On Kakeya–Nikodym averages, $L^p$-norms and lower bounds for nodal sets of eigenfunctions in higher dimensions

Blair, Matthew D., and Sogge, Christopher D.. "On Kakeya–Nikodym averages, $L^p$-norms and lower bounds for nodal sets of eigenfunctions in higher dimensions." Journal of the European Mathematical Society 017.10 (2015): 2513-2543. <http://eudml.org/doc/277705>.

@article{Blair2015,
abstract = {We extend a result of the second author [27, Theorem 1.1] to dimensions $d \ge 3$ which relates the size of $L^p$-norms of eigenfunctions for $2 < p < 2(d+1) / d-1$ to the amount of $L^2$-mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee [22] and a variable coefficient variant of an "$\epsilon $ removal lemma" of Tao and Vargas [35]. We also use Hörmander’s [20] $L^2$ oscillatory integral theorem and the Cartan–Hadamard theorem to show that, under the assumption of nonpositive curvature, the $L^2$-norm of eigenfunctions $e_\{\lambda \}$ over unit-length tubes of width $\lambda ^\{-1/2\}$ goes to zero. Using our main estimate, we deduce that, in this case, the $L^p$-norms of eigenfunctions for the above range of exponents is relatively small. As a result, we can slightly improve the known lower bounds for nodal sets in dimensions $d \ge 3$ of Colding and Minicozzi [10] in the special case of (variable) nonpositive curvature.},
author = {Blair, Matthew D., Sogge, Christopher D.},
journal = {Journal of the European Mathematical Society},
keywords = {eigenfunctions; $L^p$-norms; Kakeya averages; Laplace Beltrami operator on compact manifolds; Laplace Beltrami operator on compact manifolds; eigenfunctions; norms; Kakeya averages},
language = {eng},
number = {10},
pages = {2513-2543},
publisher = {European Mathematical Society Publishing House},
title = {On Kakeya–Nikodym averages, $L^p$-norms and lower bounds for nodal sets of eigenfunctions in higher dimensions},
url = {http://eudml.org/doc/277705},
volume = {017},
year = {2015},
}

TY - JOUR
AU - Blair, Matthew D.
AU - Sogge, Christopher D.
TI - On Kakeya–Nikodym averages, $L^p$-norms and lower bounds for nodal sets of eigenfunctions in higher dimensions
JO - Journal of the European Mathematical Society
PY - 2015
PB - European Mathematical Society Publishing House
VL - 017
IS - 10
SP - 2513
EP - 2543
AB - We extend a result of the second author [27, Theorem 1.1] to dimensions $d \ge 3$ which relates the size of $L^p$-norms of eigenfunctions for $2 < p < 2(d+1) / d-1$ to the amount of $L^2$-mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee [22] and a variable coefficient variant of an "$\epsilon $ removal lemma" of Tao and Vargas [35]. We also use Hörmander’s [20] $L^2$ oscillatory integral theorem and the Cartan–Hadamard theorem to show that, under the assumption of nonpositive curvature, the $L^2$-norm of eigenfunctions $e_{\lambda }$ over unit-length tubes of width $\lambda ^{-1/2}$ goes to zero. Using our main estimate, we deduce that, in this case, the $L^p$-norms of eigenfunctions for the above range of exponents is relatively small. As a result, we can slightly improve the known lower bounds for nodal sets in dimensions $d \ge 3$ of Colding and Minicozzi [10] in the special case of (variable) nonpositive curvature.
LA - eng
KW - eigenfunctions; $L^p$-norms; Kakeya averages; Laplace Beltrami operator on compact manifolds; Laplace Beltrami operator on compact manifolds; eigenfunctions; norms; Kakeya averages
UR - http://eudml.org/doc/277705
ER -