Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability, and superweights for 𝐿^{𝑝}-weighted Hardy inequalities

Abstract

Abstract. In this paper we give an extension of the classical Caffarelli-KohnNirenberg inequalities: we show that for 1 < p, q < ∞, 0 < r < ∞ with , where the constantMoreover, we also obtain anisotropic versions of these inequalities which can be conveniently formulated in the language of Folland and Stein's homogeneous groups. Consequently, we obtain remainder estimates for L p -weighted Hardy inequalities on homogeneous groups, which are also new in the Euclidean setting of R n . The critical Hardy inequalities of logarithmic type and uncertainty type principles on homogeneous groups are obtained. Moreover, we investigate another improved version of L p -weighted Hardy inequalities involving a distance and stability estimate. The relation between the critical and the subcritical Hardy inequalities on homogeneous groups is also investigated. We also establish sharp Hardy type inequalities in L p , 1 < p < ∞, with superweights, i.e., with the weights of the formallowing for different choices of α and β. There are two reasons why we call the appearing weights the superweights: the arbitrariness of the choice of any homogeneous quasi-norm and a wide range of parameters.

Fractional logarithmic inequalities and blow-up results with logarithmic nonlinearity on homogeneous groups

“…So by using this, (4.2) and (4.6), we obtain It means that I(u(x, t)) is decreasing functional with respect to the argument t. Let us set 19) and by Definition 4.1 we have…”

“…Recently many different versions of Caffarelli-Kohn-Nirenberg inequalities have been obtained on different Lie groups, namely, in [26] on the Heisenberg groups, in [22] and [23] on stratified groups, in [19] and [21] on (general) homogeneous groups. On the homogeneous groups a fractional analogue of Caffarelli-Kohn-Nirenberg inequality was proved in [14].…”

“…19),(4.6) and Lemma 4.3, we obtain A (t) = −2I(u) = −4J(u) + A (t) = −4J(u 0 ) + 4 (Ω) dτ + A (t). (4.22)…”

“…19),(4.20) and I(u) ≤ I(u 0 ) < 0, we obtainA (t) = A (0) − 2 t 0 I(u(x, τ ))dτ = −2I(u 0 )t ≥ 0, t ≥ 0, A(t) = −I(u 0 )t 2 ≥ 0, t ≥ 0.…”

Hardy inequalities for Landau Hamiltonian and for Baouendi-Grushin operator with Aharonov-Bohm type magnetic field. Part I

“…Weights of this type has appeared in [GM11] as well as in [RSY17], and are called the superweights due to the freedom in the choice of indices.…”