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

Matthew D. Blair; Christopher D. Sogge

Journal of the European Mathematical Society (2015)

- Volume: 017, Issue: 10, page 2513-2543
- ISSN: 1435-9855

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topBlair, 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 -

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