### Interpolation with a parameter function.

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In this paper Rothe’s classical method is extended so that it can be used to solve some linear parabolic boundary value problems in non-cylindrical domains. The corresponding existence and uniqueness theorems are proved and some further results and generalizations are discussed and applied.

We discuss the characterization of the inequality (RN+ fq u)1/q C (RN+ fp v )1/p, 0<q, p <, for monotone functions $f\ge 0$ and nonnegative weights $u$ and $v$ and $N\ge 1$. We prove a new multidimensional integral modular inequality for monotone functions. This inequality generalizes and unifies some recent results in one and several dimensions.

There are given necessary and sufficient conditions on a measure dμ(x)=w(x)dx under which the key estimates for the distribution and rearrangement of the maximal function due to Riesz, Wiener, Herz and Stein are valid. As a consequence, we obtain the equivalence of the Riesz and Wiener inequalities which seems to be new even for the Lebesgue measure. Our main tools are estimates of the distribution of the averaging function f** and a modified version of the Calderón-Zygmund decomposition. Analogous...

We prove some multi-dimensional Clarkson type inequalities for Banach spaces. The exact relations between such inequalities and the concepts of type and cotype are shown, which gives a conclusion in an extended setting to M. Milman's observation on Clarkson's inequalities and type. A similar investigation conceming the close connection between random Clarkson inequality and the corresponding concepts of type and cotype is also included. The obtained results complement, unify and generalize several...

We present a direct proof of a known result that the Hardy operator Hf(x) = 1/x ∫ f(t) dt in the space L = L(0, ∞) can be written as H = I - U, where U is a shift operator (Ue = e, n ∈ Z) for some orthonormal basis {e}. The basis {e} is constructed by using classical Laguerre polynomials. We also explain connections with the Euler differential equation of the first order y' - 1/x y = g and point out some generalizations to the case with weighted L (a, b) spaces.

We consider iteration of arithmetic and power means and discuss methods for determining their limit. These means appear naturally in connection with some problems in homogenization theory.

In this paper we present a unified approach to obtain integral representation formulas for describing the propagation of bending waves in infinite plates. The general anisotropic case is included and both new and well-known formulas are obtained in special cases (e.g. the classical Boussinesq formula). The formulas we have derived have been compared with experimental data and the coincidence is very good in all cases.

We consider a new Sobolev type function space called the space with multiweighted derivatives ${W}_{p,\overline{\alpha}}^{n}$, where $\overline{\alpha}=({\alpha}_{0},{\alpha}_{1},...,{\alpha}_{n})$, ${\alpha}_{i}\in \mathbb{R}$, $i=0,1,...,n$, and ${\parallel f\parallel}_{{W}_{p,\overline{\alpha}}^{n}}=\parallel {D}_{\overline{\alpha}}^{n}{f\parallel}_{p}+{\sum}_{i=0}^{n-1}\left|{D}_{\overline{\alpha}}^{i}f\left(1\right)\right|$, $${D}_{\overline{\alpha}}^{0}f\left(t\right)={t}^{{\alpha}_{0}}f\left(t\right),\phantom{\rule{1.0em}{0ex}}{D}_{\overline{\alpha}}^{i}f\left(t\right)={t}^{{\alpha}_{i}}\frac{\mathrm{d}}{\mathrm{d}t}{D}_{\overline{\alpha}}^{i-1}f\left(t\right),\phantom{\rule{5.0pt}{0ex}}i=1,2,...,n.$$ We establish necessary and sufficient conditions for the boundedness and compactness of the embedding ${W}_{p,\overline{\alpha}}^{n}\hookrightarrow {W}_{q,\overline{\beta}}^{m}$, when $1\le q<p<\infty $, $0\le m<n$.

We consider a quasilinear parabolic problem with time dependent coefficients oscillating rapidly in the space variable. The existence and uniqueness results are proved by using Rothe’s method combined with the technique of two-scale convergence. Moreover, we derive a concrete homogenization algorithm for giving a unique and computable approximation of the solution.

Some $q$-analysis variants of Hardy type inequalities of the form $${\int}_{0}^{b}{\left({x}^{\alpha -1}{\int}_{0}^{x}{t}^{-\alpha}f\left(t\right){\mathrm{d}}_{q}t\right)}^{p}{\mathrm{d}}_{q}x\le C{\int}_{0}^{b}{f}^{p}\left(t\right){\mathrm{d}}_{q}t$$ with sharp constant $C$ are proved and discussed. A similar result with the Riemann-Liouville operator involved is also proved. Finally, it is pointed out that by using these techniques we can also obtain some new discrete Hardy and Copson type inequalities in the classical case.

We prove and discuss some new $({H}_{p},{L}_{p})$-type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients $\{{q}_{k}:k\ge 0\}$. These results are the best possible in a special sense. As applications, some well-known as well as new results are pointed out in the theory of strong convergence of such Vilenkin-Nörlund means. To fulfil our main aims we also prove some new estimates of independent interest for the kernels of these summability results. In the special cases of general Nörlund...

We present, discuss and apply two reiteration theorems for triples of quasi-Banach function lattices. Some interpolation results for block-Lorentz spaces and triples of weighted ${L}_{p}$-spaces are proved. By using these results and a wavelet theory approach we calculate (θ,q)-spaces for triples of smooth function spaces (such as Besov spaces, Sobolev spaces, etc.). In contrast to the case of couples, for which even the scale of Besov spaces is not stable under interpolation, for triples we obtain stability...

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