Search results for "Locally integrable function"
showing 10 items of 12 documents
Some Aspects of Vector-Valued Singular Integrals
2009
Let A, B be Banach spaces and \(1 < p < \infty. \; T\) is said to be a (p, A, B)- CalderoLon–Zygmund type operator if it is of weak type (p, p), and there exist a Banach space E, a bounded bilinear map \(u: E \times A \rightarrow B,\) and a locally integrable function k from \(\mathbb{R}^n \times \mathbb{R}^n \backslash \{(x, x): x \in \mathbb{R}^n\}\) into E such that $$T\;f(x) = \int u(k(x, y), f(y))dy$$ for every A-valued simple function f and \(x \notin \; supp \; f.\)
Generalized Lebesgue points for Sobolev functions
2017
In this article, we show that a function $f\in M^{s,p}(X),$ $0<s\leq 1,$ $0<p<1,$ where $X$ is a doubling metric measure space, has generalized Lebesgue points outside a set of $\mathcal{H}^h$-Hausdorff measure zero for a suitable gauge function $h.$
Lineability of non-differentiable Pettis primitives
2014
Let \(X\) be an infinite-dimensional Banach space. In 1995, settling a long outstanding problem of Pettis, Dilworth and Girardi constructed an \(X\)-valued Pettis integrable function on \([0,1]\) whose primitive is nowhere weakly differentiable. Using their technique and some new ideas we show that \(\mathbf{ND}\), the set of strongly measurable Pettis integrable functions with nowhere weakly differentiable primitives, is lineable, i.e., there is an infinite dimensional vector space whose nonzero vectors belong to \(\mathbf{ND}\).
A Decomposition Theorem for the Fuzzy Henstock Integral
2012
We study the fuzzy Henstock and the fuzzy McShane integrals for fuzzy-number valued functions. The main purpose of this paper is to establish the following decomposition theorem: a fuzzy-number valued function is fuzzy Henstock integrable if and only if it can be represented as a sum of a fuzzy McShane integrable fuzzy-number valued function and of a fuzzy Henstock integrable fuzzy number valued function generated by a Henstock integrable function.
A Constructive Minimal Integral which Includes Lebesgue Integrable Functions and Derivatives
2000
In this paper we provide a minimal constructive integration process of Riemann type which includes the Lebesgue integral and also integrates the derivatives of differentiable functions. We provide a new solution to the classical problem of recovering a function from its derivative by integration, which, unlike the solution provided by Denjoy, Perron and many others, does not possess the generality which is not needed for this purpose.The descriptive version of the problem was treated by A. M. Bruckner, R. J. Fleissner and J. Foran in [2]. Their approach was based on the trivial observation that for the required minimal integral, a function F is the indefinite integral of f if and only if F'…
Poincaré inequalities and Steiner symmetrization
1996
A complete geometric characterization for a general Steiner symmetric domain Ω ⊂ Rn to satisfy the Poincare inequality with exponent p > n−1 is obtained and it is shown that this range of exponents is best possible. In the case where the Steiner symmetric domain is determined by revolving the graph of a Lipschitz continuous function, it is shown that the preceding characterization works for all p > 1 and furthermore for such domains a geometric characterization for a more general Sobolev–Poincare inequality to hold is given. Although the operation of Steiner symmetrization need not always preserve a Poincare inequality, a general class of domains is given for which Poincare inequalities are…
Vector-valued Hardy inequalities and B-convexity
2000
Inequalities of the form $$\sum\nolimits_{k = 0}^\infty {|\hat f(m_k )|/(k + 1) \leqslant C||f||_1 } $$ for allf∈H 1, where {m k } are special subsequences of natural numbers, are investigated in the vector-valued setting. It is proved that Hardy's inequality and the generalized Hardy inequality are equivalent for vector valued Hardy spaces defined in terms ff atoms and that they actually characterizeB-convexity. It is also shown that for 1<q<∞ and 0<α<∞ the spaceX=H(1,q,γa) consisting of analytic functions on the unit disc such that $$\int_0^1 {(1 - r)^{q\alpha - 1} M_1^q (f,r) dr< \infty } $$ satisfies the previous inequality for vector valued functions inH 1 (X), defined as the space ofX…
Relations among Henstock, McShane and Pettis integrals for multifunctions with compact convex values
2013
Fremlin (Ill J Math 38:471–479, 1994) proved that a Banach space valued function is McShane integrable if and only if it is Henstock and Pettis integrable. In this paper we prove that the result remains valid also in case of multifunctions with compact convex values being subsets of an arbitrary Banach space (see Theorem 3.4). Di Piazza and Musial (Monatsh Math 148:119–126, 2006) proved that if \(X\) is a separable Banach space, then each Henstock integrable multifunction which takes as its values convex compact subsets of \(X\) is a sum of a McShane integrable multifunction and a Henstock integrable function. Here we show that such a decomposition is true also in case of an arbitrary Banac…
Radon-Nikodym derivatives of finitely additive interval measures taking values in a Banach space with basis
2011
Let X be a Banach space with a Schauder basis {en}, and let Φ(I)= ∑n en ∫I fn(t)dt be a finitely additive interval measure on the unit interval [0, 1], where the integrals are taken in the sense of Henstock–Kurzweil. Necessary and sufficient conditions are given for Φ to be the indefinite integral of a Henstock–Kurzweil–Pettis (or Henstock, or variational Henstock) integrable function f:[0, 1] → X.
Pointwise characterizations of Hardy-Sobolev functions
2006
We establish simple pointwise characterizations of functions in the Hardy-Sobolev spaces within the range n/(n+1)<p <=1. In addition, classical Hardy inequalities are extended to the case p <= 1.