L p Smoothness on Weighted Besov–Triebel–Lizorkin Spaces in terms of Sharp Maximal Functions

It is known, in harmonic analysis theory, that maximal operators measure local smoothness of L p functions. These operators are used to study many important problems of function theory such as the embedding theorems of Sobolev type and description of Sobolev space in terms of the metric and measure. We study the Sobolev-type embedding results on weighted Besov–Triebel–Lizorkin spaces via the sharp maximal functions. The purpose of this paper is to study the extent of smoothness on weighted function spaces under the condition M α # f ∈ L p , μ , where μ is a lower doubling measure, M α # f stands for the sharp maximal function of f , and 0 ≤ α ≤ 1 is the degree of smoothness.

A Note on the Boundedness of Doob Maximal Operators on a Filtered Measure Space

Let M be the Doob maximal operator on a filtered measure space and let v be an Ap weight with 1<p<+∞. We try proving that ∥Mf∥Lp(v)≤p′[v]Ap1p−1∥f∥Lp(v), where 1/p+1/p′=1. Although we do not find an approach which gives the constant p′, we obtain that ∥Mf∥Lp(v)≤p1p−1p′[v]Ap1p−1∥f∥Lp(v), with limp→+∞p1p−1=1.

Sobolev Regularity of Multilinear Fractional Maximal Operators on Infinite Connected Graphs

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.