This is barely a post; I’m just recording a basic (but messy) bit of index arithmetic as a note-to-self.

I was going through Lieb’s 1983 Annals paper today, the argument of which uses a certain bound on the \(L^p \to L^q\) mapping for HLS on a special class of functions. This result (Lemma 2.4, p. 355 of the journal) was claimed to follow from “known results” about Lorentz spaces, but the text also included a proof (in Lieb’s description) based on “a transformation to logarithmic radial variables.”

While the result included citations, they were in the style of unpaginated, unenumerated references to three massive texts (O’Neil’s seminal Duke paper; a paper by Brezis and Wainger; and Stein & Weiss), and trawling through those works did not seem particularly appealing. So, I wanted to see if I could prove it myself, using modern machinery; I worked out the details below:

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Let’s recall the Hedberg proof of the pointwise estimate for Riesz potentials: if \(0 < \beta < d\), and we have \(1 < p, q < \infty\) satisfying $$\frac{1}{q} = \frac{1}{p} \ – \, \frac{\beta}{d},$$ then we write $$(I_{\beta}(f))(x) = \int_{|x \ – \, y| \leq t} f(y) |x \ – \, y|^{- (d \ – \, \beta)} \, dy + \int_{|x \ – \, y| > t} f(y) |x \ – \, y|^{- (d \ – \, \beta)} \, dy,$$ and estimate the first term by $$\leq (M f)(x) \|| \cdot |^{- (d \ – \, \beta)} \chi_{B_t}\|_{L^1(\mathbb{R}^d)} = c_{d, \beta} t^{\beta} (M f)(x).$$

The second term, by Hölder, is estimated as $$\leq \|f\|_{L^p(\mathbb{R}^d)} \| | \cdot |^{- (d \ – \, \beta)} \chi_{(B_t)^c} \|_{L^{p’}(\mathbb{R}^d)},$$ and because we have \(\frac{1}{p} > \frac{\beta}{d}\), we get \(\frac{1}{p’} < \frac{d \ – \, \beta}{d}\), and thus the integral becomes $$c_{p, d, \beta} \|f\|_{L^p(\mathbb{R}^d)} (t^{- (p’ (d \ – \, \beta) \ – \, d)})^{1 / p’} = c_{p, d, \beta} \|f\|_{L^p(\mathbb{R}^d)} t^{- (d / p \ – \, \beta)}.$$

Then, after discarding constants, this gives $$|(I_{\beta} f)(x)| \lesssim_{d, p, \beta} \inf_{t > 0} (t^{\beta} (M f)(x) + t^{\beta} \|f\|_{L^p(\mathbb{R}^d)} t^{- d / p}),$$ and we note in passing that \(d / p = d (1 / p) > d (\beta / d) = \beta\), so that the second power is negative while the first is positive. Equating the two and identifying \(t\), we get the bound $$|(I_{\beta} f)(x)| \lesssim ((M f)(x))^{1 \ – \, \frac{p \beta}{d}} \|f\|_{L^p(\mathbb{R}^d)}^{\frac{p \beta}{d}}.$$

Now let’s plug in our restricted-weak-type inputs: first, we have $$\|T(\chi_E)\|_{L^{\infty}(\mathbb{R}^d)} \lesssim \lambda^d(E) ^{\frac{\beta}{d}},$$ immediately.

For the other end, we’ll use the weak-type \((1, 1)\) boundedness of the maximal operator: we have that $$\{|T(\chi_E)| > t\} \subseteq \{M(\chi_E)^{1 \ – \, \frac{p \beta}{d}} \gtrsim t / \lambda^d(E)^{\frac{\beta}{d}}\},$$ and so we have $$\lambda^d(\{|T(\chi_E)| > t\}) \lesssim \bigg( \frac{\lambda^d(E)^{\frac{\beta}{d}}}{t} \bigg)^{\frac{1}{1 \ – \, \frac{p \beta}{d}}} \|\chi_E\|_{L^1(\mathbb{R}^d)},$$ and to simplify this, we note that $$1 \ – \, \frac{p \beta}{d} = p \bigg( \frac{1}{p} \ – \, \frac{\beta}{d} \bigg) = \frac{p}{q},$$ so that $$t (\lambda^d(\{|T(\chi_E)| > t\}))^{1 / (q / p)} \lesssim \lambda^d(E)^{\frac{\beta}{d}} \lambda^d(E)^{\frac{p}{q}},$$ and then we have the relationships $$\begin{align} \|T(\chi_E)\|_{L^{\infty}(\mathbb{R}^d)} & \lesssim \|\chi_E\|_{L^{\frac{d}{\beta}}(\mathbb{R}^d)}, \\ \|T(\chi_E)\|_{L^{q / p, \infty}(\mathbb{R}^d)} & \lesssim \|\chi_E\|_{L^{(\frac{\beta}{d} + \frac{p}{q})^{- 1}}(\mathbb{R}^d)}. \end{align}$$ We note in passing that \(\frac{\beta}{d} + \frac{p}{q} = 1 \ – \, (p \ – \, 1) \frac{\beta}{d} < 1\).

We now use Hunt interpolation on our (sub)linear, countably-subadditive bounded operator $$T : L^{(\frac{\beta}{d} + \frac{p}{q})^{- 1}, 1}(\mathbb{R}^d) \to L^{\frac{q}{p}, \infty}(\mathbb{R}^d),$$ $$T : L^{\frac{d}{\beta}, 1}(\mathbb{R}^d) \to L^{\infty}(\mathbb{R}^d);$$ we have that \(q / p < q < \infty\), as \(p > 1\) strictly, so we may take \(0 < \theta < 1\) ensuring $$\frac{1}{q} = \frac{1 \ – \, \theta}{q / p} + \frac{\theta}{\infty},$$ and we note that this is simply the requirement $$1 \ – \, \theta = \frac{1}{p}.$$ Then, computing the corresponding input index, we have $$\frac{1}{p_{\theta}} = \frac{1 / p}{(\frac{\beta}{d} + \frac{p}{q})^{- 1}} + \frac{1 \ -\, 1 / p}{\frac{d}{\beta}} = \frac{1}{p} \bigg( \frac{\beta}{d} + \frac{p}{q} \bigg) + \bigg(1 \ – \, \frac{1}{p} \bigg) \frac{\beta}{d},$$ which we recognize to be $$\frac{1}{q} + \frac{\beta}{d} = \frac{1}{p}.$$

So we have, for all \(1 \leq r \leq \infty\), we have for all \(f \in L^{p, r}(\mathbb{R}^d)\) $$\|I_{\beta}(f)\|_{L^{q, r}(\mathbb{R}^d)} \lesssim_{d, \beta, p, r} \|f\|_{L^{p, r}(\mathbb{R}^d)}$$ here. In particular, as \(q > p\), we can take \(r = q\) to get $$\|I_{\beta}(f)\|_{L^q(\mathbb{R}^d)} \lesssim \|f\|_{L^{p, q}(\mathbb{R}^d)} \lesssim \|f\|_{L^p(\mathbb{R}^d)}^{\frac{p}{q}} \|f\|_{L^{p, \infty}(\mathbb{R}^d)}^{1 \ – \, \frac{p}{q}},$$ where the final inequality is by Hölder on the outer \(\ell^q\) norm on the dyadic pieces. As the \(L^{p, q}\) norm is always weaker than the \(L^p\) norm for \(q > p\), this gives an improvement over the standard \(L^p \to L^q\) HLS bound.

This leads to the claimed Lemma 2.4 of Lieb (1983):

Lemma: Let \(1 < p, q < \infty\) and \(0 < \beta < d\), with the property that they are related by \(\frac{1}{q} = \frac{1}{p} \ – \, \frac{\beta}{d}\). Take any \(u \in L^p(\mathbb{R}^d)\) such that \(|u(x)| \leq K |x|^{- d / p}\) pointwise on \(\mathbb{R}^d\) for some \(K \geq 0\). Then we have $$\|| \cdot |^{- (d \ – \, \beta)} * f\|_{L^q(\mathbb{R}^d)} \lesssim_{d, p, \beta} \|f\|_{L^p(\mathbb{R}^d)}^{\frac{p}{q}} K^{1 \ – \, \frac{p}{q}}.$$

We also remark that, using the Riesz rearrangement inequality, this gives bounds for the convolution of a general \(L^p\) function \(f\) with a general weak-\(L^q\) function \(g\), in the \(L^r\) norm.