Remark on an integral inequality of the Hardy type
Some properties of strongly Wright-convex functions are presented. In particular it is shown that a function f:D → ℝ, where D is an open convex subset of an inner product space X, is strongly Wright-convex with modulus c if and only if it can be represented in the form f(x) = g(x)+a(x)+c||x||², x ∈ D, where g:D → ℝ is a convex function and a:X → ℝ is an additive function. A characterization of inner product spaces by strongly Wright-convex functions is also given.
For a function the notion of p-mean variation of order 1, is defined. It generalizes the concept of F. Riesz variation of functions on the real line ℝ¹ to ℝⁿ, n > 1. The characterisation of the Sobolev space in terms of is directly related to the characterisation of by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.
We present comparison theorems for the weighted quasi-arithmetic means and for weighted Bajraktarević means without supposing in advance that the weights are the same.
We present several continuous embeddings of the critical Besov space . We first establish a Gagliardo-Nirenberg type estimate , for 1 < p ≤ q < ∞, 1 ≤ ν < ρ ≤ ∞ and the weight function with 0 < r < n. Next, we prove the corresponding Trudinger type estimate, and obtain it in terms of the embedding , where the function Φ₀ of the weighted Besov-Orlicz space is a Young function of the exponential type. Another point of interest is to embed into the weighted Besov space with...
We consider the half-linear differential equation of the form under the assumption that is integrable on . It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as .
In this paper we consider some spaces of differentiable multifunctions, in particular the generalized Orlicz-Sobolev spaces of multifunctions, we study completeness of them, and give some theorems.
We study WDC sets, which form a substantial generalization of sets with positive reach and still admit the definition of curvature measures. Main results concern WDC sets . We prove that, for such , the distance function is a “DC aura” for , which implies that each closed locally WDC set in is a WDC set. Another consequence is that compact WDC subsets of form a Borel subset of the space of all compact sets.