The goal of this article is to study the fractal surfaces and associated fractal operator on Lebesgue spaces with respect to fractal measures. First, we show that fractal surfaces belongs to Lebesgue spaces under certain conditions. Then, we define a fractal operator on Lebesgue spaces and discuss some analytical properties of it. Moreover, we show the existence of Schauder basis of the associated fractal functions for the space In the end, we draw some graph of fractal surfaces for the various scaling factors and mention some future directions.

中文翻译：

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勒贝格空间中关于分形测度和相关分形算子的分形表面

本文的目的是研究勒贝格空间上关于分形测度的分形曲面和相关分形算子。首先，我们证明分形表面在一定条件下属于勒贝格空间。然后，我们定义了勒贝格空间上的分形算子并讨论了它的一些分析性质。此外，我们证明了空间相关分形函数的 Schauder 基的存在性。最后，我们绘制了各种比例因子的分形曲面图，并提到了一些未来的方向。