Analysis Mathematica最新文献

Boundedness conditions are found for the Hilbert transform H in Besov spaces with Muckenhoupt weights. The operator H in this situation acts on subclasses of functions from Hardy spaces. The results are obtained by representing the Hilbert transform H via Riemann–Liouville operators of fractional integration on ℝ, on the norms of images and pre-images of which independent estimates are established in the paper. Separately, a boundedness criterion is given for the transform H in weighted Besov and Triebel–Lizorkin spaces restricted to the subclass of Schwartz functions.

The approach to “locally” aggrandize Lebesgue spaces, previously suggested by the authors and based on the notion of “aggrandizer”, is combined with the usual “global” aggrandization. We study properties of such spaces including embeddings, dependence of the choice of the aggrandizer and, in particular, we discuss the question when these spaces are not new, coinciding with globally aggrandized spaces, and when they proved to be new. We study the boundedness of the maximal, singular, and maximal singular operators in the introduced spaces.

We study nuclear embeddings for function spaces of generalised smoothness defined on a bounded Lipschitz domain Ω ⊂ ℝ^d. This covers, in particular, the well-known situation for spaces of Besov and Triebel–Lizorkin spaces defined on bounded domains as well as some first results for function spaces of logarithmic smoothness. In addition, we provide some new, more general approach to compact embeddings for such function spaces, which also unifies earlier results in different settings, including also the study of their entropy numbers. Again we rely on suitable wavelet decomposition techniques and the famous Tong result (1969) about nuclear diagonal operators acting in ∓_r spaces, which we could recently extend to the vector-valued setting needed here.

We prove the following dichotomy for the spaces ℒ ^(a)_p,q,α (X, Y) of all operators T ∈ ℒ(X, Y) whose approximation numbers belong to the Lorentz-Zygmund sequence spaces ℓ_p,q(log ℓ)_α: If X and Y are infinite-dimensional Banach spaces, then the spaces ℒ ^(a)_p,q,α (X, Y) with 0 < p < ∞, 0 < q ≤ ∞ and α ∈ ℝ are all different from each other, but otherwise, if X or Y are finite-dimensional, they are all equal (to ℒ(X, Y)).

Moreover we show that the scale ({{ {cal L}_{infty ,q}^{(a)}(X,Y)} _{0, < q, < infty }}) is strictly increasing in q, where ℒ ^(a)_∈,q (X, Y) is the space of all operators in ℒ(X, Y) whose approximation numbers are in the limiting Lorentz sequence space ∓_∈,q.

We extend the two weighted norm inequalities for the potential type operators to Herz spaces. As an application of this result, we have the two weighted norm inequalities of the fractional integral operators on Herz spaces.

The paper is dedicated to the study of embeddings of the anisotropic Besov spaces (B_{p,{theta _1}, ldots ,{theta _n}}^{{beta _1}, ldots ,{beta _n}}) (ℝⁿ) into Lorentz spaces. We find the sharp asymptotic behaviour of embedding constants when some of the exponents β_k tend to 1 (β_k < 1). In particular, these results give an extension of the estimate proved by Bourgain, Brezis, and Mironescu for isotropic Besov spaces. Also, in the limit, we obtain a link with some known embeddings of anisotropic Lipschitz spaces.

One of the key results of the paper is an anisotropic type estimate of rearrangements in terms of partial moduli of continuity.

In this paper we introduce a generalization to compact subsets of certain algebraic varieties of the classical Markov inequality on the derivatives of a polynomial in terms of its own values. We also introduce an extension to such sets of a local form of the classical Markov inequality, and show the equivalence of introduced Markov and local Markov inequalities.

We consider temperate distributions on Euclidean spaces with uniformly discrete support and locally finite spectrum. We find conditions on coefficients of distributions under which they are finite sums of derivatives of generalized lattice Dirac combs. These theorems are derived from properties of families of discretely supported measures and almost periodic distributions.

The aim of this paper is to show that if the Hewitt–Stromberg pre-measures with respect to the gauge are finite, then these pre-measures have a kind of outer regularity in a general metric space X. We give also some conditions on the Hewitt–Stromberg pre-measures with respect to the gauge such that the Hewitt–Stromberg measures have an almost inner regularity on a complete separable metric space X.

By utilizing Nevanlinna theory of meromorphic functions, we characterize meromorphic solutions of the following nonlinear differential equation of the form

where n ≥ 3, t ≥ 0 and m ≥ 1 are integers, n ≥ m, P(z, f, f′, …, f^(t)) is a differential polynomial in f (z) of degree d ≤ n with small functions of f (z) as its coefficients, and α_j, P_j (j = 1, 2, …, m) are nonzero constants such that ∣α₁∣ > ∣α₂∣ > … > ∣α_m∣. Also we provide the concrete forms of the solutions of the equation above, and present some examples illustrating the sharpness of our results.