Generalizations of some classical inequalities and their applications
Weighted integral inequalities with the geometric mean operator.
Some weighted multidimensional Berwald, Thunsdorff and Borell inequalities.
Carleman's inequality – history, proofs and some new generalizations.
Weighted geometric mean inequalities over cones in .
On some fractional order Hardy inequalities.
Carlson type inequalities.
A Sawyer duality principle for radially monotone functions in .
Weight characterizations for the discrete Hardy inequality with kernel.
Some multiplicative inequalities for inner products and of the Carlson type.
On weighted inequalities with geometric mean operator generated by the Hardy-type integral transform.
Interpolation with a parameter function.
An extension of Rothe's method to non-cylindrical domains
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.
Weighted multidimensional inequalities for monotone functions
We discuss the characterization of the inequality (RN+ fq u)1/q C (RN+ fp v )1/p, 0<q, p <, for monotone functions and nonnegative weights and and . We prove a new multidimensional integral modular inequality for monotone functions. This inequality generalizes and unifies some recent results in one and several dimensions.
Distribution and rearrangement estimates of the maximal function and interpolation
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...
Clarkson type inequalities and their relations to the concepts of type and cotype.
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...
Structure of the Hardy operator related to Laguerre polynomials and the Euler differential equation.
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.
Editorials' note
A study of some constants characterizing the weighted Hardy inequality
Homogenization of quasilinear parabolic problems by the method of Rothe and two scale convergence
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.
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