Read the abstract, see who wrote it and who cites it, and get the citation in APA, MLA, Chicago or Harvard.
About 388 papers for “Locally integrable function”
Nilay Sahin BAYRAM · DergiPark (Istanbul University) · 2016
In this paper, using the concept of strong summation process, we give aKorovkin type approximation theorem for a sequence of positive linearoperators acting from $L_p,q (loc)$ into itself. We also study modulus ofcontinuity for $L_p,q (loc)$ approximation and give the rate of convergenceof these operators.
Cited by 0
Sergio Venturini · Journal of Mathematical Analysis and Applications · 2024
We prove that given a locally integrable function fon an open set of an Euclidean space the distributional derivative Xfwith respect to a locally Lipschitzian vector field Xis locally integrable if, and only if, the function fadmits a locally integrable upper gradient along the vector field X; in this case Xfcoincides with the Lie derivative LXfand |Xf|is the least upper gradient of the function f. Applications to systems of locally Lipschitzian vector fields are given
Cited by 0Open access
Sergio Venturini · arXiv (Cornell University) · 2024
We prove that given a locally integrable function $f$ on an open set of an Euclidean space the distributional derivative $Xf$ with respect to a locally Lipshitzian vector field $X$ is locally integrable if, and only if, the function $f$ admits a locally integrable upper gradient along the vector field $X$; in this case $Xf$ coincides with the Lie derivative $L_X f$ and $|Xf|$ is the least upper gradient of the function $f$. Applications to systems of locally Lipshitzian vector fields are given.
Cited by 0Open access
Peter J. Holden · Pacific Journal of Mathematics · 1990
A locally integrable function is said to be of vanishing mean oscillation (VMO) if its mean oscillation over cubes in R^ converges to zero with the volume of the cubes. We establish necessary and sufficient conditions for a locally integrable function defined on a bounded measurable set of positive measure to be the restriction to that set of a VMO function.
Cited by 19Open access
2023
The importance of the convolution product lies essentially in its regularization role. In this chapter, the authors introduce the convolution product in the space of distributions. They consider the convolution product of two integrable functions and then generalize to any two distributions by means of regular distributions. In order to introduce the convolution product of two distributions, it is most natural to define it for regular distributions, namely locally integrable functions. As for the convolution of a locally integrable function by a test function, the product of convolution of a distribution by a test function can be derived by obtaining the derivative of any of the two factors.
Cited by 0
Lucio Boccardo, Thierry Gallouët, Juan Luís Vázquez · SHILAP Revista de lepidopterología · 2001
The purpose of this paper is to prove the existence of solutions for certain types of nonlinear parabolic partial differential equations posed in the whole space, when the data are assumed to be merely locally integrable functions, without any control of their behaviour at infinity. A simple representative example of such an equation is $$ u_t-Delta u + |u|^{s-1}u=f, $$ which admits a unique globally defined weak solution $u(x,t)$ if the initial function $u(x,0)$ is a locally integrable function in $mathbb{R}^N$, $Ngeq 1$, and the second member $f$ is a locally integrable function of $xinmathbb{R}^N$ and $tin [0,T]$ whenever the exponent $s$ is larger than 1. The results extend to parabolic equations results obtained by Brezis and by the authors for elliptic equations. They have no equivalent for linear or sub-linear zero-order nonlinearities.
Cited by 29Open access
Wenchang Sun, Shifen Wang · Forum Mathematicum · 2024
Abstract We study the compactness of commutator of a locally integrable function and an oscillatory singular integral operator defined by a real-valued polynomial and a kernel function satisfying the Hölder condition. We show that such a commutator is compact from <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> {L^{p}} to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mi>q</m:mi> </m:msup> </m:math> {L^{q}} if and only if the locally integrable function is the sum of a constant and a function in certain Lebesgue space.
Cited by 0
A. Carbery, S. -y. A. Chang, J. Garnett et al. · 1925
Let ω ( x) b e a positive locally integrable weight on [0,1]. Discussed are conditions on ω necessary and sufficient for the (dyadic) Hardy-Littlewood maximal function to map L log L(wdx) into Lι(ω dx) or into weak L1. 1. Introduction, Let Mf denote the (dyadic) Hardy-Littlewood maximal function of/, for/locally integrable on Rn. That is, M/(x)=sup ±-f I/I, x&lt;=Q \\ΰ\\JQ the sup being taken over all dyadic cubes in Rn containing x. It is
Cited by 0
Bernard Harris · Proceedings of the Royal Society of Edinburgh Section A Mathematics · 1986
Synopsis We derive an asymptotic expansion for the Titchmarsh–Weyl m -function associated with the second order linear differential equation in the case where the only restriction on the real-valued function q is
Cited by 18
Colin J. Bennett · Proceedings of the American Mathematical Society · 1982
It is shown that a locally integrable function / on R" has bounded lower oscillation ( / 6 BLO) if and only if / = MF + h, where F has bounded mean oscillation ( F £ BMO) and MF < oc a.e., and h is bounded.Here, MF is a variant of the familiar Hardy-Lit tie wood maximal function: MF = supç3xQ(F) (no absolute values), where Q( F) is the mean value of F over the cube Q.We consider real-valued locally integrable functions/on R".When Q is a cube in R" with sides parallel to the coordinate axes, we denote by Q(f) the mean value of/ over Q. Define a maximal function Mf of f by
Cited by 9
Magnus - Estenes, James Serrin · 2004
s > 0 and s, p, q, a are locally integrable on [0, oo), r > 0 and r is a locally bounded measurable function on [0, oo), 1/r is locally bounded on [0, oo), b and c are locally square integrable on [0, oo). An additional assumption, specified in detail later, is satisfied if co(t, u, u', u) is a nonnegative quadratic form in (u, u', u) which is positive for u4=0. The subclass of d for which J(u) exists is ~ , the subclass of ~ for which I(I u 1) exists is cr and the subclass of c~ for which the initial conditions
Cited by 0
Takafumi Murai · Nagoya Mathematical Journal · 1984
In this paper we investigate the boundedness of Cauchy kernels. The Cauchy kernel associated with a locally integrable real-valued function θ(x ) is defined by
Cited by 3Open access
Rn Z · 2012
Abstract. A locally integrable function m(ξ, η) defined on Rn × Rn is said to be a bilinear multiplier on Rn of type (p1, p2, p3) if
Cited by 0
Victor Iluebe, Adesanmi Mogbademu · SHILAP Revista de lepidopterología · 2023
Suppose $X$ is a locally compact Hausdorff space and $\Omega \in \bigtriangleup$. If $ F $ is an interval valued function defined in $ \Omega $ with $F:\bar \Omega\rightarrow I_{\mathbb{R}}$. Suppose $F$ is Topological Henstock integrable, is $ F $ Sequential Henstock integrable? Therefore, the purpose of this paper is to provide a positive response to this query.
Cited by 0Open access
Per Åhag, Urban Cegrell, Sławomir Kołodziej et al. · arXiv (Cornell University) · 2008
We give a sufficient condition on the Monge-Ampère mass of a plurisubharmonic function $u$ for $\exp (- 2 u)$ to be locally integrable. This gives a pluripotential theoretic proof of a theorem by J-P. Demailly.
Cited by 0Open access
Liu, Hanwen · ArXiv.org · 2025
A Riemannian metric is termed a Hessian metric if in some coordinate system it can be locally represented as the Hessian quadratic form of some locally defined smooth potential function. Under very mild extra technical conditions, we first theoretically describe the potentials of flat Hessian metrics on surfaces, and then construct these potentials explicitly using methods from integrable systems.
Cited by 0Open access
Liu, Hanwen · arXiv (Cornell University) · 2025
A Riemannian metric is termed a Hessian metric if in some coordinate system it can be locally represented as the Hessian quadratic form of some locally defined smooth potential function. Under very mild extra technical conditions, we first theoretically describe the potentials of flat Hessian metrics on surfaces, and then construct these potentials explicitly using methods from integrable systems.
Cited by 0Open access
Pedro Ortega Salvador · Studia Mathematica · 1998
Let $M_{g}^{+}$ be the maximal operator defined by $M_{g}^{+}⨍(x) = sup_{h>0} (ʃ_{x}^{x+h} |⨍|g)/(ʃ_{x}^{x+h} g)$, where g is a positive locally integrable function on ℝ. Let Φ be an N-function such that both Φ and its complementary N-function satisfy $Δ_
Cited by 15Open access
Hans Detlef Hüttenbach · International Journal of Mathematical Analysis · 2021
Defining the principle of extremal action in concise mathematical terms, it is shown that this principle does not hold what it physically promises. Instead, it is shown that Lagrange functions need to be locally integrable (in an open region of space), in order that the Lagrange equations strictly apply. The principle of extremal action therefore reduces to the condition of local integrability of the Lagrange function to a (locally defined) Hamilton-Jacobi function.
Cited by 0
Elijah Liflyand · Georgian Mathematical Journal · 2014
Abstract A function f with bounded variation and integrable Hilbert transform of d f is locally absolutely continuous. We give a direct real-valued proof of this result and connect it with integrability problems for the Fourier transform of a function with bounded variation, in particular, with a Hardy–Littlewood theorem.
Cited by 1