Vasil’ev, A.: Conformal and Potential Analysis in Hele-Shaw Cells. Advances in Mathematical Fluid Mechanics. Birkhäuser Verlag, Basel (2006)
Huang, X.: Symmetry results of positive solutions of integral equations involving Riesz potential in exterior domains and in annular domains. J. Math. Anal. Appl. 427(2), 856–872 (2015). https://doi.org/10.1016/j.jmaa.2015.02.083
Huang, X., Hong, G., Li, D.: Some symmetry results for integral equations involving Wolff potential on bounded domains. Nonlinear Anal. 75(14), 5601–5611 (2012). https://doi.org/10.1016/j.na.2012.05.007
Krein, M.G., Nudel’man, A.A.: The Markov moment problem and extremal problems. American Mathematical Society, Providence, R.I. (1977). Ideas and problems of P. L. Čebyšev and A. A. Markov and their further development, Translated from the Russian by D. Louvish, Translations of Mathematical Monographs, vol. 50 (1977)
Lasserre, J.B.: Bounding the support of a measure from its marginal moments. Proc. Am. Math. Soc. 139(9), 3375–3382 (2011). https://doi.org/10.1090/S0002-9939-2011-10865-7
Lieb, E.H., Loss, M.: Analysis, Graduate Studies in Mathematics, vol. 14, 2nd edn. American Mathematical Society, Providence (2001). https://doi.org/10.1090/gsm/014
Maz’ya, V.: Sobolev spaces with applications to elliptic partial differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 342, augmented edn. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-15564-2
Maz’ya, V., Shaposhnikova, T.: On pointwise interpolation inequalities for derivatives. Math. Bohem. 124(2–3), 131–148 (1999)
Mingione, G.: Gradient potential estimates. J. Eur. Math. Soc. (JEMS) 13(2), 459–486 (2011). https://doi.org/10.4171/JEMS/258
Mingione, G.: Recent Advances in Nonlinear Potential Theory Trends in Contemporary Mathematics. Springer INdAM Ser., vol. 8, pp. 277–292. Springer, Cham (2014)
Reichel, W.: Characterization of balls by Riesz-potentials. Ann. Mat. Pura Appl. (4) 188(2), 235–245 (2009). https://doi.org/10.1007/s10231-008-0073-6
Tkachev, V.G.: Subharmonicity of higher dimensional exponential transforms. In: Ebenfelt, P., Gustafsson, B., Khavinson, D., Putinar, M. (eds) Quadrature Domains and Their Applications, Oper. Theory Adv. Appl., vol. 156, Birkhäuser, Basel (2005)
Xiao, J.: Entropy flux–electrostatic capacity–graphical mass. Proc. Am. Math. Soc. 145(2), 825–832 (2017). https://doi.org/10.1090/proc/13259