Acta Mathematica Sinica-English Series最新文献

This paper is concerned with a class of degenerate elliptic equations with rapidly oscillating coefficients in periodically perforated domains, which arises in the study of spectrum problems for uniformly elliptic equations in perforated domains. We establish a quantitative convergence rate and obtain the uniform weighted Lipschitz and W^1,p estimates.

A local L^p estimate is proved by using the σ-uniformity, which is motivated by the study of the Stein–Tomas type restriction theorems and Waring’s problem.

This paper is an offspring of the previous study on Herz spaces. A new characterization of Morrey-Herz spaces is given. As applications, the boundedness of various operators is obtained. For example, higher-order commutators generated by singular integral operators and BMO functions are proved to be bounded on Morrey-Herz spaces. The theory of Morrey-Herz-Hardy spaces is also developed.

We study embeddings between generalised Triebel–Lizorkin–Morrey spaces (cal{E}_{varphi,p,q}^{s}(mathbb{R}^{d})) and within the scales of further generalised Morrey smoothness spaces like (cal{N}_{varphi,p,q}^{s}(mathbb{R}^{d})), B ^s,φ_p,q (ℝ^d) and F ^s,φ_p,q (ℝ^d). The latter have been investigated in a recent paper by the first two authors (2023), while the embeddings of the scale (cal{N}_{varphi,p,q}^{s}(mathbb{R}^{d})) were mainly obtained in a paper of the first and last two authors (2022). Now we concentrate on the characterisation of the spaces (cal{E}_{varphi,p,q}^{s}(mathbb{R}^{d})). Our approach requires a wavelet characterisation of those spaces which we establish for the system of Daubechies’ wavelets. Then we prove necessary and sufficient conditions for the embedding (cal{E}_{varphi_{1},p_{1},q_{1}}^{s_{1}}(mathbb{R}^{d})hookrightarrowcal{E}_{varphi_{2},p_{2},q_{2}}^{s_{2}}(mathbb{R}^{d})). We can also provide some almost final answer to the question when (cal{E}_{varphi,p,q}^{s}(mathbb{R}^{d})) is embedded into C(ℝ^d), complementing our recent findings in case of (cal{N}_{varphi,p,q}^{s}(mathbb{R}^{d})).

Let ({cal{A}_{t}}_{t>0}) be a family of bounded linear operator on L²(X) where (X, d, μ) is a metric space with metric d and doubling measure μ. Assume that the family ({cal{A}_{t}}_{t>0}) satisfies suitable off-diagonal estimates from (L^{p_{0}}) to L² for some p₀ < 2. This paper aims to prove weighted bound estimates for conical square functions and g-functions associated to the family ({cal{A}_{t}}_{t>0}). Some applications such as weighted bounds for bilinear estimates associated to certain differential operators are also obtained.

We establish a pointwise property for homogeneous fractional Sobolev spaces in domains with non-empty boundary, extending a similar result of Koskela–Yang–Zhou. We use this to show that a conformal map from the unit disk onto a simply connected planar domain induces a bounded composition operator from the borderline homogeneous fractional Sobolev space of the domain into the corresponding space of the unit disk.

We introduce a bilinear extension of the so-called exotic Calderón-Zygmund operators. These kernels arise naturally from the bilinear singular integrals associated with Zygmund dilations. We show that such a class of operators satisfy a T1 theorem in the same form as the standard Calderón-Zygmund operators. However, one-parameter weighted estimates may fail in general, and unlike the linear case, we are not able to provide the end-point estimates in full generality.

We extend the (outer) measure (gamma_{cal{I}}) associated to an operator ideal (cal{I}) to a measure (gamma_{frak{J}}) for bounded bilinear operators. If (cal{I}) is surjective and closed, and (frak{J}) is the class of those bilinear operators such that (gamma_{frak{J}}(T)=0), we prove that (frak{J}) coincides with the composition bideal (cal{I}circfrak{B}). If (cal{I}) satisfies the Σ_r-condition, we establish a simple necessary and sufficient condition for an interpolated operator by the real method to belong to (frak{J}). Furthermore, if in addition (cal{I}) is symmetric, we prove a formula for the measure (gamma_{frak{J}}) of an operator interpolated by the real method. In particular, results apply to weakly compact operators.

In this paper, we establish a weighted maximal L₂ estimate of operator-valued Bochner–Riesz means. The proof is based on noncommutative square function estimates and a sharp weighted noncommutative Hardy–Littlewood maximal inequality.

In this paper we extend the duality (({overline{C_{rm comp}^{infty}({mathbb R}^{d})}}^{{rm BMO}({mathbb R}^{d})})^{ast}=H^{1}({mathbb {R}^{d}})) to generalized Campanato spaces with variable growth condition ({cal L}_{p,phi}({mathbb R}^{d})) instead of BMO(ℝ^d). We also extend the characterization of ({overline{C_{rm comp}^{infty}({mathbb R}^{d})}}^{{rm BMO}({mathbb R}^{d})}) by Uchiyama (1978) to ({overline{C_{rm comp}^{infty}({mathbb R}^{d})}}^{{cal L}_{p,phi}({mathbb R}^{d})}). Moreover, using this characterization, we prove the boundedness of singular and fractional integral operators on ({overline{C_{rm comp}^{infty}({mathbb R}^{d})}}^{{cal L}_{p,phi}({mathbb R}^{d})}). The function space ({cal L}_{p,phi}({mathbb R}^{d})) treated in this paper covers the case that it is coincide with Lip_α on one area, with BMO on another area and with the Morrey space on the other area, for example.