##
**Weighted inequalities in Fourier analysis.**
*(English)*
Zbl 0773.42008

Nonlinear analysis, function spaces and applications, Vol. 4, Proc. Spring Sch., Roudnice nad Labem/Czech. 1990, Teubner-Texte Math. 119, 42-85 (1990).

[For the entire collection see Zbl 0731.00016.]

The author gives a survey of recent progress on weighted inequalities. In 1986, he wrote a survey article describing the situation then which appeared in Teubner-Texte Math. 93, 31-53 (1986; Zbl 0619.47026). In this article he concentrates on progress since 1986. He surveys generalizations of Hardy’s inequality, weighted and measure weighted Fourier transform inequalities and applications of the above results.

He first discusses the classical results for Hardy’s inequality for \(L^ p\to L^ q\) in the cases \(1<p\leq q<+\infty\) and \(0<q<p<+\infty\), \(p\geq 1\). He then discusses more recent results applying these to the study of gradient inequalities \[ \left(\int_{\mathbb{R}^ n}| f(x)|^ q v(x)dx\right)^{1/q} \leq C\left(\int_{\mathbb{R}^ n} | x\cdot\nabla f(x)|^ p u(x)dx\right)^{1/p}, \] where it was shown by G. J. Sinnamon [Proc. R. Soc. Edinb., Sect. A 111, No. 3/4, 329-335 (1989; Zbl 0686.26004)] that this inequality holds for \(1<p=q<\infty\) if and only if \[ \sup_{x\in\mathbb{R}^ n}\left(\int^ 1_ 0 v(tx) t^{n-1} dt\right)^{1/p}\left(\int^ \infty_ 1 (u(tx)t^ n)^{1-p'}{dt\over t}\right)^{1/p'}<+\infty, \] with a different equivalence for \(0<q<p\) and \(1/p+1/p'=1\).

The second application of the classical result is obtained by studying its limit as \(p\to \infty\). If we denote by \[ Pf(x)={1\over x} \int^ x_ 0 f(t)dt\quad\text{and}\quad Q_ \alpha f(x)=\alpha x^ \alpha\int^ \infty_ x t^{-\alpha-1} f(t)dt \] then the author, Kerman and Krbec have shown that the following are equivalent: \[ \int^ \infty_ 0 u(x) e^{P(\ln f)(x)} dx\leq C_ 1 \int^ \infty_ 0 v(x) f(x)dx\tag{i} \]

\[ Pw(x)+{1\over\alpha} Q_ \alpha w(x)\leq C_ 2,\tag{ii} \]

\[ Pw(x)\in L^ \infty,\tag{iii} \]

\[ \int^ \infty_ 0 w(x) P(f^{1/p})(x)dx\leq C_ 3 \int^ \infty_ 0 f(x)dx,\;p>2,\tag{iv} \] where \(w(x)=u(x) e^{[P(\ln 1/v)(x)]}\).

He then discusses recent results of Muckenhoupt-Arino, Braverman, Neugebauer, Sawyer and Stepanov on Hardy inequalities for decreasing functions.

Attention is then turned to the situation of inequalities for the Fourier transform. The author begins by reviewing the results of Muckenhoupt, himself and Jurkat-Sampson, which had been discussed in the prior survey. Sufficient and necessary conditions are known; when the weights are monotone (in the right form), the results become necessary and sufficient. The author begins by restating a result of Benedetto, the author and the reviewer. The author and G. J. Sinnamon [Indiana Univ. Math. J. 38, No. 3, 603-628 (1989; Zbl 0668.42003)] have proved a result about monotone weights in the Muckenhoupt class \(A_ p\) that allows the result of J. J. Benedetto, the author and the reviewer [General inequalities 5, 5th Int. Conf., Oberwolfach/FRG 1986, ISNM 80, 217-232 (1987; Zbl 0625.42006)] to be stated as: if \(w\) is an even weight, increasing on \((0,+\infty)\), the following holds: \[ \left(\int_{\mathbb{R}^ n} |\widehat f(x)|^ q| x|^{n(q/p'-1)}w(1/| x|)^{q/p}dx\right)^{1/q}\leq C\left(\int_{\mathbb{R}^ n} | f(x)|^ p w(x)^ p dx\right)^{1/p}, \] \(w\in A_ p\), \(1<p\leq q\leq p'\). The advantage of this result is that the condition is testable; however, the result applies only to monotone weights. The author proves the extension of the above result to \(n\) dimensions given by himself and Sinnamon. He also discusses other necessary and sufficient conditions of greater generality, but all of which involve some monotonicity condition on at least one of the weights. There is a particularly interesting result, due to the author, in which he studies an inequality in the above form but where the function \(f\) is restricted to be decreasing with limit 0 at \(+\infty\). He shows that the inequality holds if and only if the weight satisfies the condition required for Hardy’s inequality to hold on decreasing functions. Next he surveys results on inequalities restricted to functions whose first moment is zero. There is a basic result due to C. Sadosky and R. L. Wheeden [Trans. Am. Math. Soc. 300, 521- 533 (1987; Zbl 0636.42012)] that if the weight \(w\) is in \(A_ p\), \[ \int^ \infty_{-\infty}|\widehat f(x)|^ p w(1/x)dx/| x|^ 2\leq C \int^ \infty_{-\infty}| f(x)|^ p | x|^ p w(x)dx,\quad \widehat f(0)=0. \] The author discusses results obtained jointly by him and Benedetto giving sufficient conditions for inequalities like the above but with measure weights on the left hand side.

The final section discusses sufficient conditions for restriction theorems, \[ \left(\int_{| x|=\rho}|\widehat f(x)|^ r ds\right)^{1/r} \leq C\rho^{1/r+\alpha-2/p'}\| f\|_{p,\alpha}. \] This is valid for all functions for which the right hand side is finite; some restriction theorems are given for functions satisfying moment conditions. He concludes with a proof using the Hardy inequality to give the sharp estimate for \(1< p<n\), \[ \left(\int_{\mathbb{R}^ n}\left({| f(x)|\over| x|}\right)^ p dx\right)^{1/p}\leq {p\over n- p}\left(\int_{\mathbb{R}^ n}\left({| x\cdot\nabla f(x)|\over| x|}\right)^ p dx\right)^{1/p}. \] {}.

The author gives a survey of recent progress on weighted inequalities. In 1986, he wrote a survey article describing the situation then which appeared in Teubner-Texte Math. 93, 31-53 (1986; Zbl 0619.47026). In this article he concentrates on progress since 1986. He surveys generalizations of Hardy’s inequality, weighted and measure weighted Fourier transform inequalities and applications of the above results.

He first discusses the classical results for Hardy’s inequality for \(L^ p\to L^ q\) in the cases \(1<p\leq q<+\infty\) and \(0<q<p<+\infty\), \(p\geq 1\). He then discusses more recent results applying these to the study of gradient inequalities \[ \left(\int_{\mathbb{R}^ n}| f(x)|^ q v(x)dx\right)^{1/q} \leq C\left(\int_{\mathbb{R}^ n} | x\cdot\nabla f(x)|^ p u(x)dx\right)^{1/p}, \] where it was shown by G. J. Sinnamon [Proc. R. Soc. Edinb., Sect. A 111, No. 3/4, 329-335 (1989; Zbl 0686.26004)] that this inequality holds for \(1<p=q<\infty\) if and only if \[ \sup_{x\in\mathbb{R}^ n}\left(\int^ 1_ 0 v(tx) t^{n-1} dt\right)^{1/p}\left(\int^ \infty_ 1 (u(tx)t^ n)^{1-p'}{dt\over t}\right)^{1/p'}<+\infty, \] with a different equivalence for \(0<q<p\) and \(1/p+1/p'=1\).

The second application of the classical result is obtained by studying its limit as \(p\to \infty\). If we denote by \[ Pf(x)={1\over x} \int^ x_ 0 f(t)dt\quad\text{and}\quad Q_ \alpha f(x)=\alpha x^ \alpha\int^ \infty_ x t^{-\alpha-1} f(t)dt \] then the author, Kerman and Krbec have shown that the following are equivalent: \[ \int^ \infty_ 0 u(x) e^{P(\ln f)(x)} dx\leq C_ 1 \int^ \infty_ 0 v(x) f(x)dx\tag{i} \]

\[ Pw(x)+{1\over\alpha} Q_ \alpha w(x)\leq C_ 2,\tag{ii} \]

\[ Pw(x)\in L^ \infty,\tag{iii} \]

\[ \int^ \infty_ 0 w(x) P(f^{1/p})(x)dx\leq C_ 3 \int^ \infty_ 0 f(x)dx,\;p>2,\tag{iv} \] where \(w(x)=u(x) e^{[P(\ln 1/v)(x)]}\).

He then discusses recent results of Muckenhoupt-Arino, Braverman, Neugebauer, Sawyer and Stepanov on Hardy inequalities for decreasing functions.

Attention is then turned to the situation of inequalities for the Fourier transform. The author begins by reviewing the results of Muckenhoupt, himself and Jurkat-Sampson, which had been discussed in the prior survey. Sufficient and necessary conditions are known; when the weights are monotone (in the right form), the results become necessary and sufficient. The author begins by restating a result of Benedetto, the author and the reviewer. The author and G. J. Sinnamon [Indiana Univ. Math. J. 38, No. 3, 603-628 (1989; Zbl 0668.42003)] have proved a result about monotone weights in the Muckenhoupt class \(A_ p\) that allows the result of J. J. Benedetto, the author and the reviewer [General inequalities 5, 5th Int. Conf., Oberwolfach/FRG 1986, ISNM 80, 217-232 (1987; Zbl 0625.42006)] to be stated as: if \(w\) is an even weight, increasing on \((0,+\infty)\), the following holds: \[ \left(\int_{\mathbb{R}^ n} |\widehat f(x)|^ q| x|^{n(q/p'-1)}w(1/| x|)^{q/p}dx\right)^{1/q}\leq C\left(\int_{\mathbb{R}^ n} | f(x)|^ p w(x)^ p dx\right)^{1/p}, \] \(w\in A_ p\), \(1<p\leq q\leq p'\). The advantage of this result is that the condition is testable; however, the result applies only to monotone weights. The author proves the extension of the above result to \(n\) dimensions given by himself and Sinnamon. He also discusses other necessary and sufficient conditions of greater generality, but all of which involve some monotonicity condition on at least one of the weights. There is a particularly interesting result, due to the author, in which he studies an inequality in the above form but where the function \(f\) is restricted to be decreasing with limit 0 at \(+\infty\). He shows that the inequality holds if and only if the weight satisfies the condition required for Hardy’s inequality to hold on decreasing functions. Next he surveys results on inequalities restricted to functions whose first moment is zero. There is a basic result due to C. Sadosky and R. L. Wheeden [Trans. Am. Math. Soc. 300, 521- 533 (1987; Zbl 0636.42012)] that if the weight \(w\) is in \(A_ p\), \[ \int^ \infty_{-\infty}|\widehat f(x)|^ p w(1/x)dx/| x|^ 2\leq C \int^ \infty_{-\infty}| f(x)|^ p | x|^ p w(x)dx,\quad \widehat f(0)=0. \] The author discusses results obtained jointly by him and Benedetto giving sufficient conditions for inequalities like the above but with measure weights on the left hand side.

The final section discusses sufficient conditions for restriction theorems, \[ \left(\int_{| x|=\rho}|\widehat f(x)|^ r ds\right)^{1/r} \leq C\rho^{1/r+\alpha-2/p'}\| f\|_{p,\alpha}. \] This is valid for all functions for which the right hand side is finite; some restriction theorems are given for functions satisfying moment conditions. He concludes with a proof using the Hardy inequality to give the sharp estimate for \(1< p<n\), \[ \left(\int_{\mathbb{R}^ n}\left({| f(x)|\over| x|}\right)^ p dx\right)^{1/p}\leq {p\over n- p}\left(\int_{\mathbb{R}^ n}\left({| x\cdot\nabla f(x)|\over| x|}\right)^ p dx\right)^{1/p}. \] {}.

Reviewer: R.Johnson (College Park)

### MSC:

42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |

44A05 | General integral transforms |