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تسجيل مستخدم جديدOn Hardy and Caffarelli-Kohn-Nirenberg inequalities
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We consider positive critical points of Caffarelli-Kohn-Nirenberg inequalities and prove a Liouville type result which allows us to give a complete classification of the solutions in a certain range of parameters, providing a symmetry result for posi
tive solutions. The governing operator is a weighted $p$-Laplace operator, which we consider for a general $p in (1,d)$. For $p=2$, the symmetry breaking region for extremals of Caffarelli-Kohn-Nirenberg inequalities was completely characterized in [J. Dolbeault, M. Esteban, M. Loss, Invent. Math. 44 (2016)]. Our results extend this result to a general $p$ and are optimal in some cases.
In this paper, we consider the Caffarelli-Kohn-Nirenberg (CKN) inequality: begin{eqnarray*} bigg(int_{{mathbb R}^N}|x|^{-b(p+1)}|u|^{p+1}dxbigg)^{frac{2}{p+1}}leq C_{a,b,N}int_{{mathbb R}^N}|x|^{-2a}| abla u|^2dx end{eqnarray*} where $Ngeq3$, $a<frac
{N-2}{2}$, $aleq bleq a+1$ and $p=frac{N+2(1+a-b)}{N-2(1+a-b)}$. It is well-known that up to dilations $tau^{frac{N-2}{2}-a}u(tau x)$ and scalar multiplications $Cu(x)$, the CKN inequality has a unique extremal function $W(x)$ which is positive and radially symmetric in the parameter region $b_{FS}(a)leq b<a+1$ with $a<0$ and $aleq b<a+1$ with $ageq0$ and $a+b>0$, where $b_{FS}(a)$ is the Felli-Schneider curve. We prove that in the above parameter region the following stabilities hold: begin{enumerate} item[$(1)$] quad stability of CKN inequality in the functional inequality setting $$dist_{D^{1,2}_{a}}^2(u, mathcal{Z})lesssim|u|^2_{D^{1,2}_a({mathbb R}^N)}-C_{a,b,N}^{-1}|u|^2_{L^{p+1}(|x|^{-b(p+1)},{mathbb R}^N)}$$ where $mathcal{Z}= { c W_taumid cinbbrbackslash{0}, tau>0}$; item[$(2)$]quad stability of CKN inequality in the critical point setting (in the class of nonnegative functions) begin{eqnarray*} dist_{D_a^{1,2}}(u, mathcal{Z}_0^ u)lesssimleft{aligned &Gamma(u),quad p>2text{ or } u=1, &Gamma(u)|logGamma(u)|^{frac12},quad p=2text{ and } ugeq2, &Gamma(u)^{frac{p}{2}},quad 1<p<2text{ and } ugeq2, endalignedright. end{eqnarray*} where $Gamma (u)=|div(|x|^{-a} abla u)+|x|^{-b(p+1)}|u|^{p-1}u|_{(D^{1,2}_a)^{}}$ and $$mathcal{Z}_0^ u={(W_{tau_1},W_{tau_2},cdots,W_{tau_ u})mid tau_i>0}.$$
We revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants. We also discuss the infinite sequence of power weighted Birman-Hardy-Rellich-type inequalities and derive an operator-valued version thereof.
We study the two-weighted estimate [ bigg|sum_{k=0}^na_k(x)int_0^xt^kf(t)dt|L_{q,v}(0,infty)bigg|leq c|f|L_{p,u}(0,infty)|,tag{$*$} ] where the functions $a_k(x)$ are not assumed to be positive. It is shown that for $1<pleq qleqinfty$, prov
ided that the weight $u$ satisfies the certain conditions, the estimate $(*)$ holds if and only if the estimate [ sum_{k=0}^nbigg|a_k(x)int_0^xt^kf(t)dt|L_{q,v}(0,infty)bigg| leq c|f|L_{p,u}(0,infty)|.tag{$**$} ] is fulfilled. The necessary and sufficient conditions for $(**)$ to be valid are well-known. The obtained result can be applied to the estimates of differential operators with variable coefficients in some weighted Sobolev spaces.
We prove a sharp Hardy inequality for fractional integrals for functions that are supported on a general domain. The constant is the same as the one for the half-space and hence our result settles a recent conjecture of Bogdan and Dyda.
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