Limiting weak-type behaviors for parabolic singular integrals and related operators

Limiting Weak–type Behavior for Singular Integral and Maximal Operators

The following limit result holds for the weak–type (1,1) constant of dilation-commuting singular integral operator T in Rn: for f ∈ L1(Rn), f ≥ 0, lim λ→0 λ m{x ∈ R : |Tf(x)| > λ} = 1 n ∫ Sn−1 |Ω(x)|dσ(x)‖f‖1. For the maximal operator M , the corresponding result is lim λ→0 λ m{x ∈ R : |Mf(x)| > λ} = ‖f‖1.

Bounds for Singular Fractional Integrals and Related Fourier Integral Operators

A Weak Type Estimate for Rough Singular Integrals

We obtain a weak type (1, 1) estimate for a maximal operator associated with the classical rough homogeneous singular integrals TΩ. In particular, this provides a different approach to a sparse domination for TΩ obtained recently by Conde-Alonso, Culiuc, Di Plinio and Ou [5].

General Minkowski type and related inequalities for seminormed fuzzy integrals

Minkowski type inequalities for the seminormed fuzzy integrals on abstract spaces are studied in a rather general form. Also related inequalities to Minkowski type inequality for the seminormed fuzzy integrals on abstract spaces are studied. Several examples are given to illustrate the validity of theorems. Some results on Chebyshev and Minkowski type inequalities are obtained.

Weak Type Estimates for Singular Integrals Related to a Dual Problem of Muckenhoupt–Wheeden

A well-known open problem of Muckenhoupt–Wheeden says that any Calderón–Zygmund singular integral operator T is of weak type (1,1) with respect to a couple of weights (w,Mw). In this paper, we consider a somewhat “dual” problem: sup λ>0 λw { x ∈Rn : |Tf (x)| Mw > λ } ≤ c ∫