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Weighted composition operators on variable exponent Lebesgue spaces
In this paper, we characterize the boundedness of weighted composition operators, induced by measurable transformations and complex-valued measurable...
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Measurable Chromatic Number of the Plane
As you know, the length of a segment [a, b], a < b, on the line E1 is defined as b − a. The area A of a rectangle [a1, b1] × [a2, b2], ai < bi, in...
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Lebesgue Integration
In this chapter, we are going to define the Lebesgue integral which is the main reason why we went through the foundations of measure theory in Chap....
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Characterizations for boundedness of fractional maximal function commutators in variable Lebesgue spaces on stratified groups
In this paper, the main aim is to consider the map** properties of the maximal or nonlinear commutator for the fractional maximal operator with the...
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Measurable Spaces
This chapter introduces the fundamental notions of a σ-field, of a measure on a measurable space, and of a measurable function, which will be used in...
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The Lebesgue Integral
The previous chapters concerned what one may call the basic “calculus of probability”, that is, the acquisition of the skills that suffice to deal...
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Composition Operators in Grand Lebesgue Spaces
Let Ω be an open subset of ℝ n of finite measure. Let f be a Borel measurable function from ℝ to ℝ. We prove necessary and sufficient conditions on f ...
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Composition operators on variable exponent Lebesgue spaces
We study composition operators between variable exponent Lebesgue spaces and characterize boundedness and compactness of the composition operators on...
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Measurable Functions
We will use Lebesgue’s measure theory to define Measurable functions. This class of functions is a generalization of the class of continuous...
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Classification of Measurable Functions of Several Variables and Matrix Distributions
AbstractWe consider the notion of the matrix (tensor) distribution of a measurable function of several variables. On the one hand, this is an...
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Lebesgue Integration
Riemann introduced his theory of integration in 1953 during his work on the theory of Fourier series. The theory proposed a rigorous definition of...
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The metric-valued Lebesgue differentiation theorem in measure spaces and its applications
We prove a version of the Lebesgue differentiation theorem for map**s that are defined on a measure space and take values into a metric space, with...
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Measurable Operators
This chapter presents the basic theory of measurable and \(\tau \)...
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Integration of Measurable Functions
In this chapter, we construct the Lebesgue integral of real-valued measurable functions with respect to a positive measure. After constructing the...
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On Holder’s Inequality in Lebesgue Spaces with Variable Order of Summability
In this paper, we introduce a new version of the definition of a quasinorm (in particular, a norm) in Lebesgue spaces with variable order of...
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Lebesgue Integral and Mathematical Expectation
Lebesgue’s approach to integration differs from Riemann’s by dividing the range of functions into small intervals. The derivation of the Lebesgue...
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Basis Property of the Haar System in Weighted Lebesgue Spaces with Variable Exponent
AbstractWe obtain necessary and sufficient conditions for the weight under which the Haar system is a basis in a weighted Lebesgue space with...
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Connection Between Weighted Tail, Orlicz, Grand Lorentz And Grand Lebesgue Norms
We prove that the norm of functions in a suitable Grand Lorentz space built on a measure space, equipped with sigma finite diffuse measure, coincides...
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Lebesgue spaces with variable exponent: some applications to the Navier–Stokes equations
In this article we study some problems related to the incompressible 3D Navier–Stokes equations from the point of view of Lebesgue spaces of variable...
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Measurable Functions and Random Variables
To study random variable in a more general setting, we introduce the notion of measurable functions. In probability theory, a random variable is a...