2023/02/10 by Marianna Chatzakou, Michael Ruzhansky, Chatzakou, Marianna +1 · 1 citation
Engineering · #FOS: Mathematics #Fatigue and fracture mechanics #Functional Analysis (math.FA)
paper · pdf · doi:10.48550/arxiv.2302.05126
openalex publication_date 2023/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this short note we prove the logarithmic Sobolev inequality with derivatives of fractional order on ℝn with an explicit expression for the constant. Namely, we show that for all 00 we have the inequality ∫ℝn|f(x)|2 log ( \frac|f(x)|2‖f‖2L2(ℝn)) dx+(n)/(s)(1+log a)‖f‖L2(ℝn)2≤ C(n,s,a)‖(-Δ)s/2f‖2L2(ℝn) with an explicit C(n,s,a) depending on a, the order s, and the dimension n, and investigate the behaviour of C(n,s,a) for large n. Notably, for large n and when s=1, the constant C(n,1,a) is asymptotically the same as the sharp constant of Lieb and Loss. Moreover, we prove a similar type inequality for functions f ∈ Lq(ℝn)∩ W1,p(ℝn) whenever 1