tailieunhanh - Several Hardy-type inequalities with weights related to Baouendi–Grushin operators

In this paper we shall prove several weighted Lp Hardy-type inequalities associated to the Baouendi–Grushintype operators. Since then it has attracted the attention of many mathematicians and has been comprehensively analyzed in several directions. | Turk J Math (2018) 42: 3050 – 3060 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Several Hardy-type inequalities with weights related to Baouendi–Grushin operators Abdullah YENER∗, Department of Mathematics, Faculty of Science, İstanbul Commerce University, İstanbul, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: In this paper we shall prove several weighted Lp Hardy-type inequalities associated to the Baouendi–Grushintype operators ∆γ = ∆x + |x|2γ ∆y , where ∆x and ∆y are the classical Laplace operators in the variables x ∈ Rn and y ∈ Rk , respectively, and γ is a positive real number. Key words: Baouendi–Grushin operator, weighted Hardy inequality 1. Introduction The well-known Hardy inequality in Rn , n ≥ 3, asserts that for all u ∈ C0∞ (Rn ) ( ∫ 2 Rn Though the constant ( n−2 )2 2 |∇u| dx ≥ n−2 2 )2 ∫ u2 Rn |x| 2 dx. () is sharp, equality in () is never achieved by any function u ∈ H01 (Rn ) . Hardy [13] originally discovered this inequality in 1920 for the one-dimensional case. Since then it has attracted the attention of many mathematicians and has been comprehensively analyzed in several directions; see, for example, [2, 3, 5, 10, 11, 14, 15, 21] and the references therein. The sharp Hardy inequality () arises very naturally in the study of degenerate elliptic differential operators and it was first extended in [9] by Garofalo to the Baouendi–Grushin vector fields Xi = ∂ , ∂xi i = 1, . . . , n, Yj = |x|γ ∂ , ∂yj j = 1, . . . , k, where x = (x1 , . . . , xn ) ∈ Rn , y = (y1 , . . . , yk ) ∈ Rk with n, k ≥ 1, γ > 0. To be explicit, the author in [9] proved the following Hardy inequality ( ∫ Rn+k |∇γ u|2 dxdy ≥ Q−2 2 )2 ∫ Rn+k |x|2γ u2 dxdy ρ2γ ρ2 () for every u ∈ C0∞ (Rn × Rk \ {(0, 0)}). Here, Q = n + (1 + γ)k is the homogeneous dimension, ∇γ = ( ) 1 (X1 , . . . , Xn , Y1 , . . . ,