Simpler proof of the Hardy-Littlewood-Sobolev inequality in the inhomogeneous case

The Hardy-Littlewood-Sobolev inequality is the statement that there is a $C>0$ such that $$ \tag{HLS} \lVert f\ast \lvert\cdot\rvert^{-\alpha}\rVert_p\le C\lVert f\rVert_q, $$ for all $f\in L^q(\mathbb R^d)$, where the convolution is defined as $$(f\ast \lvert\cdot\rvert^{-\alpha} )(x)= \int_{\mathbb R^d} \frac{f(y)}{\lvert x-y\rvert^{\alpha}}\, dy, $$ and the parameters satisfy the conditions $$\tag{1} 0<\alpha<d, \quad \frac1p-\frac1 q+1=\frac\alpha d.$$

Now let us define the Japanese bracket $$\langle x \rangle:= \sqrt{1+ \lvert x\rvert^2},\qquad x\in\mathbb R^d,$$ which is a non-homogeneous version of $\lvert\cdot\rvert$. The function $\langle x\rangle^{-\alpha}$ has the exact same decay at $\lvert x \rvert \to \infty$ of its homogeneous counterpart $\lvert x \rvert^{-\alpha}$, but $\langle x \rangle^{-\alpha}$ is not singular at $x=0$. By the obvious estimate $\langle x\rangle^{-\alpha}\le \lvert x\rvert^{-\alpha}$, (HLS) immediately implies its non-homogeneous version $$\tag{n-HLS} \lVert f\ast \langle\cdot\rangle^{-\alpha}\rVert_p\le C\lVert f\rVert_q, $$ under the assumptions (1).

Question. Is there a direct proof of (n-HLS) that is simpler than a proof of (HLS)?

Remark 1. The inequality (n-HLS) actually holds for $\frac1 p - \frac1q +1 \le \frac{\alpha}{d}$. However, the non-endpoint case $\frac1 p - \frac1q +1 < \frac{\alpha}{d}$ can be immediately proved by an application of the Young inequality for convolutions. Thus, the present question is concerned solely with the endpoint case (1).

In the following, the letter $C$ will always denote an irrelevant positive constant, whose value may change from line to line.

Remark 2. There are many proofs of (HLS). One that I know uses the Hardy-Littlewood maximal function. By decomposing the ball $B(x, r)$ into dyadic annuli, we obtain the local estimate $$\tag{2}\left\lvert \int_{B(x, r)}\frac{f(y)}{\lvert x-y \rvert^\alpha}dy\right\rvert \le r^{d-\alpha}\sum_{j=0}^\infty 2^{\alpha j - dj} Mf(x)= C_{d, \alpha}r^{d-\alpha}Mf(x).$$ This constant $C_{d,\alpha}$ equals $\sum 2^{(\alpha - d)j}$, which is finite because $\alpha<d$. Here $Mf$ denotes the Hardy-Littlewood maximal function $$ Mf(x):=\sup \frac1{\lvert B\rvert} \int_B \lvert f(y)\rvert\, dy,$$ where the sup is taken over all balls $B$ containing $x$.

We then estimate the tail of the convolution via the Hölder inequality; $$\tag{3} \left\lvert \int_{\mathbb R^d\setminus B(x, r)}\frac{f(y)}{\lvert x-y \rvert^\alpha}dy\right\rvert\le C r^{\frac{d}{q'}-\alpha} \lVert f\rVert_q.$$

Combining (2) and (3) gives the pointwise bound $$ \tag{4} \lvert f\ast \lvert\cdot\rvert^{-\alpha}(x)\rvert \le C\left( r^{d-\alpha} Mf(x) + r^{\frac{d}{q'}-\alpha} \lVert f\rVert_q\right).$$ Choosing the $r$ that minimises the right-hand side, then integrating and applying the Hardy-Littewood maximal estimate $\lVert Mf\rVert_q\le C\lVert f \rVert_q$, we obtain (HLS).

This procedure will of course work if $\lvert\cdot\rvert^{-\alpha}$ is replaced by $\langle\cdot\rangle^{-\alpha}$. However, I think that there shouldn't be the need for the careful dyadic analysis of equation (2). Indeed, $\langle x-y\rangle^{-\alpha}$ is not singular at $y=x$. This is why I am asking for a simpler proof.

In the given range $$\tag{1}\frac1p + 1 = \frac1q+\frac\alpha d, $$ there is no hope for a simpler proof in the non-homogeneous case.

To explain, let me consider the weak Young inequality $$\tag{2} \lVert f\ast g \rVert_p \le C\lVert f\rVert_{q_1} \lVert g\rVert_{q_2,\infty}, \quad \frac1p+1 = \frac1{q_1}+\frac1{q_2},$$ where $$\tag{3}\lVert g\rVert_{q_2,\infty} = \sup_{t>0} t \lvert \{ \lvert g\rvert >t\}\rvert^\frac1{q_2}.$$ We want to prove (n-HLS) under the assumption (1), which, according to (2) with $g=\langle\cdot\rangle^{-\alpha}$, follows from $$\tag{4} \lVert \langle \cdot \rangle^{-\alpha}\rVert_{\frac d \alpha, \infty}<\infty.$$ Now, it is clear that there is no simpler way to prove (4) than to estimate pointwise $\langle x \rangle^{-\alpha}\le \lvert x \rvert^{-\alpha}$, so that $$\lVert \langle\cdot \rangle^{-\alpha}\rVert_{\frac d \alpha, \infty} \le \ \lVert \lvert\cdot \rvert^{-\alpha}\rVert_{\frac d \alpha, \infty}<\infty,$$ which completes the proof. Thus the main step is the proof that $\lVert \lvert\cdot \rvert^{-\alpha}\rVert_{\frac d \alpha, \infty}<\infty$, which is an easy computation. Indeed, this very computation has probably been the main motivation for the definition (3).

Let me remark that the mathematics behind all this is essentially the Marcinkiewicz interpolation theorem. I have consulted the blog post of Terry Tao https://terrytao.wordpress.com/2009/03/30/245c-notes-1-interpolation-of-lp-spaces/; the weak Young inequality is Exercise 44.

In the proof that I gave in the main question, above, the weak-Lp computations are hidden behind the Hardy-Littlewood maximal estimate $\lVert Mf\rVert_p\le C \lVert f\rVert_p$, which indeed is proven via the interpolation theorem of Marcinkiewicz.