Quantitative Estimates for Square Functions with New Class of Weights

Weighted estimates for square functions associated with operators

“…This together with Equation (1.9), Corollary 2.5 and 𝐺 𝐿,𝜓 𝑓(𝑥) ≲ 𝐶 𝜆 𝑔 * 𝐿,𝜆,𝜑 𝑓(𝑥) (see [3]), we obtain Equation (1.8). □…”

“…Let ψ be an even function with $$\psi (0)=0$$ and $$\psi \in \mathcal {S}(\mathbb {R}^n)$$, where $$\mathcal {S}(\mathbb {R}^n)$$ is the class of Schwartz functions. In this paper, we consider the following square functions associated with the operator L , which were introduced in [3]. $$$$\begin{align} G_{L,\psi }f(x)={\left(\int _{0}^\infty |\psi (t\sqrt {L})f(x)|^2\frac{dt}{t}\right)}^{1/2}, \end{align}$$$$ $$$$\begin{align} S_{L,\psi }f(x)={\left(\int _{0}^\infty \int _{|x-y|<\alpha t}|\psi (t\sqrt {L})f(y)|^2\frac{dydt}{t^{n+1}}\right)}^{1/2},\alpha >0, \end{align}$$$$ …”

“…Following the work in [4], Cao et al [8] proved several kinds of weighted estimates (for Muckenhoupt weights) for square functions associated with an abstract operator. Recently, Bui et al [3] proved the quantitative weighted 𝐿 𝑝 -estimates for square functions associated with 𝐿 with new weight class, which contains the class of Muckenhoupt weights. Several natural questions arise from the work in [3]:…”

Quantitative Weighted Bounds for Littlewood-Paley Functions Generated by Fractional Heat Semigroups Related with Schrödinger Operators

“…Li et al [18] established the quantitative weighted boundedness of maximal functions, maximal heat semigroups, and fractional integral operators related to L. In 2020, Zhang and Yang [19] showed that the quantitative weighted boundedness for Littlewood-Paley functions in the Schrödinger setting. Bui et al [20] investigated the quantitative boundedness for square functions with new class of weights. In [21], Bui et al studied the quantitative estimates for some singular integrals associated with critical functions.…”

Pointwise domination and weak $L^1$ boundedness of Littlewood-Paley Operators via sparse operators

“…Motivated by these works, many authors obtained many important results (see e.g. [1,2,3,5]). Recall that a modulus of continuity is a function defined on (0, 1).…”