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The purpose of this article is to establish new lower bounds for the sums of powers of eigenvalues of the Dirichlet fractional Laplacian operator (−Δ)α/2|Ω restricted to a bounded domain Ω ⊂ ℝd with d = 2, 1 ≤ α ≤ 2 and d ≥ 3, 0 < α ≤ 2. Our main result yields a sharper lower bound, in the sense of Weyl asymptotics, for the Berezin-Li-Yau type inequality improving the previous result [S. Yildirim Yolcu and T. Yolcu, Discrete Contin. Dyn. Syst. 35(5), 2209–2225 (2015)]. Furthermore, we give a result improving the bounds for analogous elliptic operators [Y.-C. Kim, e-print arXiv:1406.5407].

Refined eigenvalue bounds on the Dirichlet fractional Laplacian