Acta Mathematica Sinica-English Series最新文献

Tree-based models have been widely applied in both academic and industrial settings due to the natural interpretability, good predictive accuracy, and high scalability. In this paper, we focus on improving the single-tree method and propose the segmented linear regression trees (SLRT) model that replaces the traditional constant leaf model with linear ones. From the parametric view, SLRT can be employed as a recursive change point detect procedure for segmented linear regression (SLR) models, which is much more efficient and flexible than the traditional grid search method. Along this way, we propose to use the conditional Kendall’s τ correlation coefficient to select the underlying change points. From the non-parametric view, we propose an efficient greedy splitting method that selects the splits by analyzing the association between residuals and each candidate split variable. Further, with the SLRT as a single-tree predictor, we propose a linear random forest approach that aggregates the SLRTs by a weighted average. Both simulation and empirical studies showed significant improvements than the CART trees and even the random forest.

In this paper, we consider the jump and variational inequalities of truncated singular integral operator with rough kernel

where the kernel (Omega in (L(log^{+}L)^{2})^{n over{n-beta}}(mathbb{S}^{n-1})) satisfies the vanishing condition and the homogeneous condition of degree 0. This kind of singular integral appears in the approximation of the surface quasi-geostrophic (SQG) equation from the generalized SQG equation. We establish the (L^p, L^q) estimate of the jump and variational inequalities of the families {T_Ω,β,ε}_ε>0 for ({1over q}={1over p}-{betaover n}) and 0 < β < 1. Moreover, one can get the L^p boundedness of the Calderón–Zygmund operator with the same kernel by letting β → 0⁺.

For s ∈ [0, 1], b ∈ ℝ and p ∈ [1, ∞), let (dot{B}_{p,infty}^{s,b}(mathbb{R}^{n})) be the logarithmic-Gagliardo–Lipschitz space, which arises as a limiting interpolation space and coincides to the classical Besov space when b = 0 and s ∈ (0, 1). In this paper, the authors study restricting principles of the Riesz potential space (cal{I}_{alpha}(dot{B}_{p,infty}^{s,b}(mathbb{R}^{n}))) into certain Radon–Campanato space.

Let Ω be a domain of ℝⁿ with n ≥ 2 and p(·) be a local Lipschitz funcion in Ω with 1 < p(x) < ∞ in Ω. We build up an interior quantitative second order Sobolev regularity for the normalized p(·)-Laplace equation −Δ ^N_p(·) u = 0 in Ω as well as the corresponding inhomogeneous equation −Δ ^N_p(·) u =f in Ω with f ∈ C⁰(Ω). In particular, given any viscosity solution u to −Δ ^N_p(·) u = 0 in Ω, we prove the following:

1. (i)

   in dimension n = 2, for any subdomain U ⋐ Ω and any β ≥ 0, one has ∣Du∣^βDu ∈ L ^2+δ_loc (U) with a quantitative upper bound, and moreover, the map ((x_{1},x_{2})rightarrowvert Duvert^{beta}(u_{x_{1}},-u_{x_{2}})) is quasiregular in U in the sense that

   $$vert D[vert Duvert^{beta};Du]vert^{2}leq-C;text{det};D[vert Duvert^{beta};Du];;;;;text{a.e.};text{in};U.$$
2. (ii)

   in dimension n ≥ 3, for any subdomain U ⋐ Ω with inf_U p(x) > 1 and (text{sup}_{U};p(x)<3+{2over{n-2}}), one has D²u ∈ L ^2+δ_loc (U) with a quantitative upper bound, and also with a pointwise upper bound

   $$vert D^{2}uvert^{2}leq-Csum_{1leq i<jleq n}[u_{x_{i}x_{j}}u_{x_{j}x_{i}}-u_{x_{i}x_{i}}u_{x_{j}x_{j}}];;;;;text{a.e};text{in};U.$$

Here constants δ > 0 and C ≥ 1 are independent of u. These extend the related results obtaind by Adamowicz–Hästö [Mappings of finite distortion and PDE with nonstandard growth. Int. Math. Res. Not. IMRN, 10, 1940–1965 (2010)] when n = 2 and β = 0.

We prove almost everywhere convergence for convolutions of locally integrable functions with shrinking L¹ dilations of a fixed integrable kernel with an integrable radially decreasing majorant. The set on which the convergence holds is an explicit subset of the Lebesgue set of the locally integrable function of full measure. This result can be viewed as an extension of the Lebesgue differentiation theorem in which the characteristic function of the unit ball is replaced by a more general kernel. We obtain a similar result for multilinear convolutions.

Let T be a bilinear vector-valued singular integral operator satisfies some mild regularity conditions, which may not fall under the scope of the theory of standard Calderón–Zygmund classes. For any (vec{b}=(b_{1},b_{2})in (text{CMO}(mathbb{R}^{n}))^{2}), let ([T,b_{j}]_{e_{j}} (j=1,2), [T,vec{b}]_{alpha}) be the commutators in the j-th entry and the iterated commutators of T, respectively. In this paper, for all p₀ > 1, ({p_{0}over 2} < p < infty), and p₀ ≤ p₁, p₂ < ∞ with 1/p = 1/p₁ + 1/p₂, we prove that ([T,b_{j}]_{e_{j}}) and ([T,vec{b}]_{alpha}) are weighted compact operators from (L^{p_{1}}(w_{1})times L^{p_{2}}(w_{2})) to (L^{p}(nu_{vec{w}})), where (vec{w}=(w_{1},w_{2})in A_{vec{p}/p_{0}}) and (nu_{vec{w}}=w_{1}^{p/p_{1}}w_{2}^{p/p_{2}}). As applications, we obtain the weighted compactness of commutators in the j-th entry and the iterated commutators of several kinds of bilinear Littlewood–Paley square operators with some mild kernel regularity, including bilinear g function, bilinear g*_λ function and bilinear Lusin’s area integral. In addition, we also get the weighted compactness of commutators in the j-th entry and the iterated commutators of bilinear Fourier multiplier operators, and bilinear square Fourier multiplier operators associated with bilinear g function, bilinear g*_λ function and bilinear Lusin’s area integral, respectively.

Let p(·): ℝⁿ → (0, ∞] be a variable exponent function satisfying the globally log-Hölder continuous condition and A a general expansive matrix on ℝⁿ. Let H ^p(·)_A (ℝⁿ) be the variable anisotropic Hardy space associated with A. In this paper, via first establishing a criterion for affirming some functions being in the space H ^p(·)_A (ℝⁿ), the authors obtain several equivalent characterizations of H ^p(·)_A (ℝⁿ) in terms of the so-called tight frame multiwavelets, which extend the well-known wavelet characterizations of classical Hardy spaces. In particular, these wavelet characterizations are shown without the help of Peetre maximal operators.

Let q ∈ (0, ∞] and φ be a Musielak–Orlicz function with uniformly lower type p ⁻_φ ∈ (0, ∞) and uniformly upper type p ⁺_φ ∈ (0, ∞). In this article, the authors establish various real-variable characterizations of the Musielak–Orlicz–Lorentz Hardy space H^φ,q(ℝⁿ), respectively, in terms of various maximal functions, finite atoms, and various Littlewood–Paley functions. As applications, the authors obtain the dual space of H^φ,q(ℝⁿ) and the summability of Fourier transforms from H^φ,q(ℝⁿ) to the Musielak–Orlicz–Lorentz space L^φ,q(ℝⁿ) when q ∈ (0, ∞) or from the Musielak–Orlicz Hardy space H^φ(ℝⁿ) to L^φ,q(ℝⁿ) in the critical case. These results are new when q ∈ (0, ∞) and also essentially improve the existing corresponding results (if any) in the case q = ∞ via removing the original assumption that φ is concave. To overcome the essential obstacles caused by both that φ may not be concave and that the boundedness of the powered Hardy–Littlewood maximal operator on associated spaces of Musielak–Orlicz spaces is still unknown, the authors make full use of the obtained atomic characterization of H^φ,q(ℝⁿ), the corresponding results related to weighted Lebesgue spaces, and the subtle relation between Musielak–Orlicz spaces and weighted Lebesgue spaces.

The dyadic representation of any singular integral operator, as an average of dyadic model operators, has found many applications. While for many purposes it is enough to have such a representation for a “suitable class” of test functions, we show that, under quite general assumptions (essentially minimal ones to make sense of the formula), the representation is actually valid for all pairs (f,g) ∈ L^p(ℝ^d) × L^p′(ℝ^d), not just test functions.

The paper deals with continuous and compact mappings generated by the Fourier transform between distinguished Besov spaces B ^s_p (ℝⁿ) = B ^s_p,p (ℝⁿ), 1 ≤ p ≤ ∞, and between Sobolev spaces H ^s_p (ℝⁿ), 1 < p < ∞. In contrast to the paper H. Triebel, Mapping properties of Fourier transforms. Z. Anal. Anwend. 41 (2022), 133–152, based mainly on embeddings between related weighted spaces, we rely on wavelet expansions, duality and interpolation of corresponding (unweighted) spaces, and (appropriately extended) Hausdorff-Young inequalities. The degree of compactness will be measured in terms of entropy numbers and approximation numbers, now using the symbiotic relationship to weighted spaces.