Weighted Fourier Inequalities in Lebesgue and Lorentz Spaces

Abstract

In this paper, we obtain sufficient conditions for the weighted Fourier-type transforms to be bounded in Lebesgue and Lorentz spaces. Two types of results are discussed. First, we review the method based on rearrangement inequalities and the corresponding Hardy’s inequalities. Second, we present Hörmander-type conditions on weights so that Fourier-type integral operators are bounded in Lebesgue and Lorentz spaces. Both restricted weak- and strong-type results are obtained. In the case of regular weights necessary and sufficient conditions are given.

Appendix A: Hardy’s Inequalities on Monotone Functions

In this section we address the characterization of the three-weight Hardy’s inequality

$$\begin{aligned} \left( \int _0^\infty \left( \nu (x)\int _0^x f(y)\mu (y)dy\right) ^q dx\right) ^{\frac{1}{q}}\leqslant C \left( \int _0^\infty [f(x)w(x)]^pdx\right) ^{\frac{1}{p}},\,\,f\in \mathfrak {M}^\downarrow , \end{aligned}$$

(6.6)

(see, e.g., [12]). Here $\mathfrak {M}^\downarrow$ is the cone of all non-increasing Lebesgue-measurable functions on $\mathbb {R}_+$. Let

$$\begin{aligned} M(t):=\int _0^t\mu , \qquad W(t):=\int _0^tw^p.\qquad \end{aligned}$$

Theorem 6.5

[12] Let C be the smallest possible constant in inequality (6.6).

(a) If $1<p\leqslant q<\infty ,$ then $C\asymp A_0+A_1,$ where

$$\begin{aligned} A_{0}=\underset{t>0}{\sup }\,\,A_{0}(t):=\underset{t>0}{\sup }\left( \int _{0}^{t}M^q\nu ^q\right) ^{\frac{1}{q}} W^{-\frac{1}{p}}(t) \end{aligned}$$

and

$$\begin{aligned} A_{1}:=\underset{t>0}{\sup }\left( \int _{t}^{\infty }\nu ^q\right) ^{\frac{1}{q} }\left( \int _{0}^{t}\left( \frac{M}{W}\right) ^{p^\prime }w^p\right) ^{\frac{1}{p^{\prime }}}. \end{aligned}$$

(b) If $0<q<p<\infty$ and $1<p<\infty ,$ then $C\asymp B_0+B_1,$ where

$$\begin{aligned} B_{0}=B_{0}(p,q):=\left( \int _{0}^{\infty }W^{-\frac{r}{p}}(t)\left( \int _{0}^{t}M^q\nu ^q\right) ^{\frac{r}{p} }M^q(t)\nu ^q(t)dt\right) ^{\frac{1}{r}} \end{aligned}$$

and

$$\begin{aligned} B_1=B_1(p,q) :=\left( \int _{0}^{\infty }\left( \int _{t}^{\infty }\nu ^q\right) ^{\frac{r}{p }}\left( \int _{0}^{t}\left( \frac{M}{W}\right) ^{p^\prime }w^p\right) ^{\frac{r}{p^{\prime }} }\nu ^q(t)dt\right) ^{\frac{1}{r}}. \end{aligned}$$

(c) If $0<q<p\leqslant 1$, then $C\asymp B_0+B_1',$ where

$$\begin{aligned} B_1'=B_1'(p,q) := \left( \int _{0}^{\infty } \left( \underset{s\in [0,t]}{\mathrm { esssup}} \frac{M^p(s)}{W(s)}\right) ^{\frac{r}{p}}\left( \int _{t}^\infty \nu ^q\right) ^{\frac{r}{p}}\nu ^q(t)dt\right) ^{\frac{1}{r}}. \end{aligned}$$

(d) If $0<p\leqslant q<\infty$ and $0<p\leqslant 1,$ then

$$\begin{aligned} C=\underset{t>0}{\sup }\,\,W^{- \frac{1}{p}}(t)\left( \int _0^\infty M^q(\min \{s,t\})\nu ^q(s)ds\right) ^{\frac{1}{q}}. \end{aligned}$$