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

Abstract

In this paper we give an extension of the classical Caffarelli-Kohn-Nirenberg inequalities: we show that for 1 > p , q > ∞ 1>p,q>\infty , 0 > r > ∞ 0>r>\infty with p + q ≥ r p+q\geq r , δ ∈ [ 0 , 1 ] ∩ [ r − q r , p r ] \delta \in [0,1]\cap \left [\frac {r-q}{r},\frac {p}{r}\right ] with δ r p + ( 1 − δ ) r q = 1 \frac {\delta r}{p}+\frac {(1-\delta )r}{q}=1 and a a , b b , c ∈ R c\in \mathbb {R} with c = δ ( a − 1 ) + b ( 1 − δ ) c=\delta (a-1)+b(1-\delta ) , and for all functions f ∈ C 0 ∞ ( R n ∖ { 0 } ) f\in C_{0}^{\infty }(\mathbb {R}^{n}\backslash \{0\}) we have ‖ | x | c f ‖ L r ( R n ) ≤ | p n − p ( 1 − a ) | δ ‖ | x | a ∇ f ‖ L p ( R n ) δ ‖ | x | b f ‖ L q ( R n ) 1 − δ \begin{equation*} \||x|^{c}f\|_{L^{r}(\mathbb {R}^{n})} \leq \left |\frac {p}{n-p(1-a)}\right |^{\delta } \left \||x|^{a}\nabla f\right \|^{\delta }_{L^{p}(\mathbb {R}^{n})} \left \||x|^{b}f\right \|^{1-\delta }_{L^{q}(\mathbb {R}^{n})} \end{equation*} for n ≠ p ( 1 − a ) n\neq p(1-a) , where the constant | p n − p ( 1 − a ) | δ \left |\frac {p}{n-p(1-a)}\right |^{\delta } is sharp for p = q p=q with a − b = 1 a-b=1 or p ≠ q p\neq q with p ( 1 − a ) + b q ≠ 0 p(1-a)+bq\neq 0 . In the critical case n = p ( 1 − a ) n=p(1-a) we have ‖ | x | c f ‖ L r ( R n ) ≤ p δ ‖ | x | a log ⁡ | x | ∇ f ‖ L p ( R n ) δ ‖ | x | b f ‖ L q ( R n ) 1 − δ . \begin{equation*} \left \||x|^{c}f\right \|_{L^{r}(\mathbb {R}^{n})} \leq p^{\delta } \left \||x|^{a}\log |x|\nabla f\right \|^{\delta }_{L^{p}(\mathbb {R}^{n})} \left \||x|^{b}f\right \|^{1-\delta }_{L^{q}(\mathbb {R}^{n})}. \end{equation*} Moreover, 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 L^{p} -weighted Hardy inequalities on homogeneous groups, which are also new in the Euclidean setting of R n \mathbb {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 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 L^{p} , 1 > p > ∞ 1>p>\infty , with superweights, i.e., with the weights of the form ( a + b | x | α ) β p | x | m \frac {(a+b|x|^{\alpha })^{\frac {\beta }{p}}}{|x|^{m}} allowing for different choices of α \alpha and β \beta . 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.