# Hardy-Littlewood-Sobolev for “componentwise product” of Riesz kernels

Let $$d\in\mathbb N$$ and $$0<\alpha. Define the Riesz kernel $$K_\alpha(x):=|x|^{\alpha-d}$$, and the associated convolution operator $$K_\alpha f(x):=\int\frac{f(y)}{|x-y|^{d-\alpha}}~dy.$$ The classical Hardy-Littlewood-Sobolev inequality states that we have the following boundedness property: for every $$q>1$$, $$\|K_\alpha f\|_q\leq C\|f\|_p$$ for some universal constant $$C>0$$ and $$1/p=1/q+\alpha/d$$.

Now let $$0<\alpha_1,\ldots,\alpha_d<1$$, and suppose instead that we consider the convolution operator given by the kernel $$K_{(\alpha_1,\ldots,\alpha_d)}(x)=\prod_{i=1}^d|x_i|^{\alpha_i-1},\qquad x=(x_1,\ldots,x_d),$$ which is what I call a "componentwise product of Riesz Kernels."

Question. Given $$q>1$$, does there exist some constant $$C>0$$ and a $$p>1$$ such that $$\|K_{(\alpha_1,\ldots,\alpha_d)} f\|_q\leq C\|f\|_p?\tag{1}$$

By a simple scaling argument, we can see that if (1) holds, then it must be for $$1/p=1/q+\sum_i\alpha_i/d$$. Indeed, letting $$f_c(x):=f(cx)$$, then $$\|f_c\|_p=c^{-d/p}\|f\|_p$$ and $$\|K_{(\alpha_1,\ldots,\alpha_d)}f_c\|_q=c^{-\sum_i\alpha_i-d/q}\|K_{(\alpha_1,\ldots,\alpha_d)}f\|_q$$.

• I would write $K_{(\alpha_1,\ldots,\alpha_d)}$ as a composition of operators $K_j$, $j = 1, \ldots, d$, defined by $K_j f(x) = \int_{-\infty}^\infty f(x - y e_j) |y|^{\alpha_j-1} dy$, where $e_j = (0, \ldots, 0, 1, 0, \ldots, 0)$ has $1$ at $j$-th coordinate. Then it seems sufficient to apply the usual H-L-S inequality to each $K_j$. Have you tried that? – Mateusz Kwaśnicki Apr 3 '20 at 21:11
• @MateuszKwaśnicki Thanks for pointing out this very simple reduction; I did not think of that. If each $K_j$ does indeed satisfies a HLS inequality of the form $\|K_jf\|_q\leq C\|f\|_p$ with $1/p=1/q+\alpha_j/d$, then we have the result! – user78370 Apr 4 '20 at 4:26

I realise I was too optimistic in my comment: no such ineqaulity holds in general. Indeed, consider $$f(x) = \prod_j f_j(x_j)$$. Then $$K_{(\alpha_1,\ldots,\alpha_d)} f(x) = \prod_j K_{\alpha_j} f_j(x_j)$$ and so $$\|f\|_p = \prod_j \|f_j\|_p , \qquad \|K_{(\alpha_1,\ldots,\alpha_d)} f\|_q = \prod_j \|K_{\alpha_j} f_j\|_q$$ So by the usual Hardy–Littlewood—Sobolev inequality (the "only if" part), in order to get the bound we are interested in, we would need $$\frac{1}{p} = \frac{1}{q} + \frac{\alpha_j}{1} ,$$ that is, $$\alpha_1 = \alpha_2 = \ldots = \alpha_n \qquad \text{and} \qquad \frac{1}{p} = \frac{1}{q} + \frac{\alpha}{d},$$ with $$\alpha = \alpha_1 + \alpha_2 + \ldots + \alpha_d$$. In this case, the "if part" of the Hardy–Littlewood—Sobolev inequality implies that indeed the desired inequality holds.
• Thanks for pointing out yet another simple observation. The result then only holds when all $\alpha_i$ are equal... – user78370 Apr 4 '20 at 9:30