A Counterexample to an Endpoint Mixed Norm Estimate of Calder\'on-Zygmund Operators

Abstract

It is known that that the endpoint mixed norm estimate $|| \, ||Tf(x,y)||_{L_{x}^{p}}||_{L_{y}^{\infty}} \lesssim || \, ||f(x,y)||_{L_{x}^{p}}||_{L_{y}^{\infty}}$ in general does not hold for Calder\'on-Zygmund operator $T$. In this article, we show that when $p=2$, even if we make the right hand side of the above estimate larger by replacing it with $ || \, ||e^{x^2+y^2}f(x,y)||_{L_{y}^{\infty}}||_{L_{x}^{\infty}} $, the estimate does not hold for the double Riesz transform given by the kernel $K(x,y)=\frac{xy}{2\pi(x^2+y^2)^{2}}$. As a consequence we will show that the mixed norm estimate $|| \, ||Tf(x,y)||_{L_x^{p}}||_{L_y^{\infty}} \lesssim|| \, ||f(x,y)||_{L_y^{\infty}}||_{L_x^{p}}$ does not hold for double Riesz transform and $p \geq 2$.