Abstract
We prove $L^{p}$-Caffarelli-Kohn-Nirenberg type inequalities on homogeneous groups, which is one of most general subclasses of nilpotent Lie groups, all with sharp constants. We also discuss some of their consequences. Already in the abelian case of $\mathbb{R}^{n}$ our results provide new insights in view of the arbitrariness of the choice of the not necessarily Euclidean quasi-norm.
Highlights
Consider the following weighted Hardy–Sobolev type inequalities due to Caffarelli–Kohn– Nirenberg [5]: for all f Î C0¥ ( n), it holds (ò ) ò 2x -pb ∣f ∣p dx n p£ Ca,b x -2a ∣ f ∣2 dx, n (1.1) where, for n 3 3-¥ < a < n - 2, 2 a £ b £ a + 1, and p= 2n + 2(b a), and, for n = 2-¥ < a < 0, a < b £ a + 1, and ba, and where x = x12 + + xn2 is the Euclidean norm
We prove Lp-Caffarelli–Kohn–Nirenberg type inequalities on homogeneous groups, which is one of most general subclasses of nilpotent Lie groups, all with sharp constants
We show that the Caffarelli–Kohn–Nirenberg inequality continues to hold in the setting of homogeneous groups
Summary
Consider the following weighted Hardy–Sobolev type inequalities due to Caffarelli–Kohn– Nirenberg [5]: for all f Î C0¥ ( n), it holds (ò ) ò 2. We show that the Caffarelli–Kohn–Nirenberg inequality continues to hold in the setting of homogeneous groups It has to hold on anisotropic n, with a number of different consequences. One of the results in [21] was that if is any homogeneous group and ∣ · ∣ is a homogeneous quasi-norm on , as an analogue of (1.9), we obtain the following generalized L p-Hardy inequality:. The critical versions of (1.9) with p = n were investigated by Ioku–Ishiwata–Ozawa [16] Their generalizations as well as a number of other critical (logarithmic) Hardy inequalities on homogeneous groups were obtained in recent works [18, 21, 22].
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