Embeddings between weighted Copson and Cesàro function spaces

In this paper, characterizations of the embeddings between weighted Copson function spaces \(Co{p_{{p_1},{q_1}}}\left( {{u_1},{v_1}} \right)\) and weighted Cesàro function spaces \(Ce{s_{{p_2},{q_2}}}\left( {{u_2},{v_2}} \right)\) are given. In particular, two-sided estimates of the optimal constant c in the inequality

$${\left( {\int_0^\infty {{{\left( {\int_0^t {f{{\left( \tau \right)}^{{p_2}}}{v_2}\left( \tau \right)d\tau } } \right)}^{{q_2}/{p_2}}}{u_2}\left( t \right)dt} } \right)^{1/{q_2}}} \leqslant c{\left( {\int_0^\infty {{{\left( {\int_t^\infty {f{{\left( \tau \right)}^{{p_1}}}{v_1}\left( \tau \right)d\tau } } \right)}^{{q_1}/{p_1}}}{u_1}\left( t \right)dt} } \right)^{1/{q_1}}},$$

where p₁, p₂, q₁, q₂ ∈ (0,∞), p₂ ≤ q₂ and u₁, u₂, v₁, v₂ are weights on (0,∞), are obtained. The most innovative part consists of the fact that possibly different parameters p₁ and p₂ and possibly different inner weights v₁ and v₂ are allowed. The proof is based on the combination of duality techniques with estimates of optimal constants of the embeddings between weighted Cesàro and Copson spaces and weighted Lebesgue spaces, which reduce the problem to the solutions of iterated Hardy-type inequalities.

     

Characterization of embeddings of Sobolev-type spaces into generalized Hölder spaces defined by Lp-modulus of smoothness

We prove a sharp estimate for the k-modulus of smoothness, modelled upon a $L^p$-Lebesgue space, of a function f in $W^k{L}^{\frac{pn}{n+kp},p}(\mathrm{\Omega })$, where Ω is a domain with minimally smooth boundary and finite Lebesgue measure, $k,n\in \mathbb{N}$, $k<n$ and $\frac{n}{n?k}<p<+\infty$. This sharp estimate is used to establish necessary and sufficient conditions for continuous embeddings of Sobolev-type spaces into generalized Hölder spaces defined by means of the k-modulus of smoothness. General results are illustrated with examples. In particular, we obtain a generalization of the classical Jawerth embeddings.     

Criteria of strong type two-weighted inequalities for fractional maximal functions

A strong type two-weight problem is solved for fractional maximal functions defined in homogeneous type general spaces. A similar problem is also solved for one-sided fractional maximal functions.     

Solution of two-weight problems for integral transforms with positive kernels

Criteria for weak and strong two-weighted inequalities are obtained for integral transforms with positive kernels.     

Criteria of weighted inequalities in Orlicz classes for maximal functions defined on homogeneous type spaces

The necessary and sufficient conditions are derived in order that a strong type weighted inequality be fulfilled in Orlicz classes for scalar and vector-valued maximal functions defined on homogeneous type spaces. A weak type problem with weights is solved for vector-valued maximal functions.     

Necessary and sufficient conditions for weighted Orlicz class inequalities for maximal functions and singular integrals. II

This paper continues the investigation of weight problems in Orlicz classes for maximal functions and singular integrals defined on homogeneous type spaces considered in [1].     

Optimal Embeddings of Bessel-Potential-Type Spaces into Generalized Hölder Spaces Involving <Emphasis Type="Italic">k</Emphasis>-Modulus of Smoothness

We establish necessary and sufficient conditions for embeddings of Bessel potential spaces H ^σ X(IR ⁿ ) with order of smoothness σ?∈?(0, n), modelled upon rearrangement invariant Banach function spaces X(IR ⁿ ), into generalized Hölder spaces (involving k-modulus of smoothness). We apply our results to the case when X(IR ⁿ ) is the Lorentz-Karamata space \(L_{p,q;b}({{\rm I\kern-.17em R}}^n)\). In particular, we are able to characterize optimal embeddings of Bessel potential spaces \(H^{\sigma}L_{p,q;b}({{\rm I\kern-.17em R}}^n)\) into generalized Hölder spaces. Applications cover both superlimiting and limiting cases. We also show that our results yield new and sharp embeddings of Sobolev-Orlicz spaces W ^k?+?1 L ^n/k(logL) ^α (IR ⁿ ) and W ^k L ^n/k(logL) ^α (IR ⁿ ) into generalized Hölder spaces.     

Boundedness of Stein’s spherical maximal function in variable Lebesgue spaces and application to the wave equation

If ${p(\cdot): \mathbb{R}^{n} {\rightarrow} (0,\infty), n\geq 3}$ , is globally log-Hölder continuous and its infimum p ^? and its supremum p ⁺ are such that ${\frac{n}{n-1} < p^{-} \leq p^{+} < p^{-} (n-1)}$ , then the spherical maximal operator (integral averages taken with respect to the (n ? 1)-dimensional surface measure) is bounded. When n = 3, the result is then interpreted as the preservation of the integrability properties of the initial velocity of propagation to the solution of the initial-value problem for the wave equation.     

Padé-Type Approximants of Markov and Meromorphic Functions

We study the question of convergence of Padé and Padé-type approximants to functions meromorphic in a domain. As an example we investigate in detail the case of functions of the formwhereμ(z) is a Markov function.