Let $K:\mathbb{R}^3\backslash\{0\}\times\mathbb{R}^3\backslash\{0\}\rightarrow\mathbb{C}$, such that $K(x,y)=K(y,x)$ and $K(x,y)=|x|^{-1}|y|^{-1}H(x,y)$, with $H$ locally bounded. Let $T$ be the (singular) integral operator with kernel $K$, i.e. $$T(f)(x)=\int_{\mathbb{R}^3}K(x,y)f(y)dy$$ Suppose that $T$ is unitary on $L^2(\mathbb{R}^3)$ and bounded from $w^{-1}L^1(\mathbb{R}^3)$ to $wL^{\infty}(\mathbb{R}^3)$, where $w(x)=1+|x|^{-1}$.

Can i deduce, by means of some interpolation tecnique, that $T$ is bounded from ${z_p}^{-1}L^p(\mathbb{R}^3)$ to ${z_p}L^q(\mathbb{R}^3)$? (with $1<p<2$, $1/p+1/q=1$and $z_p$ an appropriate weight function depending on $p$)

EDIT: As Christian pointed out, one cand find $z$ such that the operator with integral kernel $K(x,y)/(z(x)z(y))$ is bounded from $L^2$ to $L^2$ and from $L^1$ to $L^{\infty}$. Hence, a typical interpolation argument allows us to take $z_p=z$ for every $p$.

However, i'm interested about the dependence of an "optimal" $z_p$ on $p$. Hopefully, something like $z_p=1+|x|^{-\alpha(p)}$, with $\alpha(p)>0$ and decreasing to $0$ as $p\rightarrow 2$.

Thank you for any suggestion

  • $\begingroup$ If I unwrapped it correctly, this seems the same as asking if the integral operator with kernel $K(x,y)/(z(x)z(y))$ is bounded from $L^p$ to $L^q$, and of course this will hold by normal interpolation (it's not even necessary to give $z$ obscenely fast growth, but we could that too). $\endgroup$ – Christian Remling May 15 '15 at 19:30
  • $\begingroup$ @Christian: I don't fully understand your argument. Are you saying that i can choiche $z=w$ for every $p$? $\endgroup$ – Capublanca May 15 '15 at 23:41
  • $\begingroup$ No, what I had in mind was to take $z$ such that the operator with kernel $K/(z(x)z(y))$ is bounded $L^2\to L^2$ and $L^1\to L^{\infty}$ and then interpolate. I'm in fact not using the assumption with $w$. $\endgroup$ – Christian Remling May 16 '15 at 0:04
  • $\begingroup$ Ok, i understand your argument, thank you! However, i'm looking for something more precise, i.e. find the dependence on $p$ of an optimal weight $z_p$ (something like $z_p\sim 1+|x|^{-\alpha(p)}$, with $\alpha(p)\rightarrow 0$ as $p\rightarrow 2$) $\endgroup$ – Capublanca May 16 '15 at 0:29
  • $\begingroup$ Sorry, i meant $|x|^{-1}|y|^{-1}$ $\endgroup$ – Capublanca May 20 '15 at 14:55

I think what you need is in the following paper:

E. M. Stein and G. Weiss, Interpolation of operators with change of measures, Transactions of the American Mathematical Society, Vol. 87 (1958), pp. 159-172


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.