arXiv subjects

Tengfei Bai

Publications and source records attributed to Tengfei Bai.

11 recordsLinked to original sources

arXiv2026

Matrix-weighted $Z$-spaces and tent spaces:characterizations, duality and interpolation

Let $α\in\mathbb R$, $0<p<\infty$, $0<q,r\le\infty$, and $W$ be a matrix $\mathcal A_p$ weight on ${\mathbb R}^n$. We introduce the matrix-weighted $Z$-spaces $Z^{p,q,r}_α(W)$ and tent spaces $T^{p,q,r}_α(W)$ on the upper half-space and establish their characterizations using reducing operators. We then characterize homogeneous matrix-weighted Besov and Triebel--Lizorkin spaces in terms of the corresponding $Z$-spaces and tent spaces, respectively, using Schwartz kernels satisfying Tauberian and finite moment conditions. Furthermore, we establish duality for the matrix-weighted $Z$-spaces and tent spaces and obtain real and complex interpolation results for these spaces and the associated homogeneous matrix-weighted Besov and Triebel--Lizorkin spaces.

math.FA↗

arXiv2026

Mixed Bourgain-Morrey spaces and their applications to boundedness of operators

math.FA↗

arXiv2026

Boundedness of nonregular bilinear pseudodifferential operators on variable Besov and Triebel--Lizorkin spaces

In this article, we introduce two bilinear, non-regular symbol classes\linebreak $BS\mathcal B^m_δ(r,μ,v;N,λ)$ and $BS\mathcal F^m_δ(r,μ,v;N,λ)$, where $m\in\mathbb R$, $0\leqδ<1$, $0<μ,v\leq\infty$, $1\leqλ\leq2$, $r>n/μ$, and $N>2n/λ$, with $μ<\infty$ for the latter class. For $μ\geq1$, $(1-δ)r\geq n/μ$, and sufficiently large $N$, we establish boundedness of the associated bilinear pseudodifferential operators on Besov and Triebel--Lizorkin spaces with variable smoothness, integrability and summation exponents. To the best of the authors' knowledge, these operator estimates are new even on the classical Besov and Triebel--Lizorkin spaces with constant exponents. As an application, we establish the multiplication algebra property for both variable exponent spaces.

math.FA↗

arXiv2026

Preduals of weighted homogeneous Bourgain-Morrey Besov-Triebel-Lizorkin spaces associated with operators

Let $(X,ρ,μ)$ be a space of homogeneous type satisfying $μ(X) =\infty$, the doubling property and the reverse doubling condition. Let $L$ be a nonnegative self-adjoint operator on $L^2(X)$ whose heat kernel has a Gaussian upper bound. First, we study the preduals of weighted Bourgain-Morrey spaces on $X$ and their properties, such as lattice property and completeness. Then we introduce the weighted homogeneous Besov-Triebel-Lizorkin block spaces. It is shown that they are the preduals of weighted homogeneous Bourgain-Morrey Besov-Triebel-Lizorkin spaces associated with the operator $L$. Consequently, these block spaces are independent of the choice of partitions of unity.

math.FA↗

arXiv2026

Equivalent norms and $φ$-transform of matrix-weighted anisotropic Besov-type and Triebel-Lizorkin-type spaces

We introduce matrix-weighted anisotropic Besov-type and Triebel-Lizorkin-type spaces associated with an expansive matrix $A$. Inspired by the $\A_p$-dimensions of matrix weight of Bu et al. (2025), we study the properties of matrix weight associated with $A$. Using the nice properties of matrix weight, we obtain that these spaces are equivalent with their corresponding averaging spaces and establish the discrete $φ$-transform of these spaces. Finally, we introduce matrix-weighted anisotropic Triebel-Lizorkin spaces for the limiting case $p=\infty$ and obtain their $φ$-transform. The relation between matrix-weighted anisotropic Triebel-Lizorkin spaces and matrix-weighted anisotropic Besov-type and Triebel-Lizorkin-type spaces is also studied.

math.FA↗

arXiv2026

Precompactness in matrix weighted Bourgain-Morrey spaces

In this paper, we introduce matrix weighted Bourgain-Morrey spaces and obtain two sufficient conditions for precompact sets in matrix weighted Bourgain-Morrey spaces. We prove that the dyadic average operator is bounded on some matrix weighted Bourgain-Morrey spaces. With this result, we obtain the necessity for precompact sets in some matrix weighted Bourgain-Morrey spaces. The results are new even for the unweighted Bourgain-Morrey spaces.

math.FA↗

arXiv2025

On matrix weighted Bourgain-Morrey Triebel-Lizorkin spaces

We introduce the homogeneous (inhomogeneous) matrix weighted Bourgain-Morrey Triebel-Lizorkin spaces and obtain their equivalent norms. We also obtain their characterizations by Peetre type maximal functions, Lusin-area function, Littlewood-Paley $g_λ^{*}$-function, approximation, wavelet and atom. As an application, we obtain boundedness of pseudo-differential operators with symbols in the Hörmander classes and Hölder-Zygmund classes on inhomogeneous matrix weighted Bourgain-Morrey Triebel-Lizorkin spaces.

math.FA↗

arXiv2025

On the mixed Bourgain-Morrey spaces

We introduce the mixed Bourgain-Morrey spaces and obtain their preduals. The boundedness of Hardy-Littlewood maximal operator, iterated maximal operator, fractional integral operator, singular integral operator on these spaces is proved. The Littlewood-Paley theory for mixed Bourgain-Morrey spaces and their preduals are established. As applications, we consider wavelet characterizations for mixed Bourgain-Morrey spaces and a fractional chain rule in mixed Bourgain-Morrey Triebel-Lizorkin spaces. In addition, we give a description of the dual of mixed Bourgain-Morrey spaces and conclude the reflexivity of these spaces.

math.FA↗

arXiv2025

The preduals of Banach space valued Bourgain-Morrey spaces

Let $X$ be a Banach space such that there exists a Banach space $^\ast X$ and $ ( ^\ast X )^ \ast = X $. In this paper, we introduce $X$-valued Bourgain-Morrey spaces. We show that $^\ast X$-valued block spaces are the predual of $X$-valued Bourgain-Morrey spaces. We obtain the completeness, denseness and Fatou property of $^\ast X$-valued block spaces. We give a description of the dual of $X$-valued Bourgain-Morrey spaces and conclude the reflexivity of these spaces. The boundedness of powered Hardy-Littlewood maximal operator in vector valued block spaces is obtained.

math.FA↗

arXiv2025

Bourgain-Morrey-Lorentz spaces and operators on them

We introduce Bourgain-Morrey-Lorentz spaces and give a description of the predual of Bourgain-Morrey-Lorentz spaces via the block spaces. As an application of duality, we obtain the boundedness of Hardy-Littlewood maximal operator, sharp maximal operator, Calderón-Zygmund operator, fractional integral operator, commutator on Bourgain-Morrey-Lorentz spaces. Moreover, we obtain a weak Hardy factorization terms of Calderón-Zygmund operator in Bourgain-Morrey-Lorentz spaces. Using this result, we obtain a characterization of functions in $\BMO$ (the functions of ``bounded mean oscillation'') via the boundedness of commutators generated by them and a homogeneous Calderón-Zygmund operator. In the last, we show that the commutator generated by a function $b$ and a homogeneous Calderón-Zygmund operator is a compact operator on Bourgain-Morrey-Lorentz spaces if and only if $b$ is the limit of compactly supported smooth functions in $\BMO$.

math.FA↗

arXiv2025

Weighted Bourgain-Morrey-Besov type and Triebel-Lizorkin type spaces associated with operators

Let $(X,μ)$ be a space of homogeneous type satisfying $μ(X) =\infty$, the doubling property and the reverse doubling condition. Let $L$ be a nonnegative self-adjoint operator on $L^2(X)$ whose heat kernel enjoys a Gaussian upper bound. We introduce the weighted homogeneous Bourgain-Morrey-Besov type spaces and Triebel-Lizorkin type spaces associated with the operator $L$. We obtain their continuous characterizations in terms of Peetre maximal functions, noncompactly supported functional calculus, heat kernel. Atomic and molecular decompositions of weighted homogeneous Bourgain-Morrey-Besov type spaces and Triebel-Lizorkin type spaces are also given. As an application, we obtain the boundedness of the fractional power of $L$, the spectral multiplier of $L$ on Bourgain-Morrey-Besov type spaces and Triebel-Lizorkin type spaces.

math.FA↗