If p t) = p, where p ∈ (1, ∞) is a constant, by the definition of |u|p t), it is easy to get | u | p = ( ∫ 0 T | u ( t ) | p d t ) 1 ∕ p, which is the same with the usual norm in space L p. The space Lp(t)is a generalized Lebesgue space, and the space W1, p(t)is a generalized Sobolev space.
Then Rydzewska [3] conducted a similar study by changing the point to a μ-generalized Lebesgue point of f ∈ L 1 instead of a generalized Lebesgue point.
The following two examples are simple applications to a generalized Lebesgue point and μ-generalized Lebesgue point of some functions that belong to (L_{1}(mathbb{R} ^{2})).
On the other hand, if we take (alpha=frac{1}{4}) and (p=1), then the origin is also a generalized Lebesgue point.
Following this work, Gadjiev [2] proved the pointwise convergence of operators of type (1.1) at a generalized Lebesgue point and established the pertinent convergence order.
In [6, 7], Karsli obtained the pointwise convergence theorems and the rate of pointwise convergence theorems for a family of nonlinear singular integral operators at a μ-generalized Lebesgue point and a generalized Lebesgue point of f ∈ L 1 〈 a, b 〉, respectively.
Based on Taberski's study [1], Gadjiev [2] investigated both the pointwise convergence theorems and the order of pointwise convergence theorems for operators type (1) at a generalized Lebesgue point.
We give a new proof for power-type weighted Hardy inequality in the norms of generalized Lebesgue spaces.
Generally speaking, the log-Hölder condition plays a central role in harmonic analysis on variable Lebesgue and Sobolev spaces, which ensures that the Hardy-Littlewood maximal operator is still bounded within the framework of the generalized Lebesgue spaces, a mollification argument is working, variable Sobolev embedding theorem and Poincaré inequalities are available.
A generalized H∞-norm for systems with stochastic parameters and both stochastic and deterministic inputs is derived.
This allows a generalized p-norm.
