Tensor products and sums of 𝑝\mspace{1𝑚𝑢}-harmonic functions, quasiminimizers and 𝑝\mspace{1𝑚𝑢}-admissible weights

Björn, Anders; Björn, Jana

doi:10.1090/proc/14170

Tensor products and sums of $p\mspace {1mu}$-harmonic functions, quasiminimizers and $p\mspace {1mu}$-admissible weights
HTML articles powered by AMS MathViewer

by Anders Björn and Jana Björn PDF

Proc. Amer. Math. Soc. 146 (2018), 5195-5203 Request permission

Abstract:

The tensor product of two $p\mspace {1mu}$-harmonic functions is in general not $p\mspace {1mu}$-harmonic, but we show that it is a quasiminimizer. More generally, we show that the tensor product of two quasiminimizers is a quasiminimizer. Similar results are also obtained for quasisuperminimizers and for tensor sums. This is done in weighted $\mathbf {R}^n$ with $p\mspace {1mu}$-admissible weights. It is also shown that the tensor product of two $p\mspace {1mu}$-admissible measures is $p\mspace {1mu}$-admissible. This last result is generalized to metric spaces.

References

Anders Björn, A weak Kellogg property for quasiminimizers, Comment. Math. Helv. 81 (2006), no. 4, 809–825. MR 2271223, DOI 10.4171/CMH/75
Anders Björn and Jana Björn, Power-type quasiminimizers, Ann. Acad. Sci. Fenn. Math. 36 (2011), no. 1, 301–319. MR 2797698, DOI 10.5186/aasfm.2011.3619
Anders Björn and Jana Björn, Nonlinear potential theory on metric spaces, EMS Tracts in Mathematics, vol. 17, European Mathematical Society (EMS), Zürich, 2011. MR 2867756, DOI 10.4171/099
Jana Björn, Poincaré inequalities for powers and products of admissible weights, Ann. Acad. Sci. Fenn. Math. 26 (2001), no. 1, 175–188. MR 1816566
Jana Björn, Sharp exponents and a Wiener type condition for boundary regularity of quasiminimizers, Adv. Math. 301 (2016), 804–819. MR 3539390, DOI 10.1016/j.aim.2016.06.024
Jana Björn, Stephen Buckley, and Stephen Keith, Admissible measures in one dimension, Proc. Amer. Math. Soc. 134 (2006), no. 3, 703–705. MR 2180887, DOI 10.1090/S0002-9939-05-07925-6
Seng-Kee Chua and Richard L. Wheeden, Sharp conditions for weighted 1-dimensional Poincaré inequalities, Indiana Univ. Math. J. 49 (2000), no. 1, 143–175. MR 1777034, DOI 10.1512/iumj.2000.49.1754
Mariano Giaquinta and Enrico Giusti, On the regularity of the minima of variational integrals, Acta Math. 148 (1982), 31–46. MR 666107, DOI 10.1007/BF02392725
Mariano Giaquinta and Enrico Giusti, Quasiminima, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 79–107. MR 778969
Piotr Hajłasz and Pekka Koskela, Sobolev meets Poincaré, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 10, 1211–1215 (English, with English and French summaries). MR 1336257
Juha Heinonen, Tero Kilpeläinen, and Olli Martio, Nonlinear potential theory of degenerate elliptic equations, Dover Publications, Inc., Mineola, NY, 2006. Unabridged republication of the 1993 original. MR 2305115
David Jerison, The Poincaré inequality for vector fields satisfying Hörmander’s condition, Duke Math. J. 53 (1986), no. 2, 503–523. MR 850547, DOI 10.1215/S0012-7094-86-05329-9
Tero Kilpeläinen, Pekka Koskela, and Hiroaki Masaoka, Lattice property of $p$-admissible weights, Proc. Amer. Math. Soc. 143 (2015), no. 6, 2427–2437. MR 3326025, DOI 10.1090/S0002-9939-2015-12416-1
Juha Kinnunen and Olli Martio, Potential theory of quasiminimizers, Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 2, 459–490. MR 1996447
Guozhen Lu and Richard L. Wheeden, Poincaré inequalities, isoperimetric estimates, and representation formulas on product spaces, Indiana Univ. Math. J. 47 (1998), no. 1, 123–151. MR 1631545, DOI 10.1512/iumj.1998.47.1494
O. Martio and C. Sbordone, Quasiminimizers in one dimension: integrability of the derivative, inverse function and obstacle problems, Ann. Mat. Pura Appl. (4) 186 (2007), no. 4, 579–590. MR 2317779, DOI 10.1007/s10231-006-0020-3