AI 中文总结
证明了Riesz $(p,\alpha)$-变差的Pólya-Szegő原理，该变差在有界$p$-变差和Sobolev空间$W^{1,p}$之间插值，结果对某些具有分数光滑性的不可微函数也成立。
AI 中文摘要
我们证明了Riesz $(p,\alpha)$-变差的Pólya-Szegő原理，这是一种在有界$p$-变差和Sobolev空间$W^{1,p}$之间插值的分数光滑性尺度。与经典的Pólya-Szegő不等式不同，我们的结果也适用于某些具有分数光滑性的无处可微函数，包括Takagi-van der Waerden型函数和黎曼的“不可微”函数。
英文摘要
We establish a one-dimensional Pólya-Szegő principle for the Riesz $(p,α)$-variation $\mathcal{V}_p^α$, a family of functionals that interpolates between Wiener $p$-variation and the Sobolev seminorm $\|f'\|_{L^p}$. More precisely, for every measurable function $f$, we prove that its non-increasing rearrangement $f^*$ satisfies $$ \mathcal{V}_p^α(f^*)\le\mathcal{V}_p^α(f) $$ for all $1\le p<\infty$ and $0\leα\le 1-1/p$. This result contains and extends the classical variation-diminishing property of rearrangements from Sobolev spaces to a scale of spaces admitting fractional smoothness and, in particular, applies to functions that are nowhere differentiable. Our methods also yield a sharp rearrangement inequality for a modulus of continuity defined via $p$-variation, providing an analogue of a problem posed by Ul'yanov. Finally, we sketch how our inequality can be useful in the study of nonlinear Fredholm integral equations.
CommentsRevised and corrected manuscript
