Singular Integral Operators Child And Euler Equations

In chapter 1 we consider the integrality of strong maximal operator on Rⁿ. For a function f∈ L_loc(Rⁿ), its Hardy-Littlewood maximal function is defined bywhere Q is a cube with sides parallel to the coordinate axes, its strong maximal function is defined bywhere P is a rectangle with sides parallel to the coordinate axes. In addition, let M^*(f)(x) = M_n 。…。 M₁(f)(x) where Mj is the Hardy-Littlewood maximal operator on R¹ acting on the j — th coordinate X_j.It is well-known that for f with compact support in [66] Stein proved that: M(f) G ∈^L1(E) for any measurable set E of finite measure if and only if f ∈ L ln⁺ L(Rⁿ).In [38] Jessen-Marcinkiewicz-Zygmund proved that: M^*(f) ∈ L¹(E) for any measurable set E of finite measure if and only if f ∈ L(ln⁺ L)ⁿ(Rⁿ). This result can also see Fava-Gatto-Gutiérez [31]. Because M_S(f) ≤ M^*(f), if f G L(ln⁺L)ⁿ(Rⁿ), then M_S(f) ∈ L¹(E) for any measurable set E of finite measure.It was conjectured in [31] that: if Ms(f) ∈ L¹(E) for any measurable set E of finite measure,then f ∈ L(ln⁺ L)ⁿ(Rⁿ). In [1] and [35] Bagby and Gomez independently proved that there are many functions f ∈ L ln⁺ L(R²) such that M_s(f) ∈ L¹(E) for any measurable set E of finite measure.By a different way which can be easily applied to high dimensions case, we shall prove that the conjecture is also not true for n > 2. An interesting thing is that we do not need f ∈ L(ln⁺L)^n-1(Rⁿ). Precisely, we proved...