Maximal inequality in \((s,m)\)-uniform domains. (English) Zbl 1026.46022

The author deals with Sobolev spaces on bounded domains in \(\mathbb{R}^n\). A new class of bounded domains is defined. The domains are called \((s,m)\)-uniform, \(s\geq 1\), \(0<m\leq 1\), and they form a subclass of \(s\)-John domains. If a domain is bounded and its boundary is locally a graph of a \(\lambda\)-Hölder function, \(0<\lambda\leq 1\), then it is \((1/\lambda,\lambda)\)-uniform. The main theorem asserts that if \(\Omega\subset \mathbb{R}^n\) is \((s,m)\)-uniform, \(1\leq s<n/(n-1)\) and \(0<m\leq 1\), then every function \(u\) belonging to the Sobolev space \(W^{1,p}(\Omega)\) satisfies the inequality \[ |u(x)-u(y)|\leq C|x-y|^\alpha \big(\mathcal M\nabla u(x)+\mathcal M\nabla u(y)\big) \qquad x,y\in \Omega,\quad \alpha=\frac ms (n-s(n-1)) . \] Here \(\mathcal M\) denotes the Hardy-Littlewood maximal function. The inequality was previously considered by P. Hajłasz and O. Martio [J. Funct. Anal. 143, 221-246 (1997; Zbl 0877.46025)]. The result extends the earlier estimates to a wider class of domains. The paper is based on techniques developed by its forerunners.


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems


Zbl 0877.46025
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