Lower bounds of some singular integrals and their applications

Abstract

Let β > 1 and f ∈ C(R). We are interested in the lower bounds of the integral:

Equivalence of some integrals and modulus of smoothness

“…It is easy to verify that Ω is also an even, bounded, nonnegative function and R Ω = 1. Moreover, in [6], it is shown that Ω ∈ C 1 (R) and for some 0…”

“…t f (·) t 2 dt , ∀ f ∈ C 2π , h > 0.Using the last inequality they (see[6]) gave the following equivalent relation…”

“…If f ∈ C 1 2π , where C 1 2π denotes the set of functions in C 2π such that f ∈ C 2π , then the derivative of the conjugate function f can be expressed as (see [2,6])…”