加权Lebesgue空间的对偶和嵌入

本文建立了加权Lebesgue空间的对偶空间,并且给出了在这些空间之间嵌入的必要条件。     

Cauchy Problems in Weighted Lebesgue Spaces

Global solvability and asymptotics of semilinear parabolic Cauchy problems in ⁿ are considered. Following the approach of A. Mielke [15] these problems are investigated in weighted Sobolev spaces. The paper provides also a theory of second order elliptic operators in such spaces considered over ⁿ, n . In particular, the generation of analytic semigroups and the embeddings for the domains of fractional powers of elliptic operators are discussed.     

The Two-Weighted Inequalities for Sublinear Operators Generated by B Singular Integrals in Weighted Lebesgue Spaces

In this paper, the authors establish several general theorems for the boundedness of sublinear operators (B sublinear operators) satisfies the condition (1.2), generated by B singular integrals on a weighted Lebesgue spaces $L_{p,\omega,\gamma}(\mathbb{R}_{k,+}^{n})$ , where $B=\sum_{i=1}^{k} (\frac{\partial^{2}}{\partial x_{k}^{2}} + \frac{\gamma_{i}}{x_{i}}\frac{\partial}{\partial x_{i}} )$ . The condition (1.2) are satisfied by many important operators in analysis, including B maximal operator and B singular integral operators. Sufficient conditions on weighted functions ω and ω ₁ are given so that B sublinear operators satisfies the condition (1.2) are bounded from $L_{p,\omega,\gamma}(\mathbb{R}_{k,+}^{n})$ to $L_{p,\omega_{1},\gamma}(\mathbb{R}_{k,+}^{n})$ .     

Weighted Fourier Inequalities via Rearrangements

The method of using rearrangements to give sufficient conditions for Fourier inequalities between weighted Lebesgue spaces is revisited, a comparison between two known sufficient conditions is completed, and the method is extended to provide sufficient conditions for a new range of indices. When \(1<q<p<\infty \), simple conditions on weights ensure that the Fourier transform will map a weighted \(L^p\) space into a weighted \(L^q\) space. These are established in Theorems 1 and 4 of Benedetto and Heinig (J Fourier Anal Appl 9(1):1–37, 2003). The proofs apply when \(2<q<p\) and \(1<q<p<2\) but not in the remaining case, \(1<q<2<p\). Here, counterexamples are given to show that these simple conditions are no longer sufficient when \(1<q<2<p\). Also, various additional conditions are presented, any of which will restore sufficiency in that case.     

Fourier Series in Weighted Lorentz Spaces

The Fourier coefficient map is considered as an operator from a weighted Lorentz space on the circle to a weighted Lorentz sequence space. For a large range of Lorentz indices, necessary and sufficient conditions on the weights are given for the map to be bounded. In addition, new direct analogues are given for known weighted Lorentz space inequalities for the Fourier transform. Applications are given that involve Fourier coefficients of functions in LogL and more general Lorentz–Zygmund spaces.     

On Lebesgue Integrability of Fourier Transforms in Amalgam Spaces

Journal of Fourier Analysis and Applications - Let f be an element of the subspace $$(L^{q},l^{p})^{\alpha }({\mathbb {R}}^d)$$ ( $$1\le q \le \alpha \le p \le 2 $$ ) of the Wiener amalgam space...     

Trace Inequalities,Maximal Inequalities,and Weighted Fourier Transform Estimates

In the spirit of work of Kerman and Sawyer, a condition is given that is necessary and sufficient for the Fourier transform norm inequality $\Big(\int_{{\Bbb R}_d} \vert\hat{f}\vert^q d\mu\Big)^{1/q} \leq C\Big(\int_{{\Bbb R}_d} \vert f\vert^p v\Big)^{1/p}$ provided v is a radial weight for which v^?1/p is convexly decreasing and μ is a suitable measure. We also establish alternative conditions for such inequalities by proving corresponding trace type inequalities and maximal function inequalities that underlie the Fourier transform estimates. Our conditions are relatively simple to compute. Among applications we give extensions of a Sobolev restriction theorem.     

Weighted Fourier Inequalities: New Proofs and Generalizations

Fourier transform inequalities in weighted Lebesgue spaces are proved.The inequalities are generalizations of the Plancherel theorem, theyare characterized in terms of uncertainty principle relationsbetween pairs of weights, and they are put in the context of existingweighted Fourier transform inequalities. The proofs are new and relativelyelementary, and they give rise to good and explicit constants controllingthe continuity of the Fourier transform operator. The smaller the constantis,the more applicable the inequality will be in establishing weighteduncertainty principle or entropy inequalities. There are two essentiallydifferent proofs, one depending on operator theory and one depending onLorentz spaces. The results from these approaches are quantitativelycompared, leading to classical questions concerning multipliers and tonew questions concerning wavelets.     

Fine Boundary Behavior and Invariant Subspaces of Harmonically Weighted Dirichlet Spaces

We study the boundary behavior of functions in spaces of Dirichlet-type by using non-linear capacities generalizing the logarithmic capacity. We use these capacities to obtain information about the invariant subspaces of the shift operator. As an application, we prove an analogue of a conjecture of Brown and Shields when the space is weighted by the Poisson integral of a finite sum of atoms.     

Singular Integral Operators with Fixed Singularities on Weighted Lebesgue Spaces

The paper is devoted to study of singular integral operators with fixed singularities at endpoints of contours on weighted Lebesgue spaces with general Muckenhoupt weights. Compactness of certain integral operators with fixed singularities is established. The membership of singular integral operators with fixed singularities to Banach algebras of singular integral operators on weighted Lebesgue spaces with slowly oscillating Muckenhoupt weights is proved on the basis of Balakrishnans formula from the theory of strongly continuous semi-groups of closed linear operators. Symbol calculus for such operators, Fredholm criteria and index formulas are obtained.     

加权Bergman空间中函数的几个严格不等式——纪念李国平院士吴新谋教授诞辰100周年

该文给出加权Bergman空间中函数的几个严格不等式.     

Weighted Inequalities for Certain Maximal Functions in Orlicz Spaces

Let M_g be the maximal operator defined by $$M_g f\left( x \right) = \sup \frac{{\int_a^b {f\left( y \right)g\left( y \right){\text{d}}y} }}{{\int_a^b {g\left( y \right){\text{d}}y} }}$$ , where g is a positive locally integrable function on R and the supremum is taken over all intervals [a,b] such that 0≤a≤x≤b/η(b?a), here η is a non-increasing function such that η (0) = 1 and $\mathop {{\text{lim}}}\limits_{t \to {\text{ + }}\infty } \eta \left( t \right) = 0$ η (t) = 0. This maximal function was introduced by H. Aimar and L. L. Forzani [AF]. Let Φ be an N - function such that Φ and its complementary N - function satisfy Δ₂. It gives an A′_Φ(g) type characterization for the pairs of weights (u,v) such that the weak type inequality $$u\left( {\left\{ {x \in {\text{R}}\left| {M_g f\left( x \right) >\lambda } \right.} \right\}} \right) \leqslant \frac{C}{{\Phi \left( \lambda \right)}}\int_{\text{R}} {\Phi \left( {\left| f \right|v} \right)} $$ holds for every f in the Orlicz space L_Φ(v). And, there are no (nontrivial) weights w for which (w,w) satisfies the condition A′_Φ(g).     

Weighted Inequalities for Certain Maximal Functions in Orlicz Spaces

Let M_g be the maximal operator defined by

Wiener-Hopf Operators with Semi-almost Periodic Matrix Symbols on Weighted Lebesgue Spaces

Fredholm criteria and index formulas are established for Wiener-Hopf operators W(a) with semi-almost periodic matrix symbols a on weighted Lebesgue spaces where 1 < p < ∞, w belongs to a subclass of Muckenhoupt weights and . We also study the invertibility of Wiener-Hopf operators with almost periodic matrix symbols on . In the case N = 1 we also obtain a semi-Fredholm criterion for Wiener-Hopf operators with semi-almost periodic symbols and, for another subclass of weights, a Fredholm criterion for Wiener-Hopf operators with semi-periodic symbols. Work was supported by the SEP-CONACYT Project No. 25564 (México). The second author was also sponsored by the CONACYT scholarship No. 163480.     

Wiener-Hopf Operators with Slowly Oscillating Matrix Symbols on Weighted Lebesgue Spaces

Fredholm conditions and an index formula are obtained for Wiener-Hopf operators W(a) with slowly oscillating matrix symbols a on weighted Lebesgue spaces where 1 < p < ∞, w is a Muckenhoupt weight on and . The entries of matrix symbols belong to a Banach subalgebra of Fourier multipliers on that are continuous on and have, in general, different slowly oscillating asymptotics at ±∞. To define the Banach algebra SO_{p, w} of corresponding slowly oscillating functions, we apply the theory of pseudodifferential and Calderón-Zygmund operators. Established sufficient conditions become a Fredholm criterion in the case of Muckenhoupt weights with equal indices of powerlikeness, and also for Muckenhoupt weights with different indices of powerlikeness under some additional condition on p, w and a. Work was supported by the SEP-CONACYT Project No. 25564 (México). The second author was also sponsored by the CONACYT scholarship No. 163480.     

Some Modular Inequalities in Lebesgue Spaces with Variable Exponent on the Complex Plane

Mathematical Notes - We consider the modular inequalities for some linear operators on Lebesgue spaces with variable exponent on the complex plane. The main results show that the variable exponent...     

Weighted Inequalities for the Generalized Maximal Operator in Martingale Spaces

The generalized maximal operator M in martingale spaces is considered. For 1 < p ≤ q < ∞, the authors give a necessary and sufficient condition on the pair ([^(m)]\hat \mu , v) for M to be a bounded operator from martingale space L ^p (v) into L ^q ([^(m)]\hat \mu ) or weak-L ^q ([^(m)]\hat \mu ), where [^(m)]\hat \mu is a measure on Ω × ℕ and v a weight on Ω. Moreover, the similar inequalities for usual maximal operator are discussed.     

Embeddings for Weighted Spaces of Functions of Positive Smoothness on Irregular Domains into Lebesgue Spaces

For weighted spaces of functions of positive smoothness on irregular domains, embedding theorems into weighted Lebesgue spaces are proved.     

The Definition of the Fourier Transform for Weighted Inequalities

Weighted norm inequalities for the Fourier transform have a long history, motivated in part by generalizations of the differentiation formula for Fourier transforms and by applications such as Kolmogorov′s prediction theory and restriction theory. Typically, a norm inequality, || ||L^q_μ ≤ ||f||L^p_v, is established on a subspace of L¹ contained in the weighted space L^p_v, where is the Fourier transform and μ and v are weights. The problem of defining the extension of on all of L^p_v is posed and solved here for several important cases. This definition sometimes results in the ordinary distributional Fourier transform, but there are other situations which are also analyzed. Precise necessary conditions and closure theorems are inextricably related to this program, and these results are established.