Gagliardo--Nirenberg inequality for maximal functions measuring smoothness
This paper gives a Gagliardo-Nirenberg type pointwise inequality for some maximal functions measuring smoothness. The classical Gagliardo-Nirenberg inequality is known to be an interpolation inequality as \[ \|\nabla^s f\|_{L^q(\mathbb{R}^n)}\leq C_{s,k,n}\|f\|^{1-s/k}_{L^r(\mathbb{R}^n)} \|\nabla^k f\|_{L^p(\mathbb{R}^n)}^{s/k}, \] where \(0\leq s<k\), \(1\leq p,r\leq\infty\) and \(1/q=(1-s/k)/r+s/(kp)\). It is well known that there is no pointwise version of this inequality for \(f\) in general, but, with the Hardy-Littlewood maximal functions, the following pointwise Gagliardo-Nirenberg type inequality holds true: \[ M(\nabla^s f)(x)\leq C(M(f-P)(x))^{1-s/k} (M(\nabla^k f)(x))^{s/k}, \] for all functions \(f\) which locally belong to the Sobolev space \(W^k_1(\mathbb{R}^n)\) and satisfy \[ \lim_{R\to\infty}\frac1{R^s}\frac1{|B(aR,rR)|}\int_{B(aR,rR)}|f(y)|\,dy=0. \] Here \(M\) is the Hardy-Littlewood maximal operator, and \(P\) is a polynomial with degree not more than \(k\). In this paper, the authors improves these results and shows that \[ M_{s-1,p,s}f(x)\leq C(M_{s-1,p,0}f(x))^{1-s/k} (M_{k,p,k}f(x))^{s/k} \] holds for all functions \(f\in L^p_{\text{loc}}(\mathbb{R}^n)\) satisfying \[ \lim_{|Q|\to0}\frac1{|Q|^{s/n}}\inf_{\deg(\pi)<s}\left(\frac1{|Q|}\int_Q |f(y)-\pi(y)|\,dy\right)^{1/p}=0, \] where \[ M_{k,p,s}f(x):=\sup_{x\in Q}\frac1{|Q|^{s/n}}\inf_{\deg(\pi)\leq k}\left(\frac1{|Q|}\int_Q |f(y)-\pi(y)|\,dy\right)^{1/p}. \] As a corollary, the following estimate involving BMO norm is obtained: \[ M_{s-1,1,s}f(x)\leq C\|f\|_{BMO}^{1-s/k} (M_{k,1,k}f(x))^{s/k}. \]
scientific article; zbMATH DE number 6110768