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On the CaffarelliKohnNirenberg Inequalities: Sharp Constants, Existence (and Nonexistence), and Symmetry of Extremal Functions
"... Consider the following inequalities due to Caffarelli, Kohn, and Nirenberg [6]: # # # < a <(N 2)/2, a 2N/(N a)). We shall answer some fundamental questions concerning these inequalities such as the best embedding constants, the existence and nonexistence of extrem ..."
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Cited by 41 (4 self)
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Consider the following inequalities due to Caffarelli, Kohn, and Nirenberg [6]: # # # < a <(N 2)/2, a 2N/(N a)). We shall answer some fundamental questions concerning these inequalities such as the best embedding constants, the existence and nonexistence of extremal functions, and their qualitative properties. While the case a 0 has been studied extensively and a complete solution is known, little has been known for the case a < 0. Our results for the case a < 0 reveal some new phenomena which are in striking contrast with those for the case 2 are also given. c 2001 John Wiley & Sons, Inc. 1
On the best possible remaining term in the Hardy inequality
 Proc. Natl. Acad. Sci. USA 105 (2008) no
"... We give a necessary and sufficient condition on a radially symmetric potential V on Ω that makes it an admissible candidate for an improved Hardy inequality of the following form: Ω ∇u2dx − ( n−2 ..."
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Cited by 5 (3 self)
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We give a necessary and sufficient condition on a radially symmetric potential V on Ω that makes it an admissible candidate for an improved Hardy inequality of the following form: Ω ∇u2dx − ( n−2
A geometrical version of Hardy's inequality for ...
"... The aim of this article is to prove a Hardy type inequality, concerning functions in (# for some , involving the volume of# and the distance to the boundary of # The inequality is a generalization of a recently proved inequality by M.HoffmannOstenhof, T.HoffmannOstenhof and A.Laptev [9], w ..."
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Cited by 2 (1 self)
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The aim of this article is to prove a Hardy type inequality, concerning functions in (# for some , involving the volume of# and the distance to the boundary of # The inequality is a generalization of a recently proved inequality by M.HoffmannOstenhof, T.HoffmannOstenhof and A.Laptev [9], which dealt with the special case p = 2.
Analysis of degenerate elliptic operators of Gruˇsin type
, 2006
"... We analyze degenerate, secondorder, elliptic operators H in divergence form on L2(R n ×R m). We assume the coefficients are real symmetric and a1Hδ ≥ H ≥ a2Hδ for some a1, a2> 0 where Hδ = −∇x1 cδ1,δ ′ 1 (x1) ∇x1 − cδ2,δ ′ 2 (x1) ∇ 2 x2 are positive measurable functions Here x1 ∈ Rn, x2 ∈ Rm and c ..."
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We analyze degenerate, secondorder, elliptic operators H in divergence form on L2(R n ×R m). We assume the coefficients are real symmetric and a1Hδ ≥ H ≥ a2Hδ for some a1, a2> 0 where Hδ = −∇x1 cδ1,δ ′ 1 (x1) ∇x1 − cδ2,δ ′ 2 (x1) ∇ 2 x2 are positive measurable functions Here x1 ∈ Rn, x2 ∈ Rm and cδi,δ ′ i such that cδi,δ ′ i (x) behaves like xδi δ as x → 0 and x  ′ i as x → ∞ with δ1, δ ′ 1 ∈ [0, 1 〉 and δ2, δ ′ 2 ≥ 0. Our principal results state that the submarkovian semigroup St = e −tH is conservative and its kernel Kt satisfies bounds 0 ≤ Kt(x; y) ≤ a (B(x; t 1/2)  B(y; t 1/2)) −1/2 where B(x; r)  denotes the volume of the ball B(x; r) centred at x with radius r measured with respect to the Riemannian distance associated with H. The proofs depend on detailed subelliptic estimations on H, a precise characterization of the Riemannian distance and the corresponding volumes and wave equation techniques which exploit the finite speed of propagation. We discuss further implications of these bounds and give explicit examples that show the kernel is not necessarily strictly positive, nor continuous.
The Best Constant, the Nonexistence of Extremal Functions and Related Results for a HardySobolev Inequality
, 810
"... We present the best constant, the minimizing sequences and the nonexistence of extremal functions for a HardySobolev inequality. We prove that, under a proper transformation, this inequality is equivalent to the Sobolev inequality in R N. We also discuss the connection of the related functional spa ..."
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We present the best constant, the minimizing sequences and the nonexistence of extremal functions for a HardySobolev inequality. We prove that, under a proper transformation, this inequality is equivalent to the Sobolev inequality in R N. We also discuss the connection of the related functional spaces and as a result we obtain some Caffarelli Kohn Nirenberg inequalities. Our starting point is the existence of a minimizer for a Maz’ya’s inequality and indirect dependence of the Hardy inequality at the origin.
Assume the following inequality:
, 810
"... We present the best constant, the minimizing sequences and the nonexistence of extremal functions for a HardySobolev inequality. We prove that, under a proper transformation, this inequality is equivalent to the Sobolev inequality in R N. We also discuss the connection of the related functional spa ..."
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We present the best constant, the minimizing sequences and the nonexistence of extremal functions for a HardySobolev inequality. We prove that, under a proper transformation, this inequality is equivalent to the Sobolev inequality in R N. We also discuss the connection of the related functional spaces and as a result we obtain some Caffarelli Kohn Nirenberg inequalities. Our starting point is the existence of a minimizer for the Bliss ’ inequality and the indirect dependence of the Hardy inequality at the origin.
B W(x)u2 dx,
, 2008
"... We give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in R n, n ≥ 1, so that the following inequalities hold for all u ∈ C ∞ 0 (B): ..."
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We give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in R n, n ≥ 1, so that the following inequalities hold for all u ∈ C ∞ 0 (B):