Here we denote by Ẇ (C,C) denotes the “homogeneous” Sobolev space of complex valued locally integrable functions in the plane whose distributional first derivatives are in L on the plane. Integrals without specified variables are understood to be with respect to Lebesgue measure. Since ∂(f) = ∂f, Conjecture 1 is true if and only if (1.1) always holds when L(∂f, ∂f) in the integral is replaced b...

Abstract In this paper, we present the definitions of fractional integrals and derivatives a Pettis integrable function with respect to another function. This concept follows idea Stieltjes-type operators should allow us study using methods known from measure differential equations in abstract spaces. We will show that some well-known properties calculus for space Lebesgue functions also hold t...

It is more of a think sheet than mere answers. Answer to Prob. 5(a) might be wrong :) 1. Notations and Conventions a: Unless otherwise speci…ed, all notations presented here bear their generic meanings as implied or required by the contexts. b: Unless otherwise stated, minimal assumptions that are needed to induce or disprove the conclusions but are not explicitly given in the homwork problems ...

a f(x, y) dy, it is often important to know when F is differentiable and when F (x) = ∫ b a f1(x, y) dy. A sufficient condition for differentiating under the integral sign is that ∫ b a f1(x, y) dy converges uniformly; see [6, p. 260]. When we have absolute convergence, the condition |f1(x, y)| ≤ g(y) with ∫ b a g(y) dy < ∞ suffices (Weierstrass M-test and Lebesgue Dominated Convergence). If we...

Let σ be a probability Borel measure on the unit circle T and {φn} be the orthonormal polynomials with respect to σ. We say that σ is a Szegő measure, if it has an arbitrary singular part σs, and R T log σ ′dm > −∞, where σ′ is the density of the absolutely continuous part of σ, m being the normalized Lebesgue measure on T. The entropy integrals for φn are defined as n = Z T |φn| log |φn|dσ It ...

In this paper, we shall firstly illustrate why we should introduce set-valued stochastic integrals, and then we shall discuss some properties of set-valued stochastic processes and the relation between a set-valued stochastic process and its selection set. After recalling the Aumann type definition of stochastic integral, we shall introduce a new definition of Lebesgue integral of a set-valued ...

Morrey Spaces were first introduced by C.B. in 1938. space can be considered as a generalization of the Lebesgue spaces. spaces then generalized become spaces, weighted and One studies on is boundedness certain operators fractional integral. The integrals classical had been known. extensions operator was bounded purpose this study to investigate weight used Muckenhoupt class. results obtained s...

in this paper an algorithm is presented for the regularization of singular integrals with any degrees of singularity, which may be employed in all three-dimensional problems analyzed by boundary elements. the integrals in boundary integrals equations are inherently singular. for example, one can mention the integrals confronted in potential problems to evaluate the flow or the gradient of the f...

In this article, we study analogues of the van der Corput lemmas [19] involving Bessel functions. harmonic analysis, one most important estimates is lemma, which an estimate oscillatory integrals. This was first obtained by Dutch mathematician Johannes Gaultherus Corput. Van interested in behavior for large positive λ integral R b a e iλφ(x)ψ(x)dx, where φ real-valued smooth function (the phase...