Let X be a ball Banach function space on $${\mathbb R}^n$$ . In this article, under some mild assumptions about both X and the boundedness of the Hardy–Littlewood maximal operator on the associate space of the convexification of X, the authors prove that, for any locally integrable function f with $$\Vert \,|\nabla f|\,\Vert _{X}<\infty $$ , $$\begin{aligned} \sup _{\lambda \in (0,\infty )}\lambda \left\| \left| \left\{ y\in {{\mathbb {R}}}^n:\ |f(\cdot )-f(y)| >\lambda |\cdot -y|^{\frac{n}{q}+1}\right\} \right| ^{\frac{1}{q}} \right\| _X\sim \Vert \,|\nabla f|\,\Vert _X \end{aligned}$$ with the positive equivalence constants independent of f, where the index $$q\in (0,\infty )$$ is related to X and $$|\{y\in {\mathbb R}^n:\ |f(\cdot )-f(y)| >\lambda |\cdot -y|^{\frac{n}{q}+1}\}|$$ is the Lebesgue measure of the set under consideration. In particular, when $$X:=L^p({\mathbb R}^n)$$ with $$p\in [1,\infty )$$ , the above formulae hold true for any given $$q\in (0,\infty )$$ with $$n(\frac{1}{p}-\frac{1}{q})<1$$ , which when $$q=p$$ are exactly the recent surprising formulae of H. Brezis, J. Van Schaftingen, and P.-L. Yung, and which in other cases are new. This generalization has a wide range of applications and, particularly, enables the authors to establish new fractional Sobolev and new Gagliardo–Nirenberg inequalities in various function spaces, including Morrey spaces, mixed-norm Lebesgue spaces, variable Lebesgue spaces, weighted Lebesgue spaces, Orlicz spaces, Orlicz-slice (generalized amalgam) spaces, and weak Morrey spaces, and, even in all these special cases, the obtained results are new. The proofs of these results strongly depend on the Poincaré inequality, the extrapolation, the exact operator norm on $$X'$$ of the Hardy–Littlewood maximal operator, and the exquisite geometry of $${\mathbb {R}}^n.$$ 

/pdf/brezis-van-schaftingen-yung-formulae-in-ball-banach-function-2xofzphy.pdf

Brezis–Van Schaftingen–Yung formulae in ball Banach function spaces with applications to fractional Sobolev and Gagliardo–Nirenberg inequalities

New Ball Campanato-Type Function Spaces and Their Applications

We characterize entire and meromorphic solutions of certain functional equations and nonlinear partial differential equations.

On Certain Functional and Partial Differential Equations, II

Weaving Phase Retrieval and Weaving Norm Retrieval

Characteizations of woven g-frames and weaving g-frames in Hilbert spaces and C*-modules

Weaving frames are powerful tools in wireless sensor networks and pre-processing signals. In this paper, we introduce the concept of weaving for g-frames in Hilbert spaces. We first give some properties of weaving g-frames and present two necessary conditions in terms of frame bounds for weaving g-frames. Then we study the properties of weakly woven g-frames and give a sufficient condition for weaving g-frames. It is shown that weakly woven is equivalent to woven. Two sufficient conditions for weaving g-Riesz bases are given. And a weaving equivalent of an unconditional g-basis for weaving g-Riesz bases is considered. Finally, we present Paley–Wiener-type perturbation results for weaving g-frames.

On Weaving g-Frames for Hilbert Spaces

Weaving frames are powerful tools in wireless sensor networks and pre-processing signals. In this paper, we introduce the concept of weaving for g-frames in Hilbert spaces. We first give some properties of weaving g-frames and present two necessary conditions in terms of frame bounds for weaving g-frames. Then we study the properties of weakly woven g-frames and give a sufficient condition for weaving g-frames. It is shown that weakly woven is equivalent to woven. Two sufficient conditions for weaving g-Riesz bases are given. And a weaving equivalent of an unconditional g-basis for weaving g-Riesz bases is considered. Finally, we present Paley-Wiener-type perturbation results for weaving g-frames.

On weaving g-frames for Hilbert spaces

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On Weaving g-Frames for Hilbert Spaces

On weaving g-frames for Hilbert spaces