Abstract
Let G be an infinite connected graph. We introduce two kinds of multilinear fractional maximal operators on G. By assuming that the graph G satisfies certain geometric conditions, we establish the bounds for the above operators on the endpoint Sobolev spaces and Hajłasz–Sobolev spaces on G.
Highlights
In a very recent article [1], Liu and Zhang introduced the Hajłasz–Sobolev spaces on an infinite connected graph G and established the boundedness for the Hardy–Littlewood maximal operators on G and its fractional variant on the above function spaces and the endpoint Sobolev spaces
We introduce two kinds of multilinear fractional maximal operators on G and to establish the bounds for the above operators on the Hajłasz– Sobolev spaces and endpoint Sobolev spaces on G
In order to generalize results on Rn and its discrete setting to the graph setting, Liu and Xue [35] introduced the first-order Sobolev spaces on graphs and studied the Sobolev regularity of the Hardy–Littlewood maximal operator on a finite connected graph
Summary
In a very recent article [1], Liu and Zhang introduced the Hajłasz–Sobolev spaces on an infinite connected graph G and established the boundedness for the Hardy–Littlewood maximal operators on G and its fractional variant on the above function spaces and the endpoint Sobolev spaces. Let us introduce two types of multilinear fractional maximal operators on the infinite connected graphs G = (VG, EG). In order to generalize results on Rn and its discrete setting to the graph setting, Liu and Xue [35] introduced the first-order Sobolev spaces on graphs and studied the Sobolev regularity of the Hardy–Littlewood maximal operator on a finite connected graph. When G = (VG, EG) is an infinite connected graph, in [1], the authors studied the endpoint Sobolev regularity of the fractional maximal operator on G. ∏ Mκα,G ( f ) W1,q(VG) ≤ Cα,p1,...,pm,Q,B1,Q,D j=1 fj W1,pj (VG)
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