arXiv subjects

Jianxun He

Publications and source records attributed to Jianxun He.

4 recordsLinked to original sources

arXiv2024

Peetre conjecture on real interpolation spaces of Besov spaces and Grid K functional

In this paper, Peetre's conjecture about the real interpolation space of Besov space {\bf is solved completely } by using the classification of vertices of cuboids defined by {\bf wavelet coefficients and wavelet's grid structure}. Littlewood-Paley analysis provides only a decomposition of the function on the ring. We extend Lorentz's rearrangement function and Hunt's Marcinkiewicz interpolation theorem to more general cases. We use the method of calculating the topological quantity of the grid to replace the traditional methods of data classification such as gradient descent method and distributed algorithm. We developed a series of new techniques to solve this longstanding open problem. These skills make up for the deficiency of Lions-Peetre iterative theorem in dealing with strong nonlinearity. Using the properties of wavelet basis, a series of {\bf functional nonlinearities} are studied. Using the lattice property of wavelet, we study the lattice topology. By three kinds of {\bf topology nonlinearities}, we give the specific wavelet expression of K functional.

math.FA↗

arXiv2021

Well-posedness of Navier-Stokes equations established by the decaying speed of single norm

The decaying speed of a single norm more truly reflects the intrinsic harmonic analysis structure of the solution of the classical incompressible Navier-Stokes equations. No previous work has been able to establish the well-posedness under the decaying speed of a single norm with respect to time, and the previous solution space is contained in the intersection of two spaces defined by different norms. In this paper, for some separable initial space $X$, we find some new solution space which is not the subspace of $L^{\infty}(X)$. We use parametric Meyer wavelets to establish the well-posedness via the decaying speed of a single norm only, without integral norm to $t$.

math.AP↗

arXiv2018

Boundedness and compactness of commutators associated with Lipschitz functions

Let $α\in (0, 1]$, $β\in [0, n)$ and $T_{Ω,β}$ be a singular or fractional integral operator with homogeneous kernel $Ω$. In this article, a CMO type space ${\rm CMO}_α(\mathbb R^n)$ is introduced and studied. In particular, the relationship between ${\rm CMO}_α(\mathbb R^n)$ and the Lipchitz space $Lip_α(\mathbb R^n)$ is discussed. Moreover, a necessary condition of restricted boundedness of the iterated commutator $(T_{Ω,β})^m_b$ on weighted Lebesgue spaces via functions in $Lip_α(\mathbb R^n)$, and an equivalent characterization of the compactness for $(T_{Ω,β})^m_b$ via functions in ${\rm CMO}_α(\mathbb R^n)$ are obtained. Some results are new even in the unweighted setting for the first order commutators.

math.CA↗

arXiv2017

Limiting weak-type behaviors of some integral operators

In this paper, we explore the limiting weak-type behaviors of some integral operators including maximal operators, singular and fractional integral operators and maximal truncated singular integrals et al. Some optimal limiting weak-type behaviors are given, which essentially improve and extend the previous results in this topics.

math.CA↗